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Clark,∗ Creston D. Herold, James T. Merrill, Holly N. Tinkey, Wade +Rellergert, Robert Clark, Roger Brown, Wesley D. Robertson, Curtis Volin, Kara +Maller, Chris Shappert, Brian J. McMahon, Brian C. Sawyer,† and Kenton R. Brown‡ +Georgia Tech Research Institute, Atlanta, Georgia 30332, USA +Ion transport is an essential operation in some models of quantum information processing, where +fast ion shuttling with minimal motional excitation is necessary for efficient, high-fidelity quan- +tum logic. While fast and cold ion shuttling has been demonstrated, the dynamics and specific +trajectory of an ion during diabatic transport have not been studied in detail. Here we describe +a position-dependent optical deshelving technique useful for sampling an ion’s position through- +out its trajectory, and we demonstrate the technique on fast linear transport of a 40Ca+ ion in a +surface-electrode ion trap. At high speed, the trap’s electrode filters strongly distort the transport +potential waveform. With this technique, we observe deviations from the intended constant-velocity +(100 m/s) transport: we measure an average speed of 83(2) m/s and a peak speed of 251(6) m/s +over a distance of 120 µm. +Systems of trapped atomic ions represent some of the +most promising platforms for quantum information pro- +cessing, benefiting from long qubit coherence times and +from the highest operational fidelities demonstrated to +date [1–3]. In the QCCD ion-trap architecture [4], ions +are transported between various regions within the pro- +cessor to reconfigure which ions can interact at any given +step of an algorithm. +Such an architecture demands +finely tuned control of ion shuttling in order to imple- +ment reconfigurations as quickly as possible without also +degrading the fidelities of subsequent logic gates [1, 5, 6]. +Performing logic operations on the ions at the same time +as they are transported within the trap also requires well +characterized and reproducible trajectories for success +[7, 8]. +Recent work has shown that ion transport contributes +substantially to the latency of current QCCD systems [9]. +Rapid linear ion transport has been demonstrated previ- +ously [10–12] but has not been widely adopted likely due +to the demands it places on waveform control hardware +and on motional characterization and calibration. Trans- +port at slower speeds has also been achieved through +more complicated structures such as junctions of linear +sections [13–17]. While prior results have demonstrated +transport between two locations with sub-quanta mo- +tional excitation, the ion trajectory during transport has +not been measured. In fact, all prior reported transport +speeds assumed an average speed calculated from the dis- +tance between static well positions and the designed play- +back speed of the trapping voltage waveforms. Some ex- +periments have proved that an ion was transported to an +intended location, e.g. by applying a focused laser pulse +at that location [10, 12]; however, those experiments did +not verify the exact time of arrival. +In this work, we present a method to measure the loca- +tion of an ion throughout the entire arc of its transport, +∗ craig.clark@gtri.gatech.edu +† brian.sawyer@gtri.gatech.edu +‡ kenton.brown@gtri.gatech.edu +which enables us to reliably extract both instantaneous +and average velocity during transport. This is achieved +by measuring the probability for the ion to undergo a +spontaneous irreversible transition from a metastable ex- +cited state to a ground state when illuminated by a laser +beam having a spatial intensity gradient. Similar ideas +have been explored in the context of neutral atoms in +optical cavities [18, 19], as well as in the field of ultrafast +physics [20]. The method presented here contrasts with +previous schemes employing Fourier-limited coherent op- +tical interactions to extract in-flight Doppler shifts [7, 8]. +Here, we leverage the technique to show that an ion’s +trajectory deviates significantly from a simple constant- +velocity path as the trap’s electrode filters distort the ap- +plied potentials. Although the intended trajectory has a +constant speed of 100 m/s, corresponding to a displace- +ment of 120 µm (two electrodes) in 1.2 µs, the ion ac- +tually achieves a maximum speed of 251(6) m/s in the +middle of its path due to this distortion. These trajec- +tory deviations lead also to a large coherent displacement +of the ion’s motional state. +We superimpose an addi- +tional compensating sinusoidal potential onto the wave- +form to remove this displacement [12, 21] and achieve a +final transport-induced excitation of 0.7(2) quanta. +Our experimental system employs a GTRI/Honeywell +Ball Grid Array (BGA) trap [22], shown schematically in +Fig. 1a, mounted in a room-temperature ultrahigh vac- +uum chamber and confining a single 40Ca+ ion. +The +trap radiofrequency (rf) electrode is driven at 55 MHz +and realizes radial frequencies of 3.7 and 4.2 MHz. We +determine the control electrode voltages necessary to +produce axial confinement at different positions along +the trap symmetry axis using an in-house boundary ele- +ment method electrostatic solver [23]. We employ NIST- +designed PDQ digital-to-analog converters (DACs) [21] +to vary the potentials on the axial control electrodes. +Each control electrode’s potential is filtered with a two- +pole low-pass filter having a bandwidth of 608 kHz [24]. +The axial trap frequency (parallel to the axis of symme- +try) is approximately 1.7 MHz. The axial heating rate +arXiv:2301.05279v1 [quant-ph] 12 Jan 2023 + +2 +854 nm +Intensity +a) +𝑧 +𝑧� +time +Cool & +Prep +Detect +Adiabatic Transport +b) +𝒛𝒊 → 𝒛 +π +𝒛 → 𝒛𝒊 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +50 +100 +150 +200 +Deshelve Probability +Position (µm) +c) +𝒛𝒊 +𝒛𝒇 +𝑧� +Optical operations +𝑺𝟏/𝟐 +𝑫𝟓/𝟐 +𝑷𝟑/𝟐 +729 nm +854 nm +393 nm +FIG. 1. +(a) (Left) Schematic of the BGA trap with a graph of +the nominal 854 nm repumper intensity profile superimposed. +The initial and final positions of the ion for the characterized +transport are zi and zf. +(Right) Energy level diagram of +40Ca+ ion showing the shelving transition driven by 729 nm +light in red, the deshelving transition driven by 854 nm light +in orange, and spontaneous decay through emission of 393 nm +light in purple. (b) Ion trajectory calibration sequence. An +ion in a stationary potential at zi is cooled and prepared in +the S1/2 level and then shelved into the D5/2 level with a +resonant 729 nm π-pulse (shown in red in the timeline). The +ion is adiabatically transported to another position along the +trap axis, and the 854 nm beam is pulsed for 200 ns, thereby +deshelving a portion of the D5/2 population. This deshelving +fraction is in one-to-one correspondence with the local beam +intensity. The ion is shuttled adiabatically back to its initial +location zi. (c) Deshelving probability as a function of ion +position (black points) with error bars representing the 68% +confidence interval assuming binomial statistics. The red line +represents a polynomial fit to the data. +for a stationary potential is 210 quanta/s. +For Doppler cooling we use a 397 nm laser beam, nearly +resonant with the S1/2 − P1/2 transition, in combina- +tion with another beam at 866 nm which is used to re- +turn population from the D3/2 level into the cooling cy- +cle. Pulses from a beam at 729 nm, resonant with the +S1/2−D5/2 transition, coherently populate the D5/2 level +and are employed for sideband cooling and motional- +state characterization. Another beam at 854 nm is re- +sponsible for deshelving population from the D5/2 to the +S1/2 level via the intermediate P3/2 level (see Fig.1a) [25]. +We distinguish between an ion in the S1/2 level (bright) +and one in the D5/2 level (dark) via observation of state- +dependent 397 nm fluorescence as recorded on a photo- +multiplier tube. +After many experimental repetitions, +the bright and dark state populations are estimated via +maximum likelihood (see Supplemental Material of Ref. +[1]). +To characterize the ion’s position throughout the full +arc of its roughly linear trajectory, we implement a +position-sensitive optical deshelving technique [26]. For +this, we prepare the ion in the D5/2 state before transport +and then apply a pulse (200 ns, shorter than the trans- +port duration) of 854 nm light at a later time, thereby +deshelving the D5/2 level with a probability dependent on +the laser beam’s local intensity. By choosing the Gaus- +sian waist w0 of the 854 nm beam such that its intensity +varies significantly and monotonically along the ion’s tra- +jectory (w0 ∼ 100 µm), we achieve a position-dependent +deshelving probability (Pd). We can therefore invert spa- +tially monotonic measurements of this probability, ac- +quired at various times after the start of the transport, +to determine the ion’s position at these instants. The +beam is directed perpendicular to the linear transport +axis to remove velocity dependence (first-order Doppler +shifts, e.g.) from the deshelving probability. +The +above +technique +requires +calibration +of +the +deshelving probability for a given ion location using a +procedure diagrammed in Fig. 1b. For this, we Doppler +and sideband cool the ion nearly to its axial motional +ground state (¯n < 1) and prepare it in S1/2. We then +shelve it into the D5/2 level with a resonant 729 nm π- +pulse and subsequently move it adiabatically to a given +position along the trap axis (as determined from an elec- +trostatic model of the trapping potential). We illuminate +the ion with a 200 ns, 854 nm deshelving pulse. After +adiabatically returning the ion to its initial location, we +determine its state by collecting fluorescence. Repeating +this experiment 400 times each at 2 µm intervals yields +a measurement of deshelving probability Pd(z) for each +location as shown in Fig. 1c. We fit Pd(z) with a polyno- +mial (red curve of Fig. 1c) rather than with a Gaussian +to allow for possible distortion of the 854 nm laser beam +mode shape, and we ensure that Pd(z) is single-valued by +having previously displaced the beam such that its inten- +sity varies monotonically through the calibration region. +It is important that any movement of ions from one +place to another in the trap be accomplished with only a +small degree of motional excitation. For the experiments +outlined below, we design the waveform as a simple lin- +ear time interpolation of the potential minimum through +a 120 µm displacement. We choose a transport duration +(1.2 µs) that is an integer multiple of the ion’s harmonic +motional period (0.6 µs, nominally held constant during +the transport). In the absence of filter distortion, such a +waveform is expected to coherently excite ion motion at +the beginning of transport but subsequently to suppress +this excitation at the end [10, 27]. Our filter bandwidths +lie below the ion motional frequencies by design. At slow +enough speeds, the action of the filters is to smooth out + +3 +the beginning and ending accelerations, so that the ion’s +motion is not excited as much during transport as would +otherwise be the case. At faster speeds such as are used +here, filter distortion is great enough that simple scal- +ing of the waveform confinement strength can no longer +fully suppress the final excitation [27]. However, provided +that the waveform is performed identically for each ex- +perimental repetition and that the confinement remains +harmonic, the final excitation corresponds to a coherent +displacement. +We remove it by superimposing on the +transport waveform a sinusoidal rf pulse applied to four +of the trap electrodes, and we optimize the phase, fre- +quency, and amplitude of this pulse to achieve minimum +mode occupation. +With the Pd(z) calibration complete, we then trans- +port the ion through the full trajectory of 120 µm. Fig. 2a +gives a diagram of the sequence. We pulse the 854 nm +deshelving laser for 200 ns while the ion is in motion, +with a configurable delay between the start of the trans- +port waveform and the start of the pulse. The ion is then +returned adiabatically to its initial location for state de- +tection [28]. With 100 repetitions of this experiment at +each delay, we obtain a map Pd(t) of deshelving probabil- +ity in time following the start of the waveform (Fig. 2b). +To obtain the corresponding position map z(t), we in- +vert the polynomial fit of Fig. 1c to obtain z(Pd) and +then compute z(Pd(t)), producing the points in Fig. 2c. +Finally, we fit the z(t) data of Fig. 2c with the follow- +ing phenomenological expression to extract the mean and +maximum linear velocities: +z(t) = zi + +√π +2 vmaxtσ +� +erf +�t − tc +tσ +� +− erf +�t0 − tc +tσ +�� +. +(1) +Equation 1 is derived assuming that the speed follows +a Gaussian profile in time, an empirical assumption justi- +fied by its agreement with the z(t) data in Fig. 2c. Here, +zi represents the initial ion position, t0 is the initial time +(time when z(t) = zi), tσ is the 1/e temporal half-width +of the ion speed, and tc is the time of maximum speed. +We use the following standard definition for the error +function: +erf(x) = +2 +√π +� x +0 +e−y2dy. +(2) +We note that a naive estimate of the mean speed, ob- +tained from the 120 µm and 1.2 µs waveform displace- +ment and duration, would be 100 m/s. In contrast, the +fit of Fig. 2c (red curve) yields a much higher maximum +slope of 251(6) m/s. +To determine an effective mean +speed, we estimate the beginning and ending times of +the transport by determining when the ion is within a +given distance of its asymptotic positions. Such a choice +must always be made if we are to take into account the +influence of the electrode filters on the results, just as +similar cutoffs must be chosen when studying the time +response of such analog filters more generally. In partic- +ular, here we choose a distance from the fitted asymptotes +that equals the ground-state extent of the 1.7 MHz trap +potential, approximately 8 nm. This rather arbitrary de- +cision, as well as our selection of Eq. 1 as a model for ion +position, both play an outsized role in our determination +of the mean ion speed and highlight the need to define +these terms with sufficient detail in studies of ion trans- +port. With these choices we determine a mean speed of +83(2) m/s, slightly lower than the naive estimate. +To avoid additional gate errors within a quantum al- +gorithm, fast transport must not excite excessive ion mo- +tion. For trapped ions in thermal states of motion probed +in the Lamb-Dicke regime, one can measure the ratio of +the first red and first blue sideband excitations to de- +termine the mean thermal mode occupation (⟨nth⟩) [29]. +However, characterization of non-thermal (e.g. coherent) +distributions is more complicated since the ratio of first +sidebands can vary with the probe duration. Given that +fast transport can leave the ion with a large coherent exci- +tation (⟨ncoh⟩) [10, 11], we directly fit the time-dependent +excitation of the first blue sideband assuming a convolu- +tion of coherent and thermal distributions [11]. +To measure the ion’s axial motional excitation after the +full transport (diagram in Fig. 1b), we apply the same +waveform as before but do not shelve the ion with an ini- +tial π-pulse. Instead, following the ion’s adiabatic return +to its initial position, we drive the blue axial motional +sideband of the S1/2−D5/2 transition and we analyze the +dependence of S1/2 state populations PS on pulse dura- +tion (sideband flopping curves). This probability is sen- +sitive to the motional state and yields information about +both the average mode occupation and its statistical dis- +tribution [30]. Neglecting the radial modes, it is given +by +PS(t) = 1 +2 +� +1 + e−γt +∞ +� +n=0 +pn cos(2Ωn,n+1t) +� +(3) +where pn is the mode population fraction in the Fock +state |n⟩, γ is a phenomenological decoherence rate, and +Ωn,n+1 is the Rabi frequency for the first blue sideband +transition for an ion starting in |n⟩. +We use the full +expression for the first blue sideband Rabi frequency [31], +Ωn,n+1 = ηΩ0e−η2/2 +� +1 +n + 1L1 +n(η2), +(4) +where Ω0 is the optical carrier Rabi frequency, η is the +Lamb-Dicke parameter, and L1 +n is the nth associated La- +guerre polynomial of order 1. +As a simple model, we +assume that any excitation can be represented as a con- +volution of thermal and coherent contributions [11, 30], +and we fit the measured probabilities to Eq. 3 under this +assumption. +Figure 3a shows a fit to the time-dependent blue side- +band excitation, revealing a purely thermal (⟨ncoh⟩ = 0) + +4 +a) +b) +c) +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Deshelve Probability +Time (µs) +Time (µs) +Position (µm) +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +0 +50 +100 +150 +200 +time +Cool & +Prep +Detect +Transport +Optical operations + → +π +τ + → +FIG. 2. +(a) Ion trajectory measurement sequence. An ion +in a stationary potential at zi is cooled and prepared in the +S1/2 level and then shelved into the D5/2 level with a reso- +nant 729 nm π-pulse. The transport waveform to zf begins +to play, and following a delay τ the 854 nm beam is pulsed for +200 ns, thereby deshelving a portion of the D5/2 population. +The ion is shuttled adiabatically back to its initial location +zi for final detection. (b) Experimental time-dependence of +deshelving probability sampled at 10 ns intervals after the +start of the fast transport waveform; error bars represent the +68% confidence interval in state populations assuming bino- +mial statistics. +(c) Time-dependence of ion position, with +experimental data (black points) and empirical fit (red line, +see main text). Here, the experimental points are obtained by +inverting the polynomial curve in Fig. 1c with the measured +points in (b). The fit yields a maximum speed of 251(6) m/s, +while the waveform was designed with 30 samples at 40 ns +intervals to shuttle the ion across a 120 µm displacement in +1.2 µs (100 m/s). +−20 +0 +20 +Frequency (kHz) +0.4 +0.6 +0.8 +1.0 +Probability +¯nth = 1.0 ± 0.2 +0 +100 +200 +300 +400 +Time (µs) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Probability +¯nth = 1.0 ± 0.1 +¯nth = 1.0 ± 0.1 +a) +b) +FIG. 3. +Characterization of ion temperature after fast trans- +port. (a) We perform a pulse on the axial blue motional side- +band for a variable duration after the fast transport operation +described in Fig. 2. The red trace represents a fit of the data +(black points) to Eq. 3, which yields a purely thermal axial +mode excitation of 1.0(1) quanta. Here we have added to the +transport waveform a sinusoidal oscillation near the ion ax- +ial frequency with appropriate amplitude and phase to remove +transport-induced coherent excitation. (b) Red and blue side- +band lineshapes, also measured after optimized fast transport, +confirm that the ion is nearly in the ground state. Fits (solid +curves) to the data (individual points) confirm the low ion +temperature: the ratio of sideband amplitudes corresponds +to 1.0(2) quanta. +Error bars represent the 68% confidence +interval in state populations assuming binomial statistics +excitation of ⟨nth⟩ = 1.0(1). +Having determined that +the excitation is thermal, we verify the mode temper- +ature through a comparison of red- and blue-sideband +transition amplitudes [29] (Fig. 3b). This yields a post- +transport temperature of ⟨nth⟩ = 1.0(2) compared to +⟨nth⟩ = 0.3(1) measured before transport, and we con- +clude that the optimized transport induces an additional +0.7(2) quanta of motional excitation. +We note that, +without the resonant de-excitation of motion during the +transport operation, we measure an additional coherent +excitation of ⟨ncoh⟩ = 61.7(6). +In conclusion, we have developed a general method +for experimentally characterizing ion transport trajecto- + +5 +ries using position-dependent optical deshelving, and we +verified the technique in a surface-electrode ion trap by +shuttling an ion along a linear trajectory of 120 µm (two +electrode widths) with a 1.2 µs waveform. Owing to the +impact of filter distortion on the transport potentials, +the ion reaches instantaneous speeds significantly higher +than might be naively assumed from the waveform de- +sign. We characterized the final motional state using two +complementary methods to fit blue sideband flop curves +as well as red and blue sideband lineshapes. +Even at +this high speed the transport incurs only 0.7(2) quanta +of axial excitation, small enough to have minimal impact +within a quantum algorithm. +Beyond single-ion transport through linear sections, +this technique could also be applied to optimize fast +merging and separation of ions into chains [10, 32]. With +the incorporation of multiple deshelving wavelengths, the +positions of disparate ion species could be tracked simul- +taneously [9]. The method might prove particularly use- +ful when optimizing the paths of ions through junctions +of linear sections, where trajectories deviate significantly +from straight lines both horizontally and vertically. 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A 90, 033410 (2014). + diff --git a/29E4T4oBgHgl3EQf0A2b/content/tmp_files/load_file.txt b/29E4T4oBgHgl3EQf0A2b/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..871eb2ec3064a039f682241b92c6ca0d50a2f33e --- /dev/null +++ b/29E4T4oBgHgl3EQf0A2b/content/tmp_files/load_file.txt @@ -0,0 +1,694 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf,len=693 +page_content='Characterization of Fast Ion Transport via Position-Dependent Optical Deshelving Craig R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Clark,∗ Creston D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Herold, James T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Merrill, Holly N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Tinkey, Wade Rellergert, Robert Clark, Roger Brown, Wesley D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Robertson, Curtis Volin, Kara Maller, Chris Shappert, Brian J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' McMahon, Brian C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Sawyer,† and Kenton R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Brown‡ Georgia Tech Research Institute, Atlanta, Georgia 30332, USA Ion transport is an essential operation in some models of quantum information processing, where fast ion shuttling with minimal motional excitation is necessary for efficient, high-fidelity quan- tum logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' While fast and cold ion shuttling has been demonstrated, the dynamics and specific trajectory of an ion during diabatic transport have not been studied in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Here we describe a position-dependent optical deshelving technique useful for sampling an ion’s position through- out its trajectory, and we demonstrate the technique on fast linear transport of a 40Ca+ ion in a surface-electrode ion trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' At high speed, the trap’s electrode filters strongly distort the transport potential waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' With this technique, we observe deviations from the intended constant-velocity (100 m/s) transport: we measure an average speed of 83(2) m/s and a peak speed of 251(6) m/s over a distance of 120 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Systems of trapped atomic ions represent some of the most promising platforms for quantum information pro- cessing, benefiting from long qubit coherence times and from the highest operational fidelities demonstrated to date [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' In the QCCD ion-trap architecture [4], ions are transported between various regions within the pro- cessor to reconfigure which ions can interact at any given step of an algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Such an architecture demands finely tuned control of ion shuttling in order to imple- ment reconfigurations as quickly as possible without also degrading the fidelities of subsequent logic gates [1, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Performing logic operations on the ions at the same time as they are transported within the trap also requires well characterized and reproducible trajectories for success [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Recent work has shown that ion transport contributes substantially to the latency of current QCCD systems [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Rapid linear ion transport has been demonstrated previ- ously [10–12] but has not been widely adopted likely due to the demands it places on waveform control hardware and on motional characterization and calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Trans- port at slower speeds has also been achieved through more complicated structures such as junctions of linear sections [13–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' While prior results have demonstrated transport between two locations with sub-quanta mo- tional excitation, the ion trajectory during transport has not been measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' In fact, all prior reported transport speeds assumed an average speed calculated from the dis- tance between static well positions and the designed play- back speed of the trapping voltage waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Some ex- periments have proved that an ion was transported to an intended location, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' by applying a focused laser pulse at that location [10, 12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' however, those experiments did not verify the exact time of arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' In this work, we present a method to measure the loca- tion of an ion throughout the entire arc of its transport, ∗ craig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='clark@gtri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='edu † brian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='sawyer@gtri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='edu ‡ kenton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='brown@gtri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='edu which enables us to reliably extract both instantaneous and average velocity during transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' This is achieved by measuring the probability for the ion to undergo a spontaneous irreversible transition from a metastable ex- cited state to a ground state when illuminated by a laser beam having a spatial intensity gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Similar ideas have been explored in the context of neutral atoms in optical cavities [18, 19], as well as in the field of ultrafast physics [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' The method presented here contrasts with previous schemes employing Fourier-limited coherent op- tical interactions to extract in-flight Doppler shifts [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Here, we leverage the technique to show that an ion’s trajectory deviates significantly from a simple constant- velocity path as the trap’s electrode filters distort the ap- plied potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Although the intended trajectory has a constant speed of 100 m/s, corresponding to a displace- ment of 120 µm (two electrodes) in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='2 µs, the ion ac- tually achieves a maximum speed of 251(6) m/s in the middle of its path due to this distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' These trajec- tory deviations lead also to a large coherent displacement of the ion’s motional state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' We superimpose an addi- tional compensating sinusoidal potential onto the wave- form to remove this displacement [12, 21] and achieve a final transport-induced excitation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='7(2) quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Our experimental system employs a GTRI/Honeywell Ball Grid Array (BGA) trap [22], shown schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 1a, mounted in a room-temperature ultrahigh vac- uum chamber and confining a single 40Ca+ ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' The trap radiofrequency (rf) electrode is driven at 55 MHz and realizes radial frequencies of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='7 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='2 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' We determine the control electrode voltages necessary to produce axial confinement at different positions along the trap symmetry axis using an in-house boundary ele- ment method electrostatic solver [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' We employ NIST- designed PDQ digital-to-analog converters (DACs) [21] to vary the potentials on the axial control electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Each control electrode’s potential is filtered with a two- pole low-pass filter having a bandwidth of 608 kHz [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' The axial trap frequency (parallel to the axis of symme- try) is approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='7 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' The axial heating rate arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='05279v1 [quant-ph] 12 Jan 2023 2 854 nm Intensity a) 𝑧 𝑧� time Cool & Prep Detect Adiabatic Transport b) 𝒛𝒊 → 𝒛 π 𝒛 → 𝒛𝒊 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='8 1 0 50 100 150 200 Deshelve Probability Position (µm) c) 𝒛𝒊 𝒛𝒇 𝑧� Optical operations 𝑺𝟏/𝟐 𝑫𝟓/𝟐 𝑷𝟑/𝟐 729 nm 854 nm 393 nm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' (a) (Left) Schematic of the BGA trap with a graph of the nominal 854 nm repumper intensity profile superimposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' The initial and final positions of the ion for the characterized transport are zi and zf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' (Right) Energy level diagram of 40Ca+ ion showing the shelving transition driven by 729 nm light in red, the deshelving transition driven by 854 nm light in orange, and spontaneous decay through emission of 393 nm light in purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' (b) Ion trajectory calibration sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' An ion in a stationary potential at zi is cooled and prepared in the S1/2 level and then shelved into the D5/2 level with a resonant 729 nm π-pulse (shown in red in the timeline).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' The ion is adiabatically transported to another position along the trap axis, and the 854 nm beam is pulsed for 200 ns, thereby deshelving a portion of the D5/2 population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' This deshelving fraction is in one-to-one correspondence with the local beam intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' The ion is shuttled adiabatically back to its initial location zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' (c) Deshelving probability as a function of ion position (black points) with error bars representing the 68% confidence interval assuming binomial statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' The red line represents a polynomial fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' for a stationary potential is 210 quanta/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' For Doppler cooling we use a 397 nm laser beam, nearly resonant with the S1/2 − P1/2 transition, in combina- tion with another beam at 866 nm which is used to re- turn population from the D3/2 level into the cooling cy- cle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Pulses from a beam at 729 nm, resonant with the S1/2−D5/2 transition, coherently populate the D5/2 level and are employed for sideband cooling and motional- state characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Another beam at 854 nm is re- sponsible for deshelving population from the D5/2 to the S1/2 level via the intermediate P3/2 level (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='1a) [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' We distinguish between an ion in the S1/2 level (bright) and one in the D5/2 level (dark) via observation of state- dependent 397 nm fluorescence as recorded on a photo- multiplier tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' After many experimental repetitions, the bright and dark state populations are estimated via maximum likelihood (see Supplemental Material of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' To characterize the ion’s position throughout the full arc of its roughly linear trajectory, we implement a position-sensitive optical deshelving technique [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' For this, we prepare the ion in the D5/2 state before transport and then apply a pulse (200 ns, shorter than the trans- port duration) of 854 nm light at a later time, thereby deshelving the D5/2 level with a probability dependent on the laser beam’s local intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' By choosing the Gaus- sian waist w0 of the 854 nm beam such that its intensity varies significantly and monotonically along the ion’s tra- jectory (w0 ∼ 100 µm), we achieve a position-dependent deshelving probability (Pd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' We can therefore invert spa- tially monotonic measurements of this probability, ac- quired at various times after the start of the transport, to determine the ion’s position at these instants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' The beam is directed perpendicular to the linear transport axis to remove velocity dependence (first-order Doppler shifts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=') from the deshelving probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' The above technique requires calibration of the deshelving probability for a given ion location using a procedure diagrammed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' For this, we Doppler and sideband cool the ion nearly to its axial motional ground state (¯n < 1) and prepare it in S1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' We then shelve it into the D5/2 level with a resonant 729 nm π- pulse and subsequently move it adiabatically to a given position along the trap axis (as determined from an elec- trostatic model of the trapping potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' We illuminate the ion with a 200 ns, 854 nm deshelving pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' After adiabatically returning the ion to its initial location, we determine its state by collecting fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Repeating this experiment 400 times each at 2 µm intervals yields a measurement of deshelving probability Pd(z) for each location as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' We fit Pd(z) with a polyno- mial (red curve of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 1c) rather than with a Gaussian to allow for possible distortion of the 854 nm laser beam mode shape, and we ensure that Pd(z) is single-valued by having previously displaced the beam such that its inten- sity varies monotonically through the calibration region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' It is important that any movement of ions from one place to another in the trap be accomplished with only a small degree of motional excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' For the experiments outlined below, we design the waveform as a simple lin- ear time interpolation of the potential minimum through a 120 µm displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' We choose a transport duration (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='2 µs) that is an integer multiple of the ion’s harmonic motional period (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='6 µs, nominally held constant during the transport).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' In the absence of filter distortion, such a waveform is expected to coherently excite ion motion at the beginning of transport but subsequently to suppress this excitation at the end [10, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Our filter bandwidths lie below the ion motional frequencies by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' At slow enough speeds, the action of the filters is to smooth out 3 the beginning and ending accelerations, so that the ion’s motion is not excited as much during transport as would otherwise be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' At faster speeds such as are used here, filter distortion is great enough that simple scal- ing of the waveform confinement strength can no longer fully suppress the final excitation [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' However, provided that the waveform is performed identically for each ex- perimental repetition and that the confinement remains harmonic, the final excitation corresponds to a coherent displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' We remove it by superimposing on the transport waveform a sinusoidal rf pulse applied to four of the trap electrodes, and we optimize the phase, fre- quency, and amplitude of this pulse to achieve minimum mode occupation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' With the Pd(z) calibration complete, we then trans- port the ion through the full trajectory of 120 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 2a gives a diagram of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' We pulse the 854 nm deshelving laser for 200 ns while the ion is in motion, with a configurable delay between the start of the trans- port waveform and the start of the pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' The ion is then returned adiabatically to its initial location for state de- tection [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' With 100 repetitions of this experiment at each delay, we obtain a map Pd(t) of deshelving probabil- ity in time following the start of the waveform (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' To obtain the corresponding position map z(t), we in- vert the polynomial fit of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 1c to obtain z(Pd) and then compute z(Pd(t)), producing the points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Finally, we fit the z(t) data of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 2c with the follow- ing phenomenological expression to extract the mean and maximum linear velocities: z(t) = zi + √π 2 vmaxtσ � erf �t − tc tσ � − erf �t0 − tc tσ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' (1) Equation 1 is derived assuming that the speed follows a Gaussian profile in time, an empirical assumption justi- fied by its agreement with the z(t) data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Here, zi represents the initial ion position, t0 is the initial time (time when z(t) = zi), tσ is the 1/e temporal half-width of the ion speed, and tc is the time of maximum speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' We use the following standard definition for the error function: erf(x) = 2 √π � x 0 e−y2dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' (2) We note that a naive estimate of the mean speed, ob- tained from the 120 µm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='2 µs waveform displace- ment and duration, would be 100 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' In contrast, the fit of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 2c (red curve) yields a much higher maximum slope of 251(6) m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' To determine an effective mean speed, we estimate the beginning and ending times of the transport by determining when the ion is within a given distance of its asymptotic positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Such a choice must always be made if we are to take into account the influence of the electrode filters on the results, just as similar cutoffs must be chosen when studying the time response of such analog filters more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' In partic- ular, here we choose a distance from the fitted asymptotes that equals the ground-state extent of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='7 MHz trap potential, approximately 8 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' This rather arbitrary de- cision, as well as our selection of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 1 as a model for ion position, both play an outsized role in our determination of the mean ion speed and highlight the need to define these terms with sufficient detail in studies of ion trans- port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' With these choices we determine a mean speed of 83(2) m/s, slightly lower than the naive estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' To avoid additional gate errors within a quantum al- gorithm, fast transport must not excite excessive ion mo- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' For trapped ions in thermal states of motion probed in the Lamb-Dicke regime, one can measure the ratio of the first red and first blue sideband excitations to de- termine the mean thermal mode occupation (⟨nth⟩) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' However, characterization of non-thermal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' coherent) distributions is more complicated since the ratio of first sidebands can vary with the probe duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Given that fast transport can leave the ion with a large coherent exci- tation (⟨ncoh⟩) [10, 11], we directly fit the time-dependent excitation of the first blue sideband assuming a convolu- tion of coherent and thermal distributions [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' To measure the ion’s axial motional excitation after the full transport (diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 1b), we apply the same waveform as before but do not shelve the ion with an ini- tial π-pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Instead, following the ion’s adiabatic return to its initial position, we drive the blue axial motional sideband of the S1/2−D5/2 transition and we analyze the dependence of S1/2 state populations PS on pulse dura- tion (sideband flopping curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' This probability is sen- sitive to the motional state and yields information about both the average mode occupation and its statistical dis- tribution [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Neglecting the radial modes, it is given by PS(t) = 1 2 � 1 + e−γt ∞ � n=0 pn cos(2Ωn,n+1t) � (3) where pn is the mode population fraction in the Fock state |n⟩, γ is a phenomenological decoherence rate, and Ωn,n+1 is the Rabi frequency for the first blue sideband transition for an ion starting in |n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' We use the full expression for the first blue sideband Rabi frequency [31], Ωn,n+1 = ηΩ0e−η2/2 � 1 n + 1L1 n(η2), (4) where Ω0 is the optical carrier Rabi frequency, η is the Lamb-Dicke parameter, and L1 n is the nth associated La- guerre polynomial of order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' As a simple model, we assume that any excitation can be represented as a con- volution of thermal and coherent contributions [11, 30], and we fit the measured probabilities to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 3 under this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Figure 3a shows a fit to the time-dependent blue side- band excitation, revealing a purely thermal (⟨ncoh⟩ = 0) 4 a) b) c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='0 Deshelve Probability Time (µs) Time (µs) Position (µm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='50 0 50 100 150 200 time Cool & Prep Detect Transport Optical operations → π τ → FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' (a) Ion trajectory measurement sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' An ion in a stationary potential at zi is cooled and prepared in the S1/2 level and then shelved into the D5/2 level with a reso- nant 729 nm π-pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' The transport waveform to zf begins to play, and following a delay τ the 854 nm beam is pulsed for 200 ns, thereby deshelving a portion of the D5/2 population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' The ion is shuttled adiabatically back to its initial location zi for final detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' (b) Experimental time-dependence of deshelving probability sampled at 10 ns intervals after the start of the fast transport waveform;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' error bars represent the 68% confidence interval in state populations assuming bino- mial statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' (c) Time-dependence of ion position, with experimental data (black points) and empirical fit (red line, see main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Here, the experimental points are obtained by inverting the polynomial curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 1c with the measured points in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' The fit yields a maximum speed of 251(6) m/s, while the waveform was designed with 30 samples at 40 ns intervals to shuttle the ion across a 120 µm displacement in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='2 µs (100 m/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' −20 0 20 Frequency (kHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='0 Probability ¯nth = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='2 0 100 200 300 400 Time (µs) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='0 Probability ¯nth = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='1 ¯nth = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='1 a) b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Characterization of ion temperature after fast trans- port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' (a) We perform a pulse on the axial blue motional side- band for a variable duration after the fast transport operation described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' The red trace represents a fit of the data (black points) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 3, which yields a purely thermal axial mode excitation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='0(1) quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Here we have added to the transport waveform a sinusoidal oscillation near the ion ax- ial frequency with appropriate amplitude and phase to remove transport-induced coherent excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' (b) Red and blue side- band lineshapes, also measured after optimized fast transport, confirm that the ion is nearly in the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Fits (solid curves) to the data (individual points) confirm the low ion temperature: the ratio of sideband amplitudes corresponds to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='0(2) quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Error bars represent the 68% confidence interval in state populations assuming binomial statistics excitation of ⟨nth⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='0(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Having determined that the excitation is thermal, we verify the mode temper- ature through a comparison of red- and blue-sideband transition amplitudes [29] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' This yields a post- transport temperature of ⟨nth⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='0(2) compared to ⟨nth⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='3(1) measured before transport, and we con- clude that the optimized transport induces an additional 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='7(2) quanta of motional excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' We note that, without the resonant de-excitation of motion during the transport operation, we measure an additional coherent excitation of ⟨ncoh⟩ = 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='7(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' In conclusion, we have developed a general method for experimentally characterizing ion transport trajecto- 5 ries using position-dependent optical deshelving, and we verified the technique in a surface-electrode ion trap by shuttling an ion along a linear trajectory of 120 µm (two electrode widths) with a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='2 µs waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Owing to the impact of filter distortion on the transport potentials, the ion reaches instantaneous speeds significantly higher than might be naively assumed from the waveform de- sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' We characterized the final motional state using two complementary methods to fit blue sideband flop curves as well as red and blue sideband lineshapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Even at this high speed the transport incurs only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content='7(2) quanta of axial excitation, small enough to have minimal impact within a quantum algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Beyond single-ion transport through linear sections, this technique could also be applied to optimize fast merging and separation of ions into chains [10, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' With the incorporation of multiple deshelving wavelengths, the positions of disparate ion species could be tracked simul- taneously [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' The method might prove particularly use- ful when optimizing the paths of ions through junctions of linear sections, where trajectories deviate significantly from straight lines both horizontally and vertically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' With multiple deshelving beams at complementary angles one could isolate an ion’s position in all three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' This work was done in collaboration with Los Alamos National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Clark, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Tinkey, B.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' 117, 174901 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Charles Doret, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Amini, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Wright, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} 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Poschinger, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} +page_content=' A 90, 033410 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29E4T4oBgHgl3EQf0A2b/content/2301.05279v1.pdf'} diff --git a/2dE2T4oBgHgl3EQfjAeF/content/tmp_files/2301.03964v1.pdf.txt b/2dE2T4oBgHgl3EQfjAeF/content/tmp_files/2301.03964v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..52334e5c21dfb78e21998f2ac1092086c3a2a39e --- /dev/null +++ b/2dE2T4oBgHgl3EQfjAeF/content/tmp_files/2301.03964v1.pdf.txt @@ -0,0 +1,3208 @@ +Trade-offs between cost and information in cellular prediction +Age J. Tjalma,1 Vahe Galstyan,1 Jeroen Goedhart,1 Lotte Slim,1 Nils B. Becker,2 and Pieter Rein ten Wolde1, ∗ +1AMOLF, Science Park 104, 1098 XG Amsterdam, The Netherlands +2Theoretical Systems Biology, German Cancer Research Center, 69120 Heidelberg, Germany +(Dated: January 11, 2023) +Living cells can leverage correlations in environmental fluctuations to predict the future environ- +ment and mount a response ahead of time. To this end, cells need to encode the past signal into the +output of the intracellular network from which the future input is predicted. Yet, storing information +is costly while not all features of the past signal are equally informative on the future input signal. +Here, we show, for two classes of input signals, that cellular networks can reach the fundamental +bound on the predictive information as set by the information extracted from the past signal: push- +pull networks can reach this information bound for Markovian signals, while networks that take +a temporal derivative can reach the bound for predicting the future derivative of non-Markovian +signals. However, the bits of past information that are most informative about the future signal are +also prohibitively costly. As a result, the optimal system that maximizes the predictive information +for a given resource cost is, in general, not at the information bound. Applying our theory to the +chemotaxis network of Escherichia coli reveals that its adaptive kernel is optimal for predicting +future concentration changes over a broad range of background concentrations, and that the system +has been tailored to predicting these changes in shallow gradients. +Keywords: prediction, information bottleneck, sensing, resource allocation +Single-celled organisms live in a highly dynamic envi- +ronment to which they continually have to respond and +adapt. +To this end, they employ a range of response +strategies, tailored to the temporal structure of the envi- +ronmental variations. When these variations are highly +regular, such as the daily light variations, it becomes ben- +eficial to develop a clock from which the time and hence +the current and future environment can be inferred [1, 2]. +In the other limit, when the fluctuations are entirely un- +predictable, cells have no choice but to resort to either +the strategy of detect-and-respond or the bet-hedging +strategy of stochastic switching between different phe- +notypes [3]. Yet arguably the most fascinating strategy +lies in between these two extremes. When the environ- +mental fluctuations happen with some regularity, then it +becomes feasible to predict the future environment and +initiate a response ahead of time. While it is commonly +believed that only higher organisms can predict the fu- +ture, experiments have vividly demonstrated that even +single-cell organisms can leverage temporal correlations +in environmental fluctuations in order to predict, e.g., +future nutrient levels [4, 5]. +The ability to predict future signals can provide a fit- +ness benefit [6]. The capacity to anticipate changes in +oxygen levels [4], or the arrival of sugars or stress signals +[5], can increase the growth rate of single-celled organ- +isms; modeling has revealed that prediction can enhance +bacterial chemotaxis [7]. +Yet, a predict-and-anticipate +strategy is only advantageous if the cell can reliably pre- +dict the future on timescales that are longer than the +time it takes to mount a response. What fundamentally +limits the accuracy of cellular prediction remains, how- +ever, poorly understood. +∗ p.t.wolde@amolf.nl +While the cell needs to predict the future environ- +ment, it can only sense the present and remember the +past (Fig. 1A). Consequently, for a given amount of in- +formation the cell can store about the present and past +signal, there is a maximum amount of information it can +possibly have about the future [6, 8] (Fig. 1C-I). This in- +formation bound is determined by the temporal structure +of the environmental fluctuations [8, 9]. +How close cells can come to this bound depends on +the design of the intracellular biochemical network that +senses and processes the environmental signals (Fig. 1B). +To maximize the predictive power the cell must use its +memory effectively: it should extract only those charac- +teristics from the present and past signal that are most +informative about the future [7]. Whether it can do so, +is determined by the topology of the signaling network. +Moreover, like any information processing device, bio- +chemical networks require resources to be built and run. +Molecular components are needed to construct the net- +work, space is required to accommodate the components, +time is needed to process the information, and energy is +required to synthesize the components and operate the +network [10]. These resources constrain the design and +performance of any biochemical network, and the ca- +pacity to sense and process information is no exception +(Fig. 1C-II). +Cellular signaling systems provide a unique opportu- +nity for revealing the resource requirements for predic- +tion. Cells live in a highly dynamic environment, with +temporal statistics that are expected to vary markedly. +Moreover, signaling networks have distinct topologies, +which are likely tailored to the temporal statistics of the +environment [7]. In addition, for cellular systems we can +actually quantify the information processing capacity as +a function of the resources that are necessary to build +and run them—protein copies, time, and energy [10, 11]. +arXiv:2301.03964v1 [physics.bio-ph] 10 Jan 2023 + +2 +Cellular systems are thus ideal for elucidating the rela- +tionships between future and past information, system +design (i.e. network topology) and resource constraints. +Here, we derive the bound on the prediction precision as +set by the information extracted from the past signal for +two types of input signals. We will determine how close +cellular networks can come to this bound, and how this +depends on the topology of the network and the resources +to build and run it. +We find that for the two classes of input signals stud- +ied, cellular networks exists that can reach the informa- +tion bound, yet reaching the bound is exceedingly costly. +The first class of input signals consists of Markovian sig- +nals. Using the Information Bottleneck Method (IBM) +[8, 12], we first show that the system that reaches the +information bound copies the most recent input signal +into the output from which the future input is predicted. +Push-pull networks consisting of chemical modification or +GTPase cycles, which are ubiquitous in prokaryotic and +eukaryotic cells [13, 14], should be able to reach the infor- +mation bound, because they are at heart copying devices +[10, 11]. Yet, copying the most recent input into the out- +put is extremely costly, because the operating cost, as set +by the chemical power to drive the cycle, diverges at high +copying speed. More surprisingly, our results show that +the predictive and past information can be raised simul- +taneously by moving away from the information bound, +even when the operating cost is negligible: the optimal +system that maximizes the predictive information for a +given protein synthesis cost is, in general, not at the in- +formation bound. The number of bits of past information +per protein cost can be raised by increasing the integra- +tion time. While this decreases the predictive power per +bit of past information, thereby moving the system away +from the information bound, it can increase the total pre- +dictive information per protein cost. Our analysis thus +highlights that not all bits of past information are equally +costly, nor predictive. +Living cells that navigate their environment typically +experience signals with persistence as generated by their +own motion, which motivated us to study a simple class +of non-Markovian signals. Moreover, these cells can typ- +ically detect changes in the concentration over a range of +background concentrations that is orders of magnitude +larger than the change in the concentration over the ori- +entational correlation time of their movement. Our anal- +ysis reveals that in such a scenario the optimal kernel that +allows the system to reach the information bound on pre- +dicting the future input derivative is a perfectively adap- +tive, derivative-taking kernel, precisely as the bacterium +E. coli employs [15]. We again find, however, that reach- +ing the information bound is prohibitively costly. The +reason is that taking an instantaneous derivative, which +is the characteristic of the input that is most informative +about the future derivative, reduces the gain to zero be- +cause the system instantly adapts; the response becomes +thwarted by biochemical noise. The optimal system that +maximizes the predictive information under a resource +constraint thus emerges from a trade-off between taking a +derivative that is recent and one that is reliable. Finally, +our analysis reveals that the E. coli chemotaxis system +has been optimally designed to predict future concentra- +tion changes in shallow gradients. +RESULTS +We focus on cellular signaling systems that respond +linearly to changes in the input signal [11, 16–19]. These +systems not only allow for analytical results, but also +describe information transmission often remarkably well +[19–22]. The output of these systems can be written as +x(t) = +� t +−∞ +dt′k(t − t′)ℓ(t′) + ηx(t), +(1) +where k(t) is the linear response function, ℓ(t) the input +signal, and ηx(t) describes the noise in the output. We +will consider stationary signals with different temporal +correlations, obeying Gaussian statistics. +Any prediction about the future state of the environ- +ment must be based on information obtained from its +past (Fig. 1C-I). In particular, the cell needs to predict +the input ℓτ ≡ ℓ(t + τ) at a time τ into the future from +the current output x0 ≡ x(t), which itself depends on +the input signal in the past, Lp ≡ (ℓ(t), ℓ(t′), · · · ), with +t > t′ > · · · . The (qualitative) shape of the integration +kernel k(t), e.g. exponential, adaptive or oscillatory, is +determined by the topology of the signaling network [7]. +The kernel shape describes how the past signal is mapped +onto the current output, and hence which characteristics +of the past signal the cell uses to predict the future signal. +To maximize the accuracy of prediction, the cell should +extract those features that are most informative about +the future signal. These depend on the statistics of the +input signal. +Deriving the upper bound on the predictive informa- +tion as set by the past information is an optimisation +problem, which can be solved using the IBM [8]. It en- +tails the maximization of an objective function L: +max +P (x0|Lp) [L ≡ I(x0; ℓτ) − γI(x0; Lp)] . +(2) +Here, Ipred ≡ I(x0; ℓτ) is the predictive information, +which is the mutual information between the system’s +current output x0 and the future ligand concentration +ℓτ. The past information Ipast ≡ I(x0; Lp) is the mutual +information between x0 and the trajectory of past lig- +and concentrations Lp. The Lagrange multiplier γ sets +the relative cost of storing past over obtaining predic- +tive information. Given a value of γ, the objective func- +tion in Eq. 2 is maximized by optimizing the conditional +probability distribution of the output given the past in- +put trajectory, P(x0|Lp). For the linear systems consid- +ered here, this corresponds to optimizing the mapping +of the past input signal onto the current output via the + +3 +past information +predictive information +resources +inaccessible +inaccessible +I +II +information bound +A +B +C +Different input signals +maintenance +operating +Optimal networks +X +X +RL +* +time +signal +output +past info +predictive info +now +concentration +FIG. 1. Cells use biochemical networks to remember the past and predict the future. (A) Cells compress the past +input into the dynamics of the signalling network from which the future input is then predicted. (B) The optimal topology +of the network for predicting the future signal depends on the temporal statistics of the input signal. Push-pull networks, +consisting of chemical modification cycles or GTPase cycles, can optimally predict the future value of Markovian signals, with +correlation time τℓ; derivative-taking networks, like the E. coli chemotaxis system, can optimally predict the future derivative +of non-Markovian signals, with correlation time τv. The push-pull network consists of a receptor that drives a downstream +phosphorylation cycle. +The ligand binds the receptor with a correlation time τc. +The push-pull network, driven by ATP +turnover, integrates the receptor with an integration time τr. The chemotaxis system is a push-pull network, yet augmented +with negative feedback on the receptor activity via methylation on a timescale τm, as indicated by the dashed grey line. The +total resource cost consists of a maintenance cost of receptor and readout synthesis at the growth rate λ, and an operating +cost of driving the cycle. (C) The predictive information on the future signal Ipred is fundamentally bounded by how much +information Ipast it has about the past signal (panel I), which in turn is limited by the resources necessary to build and operate +the biochemical network (panel II) [6]. +integration kernel k(t). Since our model obeys Gaussian +statistics, we use the Gaussian IBM to derive the optimal +kernel kopt(t) and the information bound, defined to be +the maximum predictive information as set by the past +information [12] (see Appendix C). +Markovian signals +Optimal prediction of Markovian signals: biochemical +copying +Arguably the most elementary type of signal, albeit +perhaps the hardest to predict, is a Markovian signal. +We consider a Markovian signal ℓ(t), of which the devia- +tions δℓ(t) = ℓ(t) − ¯ℓ from its mean ¯ℓ follow an Ornstein- +Uhlenbeck (OU) process: +δ ˙ℓ = −δℓ(t)/τℓ + ηℓ(t), +(3) +where τℓ is the correlation time of the fluctuations, and +ηℓ(t) is Gaussian white noise, ⟨η(t)η(t′)⟩ = 2σ2 +ℓ/τℓ δ(t − +t′), with σ2 +ℓ the amplitude of the signal fluctuations. This +input signal obeys Gaussian statistics, characterized by +⟨δℓ(0)δℓ(t)⟩ = σ2 +ℓ exp(−t/τℓ). The optimal mapping is +therefore a linear one. Utilizing the Gaussian IBM frame- +work [12], we find that the optimal integration kernel is +given by (see Appendix C 2) +kopt(t − t′) = aδ(t − t′). +(4) +This optimal integration kernel corresponds to a signaling +system that copies the current input into the output. +This is intuitive, since for a Markovian signal there is +no additional information in the past signal that is not +already contained in the present one. The prefactor a +determines the gain ∂¯x/∂¯ℓ, which together with the noise +strength σ2 +ηx (Eq. 1) and the signal amplitude σ2 +ℓ set the +magnitude of the past and predictive information, Ipast +and Ipred, respectively (see Appendix C 1). +Fig. 2-I shows the maximum predictive information as +set by the past information. This information bound ap- +plies to any linear system that needs to predict a Marko- +vian signal. How close can biochemical systems come to +this bound? +Push-pull network can be at the information bound, yet +increase the predictive and past information by moving away +from it +Although the upper bound on the accuracy of predic- +tion is determined by the signal statistics, how close cells +can come to this bound depends on the topology of the +cellular signaling system, and the resources devoted to +building and operating it. A network motif that could +reach the information bound for Markovian signals is the +push-pull network (Fig. 2), because it is at heart a copy- +ing device: it samples the input by copying the state of + +4 +I +II +predictive information (bits) +resources +past information (bits) +FIG. 2. The optimal push-pull network is not at the +information bound. Panel I: The black line is the informa- +tion bound that maximizes the predictive information Ipred = +I(x0; ℓτ) for a given past information Ipast = I(x0; Lp). The +red curve shows Ipred against Ipast for systems in which Ipred +has been maximized for a given resource cost C = RT + XT. +The blue curve shows Ipred versus Ipast for systems where +Ipast has been maximized for a given C. Panel II shows Ipast +against C for the corresponding systems. +The forecast in- +terval is τ = τℓ. The optimization parameters are the ratio +XT/RT, τr, p and f (see Appendix E). Parameter values: +(σℓ/¯ℓ)2 = 10−2, τc/τℓ = 10−2. +the input, e.g. the ligand-binding state of a receptor or +the activation state of a kinase, into the activation state +of the output, e.g. phosphorylation state of the readout +[10, 11, 23]. +We model the push-pull network in the linear-noise +approximation: +δ ˙ +RL = bδℓ(t) − δRL(t)/τc + ηRL(t), +(5) +˙ +δx∗ = γ δRL(t) − δx∗(t)/τr + ηx(t). +(6) +Here, δRL represents the number of ligand-bound recep- +tors and δx∗ the number of modified readout molecules, +defined as deviations from their mean values; b and γ +are parameters that depend on the number of recep- +tor and readout molecules, RT and XT respectively, the +fraction of ligand-bound receptors p and active readout +molecules f; ηRL and ηx are Gaussian white noise terms +(see Appendix E). Key parameters are the correlation +time of receptor-ligand binding, τc, and the relaxation +time of x∗, τr. The latter determines for how long x∗ +carries information on the ligand-binding state of the re- +ceptor and thus sets the integration time. The readout- +modification dynamics yield an exponential integration +kernel k(t) ∝ exp(−t/τr), which in the limit τr → 0 re- +duces to a δ-function, hinting that the system may reach +the information bound. +How much information cells can extract from the past +signal depends on the resources devoted to building and +operating the network (Fig. 2-II). We define the total +resource cost to be: +C = λ(RT + XT) + c1XT∆µ/τr +(7) +The first term expresses the fact that over the course +of the cell cycle all components need to be duplicated, +which means that they have to be synthesized at a speed +that is at least the growth rate λ. The second term de- +scribes the chemical power that is necessary to run the +push-pull network [10, 11]; it depends on the flux through +the network, XT/τr, and the free-energy drop ∆µ over a +cycle, e.g. the free energy of ATP hydrolysis in the case +of a phosphorylation cycle. The coefficient c1 describes +the relative energetic cost of synthesising the components +during the cell cycle versus that of running the system. +For simplicity, we first consider the scenario that the cost +is dominated by that of protein synthesis, setting c1 → 0. +While in this scenario RT + XT is constrained, XT/RT +and other system parameters are free for optimization. +The available resources put a hard bound on the in- +formation Ipast that can be extracted from the past sig- +nal, which in turn sets a hard limit on the predictive +information Ipred (Fig. 1C). To maximize the predictive +information, it therefore seems natural to maximize the +past information Ipast for a given resource cost C. The +blue line in Fig. 2-II shows the result for the push-pull +network. We then compute the corresponding predictive +information for the systems along this line, which is the +blue line in Fig. 2-I. Strikingly, the resulting information +curve lies far below the information bound, i.e. the upper +bound on the predictive information as set by the past +information (black line, Fig. 2-I). This shows that sys- +tems that maximize past information under a resource +constraint, do not in general also maximize predictive in- +formation. It implies that not all bits of past information +are equally predictive about the future. +Precisely because not all bits of past information are +equally predictive about the future, it is paramount to +directly maximize the predictive information for a given +resource cost in order to obtain the most efficient pre- +diction device. This yields the red lines in panels I and +II in Fig. 2. +It can be seen that the predictive infor- +mation is higher while the past information is lower, as +compared to the information curves of the systems opti- +mized for maximizing the past information under a re- +source constraint (blue lines). It reflects the idea that not +all bits are equally predictive. More surprisingly, while +the bound on the predictive information as set by the +resource cost (red line panel I) is close to the bound on +the predictive information as set by the past information +(black line), it does remain lower. This is surprising, be- +cause the push-pull network is a copying device [10, 23], +which can, as we will also show below, reach the latter +bound. These two observations together imply that not +all bits of past information are equally costly. +If they +were, the cell would select under the two constraints the +same bits based on their predictive information content, +and the bound on the predictive information as set by + +5 +the resource cost would overlap with that as set by the +past information. +We thus find that not all bits of past information are +equally predictive, nor equally costly. As we show next, +it implies that the optimal information processing system +faces a trade-off between using those bits of past infor- +mation that are most informative about the future and +those that are cheapest. +Trade-off between cost and predictive power per bit +To understand the connection between predictive and +past information, and resource cost, we map out the re- +gion in the information plane that can be reached given +a resource constraint C (Fig. 3A, green region). We im- +mediately make two observations. +Firstly, the system +can indeed reach the information bound. Secondly, the +system can increase both the past and the predictive in- +formation by moving away from the bound. To elucidate +these two observations, we investigate the system along +the isocost line of C = 104, which together with the in- +formation bound envelopes the accessible region for the +maximum resource cost C ≤ 104. +Along the isocost line, +the ratio of the number +of +readout +over +receptor +molecules +is +XT/RT += +2 +� +p/(1 − p) +� +1 + τr/τc (see Appendix E 3). This can be +understood intuitively using the optimal resource alloca- +tion principle [10]. It states that in a sensing system that +employs its proteins optimally, the total number of inde- +pendent concentration measurements at the level of the +receptor during the integration time τr, RT(1 + τr/τc), +equals the number of readout molecules XT that store +these measurements, so that neither the receptors nor +the readout molecules are in excess. This design prin- +ciple specifies, for a given integration time τr, the ratio +XT/RT at which the readout molecules sample each re- +ceptor molecule roughly once every receptor correlation +time τc. +While the optimal allocation principle gives the opti- +mal ratio XT/RT of the number of readouts over recep- +tors for a given integration time τr, it does not prescribe +what the optimal integration time τropt, and hence (glob- +ally) optimal ratio Xopt +T /Ropt +T , is that maximizes Ipred for +a given resource constraint C = RT +XT. Fig. 3B shows +that as the distance θ along the isocost line is increased, +τr and hence XT/RT increase monotonically. Near the +information bound, corresponding to θ = 0, the integra- +tion time τr is zero and the number of readout molecules +equals the number of receptor molecules: XT = RT. In +this limit, the push-pull network is an instantaneous re- +sponder, with an integration kernel given by Eq. 4; only +the finite receptor correlation time τc prevents the sys- +tem from fully reaching the information bound. Yet, as +θ increases and the system moves away from the bound, +the predictive and past information first rise along the +contour, and thus with XT/RT and τr, before they even- +tually both fall. +To understand why the predictive and past informa- +tion first rise and then fall with XT/RT and τr, we note +that each readout molecule constitutes 1 physical bit and +that its binary state (phosphorylated or not) encodes at +most 1 bit of information on the ligand concentration. +The number of readout molecules XT thus sets a hard +upper bound on the sensing precision and hence the pre- +dictive information. To raise this bound, XT must be +increased. For a given resource constraint C = RT +XT, +XT can only be increased if the number of receptors RT +is simultaneously decreased. However, the cell infers the +concentration not from the readout molecules directly, +but via the receptor molecules: a readout molecule is a +sample of the receptor that provides at most 1 bit of in- +formation about the ligand-binding state of a receptor +molecule, which in turn provides at most 1 bit of infor- +mation about the input signal. To raise the lower bound +on the predictive information, the information on the in- +put must increase at both the receptor and the readout +level. +To elucidate how this can be achieved, we note that the +maximum number of independent receptor samples and +hence concentration measurements is given by N max +I += +min(XT, RT(1 + τr/τc)) [10]. For θ > 0, the system can +increase N max +I +if, and only if, XT and RT(1 + τr/τc) can +be raised simultaneously. +This can be achieved, while +obeying the constraint C = XT + RT, by decreasing RT +yet increasing τr (Fig. 3B). This is the mechanism of time +averaging, which makes it possible to increase the num- +ber of independent receptor samples [11], and explains +why both the predictive and the past information initially +increase (Fig. 3C). However, as τr is raised further, the +receptor samples become older: the readout molecules in- +creasingly reflect receptor states in the past that are less +informative about the future ligand concentration. The +collected bits of past information have become less pre- +dictive about the future (Fig. 3C). For a given resource +cost, the cell thus faces a trade-off between maximizing +the number of physical bits of past information (i.e. the +receptor samples XT) and the predictive information per +bit. This antagonism gives rise to an optimal integration +time τropt that maximizes the total predictive informa- +tion Ipred (Fig. 3C). +Interestingly, while Ipred decreases beyond τropt, the +past information Ipast first continues to rise because +N max +I +still increases. However, when the integration time +becomes longer than the input signal correlation time, +the correlation between input and output will be lost +and Ipast will fall too. +Chemical power prevents the system from reaching the +information bound +So far, we have only considered the cost of maintain- +ing the cellular system, the protein cost C = RT + XT. +Yet, running a push-pull network also requires energy. +As Eq. 7 shows, the running cost scales with the flux + +6 +A +B +C +FIG. 3. The push-pull network maximizes the predictive power under a resource constraint by moving away +from the information bound. (A) The region of accessible predictive information Ipred = I(x0; ℓτ) and past information +Ipast = I(x0; Lp) in the push-pull network under a resource constraint C ≤ (RT + XT), for the Markovian signals specified by +Eq. 3 (green). The black line is the information bound at which Ipred is maximized for a given Ipast. The push-pull network +can be at the information bound (black points), but maximizing Ipred for a resource constraint C moves the system away from +it. The red and blue lines connect, respectively, the points where Ipred and Ipast are maximized along the green isocost lines +(the contourlines of constant C); they correspond to the red and blue lines in Fig. 2, respectively. The accessible region of +Ipred and Ipast for a given C has been obtained by optimizing over τr, p, f, and XT/RT. The forecast interval is τ = τℓ. +(B) The integration time τr over the receptor correlation time τc, τr/τc, and the ratio of the number of readout and receptor +molecules, XT/RT, as a function of the distance θ along the isocost line corresponding to C = 104 in panel A; the red and +blue points denote where Ipred and Ipast are maximized along the contourline, respectively. For θ → 0, τr → 0: the system is +an instantaneous responder, which is essentially at the information boundary; as predicted by the optimal resource allocation +principle, XT = RT. The system can increase Ipred and Ipast by increasing τr and XT/RT. (C) While this decreases the +predictive information Ipred per physical bit of past information, Ipred/XT (dashed line), increasing XT/RT does increase the +number of physical bits per resource cost, XT/C (purple line). This trade-off gives rise to an optimal predictive information +per resource cost, Ipred/C (red dot on solid black line). Parameter values unless specified: (σℓ/¯ℓ)2 = 10−2, τc/τℓ = 10−2. +around the phosphorylation cycle, which is proportional +to the inverse of the integration time, τr−1. The power +thus diverges for τr → 0. Since the information bound is +reached precisely in this limit, it is clear that the chem- +ical power prevents the push-pull network from reaching +the bound (see Fig. 7 in the appendix). +Non-Markovian signals +Predicting the future change +The push-pull network can optimally predict Marko- +vian signals, yet not all signals are expected to be Marko- +vian. Especially organisms that navigate through an en- +vironment with directional persistence will sense a non- +Markovian signal, as generated by their own motion. +Moreover, when these organisms need to climb a con- +centration gradient, as E. coli during chemotaxis, then +knowing the change in the concentration is arguably more +useful than knowing the concentration itself. Indeed, it +is well known that the kernel of the E. coli chemotaxis +system detects the (relative) change in the ligand con- +centration by taking a temporal derivative of the concen- +tration [15]. However, as we will show here, the converse +statement is more subtle. If the system needs to predict +the (future) change in the signal, then the optimal ker- +nel is not necessarily one that is based on the derivative +only: in general, the optimal kernel uses a combination of +the signal value and its derivative. However, the E. coli +chemotaxis system can respond to concentrations that +vary between the dissociation constants of the inactive +and active state of the receptors, which differ by several +orders of magnitude [24]. This range of possible back- +ground concentrations is much larger than the typical +concentration change over the orientational correlation +time of the bacterium. +As our analysis below reveals, +in this regime the optimal kernel is a perfectly adaptive, +derivative-taking kernel that is insensitive to the current +signal value, precisely like that of the E. coli chemotaxis +system [15, 25–28]. Our analysis thus predicts that this +system has an adaptive kernel, because this is the opti- +mal kernel for predicting concentration derivatives over +a broad range of background concentrations. +To reveal the signal characteristics that control the +shape of the optimal integration kernel, we will consider +the family of signals that are generated by a harmonic + +7 +oscillator: +δ ˙ℓ = v(t), +(8) +˙v = −ω2 +0δℓ(t) − v(t)/τv + ηv(t), +(9) +where δℓ is the deviation of ligand concentration from its +mean ¯ℓ, v its derivative, τv a relaxation time, ηv a Gaus- +sian white noise term, and the frequency ω2 +0 = σ2 +v/σ2 +ℓ +controls the variance σ2 +ℓ of the concentration and that of +its derivative σ2 +v. +Using the IBM framework it can be shown that the +optimal encoding that allows the system to reach the +information bound, is based on a linear combination of +the current concentration ℓ(t) and its derivative v(t), such +that the output x(t) is given by (Appendix C 3): +x(t) = aδℓ(t) +σℓ ++ bv(t) +σv ++ ηx(t). +(10) +This can be understood by noting that while the signal +of Eqs. 8 and 9 is non-Markovian in the space of ℓ, it is +Markovian in ℓ and v: all the information on the future +signal is thus contained in the current concentration and +its derivative. To maximize the predictive information +Ipred = I(x0; vτ) between the current output x0 and the +future derivative of the input vτ for a given amount of +past information Ipast = I(x0; Lp), i.e to reach the infor- +mation bound for predicting the future signal derivative, +the coefficients must obey +aopt = G⟨δℓ(0)δv(τ)⟩ +σℓσv +≡ Gρℓ0vτ , +(11) +bopt = G⟨δv(0)δv(τ)⟩ +σ2v +≡ Gρv0vτ . +(12) +Here, G is the gain, which together with the noise σ2 +ηx sets +the scale of Ipred and Ipast, ρℓ0vτ is the cross-correlation +coefficient between the current concentration value ℓ0 and +the future concentration derivative vτ and ρv0vτ that be- +tween the current and future derivative (Appendix C 3). +These expressions can be understood intuitively: if the +future signal derivative that needs to be predicted is cor- +related with the current signal derivative, it is useful +to include in the prediction strategy the current signal +derivative, leading to a non-zero value of bopt. Perhaps +more surprisingly, if the future signal derivative is also +correlated with the current signal value, then the system +can enhance the prediction accuracy by also including the +current signal value, yielding a non-zero aopt. Clearly, +in general, to optimally predict the future signal change, +the system should base its prediction on both the current +signal value and its derivative. +The degree to which the systems bases its prediction on +the current value versus the current derivative depends +on the relative magnitudes of aopt and bopt, respectively. +In Appendix B 2, we show that when the concentration +change over the timescale τv, σvτv, is much smaller than +the range of possible concentrations σℓ that the bac- +terium can experience, i.e. when σvτv ≪ σℓ such that +ω0 ≪ τ −1 +v , the cross-correlation coefficient ρℓ0vτ vanishes, +such that aopt becomes zero (see Eq. 11). The optimal +kernel has become a perfectly adaptive, derivative-taking +kernel. We emphasize that while we have derived this re- +sult for the class of signals defined by Eqs. 8 and 9, the +idea is far more generic. In particular, while we do not +know the temporal structure of the ligand statistics that +E. coli experiences, we do know that it can detect con- +centration changes over a range of background concentra- +tions that is much wider that the typical concentration +change over a run, such that the correlation between the +concentration value and its future change is likely to be +very small. As our analysis shows, a perfectively adap- +tive kernel then emerges naturally from the requirement +to predict the future concentration change. +While the class of signals specified by Eqs. 8 and 9 is +arguably limited, it does describe the biologically impor- +tant regime of chemotaxis in shallow gradients. In the +limit that ω0 ≪ τv−1, Eq. 9 reduces to ˙v = −v/τv + ηv. +In shallow gradients, the stimulus only weakly affects +the swimming behavior, such that the perceived signal +is mostly determined by the intrinsic orientational dy- +namics of the bacterium in the absence of a gradient. In +this regime, the temporal statistics of the concentration +derivative v is completely determined by the steepness of +the concentration gradient g and the swimming statistics +of the bacterium in the absence of a gradient: +⟨δv(0)δv(τ)⟩ = g2¯ℓ2⟨δvx(0)δvx(τ)⟩ ≃ σ2 +vxe−τ/τvx , +(13) +where the latter is the autocorrelation function of the +(positional) velocity of the bacterium in the absence of a +gradient. It is a characteristic of the bacterium, not of +the environment, and has been measured to decay expo- +nentially with a correlation time τvx [18], precisely as our +model, with τv = τvx, predicts. This correlation time is +on the order of the typical run time of the bacterium in +the absence of a gradient, τv ∼ 0.9s [18]. +Finite resources prevent the chemotaxis system from taking +an instantaneous derivative and reaching the information +bound +The above analysis indicates that the chemotaxis sys- +tem seems ideally designed to predict the future concen- +tration change, because its integration kernel is nearly +perfectly adaptive [15, 25–28]. But how close can this +system come to the information bound for the non- +Markovian signals specified by Eqs. 8 and 9? +To address this, we consider a molecular model that +can accurately describe the response of the chemotaxis +system to a wide range of time-varying signals [29–32]. +In this model, the receptors are partitioned into clusters. +Each cluster is described via a Monod-Wyman-Changeux +model [33]. While each receptor can switch between an +active and an inactive conformational state, the energetic +cost of having different conformations in the same cluster +is prohibitively large. Each cluster is thus either active or + +8 +inactive. Ligand binding favors the inactive state while +methylation does the opposite. +Lastly, active receptor +clusters can via the associated kinase CheA phosphory- +late the downstream messenger protein CheY. +Linearizing around the steady state, we obtain: +δai(t) = αδmi(t) − βδℓ(t), +(14) +δ ˙mi = −δai(t)/(ατm) + ηmi(t), +(15) +δ ˙x∗ = γ +RT +� +i=1 +δai(t) − δx∗(t)/τr + ηx(t). +(16) +Here, δai(t) and δmi(t) are the deviations of the activ- +ity and methylation level of receptor cluster i from their +steady-state values, and RT is the total number of recep- +tor clusters; δℓ(t) and δx∗(t) are, respectively, the devi- +ations of the ligand and CheYp concentration from their +steady-state values; τm and τr are the timescales of re- +ceptor methylation and CheYp dephosphorylation; ηmi +and ηx are independent Gaussian white noise sources. In +Eq. 14, we have assumed that ligand binding is much +faster than the other timescales in the system, so that +it can be integrated out. There is therefore no need to +time average receptor-ligand binding noise, which means +that, in the absence of running costs, the optimal re- +ceptor integration time τr is zero. In what follows, we +set τr to the value measured experimentally, τr ≈ 100ms +[10, 34]. We consider the non-Markovian signals speci- +fied by Eqs. 8 and 9 in the physiologically relevant limit +ω0 → 0, such that the optimal kernel is perfectly adap- +tive, like that of E. coli. For these signals, we determine +the accessible region of Ipast and Ipred under a resource +constraint C = RT + XT (see Fig. 4) by optimizing over +the methylation time τm and the ratio of readout over +receptor molecules XT/RT. +The forecast interval τ is +set to τv, but we emphasize that the optimal design is +independent of the value of τ (see Appendix F 4). +Fig. 4A shows that the chemotaxis system is, in gen- +eral, not at the information bound that maximizes the +predictive information Ipred = I(x0; vτ) for a given past +information Ipast = I(x0; Lp). The optimal systems that +maximize Ipred under a resource constraint C, marked by +the red dots, are indeed markedly away from the infor- +mation bound. Yet, as the resource constraint is relaxed +and C is increased, the optimal system moves towards +the bound. +Panel B shows that the methylation time +τm rises along the three respective isocost lines of panel +A. It highlights that there exists an optimal methyla- +tion time τ opt +m +that maximizes the predictive information +Ipred. Moreover, τ opt +m +decreases as the resource constraint +is relaxed. +Along the respective isocost lines, XT/RT +varies only mildly (see Fig. 9 in the appendix). +These observations can be understood by noting that +the system faces a trade-off between taking a derivative +that is recent versus one that is robust. All the infor- +mation on the future derivative, which the cell aims to +predict, is contained in the current derivative of the sig- +nal; measuring the current derivative would allow the +system to reach the information bound. However, com- +puting the recent derivative is extremely costly. The cell +takes the temporal derivative of the ligand concentration +at the level of the receptor via two antagonistic reac- +tions that occur on two distinct timescales: ligand bind- +ing rapidly deactivates the receptor, while methylation +slowly reactivates it [30]. The receptor ligand-occupancy +thus encodes the current concentration, the methylation +level stores the average concentration over the past τm, +and the receptor activity reflects the difference between +the two—the temporal derivative of the signal over the +timescale τm. To obtain an instantaneous derivative, τm +must go to zero. However, this dramatically reduces the +gain; in fact, in this limit, the gain is zero, because the +receptor activity instantly adapts to the change in the +ligand concentration. Since the push-pull network down- +stream of the receptor is a device that samples the re- +ceptor stochastically [10, 36], the gain, i.e. the change in +the receptor activity due to the signal, must be raised to +lift the signal above the sampling noise. This requires a +finite methylation time τm: as we show in Appendix F 3, +the gain increases monotonically with τm. The trade-off +between a recent derivative and a reliable one gives rise +to an optimal methylation time τ opt +m +that maximizes the +predictive information for a given resource cost. +The same analysis also explains why the optimal +methylation time τ opt +m +decreases and the predictive infor- +mation increases when the resource constraint is relaxed. +The sampling noise in estimating the average receptor +activity decreases as the number of readout molecules +increases [10, 36]. A smaller gain is thus required to lift +the signal above the sampling noise. In addition, a larger +number of receptors decreases the noise in the methyla- +tion level, which also allows for a smaller gain, and hence +a smaller methylation time. These two effects together +explain why τ opt +m +decreases and Ipred increases with C. +Fig. 4A also shows that the past information Ipast = +I(x0; Lp) does not return to zero along the contourline of +constant resource cost. Along the contourline, the methy- +lation time τm rises (Fig. 4B). While the predictive infor- +mation Ipred exhibits an optimal methylation time τmopt, +the past information Ipast continues to rise with τm be- +cause the system increasingly becomes a copying device, +rather than one that takes a temporal derivative. +Comparison with experiment +To test our theory, we study the predictive power of +the E. coli chemotaxis system as a function of the steep- +ness of the ligand concentration gradient, keeping the +resource constraint at the biologically relevant value of +C = RT + XT = 104 [35]. Panel C of Fig. 4 shows Ipred +and Ipast for cells swimming in an exponential concen- +tration gradient ℓ(x) = ℓ0egx, for different values of the +gradient steepness g; along the green iso-steepness lines +τm is varied and XT/RT is optimized to maximize Ipred +and Ipast, with the red dots marking τ opt +m , while along + +9 +A +B +C +FIG. 4. Finite resources prevent chemotaxis system from reaching the information bound. (A) The region of +accessible predictive information Ipred = I(x0; vτ) and past information Ipast = I(x; Lp) for the chemotaxis system under a +resource constraint C = RT + XT, for the non-Markovian signals specified by Eqs. 8 and 9 (green). The black line shows the +information bound at which Ipred is maximized for a given Ipast. The chemotaxis system is not at the information bound, but +it does move towards it as C is increased. The red line connects the red points where Ipred is maximized for a given resource +cost C. The accessible region of Ipred and Ipast under a given resource constraint C = RT + XT is obtained by optimizing over +the methylation time τm and the ratio of readout over receptor molecules XT/RT. The forecast interval is τ = τv. (B) The +methylation time τm over the input correlation time τv as a function of the distance θ along the three respective isocost lines +shown in panel A. The methylation time τm increases along the isocost line, but there exists an optimal τm that maximizes +the predictive information, marked by the red points; θ → 0 corresponds to the origin of panel A, (Ipred, Ipast) = (0, 0); the +points where θ = 0.2 along the isocost lines of panel A are marked with a bar. As the resource constraint is relaxed (higher +C), the optimal τm decreases: the system moves towards the information bound, where it takes an instantaneous derivative, +corresponding to τr, τm → 0. (C) The contourlines of Ipred and Ipast for increasing values of the steepness g of an exponential +ligand concentration gradient ℓ(x) = ℓ0egx, keeping the total resource cost fixed at C = RT + XT = 104; τm and XT/RT +have been optimized. It is seen that the maximal predictive information Ipred under the resource constraint C (marked by +the red points) increases with the gradient steepness. The blue line shows Ipred and Ipast for the E. coli chemotaxis system +with τm = 10s and XT = RT = 5000 fixed at their measured values [35]. Our analysis predicts that this system has been +optimized to detect shallow gradients. Parameter values unless specified: τr = 100ms [10, 34]; τv = 0.9s and σ2 +v = g2¯ℓ2σ2 +vx, +with ¯ℓ = 100µM and σ2 +vx = 157.1µm2s−2 [18]; ω0 → 0; g is given in units of mm−1; in A, g = 4/mm. +the blue line τm and XT and RT are fixed at their exper- +imentally measured values [29, 30, 35]. Clearly, both the +predictive and the past information rise as the gradient +steepness g increases—a steeper concentration gradient +yields a larger change in the concentration, and thus a +stronger signal. +More interestingly, in the optimal system Ipred rises +much faster with Ipast (red line) than in the E. coli system +(blue line). A steeper gradient g yields a stronger input +signal, which raises the signal above the sampling noise +more. This allows the optimal system to take a more re- +cent derivative, with a smaller τm, which is more informa- +tive about the future. In contrast, the methylation time +τm of the E. coli chemotaxis system is fixed. As Fig. 4C +shows, this value is beneficial for detecting shallow gra- +dients, g ≲ 0.2mm−1. Moreover, in this regime, not only +Ipred but also Ipast are close to the respective values for +the optimal system. For steeper gradients Ipast becomes +much higher in the E. coli system than in the optimal +one, even though Ipred remains lower. +The bacterium +increasingly collects information that is less informative +about the future. Taken together, these results strongly +suggest that the system has been optimized to predict +future concentration changes in shallow gradients, which +necessitate a relatively long methylation time. +DISCUSSION +Cellular systems need to predict the future signal by +capitalizing on information that is contained in the past +signal. To this end, they need to encode the past sig- +nal into the dynamics of the intracellular biochemical +network from which the future input is inferred. +To +maximize the predictive information for a given amount +of information that is extracted, the cell should store +those signal characteristics that are most informative +about the future signal. For a Markovian signal obeying +an Ornstein-Uhlenbeck process this is the current signal +value, while for the non-Markovian signal corresponding +to an underdamped particle in a harmonic well, this is +the current signal value and its derivative. As we have +seen here, cellular systems are able to extract these sig- + +10 +nal characteristics: the push-pull network can copy the +current input into the output, while the chemotaxis net- +work can take an instantaneous derivative. We have thus +demonstrated that at least for two classes of signals, cel- +lular systems are in principle able to extract the most +predictive information, allowing them to reach the infor- +mation bound. +Yet, our analysis also shows that extracting the most +relevant information can be exceedingly costly. To copy +the most recent input signal into the output, the integra- +tion time of the push-pull network needs to go to zero, +which means that the chemical power diverges. More- +over, taking an instantaneous derivative reduces the gain +to zero, such that the signal is no longer lifted above +the inevitable intrinsic biochemical noise of the signalling +system. In fact, taking the chemical power cost to drive +the adaptation cycle into account [27, 37] would push the +system away from the information bound even more. +While information is a resource—the cell cannot pre- +dict the future without extracting information from the +past signal—the principal resources that have a direct +cost are time, building blocks and energy. The predic- +tive information per protein and energy cost is therefore +most likely a more relevant fitness measure than the pre- +dictive information per past information. Our analysis +reveals that, in general, it is not optimal to operate at +the information bound: cells can increase the predictive +information for a given resource constraint by moving +away from the bound. Increasing the integration time in +the push-pull network reduces the chemical power and +makes it possible to take more concentration measure- +ments per protein copy. And increasing the methylation +time in the chemotaxis system increases the gain. Both +enable the system to extract more information from the +past signal. Yet, increasing the integration time or the +methylation time also means that the information that +has been collected, is less informative about the future +signal. This interplay gives rise to an optimal integration +and methylation time, which maximize the predictive in- +formation for a given resource constraint. This argument +also explains why the respective systems move towards +the information bound when the resource constraint is +relaxed: Increasing the number of receptor and readout +molecules allows the system to take more instantaneous +concentration measurements, which makes time averag- +ing less important, thus reducing the integration time. +Increasing the number of readout molecules also reduces +the error in sampling the receptor state. This makes it +easier to detect a change in the receptor activity result- +ing from the signal, thus allowing for a smaller dynamical +gain and a shorter methylation time. +Information theory shows that the amount of transmit- +ted information depends not only on the characteristics +of the information processing system, but also on the +statistics of the input signal. While much progress has +been made in characterizing cellular signalling systems, +the statistics of the input signal is typically not known, +with a few notable exceptions [38]. +Here, we have fo- +cussed on two classes of input signals, but it seems likely +that the signals encountered by natural systems are much +more diverse. It will be interesting to extend our analy- +sis to signals with a richer temporal structure [9], and see +whether cellular systems exist that can optimally encode +these signals for prediction. +Finally, while we have analyzed the design of cellular +signaling networks to optimally predict future signals, we +have not addressed the utility of information for function +or behavior. It is clear that many functional or behavioral +tasks, like chemotaxis [18], require information, but what +the relevant bits of information are is poorly understood +[7]. Moreover, cells ultimately employ their resources— +protein copies, time, and energy—for function or behav- +ior, not for processing information per se. Here, we have +shown that maximizing predictive information under a +resource constraint, C → Ipast → Ipred, does not nec- +essarily imply maximizing past information. This hints +that optimizing a functional or behavioral task under a +resource constraint, C → Ipred → function, may not im- +ply maximizing the predictive information necessary to +carry out this task. +ACKNOWLEDGMENTS +We thank Jenny Poulton, Manuel Reinhardt, Michael +Vennettilli and Daan de Groot for many useful discus- +sions. This work is part of the Dutch Research Coun- +cil (NWO) and was performed at the research institute +AMOLF. This project has received funding from the +European Research Council (ERC) under the European +Union’s Horizon 2020 research and innovation program +(grant agreement No. 885065). + +11 +Appendix A: General +1. +Linear signalling networks +Since the systems studied in the main text have a single steady state, we will study them in the linear-noise +approximation [39]. For non-linear systems, the quality of the approximation improves with system size, but it can +already be remarkably good for systems with only 10 copies [20, 22, 40]. In the linear-noise approximation, we expand +the rate equations to first order around the steady state of the mean-field chemical rate equations, and compute the +noise at this steady state. In this approximation the network dynamics are a multidimensional Ornstein-Uhlenbeck +(OU-)process: +˙δy = Gδs(t) + J δy(t) + Bξ(t), +(A1) +where δs(t) is a length k vector of input signals and δy is the vector of all network species of length n, both +defined in terms of deviations from their mean. The vector ξ(t) describes the m independent white noise processes +associated with the m network reactions; they have zero mean, unit variance, and are delta correlated: ⟨ξi(t)⟩ = 0, +⟨ξi(t)ξj(t′)⟩ = δijδ(t − t′), with δij the Kronecker delta. The n × n matrix J is the Jacobian of the network, the n × k +signal gain matrix G describes the strength by which each signal impacts each species directly, the n × m matrix B +contains the noise strengths. The eigenvalues of the Jacobian J must be negative for the system to be stable, and we +require all signals to be stationary. +2. +Integration kernels, power spectra, and correlation functions +We continue by deriving the stationary auto-correlation matrix of a multidimensional OU-process, such as Eq. A1, +via the networks’ power spectra. The power spectrum of a real-valued random process X(t) is the squared modulus +of its Fourier transform: Sx(ω) = ⟨δ˜x(−ω)δ˜x(ω)⟩ and Sx→y(ω) = ⟨δ˜x(−ω)δ˜y(ω)⟩. Throughout this work we use +the following conventions for the Fourier transform and its inverse: F{f(t)} ≡ ˜f(ω) = +� ∞ +−∞ dtf(t) exp(−iωt) and +F−1{ ˜f(ω)} = 1/(2π) +� ∞ +−∞ dω ˜f(ω) exp(iωt) = f(t). To obtain the correlation functions from the power spectra we +invoke the Wiener-Khinchin theorem. +The general solution to Eq. A1 is +δy(t) = +� t +−∞ +dt′ eJ (t−t′) (Gδs(t′) + Bξ(t′)) , +(A2) +which shows the two contributions to the time dependent solution: that of the external signal and that of the internal +noise. The n × k matrix eJ (t−t′)G contains the integration kernels, its (i, j)th entry determines how the jth signal +affects the ith system component over time. The n × m matrix eJ (t−t′)B is similar, but contains the functions that +map the noise terms onto the system components. These matrices can be obtained by taking the Fourier transform +of Eq. A1 and solving for δ˜y(ω) +iωδ˜y(ω) = Gδ˜s(ω) + J δ˜y(ω) + B˜ξ(ω), +(A3) +δ˜y(ω) = (iωIn − J )−1 � +Gδ˜s(ω) + B˜ξ(ω) +� +. +(A4) +Using the convolution theorem to take the Fourier transform of Eq. A2, and comparing the result to Eq. A4, now +shows that F{eJ (t−t′)} = (iωIn − J )−1. We obtain for the power-spectra of the network components +Sy(ω) = ⟨δ ˜y(−ω)δ ˜y(ω)T ⟩, += G(−ω)Ss(ω)G(ω)T + |N(ω)|2, +(A5) +with the matrices of frequency dependent gains G(ω) ≡ (iωIn − J )−1G, and frequency dependent noise N(ω) ≡ +(iωIn−J )−1B. The cross terms vanish because the fluctuations of the external signal are uncorrelated from the internal +noise. Furthermore, the power spectrum of a white noise process is constant, and all the noise terms are independent +of one another, such that the spectral density of the noise vector is the identity matrix ⟨˜ξ(−ω)˜ξ(ω)T ⟩ = Im. We +also need to consider the cross-spectra between the signals and the network components, specifically we will need the +spectra from the network to the signals +Sy→s(ω) = ⟨δ ˜y(−ω)δ˜s(ω)T ⟩, += G(−ω)Ss(ω). +(A6) + +12 +From Eq. A5 and Eq. A6 we can obtain all necessary correlation functions and (co-)variances, by taking the inverse +Fourier transform of the component of interest (for a variance we can directly set t = 0). The advantage of using +this form, is that the contribution of each signal and of the noise terms appear separately. When we are for example +interested in a variance that is only caused by noise, we can omit the terms depending on the signal power spectra, +and vice versa. Moreover, the power spectra are usually simpler in form than the corresponding correlation functions. +The covariance and auto-correlation matrices can also be found by solving Eq. A2 directly in the time domain; the +solutions are shown here for completeness. For a derivation, see for example the work by Vennettilli et al. [41]. In this +case it is most convenient to include the signals as system components, we thus have a new Jacobian J ′ and a new +noise strength matrix B′ which include all network components and the signals themselves. The covariance matrix C +is then obtained by solving the Lyapunov equation +J ′C + CJ ′T + B′B′T = 0, +(A7) +and the correlation matrix is given by +C(τ) = eJ ′τC +for τ > 0. +(A8) +Appendix B: Signals and statistics +1. +Markovian signal +For the Markovian ligand concentration dynamics we use a 1-dimensional OU-process +δ ˙ℓ = −δℓ/τℓ + ηℓ(t), +(B1) +where the ligand concentration is defined in terms of the deviation from its mean δℓ = ℓ(t) − ¯ℓ. The correlation +time is give by τℓ, and the noise ηℓ(t) is derived from a unit white noise process ηℓ(t) ≡ σℓ +� +2/τℓξ(t), such that +⟨ηℓ(t)ηℓ(t′)⟩ = 2σ2 +ℓ/τℓδ(t − t′). We obtain for the steady-state auto-correlation using Eq. A7 and Eq. A8: +⟨δℓ(τ)δℓ(0)⟩ = σ2 +ℓe−τ/τℓ. +(B2) +2. +Non-Markovian signal +Not all ligand concentration trajectories encountered by cells are expected to be Markovian. For example, E. coli +swims in its environment with a speed which exhibits persistence. This leads to an auto-correlation function for the +concentrations’ derivative which does not decay instantaneously [18]. To model such a persistent signal, we use the +classical model of a particle in a harmonic well +δ ˙ℓ = v(t), +˙v = −ω2 +0δℓ(t) − v(t)/τv + ηv(t), +(B3) +where ω0 = +� +k/m, with k the spring constant and m the mass of the particle, τv is a relaxation timescale, and +ηv(t) = σv +� +2/τvξ(t), with ξ(t), as used throughout, a Gaussian white noise process of unit variance, and σv the +standard deviation of v. If the signal would obey the fluctuation-dissipation relation, then mσ2 +v = kBT, but since the +biochemical signal could very well be generated via an active process this relation may not hold. This process can be +expressed as a 2-dimensional OU-process with: +J = +� +0 +1 +−ω2 +0 +−1/τv +� +, +(B4) +B = +�0 +0 +0 +σv +� +2/τv +� +. +(B5) +We find for the covariance matrix, using Eq. A7: +C = +� +σ2 +ℓ +σℓv +σℓv +σ2 +v +� += σ2 +v +� +1/ω2 +0 +0 +0 +1 +� +. +(B6) + +13 +Using Eq. A8 we obtain the auto-correlation matrix in the overdamped regime, τ −1 +v +> 2ω0, +C(τ) = +�⟨δℓ(τ)δℓ(0)⟩ +⟨δℓ(τ)δv(0)⟩ +⟨δv(τ)δℓ(0)⟩ +⟨δv(τ)δv(0)⟩ +� +, += +� +� +� +σ2 +ℓe−µτ/2 � +cosh(ρτ) + µ +2ρ sinh(ρτ) +� +σ2 +ve−µτ/2 1 +ρ sinh(ρτ) +−σ2 +ve−µτ/2 1 +ρ sinh(ρτ) +σ2 +ve−µτ/2 � +cosh(ρτ) − µ +2ρ sinh(ρτ) +� +� +� +� , +(B7) +where ρ = +� +µ2/4 − ω2 +0, with µ = τv−1. The range of ligand concentrations which E. coli might encounter is very +large, based on the dissociation constants of the inactive and active receptor conformations, which for the Tar-MeAsp +receptor ligand combination respectively are KI +D = 18µM and KA +D = 2900µM [42, 43]. This suggests that the variance +in the ligand concentration is very large relative to that of the derivative of the ligand concentration, which is set by +the swimming behaviour of the cell. For this reason we specifically focus on the limit where ω0 → 0, which corresponds +to a vanishingly small spring constant, or a harmonic potential which becomes extremely wide. The variance in the +concentration σ2 +ℓ then diverges, the normalized correlation functions in this limit are +lim +ω0→0 +� +� +⟨δℓ(τ)δℓ(0)⟩ +σ2 +ℓ +⟨δℓ(τ)δv(0)⟩ +σℓσv +⟨δv(τ)δℓ(0)⟩ +σℓσv +⟨δv(τ)δv(0)⟩ +σ2v +� +� = +�1 +0 +0 +e−µτ +� +. +(B8) +Appendix C: Information bottleneck framework and solutions +Anticipating future environmental conditions allows for timely adaptation. +However, storing information costs +resources such as proteins, energy and time, and not all information in the past ligand concentrations will be relevant +for predicting the signal’s future state. Assuming that resources are in limited supply, this means that cells must be +efficient in which, and how much information they store. This is elegantly captured in the Information Bottleneck +Method (IBM), which describes the problem of maximizing the information on the future signal while minimizing the +information on the past signal that is stored in the network output, from which the future input is predicted [8]. The +objective function for the prediction of a variable of interest zτ ≡ z(t + τ) is: +max +P (X0|Lp) : +L = I(x0; zτ) − γI(x0; Lp). +(C1) +The value of the sensing system output at the current time t is x0 ≡ x(t). The variable of interest zτ at a future +time t + τ is the future concentration ℓτ ≡ ℓ(t + τ) for the Markovian signal, and the future concentration derivative +vτ ≡ v(t+τ) for the non-Markovian signal. Since the system of interest needs to predict one signal characteristic (either +the future signal value or its derivative), one output component is sufficient for encoding the required information, +as we describe in more detail below. The vector Lp = (δℓ(0), δℓ(−∆t), . . . , δℓ(−(N − 1)∆t))T is the past trajectory +of ligand concentrations of length N, discretized with timestep ∆t. The mutual information between the current +system output and the future property of interest is the predictive information Ipred ≡ I(x0; zτ), and the mutual +information between the current system output and the past ligand concentration trajectory is the past information +Ipast ≡ I(x0; Lp). The Lagrange multiplier γ sets the relative cost of storing past information over obtaining predictive +information. Given a value of γ, Eq. C1 is maximized by optimizing the mapping of the past ligand concentration +trajectory Lp onto the current output x0. Since, by the data processing inequality, we have Ipast ≥ Ipred, for γ = 1 +the objective function is maximized by Ipast = Ipred = 0. As γ is decreased both the past and predictive information +increase, and the parametric curve in the Ipast − Ipred plane that arises is the information bound. For γ = 0 there +is no cost to storing past information. The predictive information is then only limited by the amount of information +contained in the past about the future signal property: Ipred ≤ I(Lp; zτ). +1. +Gaussian information bottleneck +In general equation C1 can be difficult to solve, as all mappings from Lp to X0 are allowed. However, the problem +becomes analytically tractable when the joint probability distribution of Lp and zτ is a multivariate Gaussian. Here, + +14 +we follow the procedure of Chechik and coworkers to obtain this mapping [12]. In the Gaussian model, the optimal +mapping from Lp to x0 is a linear one [12] +x0 = ALp + ξ; +ξ ∼ N(0, σ2 +ξ), +(C2) +where A is a row vector which determines how strongly each entry in Lp contributes to the scalar output X0 at any +point in time. The random variable ξ is the noise added to the signal due to the stochastic nature of the mapping; it is +a Gaussian random variable independent of Lp with 0 mean and variance σ2 +ξ. Finding the optimal mapping from Lp +to x0 corresponds to finding the optimal combination of A and σ2 +ξ. It can be shown that for any pair (A, σ2 +ξ), there +exists a pair (A′, 1) which yields the same values for Ipast and Ipred after maximization of Eq. C1 [12]. Therefore, we +can set σ2 +ξ = 1 without altering the information curve. +To obtain the information bound, we rewrite Eq. C1 using the definition of the mutual information between Gaussian +random variables: +L = 1 +2 log(σ2 +x/σ2 +x|z) − γ 1 +2 log(σ2 +x/σ2 +x|L), +(C3) +with the total variance σ2 +x in the output x0, the output variance conditional on the future signal property σ2 +x|z ≡ σ2 +x|zτ , +and the output variance conditional on the complete history of ligand concentrations σ2 +x|L ≡ σ2 +x|Lp. The latter is just +the variance caused by the intrinsic noise, σ2 +x|L = σ2 +ξ = 1. The total variance in x0 can be expressed in terms of the +mapping vector A and the variance in the past signal using Eq. C2, σ2 +x = AΣLAT + 1, where ΣL ≡ ΣLp is the +covariance matrix of the past ligand concentration trajectory Lp. To express the output variance conditional on the +future signal property zτ we use the Schur complement formula, which in general form reads: +Σx|y = Σx − ΣxyΣ−1 +y Σyx, +(C4) +where Σyx = ΣT +xy. Using this formula to rewrite σ2 +x|z, and then using the linear relation from Eq. C2 again, we obtain +σ2 +x|z = AΣL|zAT + 1. +Filling in the expressions for the variances in L (Eq. C3) gives: +L = 1 +2 +� +(1 − γ) log +���AΣLAT + 1 +��� +− log +���AΣL|zAT + 1 +���� +. +(C5) +For any symmetric matrix C we have +δ +δA log |ACAT | = +� +ACAT �−1 2AC, such that we obtain for the derivative of +L to A: +δL +δA = (1 − γ) +AΣL +AΣLAT + 1 − +AΣL|z +AΣL|zAT + 1. +(C6) +In our case A is a row vector, and both denominators are thus scalars. We find the maximum of L by equating its +derivative to 0, which gives: +AΣL|zΣ−1 +L += (1 − γ)AΣL|zAT + 1 +AΣLAT + 1 A. +(C7) +For this equality to hold A must either be identically 0, or a left eigenvector of the matrix ΣL|zΣ−1 +L +with eigenvalue: +λ = (1 − γ)AΣL|zAT + 1 +AΣLAT + 1 . +(C8) +Here, we note that if the signal statistics is sufficiently rich and the prediction complexity sufficiently large (because, +for example, multiple signal characteristics need to be predicted), then the matrix ΣL|zΣ−1 +L +has multiple eigenvectors +with non-trivial eigenvalues 0 < λi < 1 [12]. This reflects the idea that storing the past information that is necessary +to enable this complex prediction task may require multiple output components, i.e. +an output vector x, where +each output component has an integration kernel given by one of the eigenvectors of ΣL|zΣ−1 +L +[12]. However, for +Markovian signals only one eigenvector with non-trivial eigenvalue 0 < λ < 1 emerges, which means that one output +component is sufficient to encode the required information. For the non-Markovian signals studied here, ΣL|zΣ−1 +L +has two eigenvectors if both the future value and its derivative need to be predicted (and z = (ℓτ, vτ)); to optimally +predict both features from the current output, two output components are then required, provided Ipast is sufficiently + +15 +large. However, here we consider the scenario that only the future derivative needs to be predicted, in which case only +one non-trivial eigenvector emerges, and one output component is sufficient for encoding the required information. +We leave the problem of predicting multiple signal features via multiple output components for future work. +We can define the optimal mapping A = ||A||ν where ν is the normalized left eigenvector of ΣL|zΣ−1 +L corresponding +to its smallest eigenvalue, 0 < λ < 1. The magnitude can be found by solving Eq. C8 for ||A||, using from Eq. C7 +that λνΣLνT = νΣL|zνT . This gives for the optimal mapping: +Aopt = +�� +1−γ−λ +ν1ΣLνT +1 λγ ν1 +for +0 < λ < 1 − γ, +0 +for +1 − γ ≤ λ ≤ 1. +(C9) +We can substitute ||A||2 = (1 − γ − λ)/(νΣLνT λγ) in the definitions for the mutual information to express them +in terms of λ and γ. For the past information we obtain: +Ipast = 1 +2 log +� +||A||2νΣLνT + 1 +� +, += 1 +2 log +�1 − γ +γ +1 − λ +λ +� +. +(C10) +And for the predictive information: +Ipred = 1 +2 log +� +||A||2νΣLνT + 1 +� +− 1 +2 log +� +||A||2νΣL|ℓτ νT + 1 +� +, += Ipast − 1 +2 log +�1 − λ +γ +� +, += 1 +2 log +�1 − γ +λ +� +. +(C11) +2. +Markovian signal +To obtain the information bound for prediction of the future ligand concentration of a Markovian signal, we need +to determine the eigenvalues and vectors of the matrix (see Eqs. C7 and C8) +W = ΣL|ℓτ Σ−1 +L . +(C12) +Using the Schur complement formula (Eq. C4) to rewrite the conditional matrix gives ΣL|ℓτ = ΣL − ΣLℓτ ΣT +Lℓτ /σ2 +ℓ. +Then defining the normalized matrices RL = ΣL/σ2 +ℓ and RLℓτ = ΣLℓτ /σ2 +ℓ we find +W = IN − RLℓτ RT +Lℓτ R−1 +L . +(C13) +where N is the length of the input trajectory Lp. The correlation matrix of the past trajectory is symmetric with +entries R(i,j) +L += exp(−|i − j|∆t/τℓ), where ∆t is the discretization timestep of the past trajectory Lp and i ranges +from 1 to N. This is a Kac-Murdock-Szeg¨o matrix, and its inverse is known: +R−1 +L += +1 +1 − e2∆t/τℓ +� +� +� +� +� +� +� +� +� +� +1 +−e−∆t/τℓ +0 +. . . +. . . +0 +−e−∆t/τℓ 1 + e−2∆t/τℓ +−e−∆t/τℓ +. . . +. . . +0 +0 +−e−∆t/τℓ +1 + e−2∆t/τℓ +... +. . . +0 +... +... +... +... +... +... +0 +. . . +. . . +−e−∆t/τℓ 1 + e−2∆t/τℓ −e−∆t/τℓ +0 +. . . +. . . +0 +−e−∆t/τℓ +1 +� +� +� +� +� +� +� +� +� +� +. +(C14) +Note that the inverse matrix is tridiagonal. The length N cross-correlation vector between past trajectory and future +concentration has entries R(i) +Lℓτ = exp(−(τ + (i − 1)∆t)/τℓ). The product of the correlation matrices is surprisingly +simple: +RLℓτ RT +Lℓτ R−1 +L += e−2τ/τℓ +� +� +� +� +� +1 +0 . . . 0 +e−∆t/τℓ +0 . . . 0 +... +... +... +... +e−(N−1)∆t/τℓ 0 . . . 0 +� +� +� +� +� . +(C15) + +16 +Using this result we can straightforwardly determine the eigenvalues, +|W − λIN| = 0, +��������� +� +� +� +� +� +1 − λ − e−2τ/τℓ +0 +. . . +0 +−e−(τ+∆t)/τℓ +1 − λ . . . +0 +... +... +... +... +−e−(τ+(N−1)∆t)/τℓ +0 +. . . 1 − λ +� +� +� +� +� +��������� += 0. +(C16) +The only contribution to the determinant comes from the diagonal, and the only nontrivial eigenvalue is thus λ = +1−e−2τ/τl. The optimal mapping is thus onto a one-dimensional scalar output x0. The corresponding left eigenvector +is given by +ν1W = (1 − e−2τ/τl)ν1, +(C17) +which holds for ν1 = +�1 0 . . . 0� +. The optimal mapping for the prediction of a one-dimensional OU-process is thus +to copy its most recent value. This agrees with intuition as for any Markovian process, all the information about the +future signal is contained in the most recent value. For a continuous input signal (rather than a discretized signal), +and a continuous integration kernel k(t) (rather than a mapping vector A), this means that the optimal integration +kernel is kopt(t) = aδ(t). +3. +Non-Markovian signal +To find the optimal mapping for the prediction of the derivative of a non-Markovian signal, based on its history of +ligand concentrations, we need to find the eigenvalues and vectors of the matrix +W = ΣL|vτ Σ−1 +L , += IN − 1 +σ2v +ΣLvτ ΣT +Lvτ Σ−1 +L . +(C18) +The covariance matrix of the past trajectory is symmetric with entries Σ(i,j) +L += ⟨δℓ(0)δℓ(|i − j|∆t)⟩ where both i and +j range from 1 to N, the past trajectory length. The covariance vector between past trajectory and future derivative +has entries Σ(i,j) +Lvτ = ⟨δℓ(0)δv(τ + (i − 1)∆t)⟩. Both the concentration auto-correlation function, and the concentration +to future derivative cross-correlation function, are shown in Eq. B7. +To better understand the optimal mapping of this signal we numerically investigate the eigenvalues of the matrix +W . For the prediction of vτ, there is only one non-trivial eigenvalue. Like for the Markovian signal, this shows that +for the prediction of the derivative of this non-Markovian signal, the optimal mapping is always onto a scalar output. +The non-trivial eigenvalue λ decreases with the discretization timestep ∆t and is minimal for ∆t → 0 Fig. 5. In this +limit, λ has the same magnitude for any N ≥ 2, see Fig. 5. A smaller eigenvalue λ corresponds to larger past and +predictive information and a larger ratio Ipred/Ipast (Eq. C10 and Eq. C11), given any value of the Lagrange multiplier +γ. For the optimal mapping we must thus have N ≥ 2 and ∆t → 0, where N sets both the past trajectory and the +mapping vector length. Because increasing the length above two does not yield an improvement in the value of λ1 we +focus on N = 2. +The fact that to reach the optimum we must have N = 2 and ∆t → 0, shows that the optimal kernel A takes an +instantaneous measurement of a combination of the most recent ligand concentration, and its derivative. This can be +understood as follows, for a trajectory of length two, the mapping vector also has length two, A = ||A||( ˆw1, ˆw2), with +� +ˆw2 +1 + ˆw2 +2 = 1. We can then express the linear mapping of Lp to x0 (Eq. C2) as: +x0 = ||A|| +� +( ˆw1 + ˆw2)δℓ(0) − ˆw2∆tδℓ(0) − δℓ(−∆t) +∆t +� ++ ξ, +(C19) +This expression shows that, as ∆t → 0, the two entries of A combine both the most recent signal value and the most +recent derivative to generate x0. This is intuitive because the signal is completely defined by its concentration and +derivative (Eq. B3). For this reason, and to obtain analytical insight into the optimal weights, we inspect the final +two entries of the past ligand concentration trajectory in the limit ∆t → 0, which defines the past signal in terms of +its most recent concentration and derivative +S0 ≡ +�δℓ(0) v(0)�T . +(C20) + +17 +0.05 +0.10 +0.15 +0.20 +0.560 +0.565 +0.570 +0.575 +0.580 +Discretization timestep Δt +Smallest eigenvalue λ +N = 2 +N = 3 +N = 4 +FIG. 5. The smallest eigenvalue of the IB matrix is minimal for N ≥ 2 and ∆t → 0. A smaller eigenvalue corresponds +to a larger ratio Ipred/Ipast for any given value of the Lagrange multiplier γ. Parameters: friction timescale τ −1 +v += 0.862s−1 as +determined in [18], prediction interval τ = τv, and ω0 = 0.4s−1 such that the system is slightly overdamped. +Because the signal is Markovian in the joint properties δℓ and v, the vector S0 contains the same information as the +trajectory Lp. The past information is now the mutual information between x0 and S0, i.e. Ipast = I(x0; S0). The +output x0 can then also be written as a projection of S0 via the alternative mapping vector ˜ +A = ||A||(ˆa,ˆb): +x0 = ||A|| +� +ˆaδℓ(0) + ˆbv(0) +� ++ ξ. +(C21) +Comparison with Eq. C19 shows how the components of ˜ +A relate back to those in A, +ˆw1 = ˆa + ˆb/∆t, +(C22) +ˆw2 = −ˆb/∆t. +(C23) +To obtain the optimal mapping vector ˜ +A the matrix of signal statistics of which the eigenvalues and -vectors need +to be determined is +W = Σs|vτ Σ−1 +s , +(C24) +with +Σs = +� +σ2 +ℓ +0 +0 +σ2 +v +� +, +(C25) +Σs|vτ = Σs − 1 +σ2v +Σsvτ ΣT +svτ , +(C26) +Σsvτ = +� +⟨δℓ(0)δv(τ)⟩ +⟨δv(0)δv(τ)⟩ +� +. +(C27) +We thus obtain +W = I − +� +� +� +⟨δℓ(0)δv(τ)⟩2 +σ2 +ℓ σ2v +⟨δℓ(0)δv(τ)⟩⟨δv(0)δv(τ)⟩ +σ4v +⟨δℓ(0)δv(τ)⟩⟨δv(0)δv(τ)⟩ +σ2 +ℓ σ2v +⟨δv(0)δv(τ)⟩2 +σ4v +� +� +� . +(C28) +This matrix has one nontrivial eigenvalue, λ = 1 − ⟨δv(0)δv(τ)⟩2 +σ4v +− ⟨δℓ(0)δv(τ)⟩2 +σ2 +ℓ σ2v +, which depends on the normalized +correlation functions between on the one hand the current concentration or derivative, and on the other hand the +future derivative. The corresponding left eigenvector is +ν1 = Q−1 � +1 +σℓ +⟨δℓ(0)δv(τ)⟩ +σℓσv +1 +σv +⟨δv(0)δv(τ)⟩ +σ2 +v +� +, +(C29) +where Q normalizes the vector.Using the linear mapping x0 = ||A||ν1S0 + ξ, and defining G ≡ ||A||/Q, shows that +the optimal output should be generated as follows +xopt +0 += G +�⟨δℓ(0)δv(τ)⟩ +σℓσv +δℓ(0) +σℓ ++ ⟨δv(0)δv(τ)⟩ +σ2v +v(0) +σv +� ++ ξ. +(C30) + +18 +Clearly, the optimal mapping depends on the (normalized) cross-correlation coefficient ρℓ0vτ ≡ ⟨δℓ(0)δv(τ)⟩/(σℓσv) +between the current concentration δℓ(0) and future derivative δv(τ), and the cross-correlation coefficient ρv0vτ between +the current derivative δv(0) and future derivative δv(τ). Indeed, to optimally predict the future derivative, the cell +should also use the current concentration and not only its current derivative. However, in the limit that the range of +concentrations sensed becomes very large, corresponding to ω0 → 0, the current concentration is no longer correlated +with the future derivative, and ρℓ0vτ → 0 (Eq. B8). In this limit, ˆa = 0 and ˆb = 1, and the kernel becomes a perfectly +adaptive, derivative-taking kernel: +lim +ω0→0 xopt +0 += ||A||v(0) + ξ. +(C31) +If we translate this back to the vector ||A||( ˆw1, ˆw2), operating on a ligand concentration trajectory Lp, the optimal +weights become ˆw1 = − ˆw2. +Appendix D: Past and predictive information for linear signalling networks +In order to address how close biochemical networks can come to the information bounds derived above, we here +describe how we obtain the past and predictive information for any linear (biochemical) network. +We then use +the resulting general expressions to compute the past and predictive information for the push-pull network and the +chemotaxis system of the main text. +For any linear network the output can be written as +δx(t) = +� t +−∞ +ds k(t − s)δℓ(s) + ηx(t). +(D1) +The mapping kernel k(t) is a property of the network and describes how the input signal is mapped onto the output. +The noise term ηx(t) is a sum of convolutions over all white noise processes in the network and corresponding network +mapping functions, see Eq. A2. The variance in the output can generally be split up in a part caused by the signal +and a part cause by the noise, and we have +σ2 +x = +� t +−∞ +ds +� t +−∞ +ds′k(t − s)k(t − s′)⟨δℓ(s)δℓ(s′)⟩ + σ2 +ηx, += σ2 +x|η + σ2 +x|L, +(D2) +where σ2 +x|η is the signal variance, i.e. all noise terms are fixed, and σ2 +x|L is the noise variance, i.e. the complete history +of the signal is fixed. Using this decomposition we find for the past information, which is the mutual information +between the current output and the complete signal history, +Ipast(x0; Lp) = 1 +2 log +� +σ2 +x +σ2 +x|L +� += 1 +2 log(1 + SNR), +(D3) +where the signal-to-noise ratio is defined as SNR = σ2 +x|η/σ2 +x|L. Using the same definition for the mutual information +when deriving the predictive information between current output and future ligand concentration, we obtain +Ipred(x0; ℓτ) = 1 +2 log +� +σ2 +x +σ2 +x|ℓτ +� +, += 1 +2 log +� +1 + +σ2 +x|η +σ2 +x|L +� +− 1 +2 log +� +1 + +σ2 +x|η − ⟨δx(0)δℓ(τ)⟩2/σ2 +ℓ +σ2 +x|L +� +, += Ipast − 1 +2 log(1 + cSNR). +(D4) +In the second line we used the Schur complement formula, Eq. C4, to decompose the variance in the output conditioned +on the future signal: σ2 +x|ℓτ = σ2 +x−⟨δx(0)δℓ(τ)⟩2/σ2 +ℓ. The quantity σ2 +x|η−⟨δx(0)δℓ(τ)⟩2/σ2 +ℓ can be understood as follows: +the first term σ2 +x|η is the contribution to the total variance of the output σ2 +x that comes from the signal variations, +while the second term quantifies the variance in the output that is correlated with the future input. The difference is +thus the variance in the output coming from the signal variations that are not correlated with the future input. The + +19 +ratio in the second logarithm can thus be understood as a conditional SNR that quantifies the part of the signal to +noise ratio that does not contain information about the future signal. This becomes more clear when considering its +form in terms of the mapping kernel and signal correlation functions. For any linear signalling network we have +σ2 +x|η − ⟨δx(0)δℓ(τ)⟩2/σ2 +ℓ = +� 0 +−∞ +ds +� 0 +−∞ +ds′k(−s)k(−s′) +� +⟨δℓ(s)δℓ(s′)⟩ − ⟨δℓ(τ)δℓ(s′)⟩⟨δℓ(τ)δℓ(s)⟩ +σ2 +ℓ +� +, +(D5) +where the term in parentheses is the conditional variance in the past signal trajectory given a future value, ΣL|ℓτ . +The form in Eq. D4 thus tells us that the predictive information is equal to the past information, minus the bits +that do not contain information about the future ligand concentration. This difference is indeed the part of the past +information that does contain predictive information about the future signal. +Although the expression above (Eq. D4) nicely relates the past and predictive information, a more straightforward +way of obtaining the predictive information is by expressing it directly in terms of the correlation between the current +output and the future ligand concentration: +Ipred(x0; ℓτ) = 1 +2 log +� +σ2 +x +σ2 +x|ℓτ +� += −1 +2 log +� +1 − ⟨δx(0)δℓ(τ)⟩2 +σ2xσ2 +ℓ +� +, +(D6) +where we again used the Schur complement formula to rewrite σ2 +x|ℓτ . Written this way we thus see that the predictive +information depends on the normalized correlation between the current network output and the future ligand con- +centration. We can simply exchange the future ligand concentration for the future derivative when considering the +chemotaxis network. +To compute the past information for linear signalling networks we use Eq. D3, and we thus need to compute the +SNR. To compute the predictive information for the prediction of a future ligand concentration, we need to compute +the ‘future correlation function’ ⟨δx(0)δℓ(τ)⟩. For the prediction of the future derivative we need ⟨δx(0)δv(τ)⟩. +Appendix E: Push-pull network +We consider a push-pull network that consists of a phosphorylation-dephosphorylation cycle downstream of a +receptor. +When bound to ligand, the receptor itself or its associated kinase, such as CheA in E. coli, catalyzes +the phosphorylation of a readout protein x, like CheY. Active readout molecules x∗ can decay spontaneously or be +deactivated by an enzyme (phosphatase), such as CheZ in E. coli. This cycle is driven by the turnover of fuel such as +ATP. We recognize that inside the living cell, the chemical driving is typically large: for example, the free energy of +ATP hydrolysis is about 20kBT, which means that the system essentially operates in the irreversible regime [10, 36]. +This system consists of the following reactions: +R + L +k+ +−−⇀ +↽−− +k− +RL +(E1) +RL + x +kf +−−→ RL + x∗ +(E2) +X∗ +kr +−−→ X +(E3) +Both the total number of receptors RT = R + RL and read-out molecules XT = X + X∗ are conserved moieties. The +chemical Langevin equations of this system are: +˙ +RL = [RT − RL(t)]ℓ(t)k+ − RL(t)k− + Bc(RL, ℓ)ξc(t), +(E4) +˙x∗ = [XT − x∗(t)]RL(t)kf − x∗(t)kr + Bx(RL, x∗)ξx(t), +(E5) +where RL is the number of bound receptors, x∗ the number of phosphorylated read-out molecules, and ξi denote +independent Gaussian white noise with unit variance, ⟨ξi(t)ξj(t′)⟩ = δijδ(t − t′). The noise strengths are Bc(RL, ℓ) = +� +(RT − RL(t))ℓ(t)k+ + RL(t)k− and Bx(RL, x∗) = +� +(XT − x∗(t))RL(t)kf + x∗(t)kr. The steady-state fraction of +ligand-bound receptors is p ≡ RL/RT = ¯ℓ/(¯ℓ+KD) with the dissociation constant KD = k−/k+, and the steady-state +fraction of phosphorylated readout molecules is f ≡ ¯x∗/XT = pRT/(pRT + kr/kf). +In the linear-noise approximation, expanding Eqs. E4 and E5 to first order around their steady state, the equations +become +δ ˙ +RL = b δℓ(t) − δRL(t)/τc + ηc(t), +(E6) +δ ˙x∗ = γ δRL(t) − δx∗(t)/τr + ηx(t). +(E7) + +20 +The parameters b = RTp(1 − p)/(¯ℓτc) and γ = XTf(1 − f)/(RTpτr) are effective rates of receptor-ligand binding +and readout phosphorylation, respectively. +The decay rate of correlations in the receptor-ligand binding state is +τc−1 = ¯ℓk+ + k−, and that of the readout phosphorylation state is τr−1 = pRTkf + kr. The rescaled white noise +processes have strengths ⟨η2 +c⟩ = B2 +c = 2RTp(1 − p)/τc and ⟨η2 +x⟩ = B2 +x = 2XTf(1 − f)/τr. +1. +Model statistics +The relevant quantity to compute the past information is the variance in the output, decomposed into the part +caused by signal variation and the part caused by noise. To compute the predictive information we further need the +correlation function between the current output and a future ligand concentration ⟨δℓ(τ)δx∗(0)⟩. These quantities +can be obtained via their Fourier transforms, as in Eq. A5 and Eq. A6. The matrices describing the properties of the +signalling network are, as defined below Eq. A1, +G = +� +b +0 +� +, +(E8) +J = +� +−τc−1 +0 +γ +−τr−1 +� +, +(E9) +B = +�� +⟨η2c⟩ +0 +0 +� +⟨η2x⟩ +� += +�� +2RTp(1 − p)/τc +0 +0 +� +2XTf(1 − f)/τr +� +. +(E10) +A useful property of the network is the matrix exponential of its Jacobian, which in Fourier space is (see Eq. A2 +and Eq. A4) +F{eJ t}(ω) = (iωI2 − J )−1, += +� +� +1 +1/τc+iω +0 +γ +(1/τc+iω)(1/τr+iω) +1 +1/τr+iω +� +� . +(E11) +We then have G(ω) = F{eJ t}(ω)G and N(ω) = F{eJ t}(ω)B, see also Eq. A4 and Eq. A5. The integration kernel that +maps the ligand concentration onto the output of the push-pull network, see Eq. D1, is given by the inverse Fourier +transform of the second entry of G(ω), which is the frequency dependent gain, ˜gℓ→x(ω), from ℓ to x: +k(t) ≡ F−1{˜gℓ→x(ω)} = bγτcτr +1 +τr − τc +� +e−t/τr − e−t/τc� +, += XTf(1 − f)(1 − p)/¯ℓ +1 +τr − τc +� +e−t/τr − e−t/τc� +, +(E12) +The so-called static gain of the network is the integral of this kernel over all time, ¯gℓ→x ≡ +� ∞ +0 +k(t)dt = XTf(1 − +f)(1 − p)/¯ℓ. +This parameter quantifies how much a step change in the input concentration changes the steady- +state level of the output: ¯gℓ→x = ∂ ¯x∗/∂¯ℓ. We will use this parameter in the statistical quantities that follow. The +static gain is also given by ¯gℓ→x = ¯gℓ→RL¯gRL→x, with ¯gℓ→RL = p(1 − p)RT/¯ℓ the static gain from ¯ℓ to RL and +¯gRL→x = f(1 − f)XT/(pRT) the static gain from RL to x∗. +We model the Markovian ligand concentration as a 1-dimensional OU process Eq. B1, which has the following +power spectrum +Sℓ(ω) = ⟨|δℓ(ω)|2⟩ = +2σ2 +ℓ/τℓ +1/τ 2 +ℓ + ω2 . +(E13) +This yields the following expression for the power spectra (see Eq. A5): +G(−ω)Sℓ(ω)G(ω)T = b2 +� +� +1 +1/τc2+ω2 +γ +1 +1/τr−iω +1 +1/τc2+ω2 +γ +1 +1/τr+iω +1 +1/τc2+ω2 γ2 +1 +1/τr2+ω2 +1 +1/τc2+ω2 +� +� +2σ2 +ℓ/τℓ +1/τ 2 +ℓ + ω2 +(E14) +|N(ω)|2 = ⟨η2 +c⟩ +� +� +� +1 +1/τc2+ω2 +γ +1 +1/τr−iω +1 +1/τc2+ω2 +γ +1 +1/τr+iω +1 +1/τc2+ω2 γ2 +1 +1/τr2+ω2 +1 +1/τc2+ω2 + ⟨η2 +x⟩ +⟨η2c⟩ +1 +1/τr2+ω2 +� +� +� +(E15) + +21 +We thus have for the power spectrum of the read-out: +Sx(ω) = ˜g2 +ℓ→x(ω)Sℓ(ω) + N 2 +x(ω) += +2b2γ2σ2 +ℓ/τℓ +(1/τr2 + ω2)(1/τc2 + ω2)(1/τ 2 +ℓ + ω2) + +γ2⟨η2 +c⟩ +(1/τr2 + ω2)(1/τc2 + ω2) + +⟨η2 +x⟩ +1/τr2 + ω2 , +(E16) +The variance in the read-out σ2 +x = 1/(2π) +� ∞ +−∞ Sx(ω) is hence given by +σ2 +x = σ2 +x|η + σ2 +x|L += ¯g2 +ℓ→x +1 + τr/τℓ + τr/τc +(1 + τc/τℓ)(1 + τr/τℓ)(1 + τr/τc)σ2 +ℓ + ¯g2 +RL→xRTp(1 − p) +1 +1 + τr/τc ++ XTf(1 − f), += ¯g2 +ℓ→x +1 + τr/τℓ + τr/τc +(1 + τc/τℓ)(1 + τr/τℓ)(1 + τr/τc) +� +�� +� +dynamical gain +σ2 +ℓ + XTf(1 − f) +� +1 + ¯gℓ→x +¯ℓ +RTp +1 +1 + τr/τc +� +, +(E17) +where ¯gRL→x = γτr = XTf(1 − f)/(RTp) is the static gain from the receptor to the readout. The expression above +gives insight into the role of the different network components in shaping the noise in the readout. It can be seen +that the contribution from the signal variance σ2 +ℓ to σ2 +x is determined by the static gain ¯g2 +ℓ→x, which is proportional +to XT, and a factor that only depends on ratios of timescales. Their product is the dynamical gain, which decreases +monotonically with τr. +The intrinsic noise in the phosphorylation state of the read-outs leads to the noise term +XTf(1 − f), which cannot be averaged out. The noise arising from ligand binding and unbinding increases with the +static gain, but can be mitigated by increasing the number of receptors or the integration time τr. The latter strategy +is what we call time-averaging. +The signal-to-noise ratio SNR = σ2 +x|η/σ2 +x|L can straightforwardly be obtained from Eq. E17. This is the quantity +that sets the magnitude of the past information, see Eq. D3. +To determine the predictive information we need +to compute the correlation function from the current output to the future ligand concentration ⟨δx(0)δℓ(τ)⟩. This +requires the cross-spectrum from output to ligand concentration, which is given by (Eq. A6) +˜gℓ→x(−ω)Sℓ(ω) = +bγ +(1/τc − iω)(1/τr − iω) +2σ2 +ℓ/τℓ +1/τ 2 +ℓ + ω2 . +(E18) +From this power spectrum we obtain the required correlation function by taking the inverse Fourier transfrom: +⟨δx(0)δℓ(τ)⟩ = F−1{˜gℓ→x(−ω)Sℓ(ω)}, += +¯gℓ→xσ2 +ℓ +(1 + τc/τℓ)(1 + τr/τℓ)e−τ/τℓ. +(E19) +This correlation function thus decays exponentially with the prediction interval τ at a rate τ −1 +ℓ +, just as the signal auto- +correlation. The (squared) correlation coefficient, which sets Ipred, is given by ⟨δx(0)δℓ(τ)⟩2/(σ2 +ℓσ2 +x) = ρ2 +ℓxe−2τ/τℓ, +with the (squared) instantaneous correlation coefficient (for convenience given as its inverse) +ρ−2 +ℓx = +¯ℓ2 +σ2 +ℓ +� +1 + τr +τℓ +�2 � +1 + τc +τℓ +�2 � +1 +XT f(1 − f)(1 − p)2 + +1 +RT p(1 − p)(1 + τr/τc) + σ2 +ℓ +¯ℓ2 +1 + τr/τℓ + τr/τc +(1 + τc/τℓ)(1 + τr/τℓ)(1 + τr/τc) +� +. +(E20) +When the right-hand-side is minimized, the correlation is thus maximized. This expression shows that increasing XT +and RT always increases the instantaneous correlation coefficient, and that the fraction of phosphorylated readout +molecules in steady state that maximizes the correlation coefficient is f = 1/2. +2. +Past and predictive information of the push-pull network +Using the quantitites computed above, we can determine both the past and the predictive information. For the +past information we use Eq. D3, whith the SNR from Eq. E17: +SNR = σ2 +x|η/σ2 +x|L = (1 − p)σ2 +ℓ +¯ℓ2 +1 + τr/τℓ + τr/τc +(1 + τc/τℓ)(1 + τr/τℓ)(1 + τr/τc) +� � +1 +XTf(1 − f)(1 − p) + +1 +RTp(1 + τr/τc) +� +. + +22 +The predictive information is a function of the correlation between the current output and the future ligand concen- +tration, Eq. D6. This correlation can be decomposed into the instantaneous correlation coefficient and an exponential +decay on the timescale of the ligand concentration fluctuations, Eq. E19. We thus obtain for the predictive information, +Ipred(x0; ℓτ) = −1 +2 log(1 − ρ2 +ℓxe−2τ/τℓ). +(E21) +The instantaneous correlation coefficient ρ2 +ℓx is given in Eq. E20. From Eq. E21 it also becomes clear that while +the value of the predictive information depends on the forecast interval τ, the optimal design of the network that +maximizes the predictive information, determined by the optimal ratio XT/RT, the optimal integration time τr, and +the optimal ligand-bound receptor fraction p, does not depend on the forecast interval τ. +3. +Optimal resource allocation +Increasing the number of receptor or readout molecules always increases the precision with which the cell can predict +a signal (see Eq. E20). However, when the total resource pool is constrained, the cell has to choose whether it makes +more receptors or more readout molecules. To find the optimal ratio of read-out to receptor molecules we, can use +the C = ART + BXT to express XT and RT in terms of the total cost C and the ratio XT/RT: +XT = C +XT/RT +A + BXT/RT +, +(E22) +RT = C +1 +A + BXT/RT +. +(E23) +The factors A and B set the cost of receptors and readout molecules, respectively. Substituting these expressions for +XT and RT into the expression for the correlation coefficient between the output and ligand concentration (Eq. E20), +setting the derivative of the resulting expression with respect to XT/RT to zero, and solving for XT/RT gives +(XT/RT)opt = +�� +1 + τr +τc +� +p +1 − p +1 +f(1 − f) +A +B , += 2 +� +p/(1 − p) +� +1 + τr/τc, +(E24) +where for the second line we used A = B = 1 and f = f opt = 1/2. This is the optimal ratio of readout to receptor +molecules in the push-pull network, given an integration time τr and a steady state fraction of ligand-bound receptors +p. +Perhaps surprisingly, this optimal ratio (XT/RT)opt maximizes, for a given τr and p, not only the predictive +information, but also the past information. This is because the ratio XT/RT determines, together with τr and p, the +interval ∆ for sampling the ligand-binding state of the receptor: when the ratio XT/RT obeys Eq. E24, the readout +molecules sample each receptor molecule roughly once every correlation time: ∆ ∼ τc [10, 36]. Eq. E24 is thus a +statement about optimally extracting the information that is encoded in the receptor-ligand binding history, both +concerning the past information and the predictive information. This is illustrated in Fig. 6. +4. +Operating costs diverge when approaching the information bound +The precision of any sensing device is limited by the resources that are devoted to it. The cost function we consider +in this work is +C = λ(RT + XT) + c1XT∆µ/τr. +(E25) +The first term is the maintenance cost; this is the cost of producing new network components at the growth rate +λ. The second term is the operating cost and describes the chemical power that is necessary to run the network; it +depends on the flux through the network, XT/τr, and the free-energy drop ∆µ over a full cycle of phosphorylation +and dephosphorylation, given by the free energy of ATP hydrolysis. The coefficient c1 describes the relative energetic +cost of synthesising the components during the cell cycle, versus that of running the system. In the main text we +consider the case where c1 → 0. Here we will investigate how close cells can come to the information bound when c1 +is finite, thus including the chemical power cost of running the network. +It is clear from Eq. E25 that for finite c1 the operating cost diverges when τr → 0. Because the optimal IBM +solutions are instantaneous, this is precisely the limit in which the network must be to reach the information bound. + +23 +0.0 +0.5 +1.0 +1.5 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +Past info (bits) +Predictive info (bits) +τr=0.01 +τr=0.2 +τr=0.5 +τr=1 +Varied XT/RT +p=0.1 +p=0.2 +p=0.4 +FIG. 6. +The past and predictive information are maximized by the same ratio XT/RT and fraction p. +The +information plane, showing the information bound in black, and the isocost line C = 104 in gray. To construct the coloured +lines in this figure the ratio XT/RT has been varied from zero to a value beyond the optimal value that maximizes Ipast and +Ipred. This is done for several values of the receptor occupancy p (p = 0.1 in red, p = 0.2 in blue, p = 0.4 in orange), and +for several values of τr (indicated in the figure). When XT/RT reaches its optimal value, both Ipast and Ipred are maximal. +When the ratio is increased further the system moves back to the origin via the same coordinates. Only the integration time +τr meaningfully distinuishes between strategies that maximize predictive or past information, or that approach the information +bound. The reason is that XT/RT, together with τr and p, control the optimal extraction of information that is encoded in +the receptor-ligand binding history, both concerning Ipast and Ipred. The gray isocost line is obtained by varying τr, while +maximizing for each τr the correlation coefficient given by Eq. E20; the latter is done by substituting Eq. E24 into Eq. E20 and +numerically optimizing the resulting expression over p. The isocost line gives the region of Ipast and Ipred that is accessible for +a given resource cost C. Parameter values are A = B = 1, f = 1/2, (σℓ/¯ℓ)2 = 10−2, τc/τℓ = 10−2. +As a consequence, when we consider the operating costs, the push-pull network can only be at the information bound +when (Ipast, Ipred) → (0, 0) or C → ∞ (Fig. 7A). The system can mitigate the operating costs by decreasing XT, +because this decreases the flux through the cycle. However, this also decreases the gain and thus, eventually, any +information transduced through the network. In the limit that both XT and τr approach zero, the system approaches +the information bound at the origin, see both Fig. 7A and B. More generally, when the running costs are taken into +account, the system time averages more (i.e., τr rises), because frequent measurements are now even more costly. Still, +τr decreases as the total resource availability C grows. +Appendix F: Chemotaxis network +The evidence is mounting that in the E. coli chemotaxis system, receptors cooperatively control the activity of the +kinase CheA [29, 44–46]. Furthermore, the kinase activity is adaptive due to the methylation of inactive receptors +[15, 47]. A widely used approach to describe the effects of receptor cooperativity and methylation on kinase activity, +has been to employ the Monod-Wyman-Changeux (MWC) model [18, 24, 29, 33, 42, 43, 48, 49]. We will follow this +approach and, more specifically, model the chemotaxis system as described by Tu and colleagues [30]. In this model, +each receptor can switch between an active and inactive conformational state. Moreover, receptors are partitioned +into clusters of equal size N. In the spirit of the MWC model, receptors within a cluster switch conformation in +concert, so that each cluster is either active or inactive [33]. Furthermore, it is assumed that receptor-ligand binding +and conformational switching are faster than the other timescales in the system. The probability for the kinase, i.e. +the receptor cluster, to be active, is then described by: +a(ℓ, m) = +1 +1 + exp(∆FT (ℓ, m)), +(F1) +where ∆FT (ℓ, m) is the total free-energy difference between the active and inactive state, which is a function of the +ligand concentration ℓ(t) and the methylation level of the cluster m(t). The simplest model adopted here assumes +a linear dependence of the total free-energy difference on the free-energy difference arising from ligand binding and +methylation: +∆FT (ℓ, m) = −∆E0 + N(∆Fℓ(ℓ) + ∆Fm(m)), +(F2) + +24 +A +B +FIG. 7. Due to diverging operating costs the push-pull network only reaches the information bound for infinite +resource availability. (A) In green, the region of accessible predictive and past information in the push-pull network under +a resource constraint C = λ(RT + XT) + c1XT∆µ/τr, with λ = 1 and c1 = 1/∆µ, corresponding to a cell doubling time of +roughly 20min [10]. The black line is the information bound; the red and blue dots mark the points where Ipred and Ipast are +maximized, respectively, under a resource constraint C; the red and blue lines connect these points, respectively, for increasing +C. The accesible region for C ≤ 104 and the isocost lines for C = 103 and C = 105 have been obtained as described under +Fig. 6. The forecast interval has been set to one signal correlation time in the future: τ = τℓ. (B) The integration time over +the receptor correlation time, τr/τc, and the ratio of the number of readout and receptor molecules, XT/RT, as a function of +the distance θ along the iscocost line for C = 104 in panel A. For θ → 0, both τr and XT go to zero, thus reducing both Ipast +and Ipred to zero. Other parameter values in both panels are f = f opt = 1/2, (σℓ/¯ℓ)2 = 10−2, τc/τℓ = 10−2. +where the free-energy difference due to ligand binding is +∆Fℓ(ℓ) = ln(1 + ℓ(t)/KI +D) − ln(1 + ℓ(t)/KA +D). +(F3) +Between the two states the cluster has an altered dissociation constant, which is denoted KI +D for the inactive state, +and KA +D for the active state. The free-energy difference due to methylation has been experimentally shown to depend +approximately linearly on the methylation level [29]: +∆Fm(m) = ˜α( ¯m − m(t)). +(F4) +We assume that inactive receptors are irreversibly methylated, and active receptors irreversibly demethylated, with +zero-order ultrasensitive kinetics [30, 31, 50]. The dynamics of the methylation level of the ith receptor cluster is then +given by: +˙mi =(1 − ai(ℓ, mi))kR − ai(ℓ, mi)kB + Bmi(ai)ξ(t), +(F5) +with B(i) +m (ai) = +� +(1 − ai(ℓ, mi))kR + ai(ℓ, mi)kB, and unit white noise ξ(t). These dynamics indeed give rise to +perfect adaptation, since from this equation we find that the steady state cluster activity is given by p ≡ ¯a = +1/(1 + kB/kR), thus indeed independent of the ligand concentration. +Finally, active receptors catalyze phosphorylation of read-out molecules, and phosphorylated read-out molecules +decay at a constant rate. We have +˙x∗ = +RT +� +i=1 +ai(t)(XT − x∗(t))kf − x∗(t)kr + Bx(ai, x∗)ξ(t), +(F6) +where RT is the total number of receptor clusters. The steady state fraction of phosphorylated read-outs is given by +f ≡ ¯x∗/XT = (1 + kr/(kfRTp))−1. + +25 +1. +Linear dynamics +We again do a first order approximation around the steady state, defining all variables in terms of deviations from +their mean: δℓ(t) = ℓ(t) − ¯ℓ, δm(t) = m(t) − ¯m and δa(t) = a(t) − p. The linear form of this model has previously +been studied in for example [30] and [31]. We obtain for the linear dynamics of the ith cluster activity +δai(t) = αδmi(t) − βδℓ(t), +(F7) +with α = ˜αNp(1 − p) and β = κNp(1 − p), with κ = (¯ℓ + KI +D)−1 − (¯ℓ + KA +D)−1. For the methylation on the ith cluster +and for the readout dynamics we then obtain, as a function of δa(t), +˙ +δmi = −δai(t)/(ατm) + ηmi(t), +(F8) +˙ +δx∗ = γ +RT +� +i=1 +δai(t) − δx∗(t)/τr + ηx(t), +(F9) +where we have introduced the relaxation times τm = (α(kR + kB))−1 for methylation and τr = (RTpkf + kr)−1 for +phosphorylation. +We have further defined the rate at which an active cluster phosphorylates the readout CheY: +γ = XTf(1 − f)/(pRTτr). Substituting the expression for δai in Eq. F7 into Eqs. F8 and F9, and expressing the +dynamics in terms of the methylation on all clusters gives +d +dt +� RT +� +i=1 +δmi +� += − +RT +� +i=1 +δmi/τm + qδℓ(t)/(ατm) + ηm(t), +(F10) +˙ +δx∗ = −δx∗(t)/τr − γqδℓ(t) + γα +RT +� +i=1 +δmi(t) + ηx(t), +(F11) +with q = RTβ (see Eq. F7 for β). The rescaled white noise ηm is the sum of the methylation noise on all receptor +clusters, ⟨η2 +m⟩ = 2RTp(1 − p)/(ατm), where we have assumed that the methylation noise on the respective receptor +clusters is independent. The phosphorylation noise has strength ⟨η2 +x⟩ = 2XTf(1 − f)/τr. +2. +Parameter values +A large body of work has studied the parameters of the MWC model for the E. coli chemotaxis system. We have +listed the parameters relevant for our model in table I. We choose the background concentration ¯ℓ to be in between +KI +D and KA +D, at ¯ℓ = 100µM. +In this work we analyze the impact of the methylation timescale τm, and the numbers of receptor clusters and +readout molecules RT and XT, on the past and predictive information. We therefore do not set them to a fixed value, +but experimental estimates are listed in table II. +3. +Model statistics +Again we take the power spectrum route to determine the variance in the network output, the SNR, and the +correlation coefficient between current output and the future signal. +We consider the system to sense the non- +Markovian ligand concentration defined in equation Eq. B3. Such a signal is characterized by both its concentration +TABLE I. Measured E. coli chemotaxis parameter values. +Parameter +Value +Source +Description +KI +D +18µM +[42, 43] +MeAsp-Tar dissociation constant inactive receptor +KA +D +2900µM +[42, 43] +MeAsp-Tar dissociation constant active receptor +N +∼ 6 +[29, 42, 43, 51] +Number of receptors per cluster +˜α +2kBT +[29] +Free energy change per added methyl group +p +1 +3, 1 +2 +[29, 43] +Steady state activity at 22◦C, 32◦C +τr +∼ 0.1s +[10, 18, 51] +Phosphorylation timescale + +26 +TABLE II. Approximate E. coli chemotaxis timescales and abundances. +Parameter +Value +Source +Description +τm +∼ 10s +[15, 18, 29] +Adaptation time +Tsr+Tar +14000, 3300 +[35] +Rich medium; RP437, OW1 strain +Tsr+Tar +24000, 37000 +[35] +Minimal medium; RP437, OW1 strain +CheY +8200, 1400 +[35] +Rich medium; RP437, OW1 strain +CheY +6300, 14000 +[35] +Minimal medium; RP437, OW1 strain +and derivative, and the (cross-)power spectra of these properties are +Ss(ω) = +� +Sℓ(ω) +Sℓ→v(ω) +Sv→ℓ(ω) +Sv(ω) +� += +� +Sℓ(ω) +iωSℓ(ω) +−iωSℓ(ω) ω2Sℓ(ω) +� +, +(F12) +with +Sℓ(ω) = +2σ2 +v/τv +(ω2 + ((2τv)−1 + ρ)2)(ω2 + ((2τv)−1 − ρ)2), +(F13) +where ρ = +� +(4τ 2v )−1 − ω2 +0. The chemotaxis signalling network is fully determined by the following matrices (Eq. A1) +G = q +� +1/(ατm) 0 +−γ +0 +� +, +(F14) +J = +� +−1/τm +0 +αγ +−1/τr +� +, +(F15) +B = +�� +⟨η2m⟩ +0 +0 +� +⟨η2x⟩ +� +. +(F16) +The Fourier transform of the matrix exponential of the Jacobian is +F{eJ t} = (iωIn − J )−1 += +� +� +1 +1/τm+iω +0 +αγ +(1/τm+iω)(1/τr+iω) +1 +1/τr+iω +� +� , +(F17) +which allows us to determine the gain matrix via G(ω) = F{eJ t}(ω)G, and the noise matrix using N(ω) = F{eJ t}(ω)B; +see also Eq. A4 and Eq. A5. +To gain more insight in the way in which the network maps the signal onto its output, we first study the integration +kernels of the system. The integration kernel from ligand concentration to output is given by the inverse Fourier +transform of element (1, 2) of the gain matrix G(ω), which is +k(t) ≡ F−1{˜gℓ→x(ω)} = κNf(1 − f)(1 − p)XT +1 +1 − τr/τm +� 1 +τm +e−τ/τm − 1 +τr +e−τ/τr +� +, +(F18) +with κ = (¯ℓ+KI +D)−1−(¯ℓ+KA +D)−1. Due to the adaptive nature of the network, the static gain from ligand concentration +to output is zero: ¯gℓ→x = +� ∞ +0 +k(t)dt = 0; the long-time response to a step change in a constant input is zero. The +kernel does indeed not change the output based on the input concentration directly, but instead takes a (time-averaged) +derivative of the input (Fig. 8A). It is therefore useful to consider the kernel that maps the signal derivative onto the +output. This kernel can be found by rearranging the expression for the output of a linear signalling network, Eq. D1. +Disregarding the noise terms and integrating by parts gives +� 0 +−∞ +k(−t)ℓ(t)dt = K(−t)ℓ(t)|0 +−∞ − +� 0 +−∞ +K(−t)v(t)dt, +(F19) + +27 +where v(t) ≡ ˙ℓ and K(t) is the primitive of k(t). To make progress we first determine K(t), +K(t) = κNf(1 − f)(1 − p)XT +1 +1 − τr/τm +� +−e−τ/τm + e−τ/τr� +. +(F20) +The form of K(t) is that of a simple exponential kernel with a delay (Fig. 8B). We thus have both K(0) = 0 and +K(∞) = 0. It is now clear that the convolution over the ligand concentration simply maps onto the convolution over +its derivative as +� 0 +−∞ +k(−t)ℓ(t)dt = − +� 0 +−∞ +K(−t)v(t)dt. +(F21) +The static gain of K(t) is ¯gv→x = +� ∞ +0 +K(t)dt = qγτrτm = κNXT(1 − p)f(1 − f)τm. The gain thus increases with the +number of receptors per cluster, N, the number of readout molecules, XT, and notably, with the adaptation time τm. +This static gain from signal derivative to network output is a useful quantity which we will use to describe the other +statistics of the network below. +0 +5 +10 +15 +20 +-5 +0 +5 +10 +15 +20 +25 +30 +Time (s) +-k(t): kernel ℓ → x* +0 +5 +10 +15 +20 +-5 +0 +5 +10 +15 +20 +25 +30 +Time (s) +K(t): kernel v → x* +0.01 +0.10 +1 +10 +100 +0.01 +1 +100 +104 +Frequency ω (s-1) +Power +τr +-1 +τm +-1 +Nx +2 +gℓ→x +2 +gℓ→x +2 /Nx +2 +A +B +C +FIG. 8. Integration kernel and power spectra. (A) The integration kernel k(t) takes a temporal derivative by weighing the +most recent signal values with an opposite sign from the preceding ones. (B) The integration kernel K(t) from the derivative +of the input concentration to the network output. The kernel K(t) is the primitive of k(t), and its static gain is proportional +to the adaptation timescale τm. (C) Frequency dependent gain ˜g2 +ℓ→x(ω), frequency dependent noise N 2 +x(ω), and their ratio, as +a function of frequency. The chemotaxis network is a band-pass filter, the frequencies that are passed through are set by τr +on the high end and τm on the low end. At low frequencies, the methylation noise dominates. Parameters used in all panels +τr = 0.1s and τm = 10s. Model parameters are ˜a = 2, N = 6, KI +D = 18µM, KA +D = 2900µM, ¯ℓ = 100µM, p = f = 0.5. +To compute the past and predictive information, we need to determine the variance in the output, the SNR, and the +correlation between the current output and the future ligand derivative. To that end we require the power spectrum +of the output, and the cross-spectrum from output to future derivative. For the power spectrum of the output we use +Eq. A5 to find +Sx(ω) = +q2γ2ω2 +(τr−2 + ω2)(τm−2 + ω2)Sℓ(ω) + +α2γ2⟨η2 +m⟩ +(τr−2 + ω2)(τm−2 + ω2) + +⟨η2 +m⟩ +τr−2 + ω2 . +(F22) +From this power spectrum we can see that the network is a band-pass filter, where the gain is maximal in the frequency +range τm−1 < ω < τr−1. Both for ω ≫ τr−1 and ω ≪ τm−1 the gain goes to 0. On long timescales the methylation +noise dominates (Fig. 8C). The cross-power spectrum between current output and future ligand derivative is given by +element (2, 2) of the matrix G(−ω)Ss(ω) which is (also see Eq. A6) +Sx→v(ω) = qγ +−ω2Sℓ(ω) +(τm−1 − iω)(τr−1 − iω). +(F23) +In the main text, we argue that the biologically relevant regime of the input signal is the limit ω0 → 0. We therefore +present below the network statistics in this limit. We start by determining the variance in the readout, via the inverse +Fourier transform of its power spectrum (Eq. F22): +lim +ω0→0 σ2 +x = ¯g2 +v→x +1 + τr/τv + τr/τm +(1 + τm/τv)(1 + τr/τv)(1 + τr/τm)σ2 +v + ¯g2 +a→xαRTp(1 − p) +1 +1 + τr/τm ++ XTf(1 − f), += ¯g2 +v→x +1 + τr/τv + τr/τm +(1 + τm/τv)(1 + τr/τv)(1 + τr/τm) +� +�� +� +dynamical gain +σ2 +v + XTf(1 − f) +� +1 + ¯gv→x +˜α(1 − p) +RTκτm +1 +1 + τr/τm +� +, +(F24) + +28 +where ¯ga→x = γτr = XTf(1−f)/(RTp) is the static gain from receptor activity to readout, and we used the definition +of α = ˜αNp(1 − p). Because there is no receptor-ligand binding noise, there is also no time averaging as in the +push-pull network (and hence no factor depending on τr/τc). There is methylation noise on a timescale τm, but +this cannot be time-averaged effectively because the integration time τr of the push-pull network is shorter than the +receptor methylation timescale τm. The methylation noise can only be averaged out significantly by increasing RT. +The contribution from the variance in the signal derivative, σ2 +v, to the output noise σ2 +x, depends on the dynamical gain, +which is the product of the static gain ¯g2 +v→v and a factor that only depends on ratios of timescales. The dynamical +gain is maximized for τr → 0 and τm → ∞, which is intuitive since subtracting a signal from an earlier one reduces +the amplification of the signal. Hence, when the system has too few XT molecules to lift the signal above the noise, +τm must be increased to raise the gain. Only when XT is sufficiently large, can τm be reduced. This allows the system +to take more recent derivatives. The signal to noise ratio SNR = σ2 +x|η/σ2 +x|L can straightforwardly be obtained from +Eq. F24. For the covariance between the current output and the future derivative we have +lim +ω0→0⟨δx(0)δv(τ)⟩ = F−1{Sx→v(ω)}, += +−¯gv→xσ2 +v +(1 + τm/τv)(1 + τr/τv)e−τ/τv. +(F25) +The variance in Eq. F24 can be used to obtain the normalized correlation function ⟨δx(0)δv(τ)⟩/(σxσv). +4. +Past and predictive information of the chemotaxis network +The past and predictive information are straightforward to compute from the quantities above. The definition of +the past information is the same as for the push-pull network, and is given by Eq. D3. The SNR is now given by, +using Eq. F24: +SNR = σ2 +x|η/σ2 +x|L = κ2Nτ 2 +mσ2 +v +1 + τr/τv + τr/τm +(1 + τm/τv)(1 + τr/τv)(1 + τr/τm) +� � +1 +NXTf(1 − f)(1 − p)2 + +˜α +RT(1 + τr/τm) +� +, +where κ = (¯ℓ + KI +D)−1 − (¯ℓ + KA +D)−1. The predictive information is found in the same manner as in Eq. D6, but now +it is a function of the correlation between the current output and the future derivative of the ligand concentration. +This correlation can be decomposed into the instantaneous correlation coefficient and an exponential decay on the +timescale of the fluctuations of the derivative of the concentration, Eq. F25. Specifically, the predictive information +is given by +Ipred(x0; vτ) = −1 +2 log(1 − ρ2 +ℓve−2τ/τv). +(F26) +The instantaneous correlation coefficient ρ2 +ℓv can be found using Eq. F25 and Eq. F24. From Eq. F26 it is clear +that just like for the push-pull network, the optimal design of the network that maximizes the predictive information, +determined by the optimal ratio XT/RT and the optimal adaptation time τm, does not depend on the forecast interval +τ. The forecast interval only affects the magnitude of the predictive information. +5. +Optimal allocation +We can determine the optimal ratio (XT/RT)opt that maximizes either the past information or the predictive infor- +mation, given all other network parameters, most notably τm. Just as for the push-pull network, we find however that +the optimal ratio (XT/RT)opt is the same regardless of whether the past or the predictive information is maximized. +This is again because the information on the future signal (be it the value or the derivative) is encoded in the receptor +occupancy, while the ratio XT/RT controls the interval by which the downstream readout samples the receptor to +estimates its occupancy. Nonetheless, the optimal methylation timescale τmopt that maximizes either the past or the +predictive information is different—maximizing predictive information requires a more recent derivative and hence a +shorter τm than obtaining past information. +Given τm and all other parameters, the optimal ratio of the number of readout molecules over receptor clusters is, + +29 +using C = RT + XT, +�XT +RT +�opt += +� +1 +α +1 +f(1 − f) +p +1 − p +� +1 + τr +τm +, += 2 +� +2/N +� +1 + τr +τm +, +(F27) +where in the second line we have used that α = ˜αNp(1 − p), and ˜α = 2, and f = p = 0.5. Because for the chemotaxis +network τr < τm the ratio τr/τm only varies between 0 and 1. For this reason, the optimal ratio (XT/RT)opt depends +only weakly on τm, and does not vary strongly along the isocost lines of Fig. 4A in the main text, see Fig. 9. +0.00 +0.05 +0.10 +0.15 +0.20 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Distance along fixed cost line θ +XT/RT +FIG. 9. 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Stock, Organiza- +tion of the Receptor-Kinase Signaling Array That Regu- +lates Escherichia coli Chemotaxis *, Journal of Biological +Chemistry 277, 36748 (2002), publisher: Elsevier. + diff --git a/2dE2T4oBgHgl3EQfjAeF/content/tmp_files/load_file.txt b/2dE2T4oBgHgl3EQfjAeF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..12f5916a48f43150488b2f4ed591096618f5605f --- /dev/null +++ b/2dE2T4oBgHgl3EQfjAeF/content/tmp_files/load_file.txt @@ -0,0 +1,1479 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf,len=1478 +page_content='Trade-offs between cost and information in cellular prediction Age J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Tjalma,1 Vahe Galstyan,1 Jeroen Goedhart,1 Lotte Slim,1 Nils B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Becker,2 and Pieter Rein ten Wolde1, ∗ 1AMOLF, Science Park 104, 1098 XG Amsterdam, The Netherlands 2Theoretical Systems Biology, German Cancer Research Center, 69120 Heidelberg, Germany (Dated: January 11, 2023) Living cells can leverage correlations in environmental fluctuations to predict the future environ- ment and mount a response ahead of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To this end, cells need to encode the past signal into the output of the intracellular network from which the future input is predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Yet, storing information is costly while not all features of the past signal are equally informative on the future input signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Here, we show, for two classes of input signals, that cellular networks can reach the fundamental bound on the predictive information as set by the information extracted from the past signal: push- pull networks can reach this information bound for Markovian signals, while networks that take a temporal derivative can reach the bound for predicting the future derivative of non-Markovian signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' However, the bits of past information that are most informative about the future signal are also prohibitively costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' As a result, the optimal system that maximizes the predictive information for a given resource cost is, in general, not at the information bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Applying our theory to the chemotaxis network of Escherichia coli reveals that its adaptive kernel is optimal for predicting future concentration changes over a broad range of background concentrations, and that the system has been tailored to predicting these changes in shallow gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Keywords: prediction, information bottleneck, sensing, resource allocation Single-celled organisms live in a highly dynamic envi- ronment to which they continually have to respond and adapt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To this end, they employ a range of response strategies, tailored to the temporal structure of the envi- ronmental variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' When these variations are highly regular, such as the daily light variations, it becomes ben- eficial to develop a clock from which the time and hence the current and future environment can be inferred [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In the other limit, when the fluctuations are entirely un- predictable, cells have no choice but to resort to either the strategy of detect-and-respond or the bet-hedging strategy of stochastic switching between different phe- notypes [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Yet arguably the most fascinating strategy lies in between these two extremes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' When the environ- mental fluctuations happen with some regularity, then it becomes feasible to predict the future environment and initiate a response ahead of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' While it is commonly believed that only higher organisms can predict the fu- ture, experiments have vividly demonstrated that even single-cell organisms can leverage temporal correlations in environmental fluctuations in order to predict, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=', future nutrient levels [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The ability to predict future signals can provide a fit- ness benefit [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The capacity to anticipate changes in oxygen levels [4], or the arrival of sugars or stress signals [5], can increase the growth rate of single-celled organ- isms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' modeling has revealed that prediction can enhance bacterial chemotaxis [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Yet, a predict-and-anticipate strategy is only advantageous if the cell can reliably pre- dict the future on timescales that are longer than the time it takes to mount a response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' What fundamentally limits the accuracy of cellular prediction remains, how- ever, poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' ∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='wolde@amolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='nl While the cell needs to predict the future environ- ment, it can only sense the present and remember the past (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 1A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Consequently, for a given amount of in- formation the cell can store about the present and past signal, there is a maximum amount of information it can possibly have about the future [6, 8] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 1C-I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This in- formation bound is determined by the temporal structure of the environmental fluctuations [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' How close cells can come to this bound depends on the design of the intracellular biochemical network that senses and processes the environmental signals (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To maximize the predictive power the cell must use its memory effectively: it should extract only those charac- teristics from the present and past signal that are most informative about the future [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Whether it can do so, is determined by the topology of the signaling network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Moreover, like any information processing device, bio- chemical networks require resources to be built and run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Molecular components are needed to construct the net- work, space is required to accommodate the components, time is needed to process the information, and energy is required to synthesize the components and operate the network [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' These resources constrain the design and performance of any biochemical network, and the ca- pacity to sense and process information is no exception (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 1C-II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Cellular signaling systems provide a unique opportu- nity for revealing the resource requirements for predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Cells live in a highly dynamic environment, with temporal statistics that are expected to vary markedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Moreover, signaling networks have distinct topologies, which are likely tailored to the temporal statistics of the environment [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In addition, for cellular systems we can actually quantify the information processing capacity as a function of the resources that are necessary to build and run them—protein copies, time, and energy [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='03964v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='bio-ph] 10 Jan 2023 2 Cellular systems are thus ideal for elucidating the rela- tionships between future and past information, system design (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' network topology) and resource constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Here, we derive the bound on the prediction precision as set by the information extracted from the past signal for two types of input signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We will determine how close cellular networks can come to this bound, and how this depends on the topology of the network and the resources to build and run it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We find that for the two classes of input signals stud- ied, cellular networks exists that can reach the informa- tion bound, yet reaching the bound is exceedingly costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The first class of input signals consists of Markovian sig- nals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Using the Information Bottleneck Method (IBM) [8, 12], we first show that the system that reaches the information bound copies the most recent input signal into the output from which the future input is predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Push-pull networks consisting of chemical modification or GTPase cycles, which are ubiquitous in prokaryotic and eukaryotic cells [13, 14], should be able to reach the infor- mation bound, because they are at heart copying devices [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Yet, copying the most recent input into the out- put is extremely costly, because the operating cost, as set by the chemical power to drive the cycle, diverges at high copying speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' More surprisingly, our results show that the predictive and past information can be raised simul- taneously by moving away from the information bound, even when the operating cost is negligible: the optimal system that maximizes the predictive information for a given protein synthesis cost is, in general, not at the in- formation bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The number of bits of past information per protein cost can be raised by increasing the integra- tion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' While this decreases the predictive power per bit of past information, thereby moving the system away from the information bound, it can increase the total pre- dictive information per protein cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Our analysis thus highlights that not all bits of past information are equally costly, nor predictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Living cells that navigate their environment typically experience signals with persistence as generated by their own motion, which motivated us to study a simple class of non-Markovian signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Moreover, these cells can typ- ically detect changes in the concentration over a range of background concentrations that is orders of magnitude larger than the change in the concentration over the ori- entational correlation time of their movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Our anal- ysis reveals that in such a scenario the optimal kernel that allows the system to reach the information bound on pre- dicting the future input derivative is a perfectively adap- tive, derivative-taking kernel, precisely as the bacterium E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli employs [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We again find, however, that reach- ing the information bound is prohibitively costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The reason is that taking an instantaneous derivative, which is the characteristic of the input that is most informative about the future derivative, reduces the gain to zero be- cause the system instantly adapts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' the response becomes thwarted by biochemical noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The optimal system that maximizes the predictive information under a resource constraint thus emerges from a trade-off between taking a derivative that is recent and one that is reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Finally, our analysis reveals that the E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli chemotaxis system has been optimally designed to predict future concentra- tion changes in shallow gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' RESULTS We focus on cellular signaling systems that respond linearly to changes in the input signal [11, 16–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' These systems not only allow for analytical results, but also describe information transmission often remarkably well [19–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The output of these systems can be written as x(t) = � t −∞ dt′k(t − t′)ℓ(t′) + ηx(t), (1) where k(t) is the linear response function, ℓ(t) the input signal, and ηx(t) describes the noise in the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We will consider stationary signals with different temporal correlations, obeying Gaussian statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Any prediction about the future state of the environ- ment must be based on information obtained from its past (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 1C-I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In particular, the cell needs to predict the input ℓτ ≡ ℓ(t + τ) at a time τ into the future from the current output x0 ≡ x(t), which itself depends on the input signal in the past, Lp ≡ (ℓ(t), ℓ(t′), · · · ), with t > t′ > · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The (qualitative) shape of the integration kernel k(t), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' exponential, adaptive or oscillatory, is determined by the topology of the signaling network [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The kernel shape describes how the past signal is mapped onto the current output, and hence which characteristics of the past signal the cell uses to predict the future signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To maximize the accuracy of prediction, the cell should extract those features that are most informative about the future signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' These depend on the statistics of the input signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Deriving the upper bound on the predictive informa- tion as set by the past information is an optimisation problem, which can be solved using the IBM [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' It en- tails the maximization of an objective function L: max P (x0|Lp) [L ≡ I(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' ℓτ) − γI(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Lp)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (2) Here, Ipred ≡ I(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' ℓτ) is the predictive information, which is the mutual information between the system’s current output x0 and the future ligand concentration ℓτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The past information Ipast ≡ I(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Lp) is the mutual information between x0 and the trajectory of past lig- and concentrations Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The Lagrange multiplier γ sets the relative cost of storing past over obtaining predic- tive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Given a value of γ, the objective func- tion in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 2 is maximized by optimizing the conditional probability distribution of the output given the past in- put trajectory, P(x0|Lp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For the linear systems consid- ered here, this corresponds to optimizing the mapping of the past input signal onto the current output via the 3 past information predictive information resources inaccessible inaccessible I II information bound A B C Different input signals maintenance operating Optimal networks X X RL time signal output past info predictive info now concentration FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Cells use biochemical networks to remember the past and predict the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (A) Cells compress the past input into the dynamics of the signalling network from which the future input is then predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (B) The optimal topology of the network for predicting the future signal depends on the temporal statistics of the input signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Push-pull networks, consisting of chemical modification cycles or GTPase cycles, can optimally predict the future value of Markovian signals, with correlation time τℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' derivative-taking networks, like the E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli chemotaxis system, can optimally predict the future derivative of non-Markovian signals, with correlation time τv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The push-pull network consists of a receptor that drives a downstream phosphorylation cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The ligand binds the receptor with a correlation time τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The push-pull network, driven by ATP turnover, integrates the receptor with an integration time τr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The chemotaxis system is a push-pull network, yet augmented with negative feedback on the receptor activity via methylation on a timescale τm, as indicated by the dashed grey line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The total resource cost consists of a maintenance cost of receptor and readout synthesis at the growth rate λ, and an operating cost of driving the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C) The predictive information on the future signal Ipred is fundamentally bounded by how much information Ipast it has about the past signal (panel I), which in turn is limited by the resources necessary to build and operate the biochemical network (panel II) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' integration kernel k(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Since our model obeys Gaussian statistics, we use the Gaussian IBM to derive the optimal kernel kopt(t) and the information bound, defined to be the maximum predictive information as set by the past information [12] (see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Markovian signals Optimal prediction of Markovian signals: biochemical copying Arguably the most elementary type of signal, albeit perhaps the hardest to predict, is a Markovian signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We consider a Markovian signal ℓ(t), of which the devia- tions δℓ(t) = ℓ(t) − ¯ℓ from its mean ¯ℓ follow an Ornstein- Uhlenbeck (OU) process: δ ˙ℓ = −δℓ(t)/τℓ + ηℓ(t), (3) where τℓ is the correlation time of the fluctuations, and ηℓ(t) is Gaussian white noise, ⟨η(t)η(t′)⟩ = 2σ2 ℓ/τℓ δ(t − t′), with σ2 ℓ the amplitude of the signal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This input signal obeys Gaussian statistics, characterized by ⟨δℓ(0)δℓ(t)⟩ = σ2 ℓ exp(−t/τℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The optimal mapping is therefore a linear one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Utilizing the Gaussian IBM frame- work [12], we find that the optimal integration kernel is given by (see Appendix C 2) kopt(t − t′) = aδ(t − t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (4) This optimal integration kernel corresponds to a signaling system that copies the current input into the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This is intuitive, since for a Markovian signal there is no additional information in the past signal that is not already contained in the present one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The prefactor a determines the gain ∂¯x/∂¯ℓ, which together with the noise strength σ2 ηx (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 1) and the signal amplitude σ2 ℓ set the magnitude of the past and predictive information, Ipast and Ipred, respectively (see Appendix C 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 2-I shows the maximum predictive information as set by the past information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This information bound ap- plies to any linear system that needs to predict a Marko- vian signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' How close can biochemical systems come to this bound?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Push-pull network can be at the information bound, yet increase the predictive and past information by moving away from it Although the upper bound on the accuracy of predic- tion is determined by the signal statistics, how close cells can come to this bound depends on the topology of the cellular signaling system, and the resources devoted to building and operating it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A network motif that could reach the information bound for Markovian signals is the push-pull network (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 2), because it is at heart a copy- ing device: it samples the input by copying the state of 4 I II predictive information (bits) resources past information (bits) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The optimal push-pull network is not at the information bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Panel I: The black line is the informa- tion bound that maximizes the predictive information Ipred = I(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' ℓτ) for a given past information Ipast = I(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Lp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The red curve shows Ipred against Ipast for systems in which Ipred has been maximized for a given resource cost C = RT + XT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The blue curve shows Ipred versus Ipast for systems where Ipast has been maximized for a given C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Panel II shows Ipast against C for the corresponding systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The forecast in- terval is τ = τℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The optimization parameters are the ratio XT/RT, τr, p and f (see Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Parameter values: (σℓ/¯ℓ)2 = 10−2, τc/τℓ = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' the input, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' the ligand-binding state of a receptor or the activation state of a kinase, into the activation state of the output, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' phosphorylation state of the readout [10, 11, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We model the push-pull network in the linear-noise approximation: δ ˙ RL = bδℓ(t) − δRL(t)/τc + ηRL(t), (5) ˙ δx∗ = γ δRL(t) − δx∗(t)/τr + ηx(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (6) Here, δRL represents the number of ligand-bound recep- tors and δx∗ the number of modified readout molecules, defined as deviations from their mean values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' b and γ are parameters that depend on the number of recep- tor and readout molecules, RT and XT respectively, the fraction of ligand-bound receptors p and active readout molecules f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' ηRL and ηx are Gaussian white noise terms (see Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Key parameters are the correlation time of receptor-ligand binding, τc, and the relaxation time of x∗, τr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The latter determines for how long x∗ carries information on the ligand-binding state of the re- ceptor and thus sets the integration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The readout- modification dynamics yield an exponential integration kernel k(t) ∝ exp(−t/τr), which in the limit τr → 0 re- duces to a δ-function, hinting that the system may reach the information bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' How much information cells can extract from the past signal depends on the resources devoted to building and operating the network (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 2-II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We define the total resource cost to be: C = λ(RT + XT) + c1XT∆µ/τr (7) The first term expresses the fact that over the course of the cell cycle all components need to be duplicated, which means that they have to be synthesized at a speed that is at least the growth rate λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The second term de- scribes the chemical power that is necessary to run the push-pull network [10, 11];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' it depends on the flux through the network, XT/τr, and the free-energy drop ∆µ over a cycle, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' the free energy of ATP hydrolysis in the case of a phosphorylation cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The coefficient c1 describes the relative energetic cost of synthesising the components during the cell cycle versus that of running the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For simplicity, we first consider the scenario that the cost is dominated by that of protein synthesis, setting c1 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' While in this scenario RT + XT is constrained, XT/RT and other system parameters are free for optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The available resources put a hard bound on the in- formation Ipast that can be extracted from the past sig- nal, which in turn sets a hard limit on the predictive information Ipred (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 1C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To maximize the predictive information, it therefore seems natural to maximize the past information Ipast for a given resource cost C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The blue line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 2-II shows the result for the push-pull network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We then compute the corresponding predictive information for the systems along this line, which is the blue line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 2-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Strikingly, the resulting information curve lies far below the information bound, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' the upper bound on the predictive information as set by the past information (black line, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 2-I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This shows that sys- tems that maximize past information under a resource constraint, do not in general also maximize predictive in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' It implies that not all bits of past information are equally predictive about the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Precisely because not all bits of past information are equally predictive about the future, it is paramount to directly maximize the predictive information for a given resource cost in order to obtain the most efficient pre- diction device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This yields the red lines in panels I and II in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' It can be seen that the predictive infor- mation is higher while the past information is lower, as compared to the information curves of the systems opti- mized for maximizing the past information under a re- source constraint (blue lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' It reflects the idea that not all bits are equally predictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' More surprisingly, while the bound on the predictive information as set by the resource cost (red line panel I) is close to the bound on the predictive information as set by the past information (black line), it does remain lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This is surprising, be- cause the push-pull network is a copying device [10, 23], which can, as we will also show below, reach the latter bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' These two observations together imply that not all bits of past information are equally costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' If they were, the cell would select under the two constraints the same bits based on their predictive information content, and the bound on the predictive information as set by 5 the resource cost would overlap with that as set by the past information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We thus find that not all bits of past information are equally predictive, nor equally costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' As we show next, it implies that the optimal information processing system faces a trade-off between using those bits of past infor- mation that are most informative about the future and those that are cheapest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Trade-off between cost and predictive power per bit To understand the connection between predictive and past information, and resource cost, we map out the re- gion in the information plane that can be reached given a resource constraint C (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 3A, green region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We im- mediately make two observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Firstly, the system can indeed reach the information bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Secondly, the system can increase both the past and the predictive in- formation by moving away from the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To elucidate these two observations, we investigate the system along the isocost line of C = 104, which together with the in- formation bound envelopes the accessible region for the maximum resource cost C ≤ 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Along the isocost line, the ratio of the number of readout over receptor molecules is XT/RT = 2 � p/(1 − p) � 1 + τr/τc (see Appendix E 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This can be understood intuitively using the optimal resource alloca- tion principle [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' It states that in a sensing system that employs its proteins optimally, the total number of inde- pendent concentration measurements at the level of the receptor during the integration time τr, RT(1 + τr/τc), equals the number of readout molecules XT that store these measurements, so that neither the receptors nor the readout molecules are in excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This design prin- ciple specifies, for a given integration time τr, the ratio XT/RT at which the readout molecules sample each re- ceptor molecule roughly once every receptor correlation time τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' While the optimal allocation principle gives the opti- mal ratio XT/RT of the number of readouts over recep- tors for a given integration time τr, it does not prescribe what the optimal integration time τropt, and hence (glob- ally) optimal ratio Xopt T /Ropt T , is that maximizes Ipred for a given resource constraint C = RT +XT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 3B shows that as the distance θ along the isocost line is increased, τr and hence XT/RT increase monotonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Near the information bound, corresponding to θ = 0, the integra- tion time τr is zero and the number of readout molecules equals the number of receptor molecules: XT = RT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In this limit, the push-pull network is an instantaneous re- sponder, with an integration kernel given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' only the finite receptor correlation time τc prevents the sys- tem from fully reaching the information bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Yet, as θ increases and the system moves away from the bound, the predictive and past information first rise along the contour, and thus with XT/RT and τr, before they even- tually both fall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To understand why the predictive and past informa- tion first rise and then fall with XT/RT and τr, we note that each readout molecule constitutes 1 physical bit and that its binary state (phosphorylated or not) encodes at most 1 bit of information on the ligand concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The number of readout molecules XT thus sets a hard upper bound on the sensing precision and hence the pre- dictive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To raise this bound, XT must be increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For a given resource constraint C = RT +XT, XT can only be increased if the number of receptors RT is simultaneously decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' However, the cell infers the concentration not from the readout molecules directly, but via the receptor molecules: a readout molecule is a sample of the receptor that provides at most 1 bit of in- formation about the ligand-binding state of a receptor molecule, which in turn provides at most 1 bit of infor- mation about the input signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To raise the lower bound on the predictive information, the information on the in- put must increase at both the receptor and the readout level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To elucidate how this can be achieved, we note that the maximum number of independent receptor samples and hence concentration measurements is given by N max I = min(XT, RT(1 + τr/τc)) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For θ > 0, the system can increase N max I if, and only if, XT and RT(1 + τr/τc) can be raised simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This can be achieved, while obeying the constraint C = XT + RT, by decreasing RT yet increasing τr (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This is the mechanism of time averaging, which makes it possible to increase the num- ber of independent receptor samples [11], and explains why both the predictive and the past information initially increase (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 3C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' However, as τr is raised further, the receptor samples become older: the readout molecules in- creasingly reflect receptor states in the past that are less informative about the future ligand concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The collected bits of past information have become less pre- dictive about the future (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 3C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For a given resource cost, the cell thus faces a trade-off between maximizing the number of physical bits of past information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' the receptor samples XT) and the predictive information per bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This antagonism gives rise to an optimal integration time τropt that maximizes the total predictive informa- tion Ipred (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 3C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Interestingly, while Ipred decreases beyond τropt, the past information Ipast first continues to rise because N max I still increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' However, when the integration time becomes longer than the input signal correlation time, the correlation between input and output will be lost and Ipast will fall too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Chemical power prevents the system from reaching the information bound So far, we have only considered the cost of maintain- ing the cellular system, the protein cost C = RT + XT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Yet, running a push-pull network also requires energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' As Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 7 shows, the running cost scales with the flux 6 A B C FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The push-pull network maximizes the predictive power under a resource constraint by moving away from the information bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (A) The region of accessible predictive information Ipred = I(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' ℓτ) and past information Ipast = I(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Lp) in the push-pull network under a resource constraint C ≤ (RT + XT), for the Markovian signals specified by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 3 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The black line is the information bound at which Ipred is maximized for a given Ipast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The push-pull network can be at the information bound (black points), but maximizing Ipred for a resource constraint C moves the system away from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The red and blue lines connect, respectively, the points where Ipred and Ipast are maximized along the green isocost lines (the contourlines of constant C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' they correspond to the red and blue lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The accessible region of Ipred and Ipast for a given C has been obtained by optimizing over τr, p, f, and XT/RT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The forecast interval is τ = τℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (B) The integration time τr over the receptor correlation time τc, τr/τc, and the ratio of the number of readout and receptor molecules, XT/RT, as a function of the distance θ along the isocost line corresponding to C = 104 in panel A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' the red and blue points denote where Ipred and Ipast are maximized along the contourline, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For θ → 0, τr → 0: the system is an instantaneous responder, which is essentially at the information boundary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' as predicted by the optimal resource allocation principle, XT = RT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The system can increase Ipred and Ipast by increasing τr and XT/RT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C) While this decreases the predictive information Ipred per physical bit of past information, Ipred/XT (dashed line), increasing XT/RT does increase the number of physical bits per resource cost, XT/C (purple line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This trade-off gives rise to an optimal predictive information per resource cost, Ipred/C (red dot on solid black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Parameter values unless specified: (σℓ/¯ℓ)2 = 10−2, τc/τℓ = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' around the phosphorylation cycle, which is proportional to the inverse of the integration time, τr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The power thus diverges for τr → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Since the information bound is reached precisely in this limit, it is clear that the chem- ical power prevents the push-pull network from reaching the bound (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 7 in the appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Non-Markovian signals Predicting the future change The push-pull network can optimally predict Marko- vian signals, yet not all signals are expected to be Marko- vian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Especially organisms that navigate through an en- vironment with directional persistence will sense a non- Markovian signal, as generated by their own motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Moreover, when these organisms need to climb a con- centration gradient, as E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli during chemotaxis, then knowing the change in the concentration is arguably more useful than knowing the concentration itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Indeed, it is well known that the kernel of the E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli chemotaxis system detects the (relative) change in the ligand con- centration by taking a temporal derivative of the concen- tration [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' However, as we will show here, the converse statement is more subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' If the system needs to predict the (future) change in the signal, then the optimal ker- nel is not necessarily one that is based on the derivative only: in general, the optimal kernel uses a combination of the signal value and its derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' However, the E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli chemotaxis system can respond to concentrations that vary between the dissociation constants of the inactive and active state of the receptors, which differ by several orders of magnitude [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This range of possible back- ground concentrations is much larger than the typical concentration change over the orientational correlation time of the bacterium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' As our analysis below reveals, in this regime the optimal kernel is a perfectly adaptive, derivative-taking kernel that is insensitive to the current signal value, precisely like that of the E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli chemotaxis system [15, 25–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Our analysis thus predicts that this system has an adaptive kernel, because this is the opti- mal kernel for predicting concentration derivatives over a broad range of background concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To reveal the signal characteristics that control the shape of the optimal integration kernel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' we will consider the family of signals that are generated by a harmonic 7 oscillator: δ ˙ℓ = v(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (8) ˙v = −ω2 0δℓ(t) − v(t)/τv + ηv(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (9) where δℓ is the deviation of ligand concentration from its mean ¯ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' v its derivative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' τv a relaxation time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' ηv a Gaus- sian white noise term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' and the frequency ω2 0 = σ2 v/σ2 ℓ controls the variance σ2 ℓ of the concentration and that of its derivative σ2 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Using the IBM framework it can be shown that the optimal encoding that allows the system to reach the information bound, is based on a linear combination of the current concentration ℓ(t) and its derivative v(t), such that the output x(t) is given by (Appendix C 3): x(t) = aδℓ(t) σℓ + bv(t) σv + ηx(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (10) This can be understood by noting that while the signal of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 8 and 9 is non-Markovian in the space of ℓ, it is Markovian in ℓ and v: all the information on the future signal is thus contained in the current concentration and its derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To maximize the predictive information Ipred = I(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' vτ) between the current output x0 and the future derivative of the input vτ for a given amount of past information Ipast = I(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Lp), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='e to reach the infor- mation bound for predicting the future signal derivative, the coefficients must obey aopt = G⟨δℓ(0)δv(τ)⟩ σℓσv ≡ Gρℓ0vτ , (11) bopt = G⟨δv(0)δv(τ)⟩ σ2v ≡ Gρv0vτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (12) Here, G is the gain, which together with the noise σ2 ηx sets the scale of Ipred and Ipast, ρℓ0vτ is the cross-correlation coefficient between the current concentration value ℓ0 and the future concentration derivative vτ and ρv0vτ that be- tween the current and future derivative (Appendix C 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' These expressions can be understood intuitively: if the future signal derivative that needs to be predicted is cor- related with the current signal derivative, it is useful to include in the prediction strategy the current signal derivative, leading to a non-zero value of bopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Perhaps more surprisingly, if the future signal derivative is also correlated with the current signal value, then the system can enhance the prediction accuracy by also including the current signal value, yielding a non-zero aopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Clearly, in general, to optimally predict the future signal change, the system should base its prediction on both the current signal value and its derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The degree to which the systems bases its prediction on the current value versus the current derivative depends on the relative magnitudes of aopt and bopt, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In Appendix B 2, we show that when the concentration change over the timescale τv, σvτv, is much smaller than the range of possible concentrations σℓ that the bac- terium can experience, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' when σvτv ≪ σℓ such that ω0 ≪ τ −1 v , the cross-correlation coefficient ρℓ0vτ vanishes, such that aopt becomes zero (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The optimal kernel has become a perfectly adaptive, derivative-taking kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We emphasize that while we have derived this re- sult for the class of signals defined by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 8 and 9, the idea is far more generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In particular, while we do not know the temporal structure of the ligand statistics that E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli experiences, we do know that it can detect con- centration changes over a range of background concentra- tions that is much wider that the typical concentration change over a run, such that the correlation between the concentration value and its future change is likely to be very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' As our analysis shows, a perfectively adap- tive kernel then emerges naturally from the requirement to predict the future concentration change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' While the class of signals specified by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 8 and 9 is arguably limited, it does describe the biologically impor- tant regime of chemotaxis in shallow gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In the limit that ω0 ≪ τv−1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 9 reduces to ˙v = −v/τv + ηv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In shallow gradients, the stimulus only weakly affects the swimming behavior, such that the perceived signal is mostly determined by the intrinsic orientational dy- namics of the bacterium in the absence of a gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In this regime, the temporal statistics of the concentration derivative v is completely determined by the steepness of the concentration gradient g and the swimming statistics of the bacterium in the absence of a gradient: ⟨δv(0)δv(τ)⟩ = g2¯ℓ2⟨δvx(0)δvx(τ)⟩ ≃ σ2 vxe−τ/τvx , (13) where the latter is the autocorrelation function of the (positional) velocity of the bacterium in the absence of a gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' It is a characteristic of the bacterium, not of the environment, and has been measured to decay expo- nentially with a correlation time τvx [18], precisely as our model, with τv = τvx, predicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This correlation time is on the order of the typical run time of the bacterium in the absence of a gradient, τv ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='9s [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Finite resources prevent the chemotaxis system from taking an instantaneous derivative and reaching the information bound The above analysis indicates that the chemotaxis sys- tem seems ideally designed to predict the future concen- tration change, because its integration kernel is nearly perfectly adaptive [15, 25–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' But how close can this system come to the information bound for the non- Markovian signals specified by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 8 and 9?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To address this, we consider a molecular model that can accurately describe the response of the chemotaxis system to a wide range of time-varying signals [29–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In this model, the receptors are partitioned into clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Each cluster is described via a Monod-Wyman-Changeux model [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' While each receptor can switch between an active and an inactive conformational state, the energetic cost of having different conformations in the same cluster is prohibitively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Each cluster is thus either active or 8 inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Ligand binding favors the inactive state while methylation does the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Lastly, active receptor clusters can via the associated kinase CheA phosphory- late the downstream messenger protein CheY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Linearizing around the steady state, we obtain: δai(t) = αδmi(t) − βδℓ(t), (14) δ ˙mi = −δai(t)/(ατm) + ηmi(t), (15) δ ˙x∗ = γ RT � i=1 δai(t) − δx∗(t)/τr + ηx(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (16) Here, δai(t) and δmi(t) are the deviations of the activ- ity and methylation level of receptor cluster i from their steady-state values, and RT is the total number of recep- tor clusters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' δℓ(t) and δx∗(t) are, respectively, the devi- ations of the ligand and CheYp concentration from their steady-state values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' τm and τr are the timescales of re- ceptor methylation and CheYp dephosphorylation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' ηmi and ηx are independent Gaussian white noise sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 14, we have assumed that ligand binding is much faster than the other timescales in the system, so that it can be integrated out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' There is therefore no need to time average receptor-ligand binding noise, which means that, in the absence of running costs, the optimal re- ceptor integration time τr is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In what follows, we set τr to the value measured experimentally, τr ≈ 100ms [10, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We consider the non-Markovian signals speci- fied by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 8 and 9 in the physiologically relevant limit ω0 → 0, such that the optimal kernel is perfectly adap- tive, like that of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For these signals, we determine the accessible region of Ipast and Ipred under a resource constraint C = RT + XT (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 4) by optimizing over the methylation time τm and the ratio of readout over receptor molecules XT/RT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The forecast interval τ is set to τv, but we emphasize that the optimal design is independent of the value of τ (see Appendix F 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 4A shows that the chemotaxis system is, in gen- eral, not at the information bound that maximizes the predictive information Ipred = I(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' vτ) for a given past information Ipast = I(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Lp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The optimal systems that maximize Ipred under a resource constraint C, marked by the red dots, are indeed markedly away from the infor- mation bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Yet, as the resource constraint is relaxed and C is increased, the optimal system moves towards the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Panel B shows that the methylation time τm rises along the three respective isocost lines of panel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' It highlights that there exists an optimal methyla- tion time τ opt m that maximizes the predictive information Ipred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Moreover, τ opt m decreases as the resource constraint is relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Along the respective isocost lines, XT/RT varies only mildly (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 9 in the appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' These observations can be understood by noting that the system faces a trade-off between taking a derivative that is recent versus one that is robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' All the infor- mation on the future derivative, which the cell aims to predict, is contained in the current derivative of the sig- nal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' measuring the current derivative would allow the system to reach the information bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' However, com- puting the recent derivative is extremely costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The cell takes the temporal derivative of the ligand concentration at the level of the receptor via two antagonistic reac- tions that occur on two distinct timescales: ligand bind- ing rapidly deactivates the receptor, while methylation slowly reactivates it [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The receptor ligand-occupancy thus encodes the current concentration, the methylation level stores the average concentration over the past τm, and the receptor activity reflects the difference between the two—the temporal derivative of the signal over the timescale τm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To obtain an instantaneous derivative, τm must go to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' However, this dramatically reduces the gain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' in fact, in this limit, the gain is zero, because the receptor activity instantly adapts to the change in the ligand concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Since the push-pull network down- stream of the receptor is a device that samples the re- ceptor stochastically [10, 36], the gain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' the change in the receptor activity due to the signal, must be raised to lift the signal above the sampling noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This requires a finite methylation time τm: as we show in Appendix F 3, the gain increases monotonically with τm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The trade-off between a recent derivative and a reliable one gives rise to an optimal methylation time τ opt m that maximizes the predictive information for a given resource cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The same analysis also explains why the optimal methylation time τ opt m decreases and the predictive infor- mation increases when the resource constraint is relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The sampling noise in estimating the average receptor activity decreases as the number of readout molecules increases [10, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A smaller gain is thus required to lift the signal above the sampling noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In addition, a larger number of receptors decreases the noise in the methyla- tion level, which also allows for a smaller gain, and hence a smaller methylation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' These two effects together explain why τ opt m decreases and Ipred increases with C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 4A also shows that the past information Ipast = I(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Lp) does not return to zero along the contourline of constant resource cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Along the contourline, the methy- lation time τm rises (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 4B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' While the predictive infor- mation Ipred exhibits an optimal methylation time τmopt, the past information Ipast continues to rise with τm be- cause the system increasingly becomes a copying device, rather than one that takes a temporal derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Comparison with experiment To test our theory, we study the predictive power of the E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli chemotaxis system as a function of the steep- ness of the ligand concentration gradient, keeping the resource constraint at the biologically relevant value of C = RT + XT = 104 [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Panel C of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 4 shows Ipred and Ipast for cells swimming in an exponential concen- tration gradient ℓ(x) = ℓ0egx, for different values of the gradient steepness g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' along the green iso-steepness lines τm is varied and XT/RT is optimized to maximize Ipred and Ipast, with the red dots marking τ opt m , while along 9 A B C FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Finite resources prevent chemotaxis system from reaching the information bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (A) The region of accessible predictive information Ipred = I(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' vτ) and past information Ipast = I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Lp) for the chemotaxis system under a resource constraint C = RT + XT, for the non-Markovian signals specified by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 8 and 9 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The black line shows the information bound at which Ipred is maximized for a given Ipast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The chemotaxis system is not at the information bound, but it does move towards it as C is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The red line connects the red points where Ipred is maximized for a given resource cost C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The accessible region of Ipred and Ipast under a given resource constraint C = RT + XT is obtained by optimizing over the methylation time τm and the ratio of readout over receptor molecules XT/RT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The forecast interval is τ = τv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (B) The methylation time τm over the input correlation time τv as a function of the distance θ along the three respective isocost lines shown in panel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The methylation time τm increases along the isocost line, but there exists an optimal τm that maximizes the predictive information, marked by the red points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' θ → 0 corresponds to the origin of panel A, (Ipred, Ipast) = (0, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' the points where θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='2 along the isocost lines of panel A are marked with a bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' As the resource constraint is relaxed (higher C), the optimal τm decreases: the system moves towards the information bound, where it takes an instantaneous derivative, corresponding to τr, τm → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C) The contourlines of Ipred and Ipast for increasing values of the steepness g of an exponential ligand concentration gradient ℓ(x) = ℓ0egx, keeping the total resource cost fixed at C = RT + XT = 104;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' τm and XT/RT have been optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' It is seen that the maximal predictive information Ipred under the resource constraint C (marked by the red points) increases with the gradient steepness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The blue line shows Ipred and Ipast for the E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli chemotaxis system with τm = 10s and XT = RT = 5000 fixed at their measured values [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Our analysis predicts that this system has been optimized to detect shallow gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Parameter values unless specified: τr = 100ms [10, 34];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' τv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='9s and σ2 v = g2¯ℓ2σ2 vx, with ¯ℓ = 100µM and σ2 vx = 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1µm2s−2 [18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' ω0 → 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' g is given in units of mm−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' in A, g = 4/mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' the blue line τm and XT and RT are fixed at their exper- imentally measured values [29, 30, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Clearly, both the predictive and the past information rise as the gradient steepness g increases—a steeper concentration gradient yields a larger change in the concentration, and thus a stronger signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' More interestingly, in the optimal system Ipred rises much faster with Ipast (red line) than in the E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli system (blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A steeper gradient g yields a stronger input signal, which raises the signal above the sampling noise more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This allows the optimal system to take a more re- cent derivative, with a smaller τm, which is more informa- tive about the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In contrast, the methylation time τm of the E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli chemotaxis system is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 4C shows, this value is beneficial for detecting shallow gra- dients, g ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='2mm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Moreover, in this regime, not only Ipred but also Ipast are close to the respective values for the optimal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For steeper gradients Ipast becomes much higher in the E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli system than in the optimal one, even though Ipred remains lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The bacterium increasingly collects information that is less informative about the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Taken together, these results strongly suggest that the system has been optimized to predict future concentration changes in shallow gradients, which necessitate a relatively long methylation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' DISCUSSION Cellular systems need to predict the future signal by capitalizing on information that is contained in the past signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To this end, they need to encode the past sig- nal into the dynamics of the intracellular biochemical network from which the future input is inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To maximize the predictive information for a given amount of information that is extracted, the cell should store those signal characteristics that are most informative about the future signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For a Markovian signal obeying an Ornstein-Uhlenbeck process this is the current signal value, while for the non-Markovian signal corresponding to an underdamped particle in a harmonic well, this is the current signal value and its derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' As we have seen here, cellular systems are able to extract these sig- 10 nal characteristics: the push-pull network can copy the current input into the output, while the chemotaxis net- work can take an instantaneous derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We have thus demonstrated that at least for two classes of signals, cel- lular systems are in principle able to extract the most predictive information, allowing them to reach the infor- mation bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Yet, our analysis also shows that extracting the most relevant information can be exceedingly costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To copy the most recent input signal into the output, the integra- tion time of the push-pull network needs to go to zero, which means that the chemical power diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' More- over, taking an instantaneous derivative reduces the gain to zero, such that the signal is no longer lifted above the inevitable intrinsic biochemical noise of the signalling system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In fact, taking the chemical power cost to drive the adaptation cycle into account [27, 37] would push the system away from the information bound even more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' While information is a resource—the cell cannot pre- dict the future without extracting information from the past signal—the principal resources that have a direct cost are time, building blocks and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The predic- tive information per protein and energy cost is therefore most likely a more relevant fitness measure than the pre- dictive information per past information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Our analysis reveals that, in general, it is not optimal to operate at the information bound: cells can increase the predictive information for a given resource constraint by moving away from the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Increasing the integration time in the push-pull network reduces the chemical power and makes it possible to take more concentration measure- ments per protein copy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' And increasing the methylation time in the chemotaxis system increases the gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Both enable the system to extract more information from the past signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Yet, increasing the integration time or the methylation time also means that the information that has been collected, is less informative about the future signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This interplay gives rise to an optimal integration and methylation time, which maximize the predictive in- formation for a given resource constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This argument also explains why the respective systems move towards the information bound when the resource constraint is relaxed: Increasing the number of receptor and readout molecules allows the system to take more instantaneous concentration measurements, which makes time averag- ing less important, thus reducing the integration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Increasing the number of readout molecules also reduces the error in sampling the receptor state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This makes it easier to detect a change in the receptor activity result- ing from the signal, thus allowing for a smaller dynamical gain and a shorter methylation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Information theory shows that the amount of transmit- ted information depends not only on the characteristics of the information processing system, but also on the statistics of the input signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' While much progress has been made in characterizing cellular signalling systems, the statistics of the input signal is typically not known, with a few notable exceptions [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Here, we have fo- cussed on two classes of input signals, but it seems likely that the signals encountered by natural systems are much more diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' It will be interesting to extend our analy- sis to signals with a richer temporal structure [9], and see whether cellular systems exist that can optimally encode these signals for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Finally, while we have analyzed the design of cellular signaling networks to optimally predict future signals, we have not addressed the utility of information for function or behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' It is clear that many functional or behavioral tasks, like chemotaxis [18], require information, but what the relevant bits of information are is poorly understood [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Moreover, cells ultimately employ their resources— protein copies, time, and energy—for function or behav- ior, not for processing information per se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Here, we have shown that maximizing predictive information under a resource constraint, C → Ipast → Ipred, does not nec- essarily imply maximizing past information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This hints that optimizing a functional or behavioral task under a resource constraint, C → Ipred → function, may not im- ply maximizing the predictive information necessary to carry out this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Jenny Poulton, Manuel Reinhardt, Michael Vennettilli and Daan de Groot for many useful discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This work is part of the Dutch Research Coun- cil (NWO) and was performed at the research institute AMOLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 885065).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 11 Appendix A: General 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Linear signalling networks Since the systems studied in the main text have a single steady state, we will study them in the linear-noise approximation [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For non-linear systems, the quality of the approximation improves with system size, but it can already be remarkably good for systems with only 10 copies [20, 22, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In the linear-noise approximation, we expand the rate equations to first order around the steady state of the mean-field chemical rate equations, and compute the noise at this steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In this approximation the network dynamics are a multidimensional Ornstein-Uhlenbeck (OU-)process: ˙δy = Gδs(t) + J δy(t) + Bξ(t), (A1) where δs(t) is a length k vector of input signals and δy is the vector of all network species of length n, both defined in terms of deviations from their mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The vector ξ(t) describes the m independent white noise processes associated with the m network reactions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' they have zero mean, unit variance, and are delta correlated: ⟨ξi(t)⟩ = 0, ⟨ξi(t)ξj(t′)⟩ = δijδ(t − t′), with δij the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The n × n matrix J is the Jacobian of the network, the n × k signal gain matrix G describes the strength by which each signal impacts each species directly, the n × m matrix B contains the noise strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The eigenvalues of the Jacobian J must be negative for the system to be stable, and we require all signals to be stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Integration kernels, power spectra, and correlation functions We continue by deriving the stationary auto-correlation matrix of a multidimensional OU-process, such as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A1, via the networks’ power spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The power spectrum of a real-valued random process X(t) is the squared modulus of its Fourier transform: Sx(ω) = ⟨δ˜x(−ω)δ˜x(ω)⟩ and Sx→y(ω) = ⟨δ˜x(−ω)δ˜y(ω)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Throughout this work we use the following conventions for the Fourier transform and its inverse: F{f(t)} ≡ ˜f(ω) = � ∞ −∞ dtf(t) exp(−iωt) and F−1{ ˜f(ω)} = 1/(2π) � ∞ −∞ dω ˜f(ω) exp(iωt) = f(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To obtain the correlation functions from the power spectra we invoke the Wiener-Khinchin theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The general solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A1 is δy(t) = � t −∞ dt′ eJ (t−t′) (Gδs(t′) + Bξ(t′)) , (A2) which shows the two contributions to the time dependent solution: that of the external signal and that of the internal noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The n × k matrix eJ (t−t′)G contains the integration kernels, its (i, j)th entry determines how the jth signal affects the ith system component over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The n × m matrix eJ (t−t′)B is similar, but contains the functions that map the noise terms onto the system components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' These matrices can be obtained by taking the Fourier transform of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A1 and solving for δ˜y(ω) iωδ˜y(ω) = Gδ˜s(ω) + J δ˜y(ω) + B˜ξ(ω), (A3) δ˜y(ω) = (iωIn − J )−1 � Gδ˜s(ω) + B˜ξ(ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (A4) Using the convolution theorem to take the Fourier transform of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A2, and comparing the result to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A4, now shows that F{eJ (t−t′)} = (iωIn − J )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We obtain for the power-spectra of the network components Sy(ω) = ⟨δ ˜y(−ω)δ ˜y(ω)T ⟩, = G(−ω)Ss(ω)G(ω)T + |N(ω)|2, (A5) with the matrices of frequency dependent gains G(ω) ≡ (iωIn − J )−1G, and frequency dependent noise N(ω) ≡ (iωIn−J )−1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The cross terms vanish because the fluctuations of the external signal are uncorrelated from the internal noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Furthermore, the power spectrum of a white noise process is constant, and all the noise terms are independent of one another, such that the spectral density of the noise vector is the identity matrix ⟨˜ξ(−ω)˜ξ(ω)T ⟩ = Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We also need to consider the cross-spectra between the signals and the network components, specifically we will need the spectra from the network to the signals Sy→s(ω) = ⟨δ ˜y(−ω)δ˜s(ω)T ⟩, = G(−ω)Ss(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (A6) 12 From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A5 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A6 we can obtain all necessary correlation functions and (co-)variances, by taking the inverse Fourier transform of the component of interest (for a variance we can directly set t = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The advantage of using this form, is that the contribution of each signal and of the noise terms appear separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' When we are for example interested in a variance that is only caused by noise, we can omit the terms depending on the signal power spectra, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Moreover, the power spectra are usually simpler in form than the corresponding correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The covariance and auto-correlation matrices can also be found by solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A2 directly in the time domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' the solutions are shown here for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For a derivation, see for example the work by Vennettilli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In this case it is most convenient to include the signals as system components, we thus have a new Jacobian J ′ and a new noise strength matrix B′ which include all network components and the signals themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The covariance matrix C is then obtained by solving the Lyapunov equation J ′C + CJ ′T + B′B′T = 0, (A7) and the correlation matrix is given by C(τ) = eJ ′τC for τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (A8) Appendix B: Signals and statistics 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Markovian signal For the Markovian ligand concentration dynamics we use a 1-dimensional OU-process δ ˙ℓ = −δℓ/τℓ + ηℓ(t), (B1) where the ligand concentration is defined in terms of the deviation from its mean δℓ = ℓ(t) − ¯ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The correlation time is give by τℓ, and the noise ηℓ(t) is derived from a unit white noise process ηℓ(t) ≡ σℓ � 2/τℓξ(t), such that ⟨ηℓ(t)ηℓ(t′)⟩ = 2σ2 ℓ/τℓδ(t − t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We obtain for the steady-state auto-correlation using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A7 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A8: ⟨δℓ(τ)δℓ(0)⟩ = σ2 ℓe−τ/τℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (B2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Non-Markovian signal Not all ligand concentration trajectories encountered by cells are expected to be Markovian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For example, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli swims in its environment with a speed which exhibits persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This leads to an auto-correlation function for the concentrations’ derivative which does not decay instantaneously [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To model such a persistent signal, we use the classical model of a particle in a harmonic well δ ˙ℓ = v(t), ˙v = −ω2 0δℓ(t) − v(t)/τv + ηv(t), (B3) where ω0 = � k/m, with k the spring constant and m the mass of the particle, τv is a relaxation timescale, and ηv(t) = σv � 2/τvξ(t), with ξ(t), as used throughout, a Gaussian white noise process of unit variance, and σv the standard deviation of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' If the signal would obey the fluctuation-dissipation relation, then mσ2 v = kBT, but since the biochemical signal could very well be generated via an active process this relation may not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This process can be expressed as a 2-dimensional OU-process with: J = � 0 1 −ω2 0 −1/τv � , (B4) B = �0 0 0 σv � 2/τv � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (B5) We find for the covariance matrix, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A7: C = � σ2 ℓ σℓv σℓv σ2 v � = σ2 v � 1/ω2 0 0 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (B6) 13 Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A8 we obtain the auto-correlation matrix in the overdamped regime, τ −1 v > 2ω0, C(τ) = �⟨δℓ(τ)δℓ(0)⟩ ⟨δℓ(τ)δv(0)⟩ ⟨δv(τ)δℓ(0)⟩ ⟨δv(τ)δv(0)⟩ � , = � � � σ2 ℓe−µτ/2 � cosh(ρτ) + µ 2ρ sinh(ρτ) � σ2 ve−µτ/2 1 ρ sinh(ρτ) −σ2 ve−µτ/2 1 ρ sinh(ρτ) σ2 ve−µτ/2 � cosh(ρτ) − µ 2ρ sinh(ρτ) � � � � , (B7) where ρ = � µ2/4 − ω2 0, with µ = τv−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The range of ligand concentrations which E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli might encounter is very large, based on the dissociation constants of the inactive and active receptor conformations, which for the Tar-MeAsp receptor ligand combination respectively are KI D = 18µM and KA D = 2900µM [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This suggests that the variance in the ligand concentration is very large relative to that of the derivative of the ligand concentration, which is set by the swimming behaviour of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For this reason we specifically focus on the limit where ω0 → 0, which corresponds to a vanishingly small spring constant, or a harmonic potential which becomes extremely wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The variance in the concentration σ2 ℓ then diverges, the normalized correlation functions in this limit are lim ω0→0 � � ⟨δℓ(τ)δℓ(0)⟩ σ2 ℓ ⟨δℓ(τ)δv(0)⟩ σℓσv ⟨δv(τ)δℓ(0)⟩ σℓσv ⟨δv(τ)δv(0)⟩ σ2v � � = �1 0 0 e−µτ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (B8) Appendix C: Information bottleneck framework and solutions Anticipating future environmental conditions allows for timely adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' However, storing information costs resources such as proteins, energy and time, and not all information in the past ligand concentrations will be relevant for predicting the signal’s future state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Assuming that resources are in limited supply, this means that cells must be efficient in which, and how much information they store.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This is elegantly captured in the Information Bottleneck Method (IBM), which describes the problem of maximizing the information on the future signal while minimizing the information on the past signal that is stored in the network output, from which the future input is predicted [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The objective function for the prediction of a variable of interest zτ ≡ z(t + τ) is: max P (X0|Lp) : L = I(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' zτ) − γI(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Lp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C1) The value of the sensing system output at the current time t is x0 ≡ x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The variable of interest zτ at a future time t + τ is the future concentration ℓτ ≡ ℓ(t + τ) for the Markovian signal, and the future concentration derivative vτ ≡ v(t+τ) for the non-Markovian signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Since the system of interest needs to predict one signal characteristic (either the future signal value or its derivative), one output component is sufficient for encoding the required information, as we describe in more detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The vector Lp = (δℓ(0), δℓ(−∆t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' , δℓ(−(N − 1)∆t))T is the past trajectory of ligand concentrations of length N, discretized with timestep ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The mutual information between the current system output and the future property of interest is the predictive information Ipred ≡ I(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' zτ), and the mutual information between the current system output and the past ligand concentration trajectory is the past information Ipast ≡ I(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Lp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The Lagrange multiplier γ sets the relative cost of storing past information over obtaining predictive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Given a value of γ, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' C1 is maximized by optimizing the mapping of the past ligand concentration trajectory Lp onto the current output x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Since, by the data processing inequality, we have Ipast ≥ Ipred, for γ = 1 the objective function is maximized by Ipast = Ipred = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' As γ is decreased both the past and predictive information increase, and the parametric curve in the Ipast − Ipred plane that arises is the information bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For γ = 0 there is no cost to storing past information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The predictive information is then only limited by the amount of information contained in the past about the future signal property: Ipred ≤ I(Lp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' zτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Gaussian information bottleneck In general equation C1 can be difficult to solve, as all mappings from Lp to X0 are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' However, the problem becomes analytically tractable when the joint probability distribution of Lp and zτ is a multivariate Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Here, 14 we follow the procedure of Chechik and coworkers to obtain this mapping [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In the Gaussian model, the optimal mapping from Lp to x0 is a linear one [12] x0 = ALp + ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' ξ ∼ N(0, σ2 ξ), (C2) where A is a row vector which determines how strongly each entry in Lp contributes to the scalar output X0 at any point in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The random variable ξ is the noise added to the signal due to the stochastic nature of the mapping;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' it is a Gaussian random variable independent of Lp with 0 mean and variance σ2 ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Finding the optimal mapping from Lp to x0 corresponds to finding the optimal combination of A and σ2 ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' It can be shown that for any pair (A, σ2 ξ), there exists a pair (A′, 1) which yields the same values for Ipast and Ipred after maximization of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' C1 [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Therefore, we can set σ2 ξ = 1 without altering the information curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To obtain the information bound, we rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' C1 using the definition of the mutual information between Gaussian random variables: L = 1 2 log(σ2 x/σ2 x|z) − γ 1 2 log(σ2 x/σ2 x|L), (C3) with the total variance σ2 x in the output x0, the output variance conditional on the future signal property σ2 x|z ≡ σ2 x|zτ , and the output variance conditional on the complete history of ligand concentrations σ2 x|L ≡ σ2 x|Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The latter is just the variance caused by the intrinsic noise, σ2 x|L = σ2 ξ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The total variance in x0 can be expressed in terms of the mapping vector A and the variance in the past signal using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' C2, σ2 x = AΣLAT + 1, where ΣL ≡ ΣLp is the covariance matrix of the past ligand concentration trajectory Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To express the output variance conditional on the future signal property zτ we use the Schur complement formula, which in general form reads: Σx|y = Σx − ΣxyΣ−1 y Σyx, (C4) where Σyx = ΣT xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Using this formula to rewrite σ2 x|z, and then using the linear relation from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' C2 again, we obtain σ2 x|z = AΣL|zAT + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Filling in the expressions for the variances in L (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' C3) gives: L = 1 2 � (1 − γ) log ���AΣLAT + 1 ��� − log ���AΣL|zAT + 1 ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C5) For any symmetric matrix C we have δ δA log |ACAT | = � ACAT �−1 2AC, such that we obtain for the derivative of L to A: δL δA = (1 − γ) AΣL AΣLAT + 1 − AΣL|z AΣL|zAT + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C6) In our case A is a row vector, and both denominators are thus scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We find the maximum of L by equating its derivative to 0, which gives: AΣL|zΣ−1 L = (1 − γ)AΣL|zAT + 1 AΣLAT + 1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C7) For this equality to hold A must either be identically 0, or a left eigenvector of the matrix ΣL|zΣ−1 L with eigenvalue: λ = (1 − γ)AΣL|zAT + 1 AΣLAT + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C8) Here, we note that if the signal statistics is sufficiently rich and the prediction complexity sufficiently large (because, for example, multiple signal characteristics need to be predicted), then the matrix ΣL|zΣ−1 L has multiple eigenvectors with non-trivial eigenvalues 0 < λi < 1 [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This reflects the idea that storing the past information that is necessary to enable this complex prediction task may require multiple output components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' an output vector x, where each output component has an integration kernel given by one of the eigenvectors of ΣL|zΣ−1 L [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' However, for Markovian signals only one eigenvector with non-trivial eigenvalue 0 < λ < 1 emerges, which means that one output component is sufficient to encode the required information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For the non-Markovian signals studied here, ΣL|zΣ−1 L has two eigenvectors if both the future value and its derivative need to be predicted (and z = (ℓτ, vτ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' to optimally predict both features from the current output, two output components are then required, provided Ipast is sufficiently 15 large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' However, here we consider the scenario that only the future derivative needs to be predicted, in which case only one non-trivial eigenvector emerges, and one output component is sufficient for encoding the required information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We leave the problem of predicting multiple signal features via multiple output components for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We can define the optimal mapping A = ||A||ν where ν is the normalized left eigenvector of ΣL|zΣ−1 L corresponding to its smallest eigenvalue, 0 < λ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The magnitude can be found by solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' C8 for ||A||, using from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' C7 that λνΣLνT = νΣL|zνT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This gives for the optimal mapping: Aopt = �� 1−γ−λ ν1ΣLνT 1 λγ ν1 for 0 < λ < 1 − γ, 0 for 1 − γ ≤ λ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C9) We can substitute ||A||2 = (1 − γ − λ)/(νΣLνT λγ) in the definitions for the mutual information to express them in terms of λ and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For the past information we obtain: Ipast = 1 2 log � ||A||2νΣLνT + 1 � , = 1 2 log �1 − γ γ 1 − λ λ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C10) And for the predictive information: Ipred = 1 2 log � ||A||2νΣLνT + 1 � − 1 2 log � ||A||2νΣL|ℓτ νT + 1 � , = Ipast − 1 2 log �1 − λ γ � , = 1 2 log �1 − γ λ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C11) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Markovian signal To obtain the information bound for prediction of the future ligand concentration of a Markovian signal, we need to determine the eigenvalues and vectors of the matrix (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' C7 and C8) W = ΣL|ℓτ Σ−1 L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C12) Using the Schur complement formula (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' C4) to rewrite the conditional matrix gives ΣL|ℓτ = ΣL − ΣLℓτ ΣT Lℓτ /σ2 ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Then defining the normalized matrices RL = ΣL/σ2 ℓ and RLℓτ = ΣLℓτ /σ2 ℓ we find W = IN − RLℓτ RT Lℓτ R−1 L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C13) where N is the length of the input trajectory Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The correlation matrix of the past trajectory is symmetric with entries R(i,j) L = exp(−|i − j|∆t/τℓ), where ∆t is the discretization timestep of the past trajectory Lp and i ranges from 1 to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This is a Kac-Murdock-Szeg¨o matrix, and its inverse is known: R−1 L = 1 1 − e2∆t/τℓ � � � � � � � � � � 1 −e−∆t/τℓ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 0 −e−∆t/τℓ 1 + e−2∆t/τℓ −e−∆t/τℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 0 0 −e−∆t/τℓ 1 + e−2∆t/τℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' −e−∆t/τℓ 1 + e−2∆t/τℓ −e−∆t/τℓ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 0 −e−∆t/τℓ 1 � � � � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C14) Note that the inverse matrix is tridiagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The length N cross-correlation vector between past trajectory and future concentration has entries R(i) Lℓτ = exp(−(τ + (i − 1)∆t)/τℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The product of the correlation matrices is surprisingly simple: RLℓτ RT Lℓτ R−1 L = e−2τ/τℓ � � � � � 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 0 e−∆t/τℓ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' e−(N−1)∆t/τℓ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 0 � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C15) 16 Using this result we can straightforwardly determine the eigenvalues, |W − λIN| = 0, ��������� � � � � � 1 − λ − e−2τ/τℓ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 0 −e−(τ+∆t)/τℓ 1 − λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' −e−(τ+(N−1)∆t)/τℓ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 1 − λ � � � � � ��������� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C16) The only contribution to the determinant comes from the diagonal, and the only nontrivial eigenvalue is thus λ = 1−e−2τ/τl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The optimal mapping is thus onto a one-dimensional scalar output x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The corresponding left eigenvector is given by ν1W = (1 − e−2τ/τl)ν1, (C17) which holds for ν1 = �1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 0� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The optimal mapping for the prediction of a one-dimensional OU-process is thus to copy its most recent value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This agrees with intuition as for any Markovian process, all the information about the future signal is contained in the most recent value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For a continuous input signal (rather than a discretized signal), and a continuous integration kernel k(t) (rather than a mapping vector A), this means that the optimal integration kernel is kopt(t) = aδ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Non-Markovian signal To find the optimal mapping for the prediction of the derivative of a non-Markovian signal, based on its history of ligand concentrations, we need to find the eigenvalues and vectors of the matrix W = ΣL|vτ Σ−1 L , = IN − 1 σ2v ΣLvτ ΣT Lvτ Σ−1 L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C18) The covariance matrix of the past trajectory is symmetric with entries Σ(i,j) L = ⟨δℓ(0)δℓ(|i − j|∆t)⟩ where both i and j range from 1 to N, the past trajectory length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The covariance vector between past trajectory and future derivative has entries Σ(i,j) Lvτ = ⟨δℓ(0)δv(τ + (i − 1)∆t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Both the concentration auto-correlation function, and the concentration to future derivative cross-correlation function, are shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' B7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To better understand the optimal mapping of this signal we numerically investigate the eigenvalues of the matrix W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For the prediction of vτ, there is only one non-trivial eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Like for the Markovian signal, this shows that for the prediction of the derivative of this non-Markovian signal, the optimal mapping is always onto a scalar output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The non-trivial eigenvalue λ decreases with the discretization timestep ∆t and is minimal for ∆t → 0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In this limit, λ has the same magnitude for any N ≥ 2, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A smaller eigenvalue λ corresponds to larger past and predictive information and a larger ratio Ipred/Ipast (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' C10 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' C11), given any value of the Lagrange multiplier γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For the optimal mapping we must thus have N ≥ 2 and ∆t → 0, where N sets both the past trajectory and the mapping vector length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Because increasing the length above two does not yield an improvement in the value of λ1 we focus on N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The fact that to reach the optimum we must have N = 2 and ∆t → 0, shows that the optimal kernel A takes an instantaneous measurement of a combination of the most recent ligand concentration, and its derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This can be understood as follows, for a trajectory of length two, the mapping vector also has length two, A = ||A||( ˆw1, ˆw2), with � ˆw2 1 + ˆw2 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We can then express the linear mapping of Lp to x0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' C2) as: x0 = ||A|| � ( ˆw1 + ˆw2)δℓ(0) − ˆw2∆tδℓ(0) − δℓ(−∆t) ∆t � + ξ, (C19) This expression shows that, as ∆t → 0, the two entries of A combine both the most recent signal value and the most recent derivative to generate x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This is intuitive because the signal is completely defined by its concentration and derivative (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' B3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For this reason, and to obtain analytical insight into the optimal weights, we inspect the final two entries of the past ligand concentration trajectory in the limit ∆t → 0, which defines the past signal in terms of its most recent concentration and derivative S0 ≡ �δℓ(0) v(0)�T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C20) 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='560 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='565 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='570 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='580 Discretization timestep Δt Smallest eigenvalue λ N = 2 N = 3 N = 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The smallest eigenvalue of the IB matrix is minimal for N ≥ 2 and ∆t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A smaller eigenvalue corresponds to a larger ratio Ipred/Ipast for any given value of the Lagrange multiplier γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Parameters: friction timescale τ −1 v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='862s−1 as determined in [18], prediction interval τ = τv, and ω0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='4s−1 such that the system is slightly overdamped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Because the signal is Markovian in the joint properties δℓ and v, the vector S0 contains the same information as the trajectory Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The past information is now the mutual information between x0 and S0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Ipast = I(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' S0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The output x0 can then also be written as a projection of S0 via the alternative mapping vector ˜ A = ||A||(ˆa,ˆb): x0 = ||A|| � ˆaδℓ(0) + ˆbv(0) � + ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C21) Comparison with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' C19 shows how the components of ˜ A relate back to those in A, ˆw1 = ˆa + ˆb/∆t, (C22) ˆw2 = −ˆb/∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C23) To obtain the optimal mapping vector ˜ A the matrix of signal statistics of which the eigenvalues and -vectors need to be determined is W = Σs|vτ Σ−1 s , (C24) with Σs = � σ2 ℓ 0 0 σ2 v � , (C25) Σs|vτ = Σs − 1 σ2v Σsvτ ΣT svτ , (C26) Σsvτ = � ⟨δℓ(0)δv(τ)⟩ ⟨δv(0)δv(τ)⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C27) We thus obtain W = I − � � � ⟨δℓ(0)δv(τ)⟩2 σ2 ℓ σ2v ⟨δℓ(0)δv(τ)⟩⟨δv(0)δv(τ)⟩ σ4v ⟨δℓ(0)δv(τ)⟩⟨δv(0)δv(τ)⟩ σ2 ℓ σ2v ⟨δv(0)δv(τ)⟩2 σ4v � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C28) This matrix has one nontrivial eigenvalue, λ = 1 − ⟨δv(0)δv(τ)⟩2 σ4v − ⟨δℓ(0)δv(τ)⟩2 σ2 ℓ σ2v , which depends on the normalized correlation functions between on the one hand the current concentration or derivative, and on the other hand the future derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The corresponding left eigenvector is ν1 = Q−1 � 1 σℓ ⟨δℓ(0)δv(τ)⟩ σℓσv 1 σv ⟨δv(0)δv(τ)⟩ σ2 v � , (C29) where Q normalizes the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='Using the linear mapping x0 = ||A||ν1S0 + ξ, and defining G ≡ ||A||/Q, shows that the optimal output should be generated as follows xopt 0 = G �⟨δℓ(0)δv(τ)⟩ σℓσv δℓ(0) σℓ + ⟨δv(0)δv(τ)⟩ σ2v v(0) σv � + ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C30) 18 Clearly, the optimal mapping depends on the (normalized) cross-correlation coefficient ρℓ0vτ ≡ ⟨δℓ(0)δv(τ)⟩/(σℓσv) between the current concentration δℓ(0) and future derivative δv(τ), and the cross-correlation coefficient ρv0vτ between the current derivative δv(0) and future derivative δv(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Indeed, to optimally predict the future derivative, the cell should also use the current concentration and not only its current derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' However, in the limit that the range of concentrations sensed becomes very large, corresponding to ω0 → 0, the current concentration is no longer correlated with the future derivative, and ρℓ0vτ → 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' B8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In this limit, ˆa = 0 and ˆb = 1, and the kernel becomes a perfectly adaptive, derivative-taking kernel: lim ω0→0 xopt 0 = ||A||v(0) + ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C31) If we translate this back to the vector ||A||( ˆw1, ˆw2), operating on a ligand concentration trajectory Lp, the optimal weights become ˆw1 = − ˆw2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Appendix D: Past and predictive information for linear signalling networks In order to address how close biochemical networks can come to the information bounds derived above, we here describe how we obtain the past and predictive information for any linear (biochemical) network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We then use the resulting general expressions to compute the past and predictive information for the push-pull network and the chemotaxis system of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For any linear network the output can be written as δx(t) = � t −∞ ds k(t − s)δℓ(s) + ηx(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (D1) The mapping kernel k(t) is a property of the network and describes how the input signal is mapped onto the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The noise term ηx(t) is a sum of convolutions over all white noise processes in the network and corresponding network mapping functions, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The variance in the output can generally be split up in a part caused by the signal and a part cause by the noise, and we have σ2 x = � t −∞ ds � t −∞ ds′k(t − s)k(t − s′)⟨δℓ(s)δℓ(s′)⟩ + σ2 ηx, = σ2 x|η + σ2 x|L, (D2) where σ2 x|η is the signal variance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' all noise terms are fixed, and σ2 x|L is the noise variance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' the complete history of the signal is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Using this decomposition we find for the past information, which is the mutual information between the current output and the complete signal history, Ipast(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Lp) = 1 2 log � σ2 x σ2 x|L � = 1 2 log(1 + SNR), (D3) where the signal-to-noise ratio is defined as SNR = σ2 x|η/σ2 x|L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Using the same definition for the mutual information when deriving the predictive information between current output and future ligand concentration, we obtain Ipred(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' ℓτ) = 1 2 log � σ2 x σ2 x|ℓτ � , = 1 2 log � 1 + σ2 x|η σ2 x|L � − 1 2 log � 1 + σ2 x|η − ⟨δx(0)δℓ(τ)⟩2/σ2 ℓ σ2 x|L � , = Ipast − 1 2 log(1 + cSNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (D4) In the second line we used the Schur complement formula, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' C4, to decompose the variance in the output conditioned on the future signal: σ2 x|ℓτ = σ2 x−⟨δx(0)δℓ(τ)⟩2/σ2 ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The quantity σ2 x|η−⟨δx(0)δℓ(τ)⟩2/σ2 ℓ can be understood as follows: the first term σ2 x|η is the contribution to the total variance of the output σ2 x that comes from the signal variations, while the second term quantifies the variance in the output that is correlated with the future input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The difference is thus the variance in the output coming from the signal variations that are not correlated with the future input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The 19 ratio in the second logarithm can thus be understood as a conditional SNR that quantifies the part of the signal to noise ratio that does not contain information about the future signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This becomes more clear when considering its form in terms of the mapping kernel and signal correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For any linear signalling network we have σ2 x|η − ⟨δx(0)δℓ(τ)⟩2/σ2 ℓ = � 0 −∞ ds � 0 −∞ ds′k(−s)k(−s′) � ⟨δℓ(s)δℓ(s′)⟩ − ⟨δℓ(τ)δℓ(s′)⟩⟨δℓ(τ)δℓ(s)⟩ σ2 ℓ � , (D5) where the term in parentheses is the conditional variance in the past signal trajectory given a future value, ΣL|ℓτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' D4 thus tells us that the predictive information is equal to the past information, minus the bits that do not contain information about the future ligand concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This difference is indeed the part of the past information that does contain predictive information about the future signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Although the expression above (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' D4) nicely relates the past and predictive information, a more straightforward way of obtaining the predictive information is by expressing it directly in terms of the correlation between the current output and the future ligand concentration: Ipred(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' ℓτ) = 1 2 log � σ2 x σ2 x|ℓτ � = −1 2 log � 1 − ⟨δx(0)δℓ(τ)⟩2 σ2xσ2 ℓ � , (D6) where we again used the Schur complement formula to rewrite σ2 x|ℓτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Written this way we thus see that the predictive information depends on the normalized correlation between the current network output and the future ligand con- centration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We can simply exchange the future ligand concentration for the future derivative when considering the chemotaxis network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To compute the past information for linear signalling networks we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' D3, and we thus need to compute the SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To compute the predictive information for the prediction of a future ligand concentration, we need to compute the ‘future correlation function’ ⟨δx(0)δℓ(τ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For the prediction of the future derivative we need ⟨δx(0)δv(τ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Appendix E: Push-pull network We consider a push-pull network that consists of a phosphorylation-dephosphorylation cycle downstream of a receptor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' When bound to ligand, the receptor itself or its associated kinase, such as CheA in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli, catalyzes the phosphorylation of a readout protein x, like CheY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Active readout molecules x∗ can decay spontaneously or be deactivated by an enzyme (phosphatase), such as CheZ in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This cycle is driven by the turnover of fuel such as ATP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We recognize that inside the living cell, the chemical driving is typically large: for example, the free energy of ATP hydrolysis is about 20kBT, which means that the system essentially operates in the irreversible regime [10, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This system consists of the following reactions: R + L k+ −−⇀ ↽−− k− RL (E1) RL + x kf −−→ RL + x∗ (E2) X∗ kr −−→ X (E3) Both the total number of receptors RT = R + RL and read-out molecules XT = X + X∗ are conserved moieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The chemical Langevin equations of this system are: ˙ RL = [RT − RL(t)]ℓ(t)k+ − RL(t)k− + Bc(RL, ℓ)ξc(t), (E4) ˙x∗ = [XT − x∗(t)]RL(t)kf − x∗(t)kr + Bx(RL, x∗)ξx(t), (E5) where RL is the number of bound receptors, x∗ the number of phosphorylated read-out molecules, and ξi denote independent Gaussian white noise with unit variance, ⟨ξi(t)ξj(t′)⟩ = δijδ(t − t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The noise strengths are Bc(RL, ℓ) = � (RT − RL(t))ℓ(t)k+ + RL(t)k− and Bx(RL, x∗) = � (XT − x∗(t))RL(t)kf + x∗(t)kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The steady-state fraction of ligand-bound receptors is p ≡ RL/RT = ¯ℓ/(¯ℓ+KD) with the dissociation constant KD = k−/k+, and the steady-state fraction of phosphorylated readout molecules is f ≡ ¯x∗/XT = pRT/(pRT + kr/kf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In the linear-noise approximation, expanding Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' E4 and E5 to first order around their steady state, the equations become δ ˙ RL = b δℓ(t) − δRL(t)/τc + ηc(t), (E6) δ ˙x∗ = γ δRL(t) − δx∗(t)/τr + ηx(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (E7) 20 The parameters b = RTp(1 − p)/(¯ℓτc) and γ = XTf(1 − f)/(RTpτr) are effective rates of receptor-ligand binding and readout phosphorylation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The decay rate of correlations in the receptor-ligand binding state is τc−1 = ¯ℓk+ + k−, and that of the readout phosphorylation state is τr−1 = pRTkf + kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The rescaled white noise processes have strengths ⟨η2 c⟩ = B2 c = 2RTp(1 − p)/τc and ⟨η2 x⟩ = B2 x = 2XTf(1 − f)/τr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Model statistics The relevant quantity to compute the past information is the variance in the output, decomposed into the part caused by signal variation and the part caused by noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To compute the predictive information we further need the correlation function between the current output and a future ligand concentration ⟨δℓ(τ)δx∗(0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' These quantities can be obtained via their Fourier transforms, as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A5 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The matrices describing the properties of the signalling network are, as defined below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A1, G = � b 0 � , (E8) J = � −τc−1 0 γ −τr−1 � , (E9) B = �� ⟨η2c⟩ 0 0 � ⟨η2x⟩ � = �� 2RTp(1 − p)/τc 0 0 � 2XTf(1 − f)/τr � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (E10) A useful property of the network is the matrix exponential of its Jacobian, which in Fourier space is (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A2 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A4) F{eJ t}(ω) = (iωI2 − J )−1, = � � 1 1/τc+iω 0 γ (1/τc+iω)(1/τr+iω) 1 1/τr+iω � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (E11) We then have G(ω) = F{eJ t}(ω)G and N(ω) = F{eJ t}(ω)B, see also Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A4 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The integration kernel that maps the ligand concentration onto the output of the push-pull network, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' D1, is given by the inverse Fourier transform of the second entry of G(ω), which is the frequency dependent gain, ˜gℓ→x(ω), from ℓ to x: k(t) ≡ F−1{˜gℓ→x(ω)} = bγτcτr 1 τr − τc � e−t/τr − e−t/τc� , = XTf(1 − f)(1 − p)/¯ℓ 1 τr − τc � e−t/τr − e−t/τc� , (E12) The so-called static gain of the network is the integral of this kernel over all time, ¯gℓ→x ≡ � ∞ 0 k(t)dt = XTf(1 − f)(1 − p)/¯ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This parameter quantifies how much a step change in the input concentration changes the steady- state level of the output: ¯gℓ→x = ∂ ¯x∗/∂¯ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We will use this parameter in the statistical quantities that follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The static gain is also given by ¯gℓ→x = ¯gℓ→RL¯gRL→x, with ¯gℓ→RL = p(1 − p)RT/¯ℓ the static gain from ¯ℓ to RL and ¯gRL→x = f(1 − f)XT/(pRT) the static gain from RL to x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We model the Markovian ligand concentration as a 1-dimensional OU process Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' B1, which has the following power spectrum Sℓ(ω) = ⟨|δℓ(ω)|2⟩ = 2σ2 ℓ/τℓ 1/τ 2 ℓ + ω2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (E13) This yields the following expression for the power spectra (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A5): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='G(−ω)Sℓ(ω)G(ω)T = b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1/τc2+ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1/τr−iω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1/τc2+ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1/τr+iω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1/τc2+ω2 γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1/τr2+ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1/τc2+ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='2σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='ℓ/τℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1/τ 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='ℓ + ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='(E14) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='|N(ω)|2 = ⟨η2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='c⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1/τc2+ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1/τr−iω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1/τc2+ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1/τr+iω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1/τc2+ω2 γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1/τr2+ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1/τc2+ω2 + ⟨η2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='x⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='⟨η2c⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1/τr2+ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='(E15) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='We thus have for the power spectrum of the read-out: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='Sx(ω) = ˜g2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='ℓ→x(ω)Sℓ(ω) + N 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='x(ω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='2b2γ2σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='ℓ/τℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='(1/τr2 + ω2)(1/τc2 + ω2)(1/τ 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='ℓ + ω2) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='γ2⟨η2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='c⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='(1/τr2 + ω2)(1/τc2 + ω2) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='⟨η2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='x⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1/τr2 + ω2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (E16) The variance in the read-out σ2 x = 1/(2π) � ∞ −∞ Sx(ω) is hence given by σ2 x = σ2 x|η + σ2 x|L = ¯g2 ℓ→x 1 + τr/τℓ + τr/τc (1 + τc/τℓ)(1 + τr/τℓ)(1 + τr/τc)σ2 ℓ + ¯g2 RL→xRTp(1 − p) 1 1 + τr/τc + XTf(1 − f),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' = ¯g2 ℓ→x 1 + τr/τℓ + τr/τc (1 + τc/τℓ)(1 + τr/τℓ)(1 + τr/τc) � �� � dynamical gain σ2 ℓ + XTf(1 − f) � 1 + ¯gℓ→x ¯ℓ RTp 1 1 + τr/τc � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (E17) where ¯gRL→x = γτr = XTf(1 − f)/(RTp) is the static gain from the receptor to the readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The expression above gives insight into the role of the different network components in shaping the noise in the readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' It can be seen that the contribution from the signal variance σ2 ℓ to σ2 x is determined by the static gain ¯g2 ℓ→x, which is proportional to XT, and a factor that only depends on ratios of timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Their product is the dynamical gain, which decreases monotonically with τr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The intrinsic noise in the phosphorylation state of the read-outs leads to the noise term XTf(1 − f), which cannot be averaged out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The noise arising from ligand binding and unbinding increases with the static gain, but can be mitigated by increasing the number of receptors or the integration time τr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The latter strategy is what we call time-averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The signal-to-noise ratio SNR = σ2 x|η/σ2 x|L can straightforwardly be obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' E17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This is the quantity that sets the magnitude of the past information, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To determine the predictive information we need to compute the correlation function from the current output to the future ligand concentration ⟨δx(0)δℓ(τ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This requires the cross-spectrum from output to ligand concentration, which is given by (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A6) ˜gℓ→x(−ω)Sℓ(ω) = bγ (1/τc − iω)(1/τr − iω) 2σ2 ℓ/τℓ 1/τ 2 ℓ + ω2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (E18) From this power spectrum we obtain the required correlation function by taking the inverse Fourier transfrom: ⟨δx(0)δℓ(τ)⟩ = F−1{˜gℓ→x(−ω)Sℓ(ω)}, = ¯gℓ→xσ2 ℓ (1 + τc/τℓ)(1 + τr/τℓ)e−τ/τℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (E19) This correlation function thus decays exponentially with the prediction interval τ at a rate τ −1 ℓ , just as the signal auto- correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The (squared) correlation coefficient, which sets Ipred, is given by ⟨δx(0)δℓ(τ)⟩2/(σ2 ℓσ2 x) = ρ2 ℓxe−2τ/τℓ, with the (squared) instantaneous correlation coefficient (for convenience given as its inverse) ρ−2 ℓx = ¯ℓ2 σ2 ℓ � 1 + τr τℓ �2 � 1 + τc τℓ �2 � 1 XT f(1 − f)(1 − p)2 + 1 RT p(1 − p)(1 + τr/τc) + σ2 ℓ ¯ℓ2 1 + τr/τℓ + τr/τc (1 + τc/τℓ)(1 + τr/τℓ)(1 + τr/τc) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (E20) When the right-hand-side is minimized, the correlation is thus maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This expression shows that increasing XT and RT always increases the instantaneous correlation coefficient, and that the fraction of phosphorylated readout molecules in steady state that maximizes the correlation coefficient is f = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Past and predictive information of the push-pull network Using the quantitites computed above, we can determine both the past and the predictive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For the past information we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' D3, whith the SNR from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' E17: SNR = σ2 x|η/σ2 x|L = (1 − p)σ2 ℓ ¯ℓ2 1 + τr/τℓ + τr/τc (1 + τc/τℓ)(1 + τr/τℓ)(1 + τr/τc) � � 1 XTf(1 − f)(1 − p) + 1 RTp(1 + τr/τc) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 22 The predictive information is a function of the correlation between the current output and the future ligand concen- tration, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' D6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This correlation can be decomposed into the instantaneous correlation coefficient and an exponential decay on the timescale of the ligand concentration fluctuations, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' E19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We thus obtain for the predictive information, Ipred(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' ℓτ) = −1 2 log(1 − ρ2 ℓxe−2τ/τℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (E21) The instantaneous correlation coefficient ρ2 ℓx is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' E20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' E21 it also becomes clear that while the value of the predictive information depends on the forecast interval τ, the optimal design of the network that maximizes the predictive information, determined by the optimal ratio XT/RT, the optimal integration time τr, and the optimal ligand-bound receptor fraction p, does not depend on the forecast interval τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Optimal resource allocation Increasing the number of receptor or readout molecules always increases the precision with which the cell can predict a signal (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' E20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' However, when the total resource pool is constrained, the cell has to choose whether it makes more receptors or more readout molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To find the optimal ratio of read-out to receptor molecules we, can use the C = ART + BXT to express XT and RT in terms of the total cost C and the ratio XT/RT: XT = C XT/RT A + BXT/RT , (E22) RT = C 1 A + BXT/RT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (E23) The factors A and B set the cost of receptors and readout molecules, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Substituting these expressions for XT and RT into the expression for the correlation coefficient between the output and ligand concentration (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' E20), setting the derivative of the resulting expression with respect to XT/RT to zero, and solving for XT/RT gives (XT/RT)opt = �� 1 + τr τc � p 1 − p 1 f(1 − f) A B , = 2 � p/(1 − p) � 1 + τr/τc, (E24) where for the second line we used A = B = 1 and f = f opt = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This is the optimal ratio of readout to receptor molecules in the push-pull network, given an integration time τr and a steady state fraction of ligand-bound receptors p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Perhaps surprisingly, this optimal ratio (XT/RT)opt maximizes, for a given τr and p, not only the predictive information, but also the past information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This is because the ratio XT/RT determines, together with τr and p, the interval ∆ for sampling the ligand-binding state of the receptor: when the ratio XT/RT obeys Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' E24, the readout molecules sample each receptor molecule roughly once every correlation time: ∆ ∼ τc [10, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' E24 is thus a statement about optimally extracting the information that is encoded in the receptor-ligand binding history, both concerning the past information and the predictive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Operating costs diverge when approaching the information bound The precision of any sensing device is limited by the resources that are devoted to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The cost function we consider in this work is C = λ(RT + XT) + c1XT∆µ/τr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (E25) The first term is the maintenance cost;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' this is the cost of producing new network components at the growth rate λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The second term is the operating cost and describes the chemical power that is necessary to run the network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' it depends on the flux through the network, XT/τr, and the free-energy drop ∆µ over a full cycle of phosphorylation and dephosphorylation, given by the free energy of ATP hydrolysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The coefficient c1 describes the relative energetic cost of synthesising the components during the cell cycle, versus that of running the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In the main text we consider the case where c1 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Here we will investigate how close cells can come to the information bound when c1 is finite, thus including the chemical power cost of running the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' It is clear from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' E25 that for finite c1 the operating cost diverges when τr → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Because the optimal IBM solutions are instantaneous, this is precisely the limit in which the network must be to reach the information bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='10 Past info (bits) Predictive info (bits) τr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='01 τr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='2 τr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='5 τr=1 Varied XT/RT p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1 p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='2 p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The past and predictive information are maximized by the same ratio XT/RT and fraction p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The information plane, showing the information bound in black, and the isocost line C = 104 in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To construct the coloured lines in this figure the ratio XT/RT has been varied from zero to a value beyond the optimal value that maximizes Ipast and Ipred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This is done for several values of the receptor occupancy p (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1 in red, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='2 in blue, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='4 in orange), and for several values of τr (indicated in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' When XT/RT reaches its optimal value, both Ipast and Ipred are maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' When the ratio is increased further the system moves back to the origin via the same coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Only the integration time τr meaningfully distinuishes between strategies that maximize predictive or past information, or that approach the information bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The reason is that XT/RT, together with τr and p, control the optimal extraction of information that is encoded in the receptor-ligand binding history, both concerning Ipast and Ipred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The gray isocost line is obtained by varying τr, while maximizing for each τr the correlation coefficient given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' E20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' the latter is done by substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' E24 into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' E20 and numerically optimizing the resulting expression over p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The isocost line gives the region of Ipast and Ipred that is accessible for a given resource cost C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Parameter values are A = B = 1, f = 1/2, (σℓ/¯ℓ)2 = 10−2, τc/τℓ = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' As a consequence, when we consider the operating costs, the push-pull network can only be at the information bound when (Ipast, Ipred) → (0, 0) or C → ∞ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 7A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The system can mitigate the operating costs by decreasing XT, because this decreases the flux through the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' However, this also decreases the gain and thus, eventually, any information transduced through the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In the limit that both XT and τr approach zero, the system approaches the information bound at the origin, see both Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 7A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' More generally, when the running costs are taken into account, the system time averages more (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=', τr rises), because frequent measurements are now even more costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Still, τr decreases as the total resource availability C grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Appendix F: Chemotaxis network The evidence is mounting that in the E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli chemotaxis system, receptors cooperatively control the activity of the kinase CheA [29, 44–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Furthermore, the kinase activity is adaptive due to the methylation of inactive receptors [15, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A widely used approach to describe the effects of receptor cooperativity and methylation on kinase activity, has been to employ the Monod-Wyman-Changeux (MWC) model [18, 24, 29, 33, 42, 43, 48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We will follow this approach and, more specifically, model the chemotaxis system as described by Tu and colleagues [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In this model, each receptor can switch between an active and inactive conformational state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Moreover, receptors are partitioned into clusters of equal size N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In the spirit of the MWC model, receptors within a cluster switch conformation in concert, so that each cluster is either active or inactive [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Furthermore, it is assumed that receptor-ligand binding and conformational switching are faster than the other timescales in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The probability for the kinase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' the receptor cluster, to be active, is then described by: a(ℓ, m) = 1 1 + exp(∆FT (ℓ, m)), (F1) where ∆FT (ℓ, m) is the total free-energy difference between the active and inactive state, which is a function of the ligand concentration ℓ(t) and the methylation level of the cluster m(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The simplest model adopted here assumes a linear dependence of the total free-energy difference on the free-energy difference arising from ligand binding and methylation: ∆FT (ℓ, m) = −∆E0 + N(∆Fℓ(ℓ) + ∆Fm(m)), (F2) 24 A B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Due to diverging operating costs the push-pull network only reaches the information bound for infinite resource availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (A) In green, the region of accessible predictive and past information in the push-pull network under a resource constraint C = λ(RT + XT) + c1XT∆µ/τr, with λ = 1 and c1 = 1/∆µ, corresponding to a cell doubling time of roughly 20min [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The black line is the information bound;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' the red and blue dots mark the points where Ipred and Ipast are maximized, respectively, under a resource constraint C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' the red and blue lines connect these points, respectively, for increasing C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The accesible region for C ≤ 104 and the isocost lines for C = 103 and C = 105 have been obtained as described under Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The forecast interval has been set to one signal correlation time in the future: τ = τℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (B) The integration time over the receptor correlation time, τr/τc, and the ratio of the number of readout and receptor molecules, XT/RT, as a function of the distance θ along the iscocost line for C = 104 in panel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For θ → 0, both τr and XT go to zero, thus reducing both Ipast and Ipred to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Other parameter values in both panels are f = f opt = 1/2, (σℓ/¯ℓ)2 = 10−2, τc/τℓ = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' where the free-energy difference due to ligand binding is ∆Fℓ(ℓ) = ln(1 + ℓ(t)/KI D) − ln(1 + ℓ(t)/KA D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (F3) Between the two states the cluster has an altered dissociation constant, which is denoted KI D for the inactive state, and KA D for the active state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The free-energy difference due to methylation has been experimentally shown to depend approximately linearly on the methylation level [29]: ∆Fm(m) = ˜α( ¯m − m(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (F4) We assume that inactive receptors are irreversibly methylated, and active receptors irreversibly demethylated, with zero-order ultrasensitive kinetics [30, 31, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The dynamics of the methylation level of the ith receptor cluster is then given by: ˙mi =(1 − ai(ℓ, mi))kR − ai(ℓ, mi)kB + Bmi(ai)ξ(t), (F5) with B(i) m (ai) = � (1 − ai(ℓ, mi))kR + ai(ℓ, mi)kB, and unit white noise ξ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' These dynamics indeed give rise to perfect adaptation, since from this equation we find that the steady state cluster activity is given by p ≡ ¯a = 1/(1 + kB/kR), thus indeed independent of the ligand concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Finally, active receptors catalyze phosphorylation of read-out molecules, and phosphorylated read-out molecules decay at a constant rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We have ˙x∗ = RT � i=1 ai(t)(XT − x∗(t))kf − x∗(t)kr + Bx(ai, x∗)ξ(t), (F6) where RT is the total number of receptor clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The steady state fraction of phosphorylated read-outs is given by f ≡ ¯x∗/XT = (1 + kr/(kfRTp))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Linear dynamics We again do a first order approximation around the steady state, defining all variables in terms of deviations from their mean: δℓ(t) = ℓ(t) − ¯ℓ, δm(t) = m(t) − ¯m and δa(t) = a(t) − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The linear form of this model has previously been studied in for example [30] and [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We obtain for the linear dynamics of the ith cluster activity δai(t) = αδmi(t) − βδℓ(t), (F7) with α = ˜αNp(1 − p) and β = κNp(1 − p), with κ = (¯ℓ + KI D)−1 − (¯ℓ + KA D)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For the methylation on the ith cluster and for the readout dynamics we then obtain, as a function of δa(t), ˙ δmi = −δai(t)/(ατm) + ηmi(t), (F8) ˙ δx∗ = γ RT � i=1 δai(t) − δx∗(t)/τr + ηx(t), (F9) where we have introduced the relaxation times τm = (α(kR + kB))−1 for methylation and τr = (RTpkf + kr)−1 for phosphorylation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We have further defined the rate at which an active cluster phosphorylates the readout CheY: γ = XTf(1 − f)/(pRTτr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Substituting the expression for δai in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' F7 into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' F8 and F9, and expressing the dynamics in terms of the methylation on all clusters gives d dt � RT � i=1 δmi � = − RT � i=1 δmi/τm + qδℓ(t)/(ατm) + ηm(t), (F10) ˙ δx∗ = −δx∗(t)/τr − γqδℓ(t) + γα RT � i=1 δmi(t) + ηx(t), (F11) with q = RTβ (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' F7 for β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The rescaled white noise ηm is the sum of the methylation noise on all receptor clusters, ⟨η2 m⟩ = 2RTp(1 − p)/(ατm), where we have assumed that the methylation noise on the respective receptor clusters is independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The phosphorylation noise has strength ⟨η2 x⟩ = 2XTf(1 − f)/τr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Parameter values A large body of work has studied the parameters of the MWC model for the E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli chemotaxis system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We have listed the parameters relevant for our model in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We choose the background concentration ¯ℓ to be in between KI D and KA D, at ¯ℓ = 100µM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' In this work we analyze the impact of the methylation timescale τm, and the numbers of receptor clusters and readout molecules RT and XT, on the past and predictive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We therefore do not set them to a fixed value, but experimental estimates are listed in table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Model statistics Again we take the power spectrum route to determine the variance in the network output, the SNR, and the correlation coefficient between current output and the future signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We consider the system to sense the non- Markovian ligand concentration defined in equation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Such a signal is characterized by both its concentration TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Measured E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli chemotaxis parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Parameter Value Source Description KI D 18µM [42, 43] MeAsp-Tar dissociation constant inactive receptor KA D 2900µM [42, 43] MeAsp-Tar dissociation constant active receptor N ∼ 6 [29, 42, 43, 51] Number of receptors per cluster ˜α 2kBT [29] Free energy change per added methyl group p 1 3, 1 2 [29, 43] Steady state activity at 22◦C, 32◦C τr ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1s [10, 18, 51] Phosphorylation timescale 26 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Approximate E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' coli chemotaxis timescales and abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Parameter Value Source Description τm ∼ 10s [15, 18, 29] Adaptation time Tsr+Tar 14000, 3300 [35] Rich medium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' RP437, OW1 strain Tsr+Tar 24000, 37000 [35] Minimal medium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' RP437, OW1 strain CheY 8200, 1400 [35] Rich medium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' RP437, OW1 strain CheY 6300, 14000 [35] Minimal medium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' RP437, OW1 strain and derivative, and the (cross-)power spectra of these properties are Ss(ω) = � Sℓ(ω) Sℓ→v(ω) Sv→ℓ(ω) Sv(ω) � = � Sℓ(ω) iωSℓ(ω) −iωSℓ(ω) ω2Sℓ(ω) � , (F12) with Sℓ(ω) = 2σ2 v/τv (ω2 + ((2τv)−1 + ρ)2)(ω2 + ((2τv)−1 − ρ)2), (F13) where ρ = � (4τ 2v )−1 − ω2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The chemotaxis signalling network is fully determined by the following matrices (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A1) G = q � 1/(ατm) 0 −γ 0 � , (F14) J = � −1/τm 0 αγ −1/τr � , (F15) B = �� ⟨η2m⟩ 0 0 � ⟨η2x⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (F16) The Fourier transform of the matrix exponential of the Jacobian is F{eJ t} = (iωIn − J )−1 = � � 1 1/τm+iω 0 αγ (1/τm+iω)(1/τr+iω) 1 1/τr+iω � � , (F17) which allows us to determine the gain matrix via G(ω) = F{eJ t}(ω)G, and the noise matrix using N(ω) = F{eJ t}(ω)B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' see also Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A4 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To gain more insight in the way in which the network maps the signal onto its output, we first study the integration kernels of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The integration kernel from ligand concentration to output is given by the inverse Fourier transform of element (1, 2) of the gain matrix G(ω), which is k(t) ≡ F−1{˜gℓ→x(ω)} = κNf(1 − f)(1 − p)XT 1 1 − τr/τm � 1 τm e−τ/τm − 1 τr e−τ/τr � , (F18) with κ = (¯ℓ+KI D)−1−(¯ℓ+KA D)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Due to the adaptive nature of the network, the static gain from ligand concentration to output is zero: ¯gℓ→x = � ∞ 0 k(t)dt = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' the long-time response to a step change in a constant input is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The kernel does indeed not change the output based on the input concentration directly, but instead takes a (time-averaged) derivative of the input (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 8A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' It is therefore useful to consider the kernel that maps the signal derivative onto the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This kernel can be found by rearranging the expression for the output of a linear signalling network, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Disregarding the noise terms and integrating by parts gives � 0 −∞ k(−t)ℓ(t)dt = K(−t)ℓ(t)|0 −∞ − � 0 −∞ K(−t)v(t)dt, (F19) 27 where v(t) ≡ ˙ℓ and K(t) is the primitive of k(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To make progress we first determine K(t), K(t) = κNf(1 − f)(1 − p)XT 1 1 − τr/τm � −e−τ/τm + e−τ/τr� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (F20) The form of K(t) is that of a simple exponential kernel with a delay (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 8B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We thus have both K(0) = 0 and K(∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' It is now clear that the convolution over the ligand concentration simply maps onto the convolution over its derivative as � 0 −∞ k(−t)ℓ(t)dt = − � 0 −∞ K(−t)v(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (F21) The static gain of K(t) is ¯gv→x = � ∞ 0 K(t)dt = qγτrτm = κNXT(1 − p)f(1 − f)τm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The gain thus increases with the number of receptors per cluster, N, the number of readout molecules, XT, and notably, with the adaptation time τm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This static gain from signal derivative to network output is a useful quantity which we will use to describe the other statistics of the network below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 0 5 10 15 20 5 0 5 10 15 20 25 30 Time (s) k(t): kernel ℓ → x* 0 5 10 15 20 5 0 5 10 15 20 25 30 Time (s) K(t): kernel v → x* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='10 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='01 1 100 104 Frequency ω (s-1) Power τr 1 τm 1 Nx 2 gℓ→x 2 gℓ→x 2 /Nx 2 A B C FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Integration kernel and power spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (A) The integration kernel k(t) takes a temporal derivative by weighing the most recent signal values with an opposite sign from the preceding ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (B) The integration kernel K(t) from the derivative of the input concentration to the network output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The kernel K(t) is the primitive of k(t), and its static gain is proportional to the adaptation timescale τm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (C) Frequency dependent gain ˜g2 ℓ→x(ω), frequency dependent noise N 2 x(ω), and their ratio, as a function of frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The chemotaxis network is a band-pass filter, the frequencies that are passed through are set by τr on the high end and τm on the low end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' At low frequencies, the methylation noise dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Parameters used in all panels τr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1s and τm = 10s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Model parameters are ˜a = 2, N = 6, KI D = 18µM, KA D = 2900µM, ¯ℓ = 100µM, p = f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To compute the past and predictive information, we need to determine the variance in the output, the SNR, and the correlation between the current output and the future ligand derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' To that end we require the power spectrum of the output, and the cross-spectrum from output to future derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For the power spectrum of the output we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A5 to find Sx(ω) = q2γ2ω2 (τr−2 + ω2)(τm−2 + ω2)Sℓ(ω) + α2γ2⟨η2 m⟩ (τr−2 + ω2)(τm−2 + ω2) + ⟨η2 m⟩ τr−2 + ω2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (F22) From this power spectrum we can see that the network is a band-pass filter, where the gain is maximal in the frequency range τm−1 < ω < τr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Both for ω ≫ τr−1 and ω ≪ τm−1 the gain goes to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' On long timescales the methylation noise dominates (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 8C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The cross-power spectrum between current output and future ligand derivative is given by element (2, 2) of the matrix G(−ω)Ss(ω) which is (also see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' A6) Sx→v(ω) = qγ −ω2Sℓ(ω) (τm−1 − iω)(τr−1 − iω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (F23) In the main text, we argue that the biologically relevant regime of the input signal is the limit ω0 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We therefore present below the network statistics in this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' We start by determining the variance in the readout, via the inverse Fourier transform of its power spectrum (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' F22): lim ω0→0 σ2 x = ¯g2 v→x 1 + τr/τv + τr/τm (1 + τm/τv)(1 + τr/τv)(1 + τr/τm)σ2 v + ¯g2 a→xαRTp(1 − p) 1 1 + τr/τm + XTf(1 − f), = ¯g2 v→x 1 + τr/τv + τr/τm (1 + τm/τv)(1 + τr/τv)(1 + τr/τm) � �� � dynamical gain σ2 v + XTf(1 − f) � 1 + ¯gv→x ˜α(1 − p) RTκτm 1 1 + τr/τm � , (F24) 28 where ¯ga→x = γτr = XTf(1−f)/(RTp) is the static gain from receptor activity to readout, and we used the definition of α = ˜αNp(1 − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Because there is no receptor-ligand binding noise, there is also no time averaging as in the push-pull network (and hence no factor depending on τr/τc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' There is methylation noise on a timescale τm, but this cannot be time-averaged effectively because the integration time τr of the push-pull network is shorter than the receptor methylation timescale τm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The methylation noise can only be averaged out significantly by increasing RT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The contribution from the variance in the signal derivative, σ2 v, to the output noise σ2 x, depends on the dynamical gain, which is the product of the static gain ¯g2 v→v and a factor that only depends on ratios of timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The dynamical gain is maximized for τr → 0 and τm → ∞, which is intuitive since subtracting a signal from an earlier one reduces the amplification of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Hence, when the system has too few XT molecules to lift the signal above the noise, τm must be increased to raise the gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Only when XT is sufficiently large, can τm be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This allows the system to take more recent derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The signal to noise ratio SNR = σ2 x|η/σ2 x|L can straightforwardly be obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' F24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For the covariance between the current output and the future derivative we have lim ω0→0⟨δx(0)δv(τ)⟩ = F−1{Sx→v(ω)}, = −¯gv→xσ2 v (1 + τm/τv)(1 + τr/τv)e−τ/τv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (F25) The variance in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' F24 can be used to obtain the normalized correlation function ⟨δx(0)δv(τ)⟩/(σxσv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Past and predictive information of the chemotaxis network The past and predictive information are straightforward to compute from the quantities above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The definition of the past information is the same as for the push-pull network, and is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The SNR is now given by, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' F24: SNR = σ2 x|η/σ2 x|L = κ2Nτ 2 mσ2 v 1 + τr/τv + τr/τm (1 + τm/τv)(1 + τr/τv)(1 + τr/τm) � � 1 NXTf(1 − f)(1 − p)2 + ˜α RT(1 + τr/τm) � , where κ = (¯ℓ + KI D)−1 − (¯ℓ + KA D)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The predictive information is found in the same manner as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' D6, but now it is a function of the correlation between the current output and the future derivative of the ligand concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This correlation can be decomposed into the instantaneous correlation coefficient and an exponential decay on the timescale of the fluctuations of the derivative of the concentration, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' F25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Specifically, the predictive information is given by Ipred(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' vτ) = −1 2 log(1 − ρ2 ℓve−2τ/τv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' (F26) The instantaneous correlation coefficient ρ2 ℓv can be found using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' F25 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' F24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' F26 it is clear that just like for the push-pull network, the optimal design of the network that maximizes the predictive information, determined by the optimal ratio XT/RT and the optimal adaptation time τm, does not depend on the forecast interval τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The forecast interval only affects the magnitude of the predictive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Optimal allocation We can determine the optimal ratio (XT/RT)opt that maximizes either the past information or the predictive infor- mation, given all other network parameters, most notably τm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Just as for the push-pull network, we find however that the optimal ratio (XT/RT)opt is the same regardless of whether the past or the predictive information is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' This is again because the information on the future signal (be it the value or the derivative) is encoded in the receptor occupancy, while the ratio XT/RT controls the interval by which the downstream readout samples the receptor to estimates its occupancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Nonetheless, the optimal methylation timescale τmopt that maximizes either the past or the predictive information is different—maximizing predictive information requires a more recent derivative and hence a shorter τm than obtaining past information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Given τm and all other parameters, the optimal ratio of the number of readout molecules over receptor clusters is, 29 using C = RT + XT, �XT RT �opt = � 1 α 1 f(1 − f) p 1 − p � 1 + τr τm , = 2 � 2/N � 1 + τr τm , (F27) where in the second line we have used that α = ˜αNp(1 − p), and ˜α = 2, and f = p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Because for the chemotaxis network τr < τm the ratio τr/τm only varies between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' For this reason, the optimal ratio (XT/RT)opt depends only weakly on τm, and does not vary strongly along the isocost lines of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 4A in the main text, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='0 Distance along fixed cost line θ XT/RT FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The optimal allocation ratio XT/RT varies only slightly along the isocost lines of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 4A in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The optimal ratio XT/RT as a function of the distance θ along the isocost lines of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' 4A of the main text;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' dotted line C = 102, solid line C = 104, dashed line C = 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' The red dots mark the points where the predictive information is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Along the isocost lines XT/RT varies much more weakly than for the push-pull network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' for resource availability C ≤ 104 the ratio is almost constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Parameters used g = 4mm−1, τr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content='1s, KI D = 18µM, KA D = 2900µM, N = 6, ˜α = 2, p = f = 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} +page_content=' Stock, Organiza- tion of the Receptor-Kinase Signaling Array That Regu- lates Escherichia coli Chemotaxis *, Journal of Biological Chemistry 277, 36748 (2002), publisher: Elsevier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQfjAeF/content/2301.03964v1.pdf'} diff --git a/3NAzT4oBgHgl3EQf9P6r/vector_store/index.pkl b/3NAzT4oBgHgl3EQf9P6r/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..7ce6d0ee07cd7fdafc519400834890e1516d00b1 --- /dev/null +++ b/3NAzT4oBgHgl3EQf9P6r/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:02b10055ca8e0c10b5510da5b3c1c64fdd235766cf4f4715612beb1e29a0206a +size 145904 diff --git a/3dE1T4oBgHgl3EQf5wXA/content/tmp_files/2301.03516v1.pdf.txt b/3dE1T4oBgHgl3EQf5wXA/content/tmp_files/2301.03516v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5cef3c374d929820c79e0928332807ef2f6e3d9f --- /dev/null +++ b/3dE1T4oBgHgl3EQf5wXA/content/tmp_files/2301.03516v1.pdf.txt @@ -0,0 +1,671 @@ +Coherent Stokes Raman scattering microscopy +(CSRS) +SANDRO HEUKE1,* AND HERVÉ RIGNEAULT1,* +1Aix Marseille Univ, CNRS, Centrale Marseille, Turing Center for Living Systems, Institut Fresnel, +Marseille, France. +*Corresponding authors: Sandro.Heuke@fresnel.fr & herve.rigneault@fresnel.fr +Abstract: +We report the first implementation of laser scanning Coherent Stokes Raman +scattering (CSRS - pronounced "sCiSsoRS") microscopy. To overcome the major challenge in +CSRS imaging, we show how to suppress the fluorescence background by narrow bandpass +filter and a lock-in based demodulation. Near background free CSRS imaging of polymer beads, +human skin, onion cells, avocado flesh and the wing disc of a drosphila larva are presented. +Finally, we explain and demonstrate numerically that CSRS solves a major obstacle of other +coherent Raman techniques by sending a significant part (up to 100%) of the CSRS photons into +the backward direction under tight focusing conditions. We believe that this discovery will pave +the way for numerous technological advances, e.g. in epi-detected coherent Raman multi-focus +imaging, real-time laser scanning based spectroscopy or efficient endoscopy. +© 2023 Optica Publishing Group under the terms of the Optica Publishing Group Publishing Agreement +1. +Introduction +Conventional bright-field microscopy provides information about the refractive index and +absorption properties, but cannot elucidate the sample’s chemical composition. +Infra-red +absorption and linear Raman scattering retrieve the chemical fingerprint [1,2], but are incompatible +with high spatial resolution or real-time imaging. Coherent Raman imaging (CRI) fills this +technological gab joining a chemical bond specific contrast with signal levels that permit +video-rate image acquisition. Well established CRI microscopy techniques are the coherent +anti-Stokes Raman scattering (CARS) [3, 4] and stimulated Raman scattering (SRS) [5–7]. +CARS owes its wide-range application to the blue-shifted anti-Stokes radiation which greatly +facilitates its separation from linear fluorescence. When working with near infra-red excitation +wavelength, the blue-shifted CARS radiation is readily detected using photo-election multiplier +tubes (PMT) of standard laser scanning microscopes. SRS’s popularity arises from the homodyne +signal amplification that frees SRS images from an omnipresent non-resonant four-wave-mixing +background and allows for measurements under daylight conditions. +Overshadowed by CARS and SRS until now, there exists a 3rd four-wave-mixing process termed +coherent Stokes Raman scattering (CSRS, "Scissors") [8–10] which is always appearing within +any CARS or SRS experiment and provides near identical mapping of molecular oscillators [11] - +see Fig.1. In analogy to the Stokes emission in linear Raman microscopy, the CSRS radiation +(2𝜔𝑆 − 𝜔𝑝) is red-shifted with respect to the excitation frequencies of the pump (𝜔𝑝) and +Stokes beams (𝜔𝑆). Surprisingly, CSRS was not yet implemented for laser scanning microscopy. +Presumably, this neglect must be attributed to the high degree of resemblance of CARS and +CSRS spectra [11] rendering CSRS - prima facie - to be either CARS with an added fluorescence +background when working with visible light sources or, using near infra-red (NIR) excitation, +CARS with a radiation wavelength offside high quantum yields of common detectors. CSRS +provides, however, some unique properties that are of high interest for imaging. (1) The CSRS +spectrum differs from CARS in the presence of accessible electronic resonances. For example, +pre-resonant CSRS will offer complementary information in application to alkyne-labeled +dyes [12] and standard dyes used in microbiology [13]. (2) The red-shifted radiation of CSRS +arXiv:2301.03516v1 [physics.optics] 9 Jan 2023 + +Fig. 1. Coherent Raman imaging techniques in energy diagrams, relative radiation +wavelength and energy conservation under plane-wave illumination. +imaging becomes an advantage for UV or near-UV excitation where CARS photons [14] would +be too far blue-shifted to be detected efficiently while any SRS image [15] is likely to be +compromised by various artifacts such as multi-photon absorption [16,17]. Thus, UV excited +CSRS holds the potential to achieve the highest possible spatial resolution (𝜆Stokes/[ +√ +8𝑁 𝐴]) +in coherent Raman imaging. (3) NIR-excitation wavelength combined with CSRS may allow +for deeper tissue imaging due to the reduced scattering and absorption of its radiation [18]. (4) +Last but most important: Due to a modified phase-matching geometry, CSRS microscopy can +be configured to radiate more light into the backward direction which will add game-changing +benefits for the investigation of thick samples, real-time spectroscopy, multi-focus imaging and +endoscopy [19]. Within this contribution, we want to open up the field of laser scanning CSRS +imaging by demonstrating CSRS microscopy within the visible excitation spectrum. To remove +the major obstacle, we will show how linear fluorescence can be suppressed by a set of bandpass +filter and nearly nullified in combination with a lock-in based detection scheme as a premise +for near-UV excited CSRS imaging with a lateral resolution < 100 nm. Furthermore, we shall +investigate numerically CSRS’ spatial radiation behavior under NIR excitation paving the way +towards CSRS experiments with an efficient epi-detection. +2. +Experimental result and discussion +The CSRS signal of biomedical samples is readily overwhelmed by linear fluorescence. Time- +gating [20], a time-resolved detection using streak cameras [21] or polarization filtration can be +used to reduce or suppress any fluorescence signal. These methods require, however, either a +substantial alteration of standard coherent Raman microscopes or do not work in the presence of +large quantities of fluorescence light. Here, we exploit the fact that the CSRS is spectrally narrow +under ps-excitation. Thus, the majority of fluorescence is readily suppressed by the choice of a +the narrow-band filter. Filters with a spectral width below < 1nm are commercially available but +the selection of a specific center wavelength requires expensive costume solutions. This is the +reason why we use a combination of two inexpensive bandpass filter with a width of about 15 nm, +but different center wavelength. In a addition, we fine-tune the filter transmission by a tilt (<20◦) +with respect to the incident beam. Thus, two tilt-adjusted bandfilter create a sharp transmission +line (FWHM<3nm) for the CSRS signal while rejecting significant parts of the autofluorescence. + +CARS +SRS +CSRS +kas+ks=kp1+kp2 +111 +CARS +ks1+kp1=ks2+kp2 +SRS +dwnd +Stokes +SRS +kcs+k,=ks1+ks2 +Intensity +CARS +CSRS +CSRS +2Fig. 2. CSRS experimental implementation and characterization. Bottom left: The +CSRS signal is separated from fluorescence by means of 2 angle-tuned narrow bandpass +filter. Bottom right: Additional suppression of fluorescence is achieved by intensity +modulating the Stokes and pump beam at the radio frequencies f1 and f2, respectively. +Fluorescence free CSRS signal is obtained at f1-f2. Right center: Time separation of +the pump and Stokes pulses as well as blocking the excitation highlights the superior +suppression of fluoresence background at the demodulation frequency f1-f2 compared +to CSRS signal obtained at f1, f2 or the DC frequency. Top right: The intensity +profile at the interface of a PMMA bead and olive oil indicates a lateral resolution +of <400nm. Top left: scheme of the CSRS experiment. 1 Yb-fiber laser, 2 optical +parametric oscillator (OPO), 3 Second harmonic generation (SHG), 4 acousto-optic +modulator (AOM), Laser scanning microscope (LSM), 6 photo-electron multiplier +(PMT), 7 Lock-in amplifier. + +IcSRS +PSF. +CSRS demodulated at +20μm +DC, f1, f2, f1-f2 +PMMA bead +<400nm +G +5 μm +88 +0 +2 +( +1 +3 +x / μm +④ f2 +Lock-in +DC +f1 +3 +f2 +f1-f2 +lp=0 +④f1 +Is=0 +△t>>3ps +T +CSRS +Intensity +ump2 +f1-f2 +Fluorescence +SRS +concefiFig. 3. LSM-CSRS at 2850 cm−1. The left and right column show the CSRS image +demodulated at the frequencies f1-f2 = 1.47 MHz and 0 Hz (DC). To estimate the +remaining fluorescence level, images without temporal overlap of the pump and Stokes +pulses are displayed to the right. a) Mixture of polystyrene (PS, 30µm) and Poly-methyl- +methacrylate (PMMA, 20µm) beads in olive oil. b) and c) Epithelium and dermis of a +20µm thick human skin section d) Cells of an onion. e) Lipid droplets within the flesh +of an avocado. d) Wing disc of a Drosophila larva. The white and blue scale bar equals +20µm and 5µm, respectively. + +CSRS at f1-f2 +△t >> 3ps +CSRS at DC +△t >> 3ps +PS +PMMA +olive.oil +Human skin +Epithelium +Dermis +Onion +avocado +Drosophila larva +wing disc +notum +hinge +pouchAs a second method for fluorescence discrimination, we take advantage of CSRS intensity +dependence on both excitation colors while linear fluorescence follows either the intensity of +the pump or the Stokes laser. Consequently, modulating the pump and Stokes beams at f1 and +f2 while demodulation the signal at f1-f2 (or f1+f2) yields exclusively nonlinear signals that +depend on both excitation colors. The f1-f2 demodulation, therefore, also discriminates the CSRS +signal against 2-photon excited fluorescence (2PEF) under single-color excitation. It shall be +noted that the double modulation is also sensitive to two-color 2-photon fluorescence (2C-2PEF). +Nevertheless, we will find experimentally, that the emission strength of native 2C-2PEF is +negligible within our CSRS approach. +For the experimental implementation of CSRS into laser scanning microscopy, we chose visible +excitation wavelengths at 445nm (pump) and 515nm (Stokes) for the following reasons: (1) +CSRS under near UV excitation is a potentially important application area since the CARS signal +falls into the UV range while SRS artifacts are increased due the high concentration of matching +chromophores. (2) The red-shifted CSRS radiation is readily detected by ordinary PMTs. (3) +Fluorescence artifacts are enhance compared to a near infra-red (NIR) excitation. Thus, our +approach will be viable as well for CSRS under NIR excitation, if pure CSRS signals can be +obtained under VIS excitation. The experimental implementation, the spectral filtration and +the double modulation are schematically shown in Fig.2a. Our implementation resembles a +standard SRS setup with the difference that we use visible excitation wavelengths, we modulate +not one but both beams and the photo-diode is replaced by a PMT which is connected to a +lock-in amplifier. More information about the setup can be found within the part Methods: +Experimental setup. To quantify the level of fluorescence rejection, we investigated the signal +of native olive oil at 2850 cm−1 when blocking the Stokes or pump beams and when the +temporal pulse overlap is removed. The output signal of the lock-in is plotted as functions +of the demodulation frequencies at 0 Hz (DC), f1, f2 and f1-f2 in Fig.2. It can be observed +that the DC channel contains significant amounts of fluorescence while this artifact is already +reduced within the channels f1 and f2. Nevertheless, only the difference frequency channel at +f1-f2 becomes dark, when the excitation pulses do not overlap in time. In a second experiment, +we imaged the interface of olive oil and a 20µm sized Plexiglas (PMMA) bead to obtain an +estimation of the lateral resolution for an excitation objective featuring an NA of 1.45 - see +Fig.2. From this "knife-edge" CSRS intensity profile, we can infer a lateral resolution below +400nm. The difference to the expected 𝜆𝑆𝑡𝑜𝑘𝑒𝑠/[ +√ +8𝑁 𝐴]= 515nm/[ +√ +81.49]=120nm can be +attributed to underfilling of the excitation objective lens and the bent oil/bead interface. Having +confirmed a high-resolved, fluorescence-free CSRS image contrast, we investigated the suitability +of LSM-CSRS for vibrational imaging of various objects featuring non-negligible fluorescence +levels. Within Fig. 3, we show the CSRS images of test and biomedical samples demodulated at +the DC and f1-f2 frequencies for (non-)overlapping pump and Stokes pulses. The images were +organized along the ratio of the CSRS to fluorescence signal starting from the highest at the +top. Comparing the DC and f1-f2 images in Fig. 3a, it obvious that a narrow spectral filtering is +already sufficient for CSRS imaging of polymer beads in oil. The first artifacts become visible for +the DC CSRS images of the epithelium and dermis of a 20µm thick section of human skin - see +Figs. 3b and c. For the epithelium, a pronounced fluorescence artifact arises from melanin within +the Stratum basale. Artifacts within the Dermis can be attributed to the auto-fluorescence of +collagen and elastin [22]. The quantity of fluorescence observed within the DC channel increases +stepwise further for CSRS imaging of onion cells, lipid droplets within the flesh of an avocado +and the wing disc of a Drosophilia larva. From the second row of Fig. 3, it is reconfirmed that +almost no fluorescence is leaking into the f1-f2 CSRS channel as an important condition for the +estimation of the true concentration of the targeted molecular group. The origin of fluorescence +for these 3 samples, however, cannot be attributed with certainty, but might arise from NADH, +flavins and chlorophyll. + +In a broader context, we would like to point out that other nonlinear microscopy techniques would +also greatly benefit from the narrow-band filter plus demodulation combination for rejection of +spurious background signals. For example, the 2PEF signal of chlorophyll in plant leaves readily +overwhelms any CARS or second harmonic generation (SHG) image contrast even under NIR +excitation. A double modulation of the excitation combined with a lock-in based demodulation +will purify the signal, reduce the sensitivity against other light sources such as room light +and reestablish the reliability of the following image analysis. Having removed the why-not +argument for the CSRS image contrast, we shall introduce in the next section a non-intuitive +but game-changing argument for CSRS microscopy : the increased backwards radiation as the +prerequisite of an effective epi-CSRS detection. +3. +Numerical results +In this section, we shall show and explain CSRS’ superior backward radiation properties. Before +entering into the calculations, we want to consider CSRS from a heuristic viewpoint investigating +the momentum conservation laws for CSRS and compare it to CARS. Under plane illumination, +the momentum conservation laws can be written as K = k𝑝 − k𝑆 + k𝑝 − k𝑎𝑆 for CARS [23] and +K = k𝑆 − k𝑝 + k𝑆 − k𝑐𝑆 for CSRS with K, k𝑝, k𝑆, k𝑎𝑆 and k𝑐𝑆 representing the wavevectors +of the object, the pump(probe) and Stokes beam as well as the anti-Stokes and coherent Stokes +radiation, respectively. Note that for homogeneous samples these laws are also referred to as +phase-matching condition and simplify to k𝑝 + k𝑝 = k𝑆 + k𝑎𝑆 (CARS) and k𝑆 + k𝑆 = k𝑝 + k𝑐𝑆 +(CSRS). Under focusing conditions, the single wavevectors are replaced by the distribution of +incident wavevectors which are distributed over a cap of a sphere. To identify those object +frequencies (K) that are effectively probed, every combination of excitation and emission +wavevector must be identified. This operation is equivalent to the convolution of the caps of +the illumination and detection Ewald spheres. Neglecting polarization effects, the result of this +convolution (simplified to 3 points per arc) is shown in 2D within Fig. 4a. +Evidently, there exist no vector combination for epi-scattered CARS photons which would +cover the origin K(0,0,0) of the object space. Thus, a homogeneous sample, such as olive oil, +does not provide any backward radiation. On the contrary, structures that feature high object +frequencies, such as small polymer beads or layered materials, generate Epi-CARS radiation. +In the past, Epi-CARS was occasionally considered to be a size selective contrast that would +highlight exclusively small objects [24]. While this statement holds for the majority of biomedical +samples, there do exist large structures, e.g. multi-layered lipids in vesicles that also emit a strong +CARS radiation into the backward direction. Hence, it is more appropriate to refer to Epi-CARS +as a technique that probes high object frequencies instead of been considered as size selective. +Switching the detection wavelength to the red-shifted coherent Stokes radiation changes the +covered object support significantly and includes now the origin at K(0,0,0). Due to the reduced +size of the detection wavevector (|k𝑐𝑆| ≪ |k𝑎𝑆|) and the pump vector entering as complex +conjugated, see Eq. 3, it is now possible to find vector combinations that cover the origin +at K(0,0,0). Consequently, even a homogeneous object will radiate considerable amounts of +Epi-CSRS. Nevertheless, since the the centroid of the Epi-CSRS object support, i.e. the gray +cloud within Fig. 4a, does not coincidence with the K-space origin, Epi-CSRS images will also +highlight objects containing higher frequencies. +To address the question of how to increase the ratio of Epi versus forward Epi-CSRS, and which +object frequencies are most efficiently probed using Epi-CSRS, we performed finite element +simulations whose results are summarized in Fig. 4b-e. The equations implemented numerically +as well as important parameters are found in the annex - numerical calculation. From the +momentum conservation law and the vector diagrams in Fig. 4a, it is readily comprehensible +that a larger wavelength difference in between the pump and coherent Stokes wavelength relaxes +greatly the necessity for extreme incident illumination angles of the Stokes beam. Furthermore, + +Fig. 4. Object frequency support and radiation behavior of CSRS versus CARS. a) +The object K-support for Epi-CSRS(CARS) is found by convolving the illumination +Ewald spheres of the Stokes (pump), pump (Stokes), and Stokes (probe) with the cap +of detection Ewald sphere at (anti-)Stokes frequency. Note that vector combinations +covering the frequency of a homogeneous sample K(0,0,0) are only found for CSRS +but not for CARS. A single wavevector combination that phase-matches K(0,0,0) is +highlighted to the left while a similar approach for CARS leads to a large phase-mismatch +(ΔK). b) CSRS and CARS radiation behavior of a homogeneous sample under standard +illumination condition, i.e. the pump and Stokes beam fill the objective aperture +homogeneously (𝜃𝑚𝑎𝑥=80◦). c) same as in b) but with an annular pupil filter applied +to the Stokes beam for CSRS covering 50% of area of the objective back-aperture. For +an equitable comparison with CARS, the same pupil filter was applied to the pump +beam. d) same as for b) but the homogeneous sample was replaced by a frequency +object whose scatter density is described as 1 + cos(2𝜋𝑧/𝜆𝑜) and 𝜆𝑜=1µm. e) Plot of +the ratio of backward/forward radiation (Rb/f) as a function of the object frequency 𝜆𝑜. + +Ks1 +Ks2 +kcs +Kp +(a) +个 Kz +个Kz +个Kz +Epi +Ks1 +Epi-CSRS +Ks2 +Kcs +0 +0 +0 +0 +0 +Kx +0 +Kx +Kp1 +ks +Kp2 +kas +Epi +个Kz +个 Kz +个 Kz +Epi-CARS +0 +Kp2 +AK +kas +0 +0 +o +Kx +0 +Kx +0 +(b) +(c) +(d) +(e) +Rb/f = 0.79*10-2 +Rb/f = 1.54 +Rp/f = 0.25 +% +30 +ratio forward/backward in +5N +IN +z +1.5 +20 +CSRS +0.5 +10 +5 +5 +0.5 +20 +0.66 +0.5 +x +y +y +y +Rb/f = 0.18*10-3 +Rb/f = 0.33*10-2 +z +z +z +1.5 +1.5 +4 +CARS +0.5 - +2 +0.5 +1 +20 +0.66 +0.5 +y +y +2。 / μm +Stokes +pump +Stokes +pump +z +z +50% +"standard CRS" +pupil filtering +frequency objectsince most of the coherent Raman experiments apply NIR instead of VIS excitation wavelength, +we used for within our simulations the wavelength 𝜆𝑝 = 797𝑛𝑚 and 𝜆𝑆 = 1030𝑛𝑚 which matches +the most commonly targeted Raman shift in CRI imaging at 2850cm−1. For these conditions, the +coherent Stokes radiation will be observed at 𝜆𝑐𝑆 = 1450𝑛𝑚. It shall be noted that our results +equally apply for the visible excitation wavelength with gently higher excitation angle or thinner +annular masks. +To start with, we computed the radiation pattern of CSRS and CARS of a homogeneous object +using an NA of 1.49 (oil immersion) corresponding to a maximum illumination angle of 80◦. +From Fig. 4b, it is evident that both CARS and CSRS are predominately forward directed +though the CSRS’ radiation distribution features a larger radiation cone. Considering the ratio of +backward versus forward directed photons Rb/ 𝑓 , we find numerically that less than 1 photon in +105 is backward directed for CARS. Note that the momentum conservation actually law predicts +Rb/ 𝑓 =0 for CARS. Thus, the resulting deviation must be attributed to the finite number of voxels +of the numerical model. For CSRS, Rb/ 𝑓 increase dramatically to about 1 in 100 photons. +Since common surfaces within biomedical samples scatter more than 1%, we have to assume, +however, that also epi detected CSRS will be just forward generated CSRS that was redirected +by linear scattering at an interface. Still, using a confocal detection, i.e. a pinhole in front of +the detector placed at the conjugated plane of the excitation focus, might already yield true +Epi-CSRS images of homogeneous samples where Epi-CARS images would remain dark. To +find an approach that increases the proportion of CSRS’ epi radiation, we shall consider the +CSRS vector diagram matching K(0,0,0) on the left of Fig. 4a. The ratio of backward versus +forward radiation is readily increased by reducing the impact of vectors combinations probing +higher frequencies and favoring those covering the origin. This boost of epi-CSRS radiation +can be achieved using an annular illumination of the Stokes beam. Experimentally, such an +annular illumination is generated, without power-loss, using 2 axicons within the Stokes beam +path [25,26]. Numerically, we restricted the incident angles for the Stokes between 𝜃𝑚𝑖𝑛=56.5◦ +and 𝜃𝑚𝑎𝑥=80◦, which corresponds to covering 50% of the area of the objective lens’ back-focal +plane. With this pupil filtering, the radio of backward to forward radiation increased for CARS to +2 in 104 photons while the majority of all CSRS radiation is backward directed (Rb/ 𝑓 =1.5) when +focusing the pump and Stokes beam into a homogeneous object - see Fig. 4c. +As a second important result from the heuristic derivation of CSRS’ object support, we found that +the presence of high object frequencies increases the amount of backward radiation. To confirm +this prediction, we investigated in Fig. 4d and e an object whose nonlinear scatterer density, +i.e. concentration of molecular groups, is modulated along the optical axis as 1+cos(𝐾𝑧𝑧) with +𝐾𝑧 = 2𝜋/𝜆𝑜 being the object frequency. As an example, Fig. 4d outlines the radiation behavior +of a wave-like structured object with K𝑧=2𝜋/1µm. It is found that R𝑏/ 𝑓 increases to one forth +for Epi-CSRS while Epi-CARS remains negligible weak. To identify those object frequencies +which are most efficiently probed by Epi-CSRS, we computed R𝑏/ 𝑓 as a function of K𝑧. From +Fig. 4e, we find that Epi-CSRS peaks at K𝑧=2𝜋/1µm whereas Epi-CARS R𝑏/ 𝑓 still increases at +𝐾𝑧 = 2𝜋/0.25µm. It shall be note that the Rb/ 𝑓 never reaches 1 which arise from the 1+ within +the definition of the frequency object (1+cos(𝐾𝑧𝑧)). The 1+ implies that the wave-object always +features twice the amplitude at K(0,0,0), which corresponds to a homogeneous predominantly +forward scattering object, compared to the scatterer density modulation K(0,0,±K𝑧). Our +simulation results in a nutshell: we have found that CSRS features a non-negligible backward +radiation from a homogenous sample under tight-focusing conditions while this is not the case for +CARS. The amount of backward radiated CSRS can be enhanced by increasing the illumination +power of the Stokes beam with high incident angles. Furthermore, the natural structure of +biomedical samples, which are usually not homogeneous, will also elevate the CSRS backward +radiation. + +Conclusion +We have demonstrated the first laser scanning microscopy CSRS experiment. As the major +challenge, we were able to reduce the fluorescence background significantly using a pair of tilted +bandpass filter. The remaining fluorescence contribution is removed by intensity modulating the +Stokes and pump beams at the radio frequencies f1 and f2 and a lock-in based demodulation of +the CSRS signal. Taking advantage of CSRS’ characteristic dependence on both excitation colors, +the best fluoresence background suppression is obtained when demodulating the CSRS signal at +f1-f2. Background-free LSM-CSRS imaging was demonstrated for samples of decreasing ratio +of CSRS to fluoresence signal, namely: polymer beads, the epithelium and dermis of human +skin, onion cells, avocado flesh and the wing disc of a Drosophila larva. Having removed the +major obstacle for CSRS imaging, we introduced and quantified numerically the major interest of +CSRS which is its unique backward radiation property in combination with high NA objective +lenses. CSRS’ backward radiation and its distinction from CARS is readily understood from the +momentum conservation laws when considering all incident k-vectors forming the excitation focal +spots. Using dynadic Green functions, we show numerically that the CSRS is predominantly +forward directed for a homogeneous object, but the backward CSRS contribution rises to 1/4 for +objects that are structured axially. Moreover, backward CSRS signal can even dominate forward +CSRS (up to 100%) if an annular Stokes illumination is applied. With an efficient Epi-CARS +radiation at hand, various coherent Raman experiments become feasible which were impossible +before. Just to name a few: Epi-detected confocal multi-focus CSRS; Epi-detected LSM-CSRS +with a spectrometer at the descanned position; Epi-detected CSRS image scanning microscopy. +Thus, we believe that this contribution is just the first milestone in CSRS microscopy with many +others to follow. +Methods: Experimental setup +A Yb-based fiber laser (APE Emerald engine, 80 MHz, 2–3 ps) is frequency doubled yielding 7 W +of 515 nm output power. Parts of the emissions is used directly as Stokes beam to drive the CSRS +process. The major part (4 W) of the 515 nm is employed to pump an optical parametric oscillator +(OPO, APE Emerald). The OPO’s signal beam is tunable to 660-950 nm and coupled into an +external SHG unit. The latter generates up to 50 mW within the spectral range of 330-475 nm +serving as pump beam for the CSRS four wave mixing. Thus, the 330-475 nm pump combined +with the 515 nm Stokes beam allows addressing a Raman shift range from 1630-11000cm−1. The +pump and Stokes beams are superimposed in space and time via a dichroic beam splitter (Semrock, +FF470-Di01-25x36) and a delay stage. Both beams are coupled into a home-built laser scanning +microscope and focused by a 40x water objective lens (Nikon, Plan, NA = 1.15, immersion: +water) into the sample. The excitation objective lens was replaced for a 60x objective (Nikon, Plan +Apo TIRF, NA 1.45, immersion:oil) to generate the bead-oil interface image within Fig. 2. The +CSRS radiation is collected by a condenser lens (Nikon, Achr-Apl, NA 1.4) in forward direction, +spectrally separated from the broadband fluorescence background by means of 2 tilted bandpass +filter (Semrock FF01-620/14-25 + FF01-605/15-25) and detected by a photo-electron multiplier +(PMT, Thorlabs, PMT1001). For an enhanced suppression of the linear fluorescence background, +2 acousto-optic modulators (AOM, AA, MT200-A0.5-VIS) were applied to modulate the intensity +of the Stokes and pump beams and at the frequencies f1 = 2.28 MHz and f2 = 3.75 MHz, +respectively. The PMT output was demodulated simultaneously at the DC frequency, f1, f2 and +at f1-f2 = 1.47 MHz using a lock-in amplifier (Zürich instruments, HF2LI). The lock-in time +constant was set to 30 µs. All CSRS-images shown were recorded with a pixel dwell time of +40 µs. + +Annex - numerical calculation +In the following, we shall summarize the equations used to generate Fig. 4b-e. The meaning of +the variables is summarized in Fig. 5. +The focused field at the sample is given by the angular spectrum representation [27]: +��������� +𝐸𝑥(𝜌, 𝜙, 𝑧) +𝐸𝑦(𝜌, 𝜙, 𝑧) +𝐸𝑧(𝜌, 𝜙, 𝑧) +��������� += 𝑖𝑘 𝑓 +2 exp(−𝑖𝑘 𝑓 ) +��������� +𝐼00 + 𝐼02 cos(2𝜙) +𝐼02 sin(2𝜙) +−𝑖2𝐼01 cos(𝜙) +��������� +(1) +Here 𝑓 denotes the focal length of the objective lens and the integrals 𝐼0𝑚 are provided by +𝐼0𝑚 = +∫ +𝜃𝑚𝑎𝑥 +𝜃𝑚𝑖𝑛 +𝐸𝑖𝑛𝑐(𝜃) sin(𝜃)[cos(𝜃)]1/2𝑔𝑚(𝜃)Jm[𝑘𝜌 sin(𝜃)]d𝜃 +(2) +where 𝑔𝑚 equals 1 + cos(𝜃), sin(𝜃) and 1 − cos(𝜃) for 𝑚 = 0, 1, 2, respectively. 𝐽𝑚 is the +𝑚𝑡ℎ order Bessel function while 𝐸𝑖𝑛𝑐 is the incoming electric field which we assumed to be +x-polarized and constant within the (annular) aperture angles 𝜃𝑚𝑖𝑛 ≤ 𝜃 ≤ 𝜃𝑚𝑎𝑥. The nonlinear +polarization at anti-Stokes and coherent Stokes wavelength is given by: +𝑃(3) +𝑎𝑆,𝑎(𝑟) = 3𝜒(3) +𝑎𝑏𝑐𝑑(𝑟)𝐸 𝑝,𝑏𝐸∗ +𝑆,𝑐𝐸 𝑝,𝑑 +𝑃(3) +𝑐𝑆,𝑎(𝑟) = 3𝜒(3) +𝑎𝑏𝑐𝑑(𝑟)𝐸𝑆,𝑏𝐸∗ +𝑝,𝑐𝐸𝑆,𝑑 +(3) +Fig. 5. Declaration of variables +Where a,b,c,d represent the polarization +coordinates x, y or z. Using an x-polarized ex- +citation, it was noticed that 𝜒(3) +𝑥𝑥𝑥𝑥 dominates +all other tensor components even under tight +focusing conditions while filling the objective +lens homogeneously [27]. Nevertheless, for +the generation of Fig. 4c an annular mask with +𝜃𝑚𝑖𝑛 = 56.5◦ and 𝜃𝑚𝑎𝑥 = 80◦ was applied +which does necessitate the inclusion of other +tensor elements. For simplicity, we consider +here only isotropic samples reducing the 81 +susceptibility tensor elements to 21 which +are nonzero [28]. +Within isotropic media, +these nonzero elements follow certain sym- +metry rules which are, 𝜒1111 = 𝜒2222 = 𝜒3333, +𝜒1122 = 𝜒1133 = 𝜒2211 = 𝜒2233 = 𝜒3311 = +𝜒3322, 𝜒1212 = 𝜒1313 = 𝜒2323 = 𝜒2121 = 𝜒3131 = 𝜒3232, 𝜒1221 = 𝜒1331 = 𝜒2112 = 𝜒2332 = 𝜒3113 = +𝜒3223. Further, it applies 𝜒1111 = 𝜒1122 + 𝜒1212 + 𝜒1221 [28]. Within our simulations we were +setting 𝜒1122 = 𝜒1212 = 𝜒1221 = 1 and, hence, 𝜒1111 = 3. The nonlinear far-field radiation +distributions is obtained using a dyadic Green function approach: + +��������� +𝐸𝑞,𝑅(𝑅, Θ, Φ) +𝐸𝑞,Θ(𝑅, Θ, Φ) +𝐸𝑞,Φ(𝑅, Θ, Φ) +��������� += − +𝜔2 +𝑞 +𝑐2 +exp(𝑖𝑘𝑞|𝑅|) +|𝑅| +∭ ∞ +−∞ +𝜌d𝜌d𝜙d𝑧 exp(𝑖𝑘𝑞rR) +|𝑅| +× +��������� +0 +0 +0 +cos(Θ) cos(Φ) +cos(Θ) sin(Φ) +− sin(Θ) +− sin(Φ) +cos(Φ) +0 +��������� +��������� +𝑃(3) +𝑞,𝑥(r) +𝑃(3) +𝑞,𝑦(r) +𝑃(3) +𝑞,𝑧(r) +��������� +(4) +where q is replaced by aS or cS to calculate either the anti-Stokes or coherent Stokes radiation. +Within the simulations, we segmented the focal area into (121×121×121≈) 1.77 mio elements +of a width of 50 nm equally spaced into the x, y and z direction. The far-field radiation sphere +was discretized into (ΔΘ=1◦, ΔΦ=2◦) 32400 elements. The coherent (anti-)Stokes radiation +was qualified as either forward or backward directed if falling into the range Θ.. 0-80◦ or Θ.. +100-180◦, respectively. +Funding Information +We acknowledge financial support from the Centre National de la Recherche Scientifique (CNRS), +Aix-Marseille University (A-M-AAP-ID-17-13-170228-15.22-RIGNEAULT), A*Midex (ANR- +11-IDEX-0001-02), Cancéropôle Provence-Alpes Côte d’Azur, French National Cancer institute +(INCa), Région Sud, ANR grants (ANR-10-INSB-04-01, ANR-11-INSB-0006, ANR-16-CONV- +0001), INSERM PC201508 and 18CP128-00. +Data availability +The data that support the findings of this study are available from the corresponding author upon +reasonable request. +Disclosure +The authors declare no conflict of interest. +References +1. +A. Bunaciu, S. Fleschin, and H. Aboul-Enein, “Infrared microspectroscopy applications - review,” Curr. Anal. Chem. +10, 132–139 (2013). +2. +K. A. Antonio and Z. D. 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Xie, eds., Coherent Raman Scattering Microscopy (Series in Cellular and Clinical Imaging) +(CRC Press, 2016). + diff --git a/3dE1T4oBgHgl3EQf5wXA/content/tmp_files/load_file.txt b/3dE1T4oBgHgl3EQf5wXA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d0bc0a884c6f3ef1ac9469687b031c9384b0caf0 --- /dev/null +++ b/3dE1T4oBgHgl3EQf5wXA/content/tmp_files/load_file.txt @@ -0,0 +1,567 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf,len=566 +page_content='Coherent Stokes Raman scattering microscopy (CSRS) SANDRO HEUKE1,* AND HERVÉ RIGNEAULT1,* 1Aix Marseille Univ, CNRS, Centrale Marseille, Turing Center for Living Systems, Institut Fresnel, Marseille, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Corresponding authors: Sandro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='Heuke@fresnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='fr & herve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='rigneault@fresnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='fr Abstract: We report the first implementation of laser scanning Coherent Stokes Raman scattering (CSRS - pronounced "sCiSsoRS") microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' To overcome the major challenge in CSRS imaging, we show how to suppress the fluorescence background by narrow bandpass filter and a lock-in based demodulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Near background free CSRS imaging of polymer beads, human skin, onion cells, avocado flesh and the wing disc of a drosphila larva are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Finally, we explain and demonstrate numerically that CSRS solves a major obstacle of other coherent Raman techniques by sending a significant part (up to 100%) of the CSRS photons into the backward direction under tight focusing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' We believe that this discovery will pave the way for numerous technological advances, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' in epi-detected coherent Raman multi-focus imaging, real-time laser scanning based spectroscopy or efficient endoscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' © 2023 Optica Publishing Group under the terms of the Optica Publishing Group Publishing Agreement 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Introduction Conventional bright-field microscopy provides information about the refractive index and absorption properties, but cannot elucidate the sample’s chemical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Infra-red absorption and linear Raman scattering retrieve the chemical fingerprint [1,2], but are incompatible with high spatial resolution or real-time imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Coherent Raman imaging (CRI) fills this technological gab joining a chemical bond specific contrast with signal levels that permit video-rate image acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Well established CRI microscopy techniques are the coherent anti-Stokes Raman scattering (CARS) [3, 4] and stimulated Raman scattering (SRS) [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' CARS owes its wide-range application to the blue-shifted anti-Stokes radiation which greatly facilitates its separation from linear fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' When working with near infra-red excitation wavelength, the blue-shifted CARS radiation is readily detected using photo-election multiplier tubes (PMT) of standard laser scanning microscopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' SRS’s popularity arises from the homodyne signal amplification that frees SRS images from an omnipresent non-resonant four-wave-mixing background and allows for measurements under daylight conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Overshadowed by CARS and SRS until now, there exists a 3rd four-wave-mixing process termed coherent Stokes Raman scattering (CSRS, "Scissors") [8–10] which is always appearing within any CARS or SRS experiment and provides near identical mapping of molecular oscillators [11] - see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' In analogy to the Stokes emission in linear Raman microscopy, the CSRS radiation (2𝜔𝑆 − 𝜔𝑝) is red-shifted with respect to the excitation frequencies of the pump (𝜔𝑝) and Stokes beams (𝜔𝑆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Surprisingly, CSRS was not yet implemented for laser scanning microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Presumably, this neglect must be attributed to the high degree of resemblance of CARS and CSRS spectra [11] rendering CSRS - prima facie - to be either CARS with an added fluorescence background when working with visible light sources or, using near infra-red (NIR) excitation, CARS with a radiation wavelength offside high quantum yields of common detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' CSRS provides, however, some unique properties that are of high interest for imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' (1) The CSRS spectrum differs from CARS in the presence of accessible electronic resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' For example, pre-resonant CSRS will offer complementary information in application to alkyne-labeled dyes [12] and standard dyes used in microbiology [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' (2) The red-shifted radiation of CSRS arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='03516v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='optics] 9 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Coherent Raman imaging techniques in energy diagrams, relative radiation wavelength and energy conservation under plane-wave illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' imaging becomes an advantage for UV or near-UV excitation where CARS photons [14] would be too far blue-shifted to be detected efficiently while any SRS image [15] is likely to be compromised by various artifacts such as multi-photon absorption [16,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Thus, UV excited CSRS holds the potential to achieve the highest possible spatial resolution (𝜆Stokes/[ √ 8𝑁 𝐴]) in coherent Raman imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' (3) NIR-excitation wavelength combined with CSRS may allow for deeper tissue imaging due to the reduced scattering and absorption of its radiation [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' (4) Last but most important: Due to a modified phase-matching geometry, CSRS microscopy can be configured to radiate more light into the backward direction which will add game-changing benefits for the investigation of thick samples, real-time spectroscopy, multi-focus imaging and endoscopy [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Within this contribution, we want to open up the field of laser scanning CSRS imaging by demonstrating CSRS microscopy within the visible excitation spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' To remove the major obstacle, we will show how linear fluorescence can be suppressed by a set of bandpass filter and nearly nullified in combination with a lock-in based detection scheme as a premise for near-UV excited CSRS imaging with a lateral resolution < 100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Furthermore, we shall investigate numerically CSRS’ spatial radiation behavior under NIR excitation paving the way towards CSRS experiments with an efficient epi-detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Experimental result and discussion The CSRS signal of biomedical samples is readily overwhelmed by linear fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Time- gating [20], a time-resolved detection using streak cameras [21] or polarization filtration can be used to reduce or suppress any fluorescence signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' These methods require, however, either a substantial alteration of standard coherent Raman microscopes or do not work in the presence of large quantities of fluorescence light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Here, we exploit the fact that the CSRS is spectrally narrow under ps-excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Thus, the majority of fluorescence is readily suppressed by the choice of a the narrow-band filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Filters with a spectral width below < 1nm are commercially available but the selection of a specific center wavelength requires expensive costume solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' This is the reason why we use a combination of two inexpensive bandpass filter with a width of about 15 nm, but different center wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' In a addition, we fine-tune the filter transmission by a tilt (<20◦) with respect to the incident beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Thus, two tilt-adjusted bandfilter create a sharp transmission line (FWHM<3nm) for the CSRS signal while rejecting significant parts of the autofluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' CARS SRS CSRS kas+ks=kp1+kp2 111 CARS ks1+kp1=ks2+kp2 SRS dwnd Stokes SRS kcs+k,=ks1+ks2 Intensity CARS CSRS CSRS 2Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' CSRS experimental implementation and characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Bottom left: The CSRS signal is separated from fluorescence by means of 2 angle-tuned narrow bandpass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Bottom right: Additional suppression of fluorescence is achieved by intensity modulating the Stokes and pump beam at the radio frequencies f1 and f2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Fluorescence free CSRS signal is obtained at f1-f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Right center: Time separation of the pump and Stokes pulses as well as blocking the excitation highlights the superior suppression of fluoresence background at the demodulation frequency f1-f2 compared to CSRS signal obtained at f1, f2 or the DC frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Top right: The intensity profile at the interface of a PMMA bead and olive oil indicates a lateral resolution of <400nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Top left: scheme of the CSRS experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 1 Yb-fiber laser, 2 optical parametric oscillator (OPO), 3 Second harmonic generation (SHG), 4 acousto-optic modulator (AOM), Laser scanning microscope (LSM), 6 photo-electron multiplier (PMT), 7 Lock-in amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' IcSRS PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' CSRS demodulated at 20μm DC, f1, f2, f1-f2 PMMA bead <400nm G 5 μm 88 0 2 ( 1 3 x / μm ④ f2 Lock-in DC f1 3 f2 f1-f2 lp=0 ④f1 Is=0 △t>>3ps T CSRS Intensity ump2 f1-f2 Fluorescence SRS concefiFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' LSM-CSRS at 2850 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The left and right column show the CSRS image demodulated at the frequencies f1-f2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='47 MHz and 0 Hz (DC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' To estimate the remaining fluorescence level, images without temporal overlap of the pump and Stokes pulses are displayed to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' a) Mixture of polystyrene (PS, 30µm) and Poly-methyl- methacrylate (PMMA, 20µm) beads in olive oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' b) and c) Epithelium and dermis of a 20µm thick human skin section d) Cells of an onion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' e) Lipid droplets within the flesh of an avocado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' d) Wing disc of a Drosophila larva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The white and blue scale bar equals 20µm and 5µm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' CSRS at f1-f2 △t >> 3ps CSRS at DC △t >> 3ps PS PMMA olive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='oil Human skin Epithelium Dermis Onion avocado Drosophila larva wing disc notum hinge pouchAs a second method for fluorescence discrimination, we take advantage of CSRS intensity dependence on both excitation colors while linear fluorescence follows either the intensity of the pump or the Stokes laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Consequently, modulating the pump and Stokes beams at f1 and f2 while demodulation the signal at f1-f2 (or f1+f2) yields exclusively nonlinear signals that depend on both excitation colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The f1-f2 demodulation, therefore, also discriminates the CSRS signal against 2-photon excited fluorescence (2PEF) under single-color excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' It shall be noted that the double modulation is also sensitive to two-color 2-photon fluorescence (2C-2PEF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Nevertheless, we will find experimentally, that the emission strength of native 2C-2PEF is negligible within our CSRS approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' For the experimental implementation of CSRS into laser scanning microscopy, we chose visible excitation wavelengths at 445nm (pump) and 515nm (Stokes) for the following reasons: (1) CSRS under near UV excitation is a potentially important application area since the CARS signal falls into the UV range while SRS artifacts are increased due the high concentration of matching chromophores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' (2) The red-shifted CSRS radiation is readily detected by ordinary PMTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' (3) Fluorescence artifacts are enhance compared to a near infra-red (NIR) excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Thus, our approach will be viable as well for CSRS under NIR excitation, if pure CSRS signals can be obtained under VIS excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The experimental implementation, the spectral filtration and the double modulation are schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Our implementation resembles a standard SRS setup with the difference that we use visible excitation wavelengths, we modulate not one but both beams and the photo-diode is replaced by a PMT which is connected to a lock-in amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' More information about the setup can be found within the part Methods: Experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' To quantify the level of fluorescence rejection, we investigated the signal of native olive oil at 2850 cm−1 when blocking the Stokes or pump beams and when the temporal pulse overlap is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The output signal of the lock-in is plotted as functions of the demodulation frequencies at 0 Hz (DC), f1, f2 and f1-f2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' It can be observed that the DC channel contains significant amounts of fluorescence while this artifact is already reduced within the channels f1 and f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Nevertheless, only the difference frequency channel at f1-f2 becomes dark, when the excitation pulses do not overlap in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' In a second experiment, we imaged the interface of olive oil and a 20µm sized Plexiglas (PMMA) bead to obtain an estimation of the lateral resolution for an excitation objective featuring an NA of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='45 - see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' From this "knife-edge" CSRS intensity profile, we can infer a lateral resolution below 400nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The difference to the expected 𝜆𝑆𝑡𝑜𝑘𝑒𝑠/[ √ 8𝑁 𝐴]= 515nm/[ √ 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='49]=120nm can be attributed to underfilling of the excitation objective lens and the bent oil/bead interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Having confirmed a high-resolved, fluorescence-free CSRS image contrast, we investigated the suitability of LSM-CSRS for vibrational imaging of various objects featuring non-negligible fluorescence levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Within Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 3, we show the CSRS images of test and biomedical samples demodulated at the DC and f1-f2 frequencies for (non-)overlapping pump and Stokes pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The images were organized along the ratio of the CSRS to fluorescence signal starting from the highest at the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Comparing the DC and f1-f2 images in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 3a, it obvious that a narrow spectral filtering is already sufficient for CSRS imaging of polymer beads in oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The first artifacts become visible for the DC CSRS images of the epithelium and dermis of a 20µm thick section of human skin - see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 3b and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' For the epithelium, a pronounced fluorescence artifact arises from melanin within the Stratum basale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Artifacts within the Dermis can be attributed to the auto-fluorescence of collagen and elastin [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The quantity of fluorescence observed within the DC channel increases stepwise further for CSRS imaging of onion cells, lipid droplets within the flesh of an avocado and the wing disc of a Drosophilia larva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' From the second row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 3, it is reconfirmed that almost no fluorescence is leaking into the f1-f2 CSRS channel as an important condition for the estimation of the true concentration of the targeted molecular group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The origin of fluorescence for these 3 samples, however, cannot be attributed with certainty, but might arise from NADH, flavins and chlorophyll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' In a broader context, we would like to point out that other nonlinear microscopy techniques would also greatly benefit from the narrow-band filter plus demodulation combination for rejection of spurious background signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' For example, the 2PEF signal of chlorophyll in plant leaves readily overwhelms any CARS or second harmonic generation (SHG) image contrast even under NIR excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' A double modulation of the excitation combined with a lock-in based demodulation will purify the signal, reduce the sensitivity against other light sources such as room light and reestablish the reliability of the following image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Having removed the why-not argument for the CSRS image contrast, we shall introduce in the next section a non-intuitive but game-changing argument for CSRS microscopy : the increased backwards radiation as the prerequisite of an effective epi-CSRS detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Numerical results In this section, we shall show and explain CSRS’ superior backward radiation properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Before entering into the calculations, we want to consider CSRS from a heuristic viewpoint investigating the momentum conservation laws for CSRS and compare it to CARS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Under plane illumination, the momentum conservation laws can be written as K = k𝑝 − k𝑆 + k𝑝 − k𝑎𝑆 for CARS [23] and K = k𝑆 − k𝑝 + k𝑆 − k𝑐𝑆 for CSRS with K, k𝑝, k𝑆, k𝑎𝑆 and k𝑐𝑆 representing the wavevectors of the object, the pump(probe) and Stokes beam as well as the anti-Stokes and coherent Stokes radiation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Note that for homogeneous samples these laws are also referred to as phase-matching condition and simplify to k𝑝 + k𝑝 = k𝑆 + k𝑎𝑆 (CARS) and k𝑆 + k𝑆 = k𝑝 + k𝑐𝑆 (CSRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Under focusing conditions, the single wavevectors are replaced by the distribution of incident wavevectors which are distributed over a cap of a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' To identify those object frequencies (K) that are effectively probed, every combination of excitation and emission wavevector must be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' This operation is equivalent to the convolution of the caps of the illumination and detection Ewald spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Neglecting polarization effects, the result of this convolution (simplified to 3 points per arc) is shown in 2D within Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Evidently, there exist no vector combination for epi-scattered CARS photons which would cover the origin K(0,0,0) of the object space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Thus, a homogeneous sample, such as olive oil, does not provide any backward radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' On the contrary, structures that feature high object frequencies, such as small polymer beads or layered materials, generate Epi-CARS radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' In the past, Epi-CARS was occasionally considered to be a size selective contrast that would highlight exclusively small objects [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' While this statement holds for the majority of biomedical samples, there do exist large structures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' multi-layered lipids in vesicles that also emit a strong CARS radiation into the backward direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Hence, it is more appropriate to refer to Epi-CARS as a technique that probes high object frequencies instead of been considered as size selective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Switching the detection wavelength to the red-shifted coherent Stokes radiation changes the covered object support significantly and includes now the origin at K(0,0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Due to the reduced size of the detection wavevector (|k𝑐𝑆| ≪ |k𝑎𝑆|) and the pump vector entering as complex conjugated, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 3, it is now possible to find vector combinations that cover the origin at K(0,0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Consequently, even a homogeneous object will radiate considerable amounts of Epi-CSRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Nevertheless, since the the centroid of the Epi-CSRS object support, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' the gray cloud within Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 4a, does not coincidence with the K-space origin, Epi-CSRS images will also highlight objects containing higher frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' To address the question of how to increase the ratio of Epi versus forward Epi-CSRS, and which object frequencies are most efficiently probed using Epi-CSRS, we performed finite element simulations whose results are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 4b-e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The equations implemented numerically as well as important parameters are found in the annex - numerical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' From the momentum conservation law and the vector diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 4a, it is readily comprehensible that a larger wavelength difference in between the pump and coherent Stokes wavelength relaxes greatly the necessity for extreme incident illumination angles of the Stokes beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Furthermore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Object frequency support and radiation behavior of CSRS versus CARS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' a) The object K-support for Epi-CSRS(CARS) is found by convolving the illumination Ewald spheres of the Stokes (pump), pump (Stokes), and Stokes (probe) with the cap of detection Ewald sphere at (anti-)Stokes frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Note that vector combinations covering the frequency of a homogeneous sample K(0,0,0) are only found for CSRS but not for CARS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' A single wavevector combination that phase-matches K(0,0,0) is highlighted to the left while a similar approach for CARS leads to a large phase-mismatch (ΔK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' b) CSRS and CARS radiation behavior of a homogeneous sample under standard illumination condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' the pump and Stokes beam fill the objective aperture homogeneously (𝜃𝑚𝑎𝑥=80◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' c) same as in b) but with an annular pupil filter applied to the Stokes beam for CSRS covering 50% of area of the objective back-aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' For an equitable comparison with CARS, the same pupil filter was applied to the pump beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' d) same as for b) but the homogeneous sample was replaced by a frequency object whose scatter density is described as 1 + cos(2𝜋𝑧/𝜆𝑜) and 𝜆𝑜=1µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' e) Plot of the ratio of backward/forward radiation (Rb/f) as a function of the object frequency 𝜆𝑜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Ks1 Ks2 kcs Kp (a) 个 Kz 个Kz 个Kz Epi Ks1 Epi-CSRS Ks2 Kcs 0 0 0 0 0 Kx 0 Kx Kp1 ks Kp2 kas Epi 个Kz 个 Kz 个 Kz Epi-CARS 0 Kp2 AK kas 0 0 o Kx 0 Kx 0 (b) (c) (d) (e) Rb/f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='79*10-2 Rb/f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='54 Rp/f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='25 % 30 ratio forward/backward in 5N IN z 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='5 20 CSRS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='5 10 5 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='5 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='5 x y y y Rb/f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='18*10-3 Rb/f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='33*10-2 z z z 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='5 4 CARS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='5 - 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='5 1 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='5 y y 2。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' / μm Stokes pump Stokes pump z z 50% "standard CRS" pupil filtering frequency objectsince most of the coherent Raman experiments apply NIR instead of VIS excitation wavelength, we used for within our simulations the wavelength 𝜆𝑝 = 797𝑛𝑚 and 𝜆𝑆 = 1030𝑛𝑚 which matches the most commonly targeted Raman shift in CRI imaging at 2850cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' For these conditions, the coherent Stokes radiation will be observed at 𝜆𝑐𝑆 = 1450𝑛𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' It shall be noted that our results equally apply for the visible excitation wavelength with gently higher excitation angle or thinner annular masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' To start with, we computed the radiation pattern of CSRS and CARS of a homogeneous object using an NA of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='49 (oil immersion) corresponding to a maximum illumination angle of 80◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 4b, it is evident that both CARS and CSRS are predominately forward directed though the CSRS’ radiation distribution features a larger radiation cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Considering the ratio of backward versus forward directed photons Rb/ 𝑓 , we find numerically that less than 1 photon in 105 is backward directed for CARS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Note that the momentum conservation actually law predicts Rb/ 𝑓 =0 for CARS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Thus, the resulting deviation must be attributed to the finite number of voxels of the numerical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' For CSRS, Rb/ 𝑓 increase dramatically to about 1 in 100 photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Since common surfaces within biomedical samples scatter more than 1%, we have to assume, however, that also epi detected CSRS will be just forward generated CSRS that was redirected by linear scattering at an interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Still, using a confocal detection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' a pinhole in front of the detector placed at the conjugated plane of the excitation focus, might already yield true Epi-CSRS images of homogeneous samples where Epi-CARS images would remain dark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' To find an approach that increases the proportion of CSRS’ epi radiation, we shall consider the CSRS vector diagram matching K(0,0,0) on the left of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The ratio of backward versus forward radiation is readily increased by reducing the impact of vectors combinations probing higher frequencies and favoring those covering the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' This boost of epi-CSRS radiation can be achieved using an annular illumination of the Stokes beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Experimentally, such an annular illumination is generated, without power-loss, using 2 axicons within the Stokes beam path [25,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Numerically, we restricted the incident angles for the Stokes between 𝜃𝑚𝑖𝑛=56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='5◦ and 𝜃𝑚𝑎𝑥=80◦, which corresponds to covering 50% of the area of the objective lens’ back-focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' With this pupil filtering, the radio of backward to forward radiation increased for CARS to 2 in 104 photons while the majority of all CSRS radiation is backward directed (Rb/ 𝑓 =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='5) when focusing the pump and Stokes beam into a homogeneous object - see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' As a second important result from the heuristic derivation of CSRS’ object support, we found that the presence of high object frequencies increases the amount of backward radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' To confirm this prediction, we investigated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 4d and e an object whose nonlinear scatterer density, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' concentration of molecular groups, is modulated along the optical axis as 1+cos(𝐾𝑧𝑧) with 𝐾𝑧 = 2𝜋/𝜆𝑜 being the object frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' As an example, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 4d outlines the radiation behavior of a wave-like structured object with K𝑧=2𝜋/1µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' It is found that R𝑏/ 𝑓 increases to one forth for Epi-CSRS while Epi-CARS remains negligible weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' To identify those object frequencies which are most efficiently probed by Epi-CSRS, we computed R𝑏/ 𝑓 as a function of K𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 4e, we find that Epi-CSRS peaks at K𝑧=2𝜋/1µm whereas Epi-CARS R𝑏/ 𝑓 still increases at 𝐾𝑧 = 2𝜋/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='25µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' It shall be note that the Rb/ 𝑓 never reaches 1 which arise from the 1+ within the definition of the frequency object (1+cos(𝐾𝑧𝑧)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The 1+ implies that the wave-object always features twice the amplitude at K(0,0,0), which corresponds to a homogeneous predominantly forward scattering object, compared to the scatterer density modulation K(0,0,±K𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Our simulation results in a nutshell: we have found that CSRS features a non-negligible backward radiation from a homogenous sample under tight-focusing conditions while this is not the case for CARS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The amount of backward radiated CSRS can be enhanced by increasing the illumination power of the Stokes beam with high incident angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Furthermore, the natural structure of biomedical samples, which are usually not homogeneous, will also elevate the CSRS backward radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Conclusion We have demonstrated the first laser scanning microscopy CSRS experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' As the major challenge, we were able to reduce the fluorescence background significantly using a pair of tilted bandpass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The remaining fluorescence contribution is removed by intensity modulating the Stokes and pump beams at the radio frequencies f1 and f2 and a lock-in based demodulation of the CSRS signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Taking advantage of CSRS’ characteristic dependence on both excitation colors, the best fluoresence background suppression is obtained when demodulating the CSRS signal at f1-f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Background-free LSM-CSRS imaging was demonstrated for samples of decreasing ratio of CSRS to fluoresence signal, namely: polymer beads, the epithelium and dermis of human skin, onion cells, avocado flesh and the wing disc of a Drosophila larva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Having removed the major obstacle for CSRS imaging, we introduced and quantified numerically the major interest of CSRS which is its unique backward radiation property in combination with high NA objective lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' CSRS’ backward radiation and its distinction from CARS is readily understood from the momentum conservation laws when considering all incident k-vectors forming the excitation focal spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Using dynadic Green functions, we show numerically that the CSRS is predominantly forward directed for a homogeneous object, but the backward CSRS contribution rises to 1/4 for objects that are structured axially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Moreover, backward CSRS signal can even dominate forward CSRS (up to 100%) if an annular Stokes illumination is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' With an efficient Epi-CARS radiation at hand, various coherent Raman experiments become feasible which were impossible before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Just to name a few: Epi-detected confocal multi-focus CSRS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Epi-detected LSM-CSRS with a spectrometer at the descanned position;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Epi-detected CSRS image scanning microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Thus, we believe that this contribution is just the first milestone in CSRS microscopy with many others to follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Methods: Experimental setup A Yb-based fiber laser (APE Emerald engine, 80 MHz, 2–3 ps) is frequency doubled yielding 7 W of 515 nm output power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Parts of the emissions is used directly as Stokes beam to drive the CSRS process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The major part (4 W) of the 515 nm is employed to pump an optical parametric oscillator (OPO, APE Emerald).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The OPO’s signal beam is tunable to 660-950 nm and coupled into an external SHG unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The latter generates up to 50 mW within the spectral range of 330-475 nm serving as pump beam for the CSRS four wave mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Thus, the 330-475 nm pump combined with the 515 nm Stokes beam allows addressing a Raman shift range from 1630-11000cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The pump and Stokes beams are superimposed in space and time via a dichroic beam splitter (Semrock, FF470-Di01-25x36) and a delay stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Both beams are coupled into a home-built laser scanning microscope and focused by a 40x water objective lens (Nikon, Plan, NA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='15, immersion: water) into the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The excitation objective lens was replaced for a 60x objective (Nikon, Plan Apo TIRF, NA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='45, immersion:oil) to generate the bead-oil interface image within Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The CSRS radiation is collected by a condenser lens (Nikon, Achr-Apl, NA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='4) in forward direction, spectrally separated from the broadband fluorescence background by means of 2 tilted bandpass filter (Semrock FF01-620/14-25 + FF01-605/15-25) and detected by a photo-electron multiplier (PMT, Thorlabs, PMT1001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' For an enhanced suppression of the linear fluorescence background, 2 acousto-optic modulators (AOM, AA, MT200-A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='5-VIS) were applied to modulate the intensity of the Stokes and pump beams and at the frequencies f1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='28 MHz and f2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='75 MHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The PMT output was demodulated simultaneously at the DC frequency, f1, f2 and at f1-f2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='47 MHz using a lock-in amplifier (Zürich instruments, HF2LI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The lock-in time constant was set to 30 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' All CSRS-images shown were recorded with a pixel dwell time of 40 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Annex - numerical calculation In the following, we shall summarize the equations used to generate Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 4b-e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The meaning of the variables is summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The focused field at the sample is given by the angular spectrum representation [27]: ��������� 𝐸𝑥(𝜌, 𝜙, 𝑧) 𝐸𝑦(𝜌, 𝜙, 𝑧) 𝐸𝑧(𝜌, 𝜙, 𝑧) ��������� = 𝑖𝑘 𝑓 2 exp(−𝑖𝑘 𝑓 ) ��������� 𝐼00 + 𝐼02 cos(2𝜙) 𝐼02 sin(2𝜙) −𝑖2𝐼01 cos(𝜙) ��������� (1) Here 𝑓 denotes the focal length of the objective lens and the integrals 𝐼0𝑚 are provided by 𝐼0𝑚 = ∫ 𝜃𝑚𝑎𝑥 𝜃𝑚𝑖𝑛 𝐸𝑖𝑛𝑐(𝜃) sin(𝜃)[cos(𝜃)]1/2𝑔𝑚(𝜃)Jm[𝑘𝜌 sin(𝜃)]d𝜃 (2) where 𝑔𝑚 equals 1 + cos(𝜃), sin(𝜃) and 1 − cos(𝜃) for 𝑚 = 0, 1, 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 𝐽𝑚 is the 𝑚𝑡ℎ order Bessel function while 𝐸𝑖𝑛𝑐 is the incoming electric field which we assumed to be x-polarized and constant within the (annular) aperture angles 𝜃𝑚𝑖𝑛 ≤ 𝜃 ≤ 𝜃𝑚𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The nonlinear polarization at anti-Stokes and coherent Stokes wavelength is given by: 𝑃(3) 𝑎𝑆,𝑎(𝑟) = 3𝜒(3) 𝑎𝑏𝑐𝑑(𝑟)𝐸 𝑝,𝑏𝐸∗ 𝑆,𝑐𝐸 𝑝,𝑑 𝑃(3) 𝑐𝑆,𝑎(𝑟) = 3𝜒(3) 𝑎𝑏𝑐𝑑(𝑟)𝐸𝑆,𝑏𝐸∗ 𝑝,𝑐𝐸𝑆,𝑑 (3) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Declaration of variables Where a,b,c,d represent the polarization coordinates x, y or z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Using an x-polarized ex- citation, it was noticed that 𝜒(3) 𝑥𝑥𝑥𝑥 dominates all other tensor components even under tight focusing conditions while filling the objective lens homogeneously [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Nevertheless, for the generation of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' 4c an annular mask with 𝜃𝑚𝑖𝑛 = 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='5◦ and 𝜃𝑚𝑎𝑥 = 80◦ was applied which does necessitate the inclusion of other tensor elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' For simplicity, we consider here only isotropic samples reducing the 81 susceptibility tensor elements to 21 which are nonzero [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Within isotropic media, these nonzero elements follow certain sym- metry rules which are, 𝜒1111 = 𝜒2222 = 𝜒3333, 𝜒1122 = 𝜒1133 = 𝜒2211 = 𝜒2233 = 𝜒3311 = 𝜒3322, 𝜒1212 = 𝜒1313 = 𝜒2323 = 𝜒2121 = 𝜒3131 = 𝜒3232, 𝜒1221 = 𝜒1331 = 𝜒2112 = 𝜒2332 = 𝜒3113 = 𝜒3223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Further, it applies 𝜒1111 = 𝜒1122 + 𝜒1212 + 𝜒1221 [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Within our simulations we were setting 𝜒1122 = 𝜒1212 = 𝜒1221 = 1 and, hence, 𝜒1111 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The nonlinear far-field radiation distributions is obtained using a dyadic Green function approach: ��������� 𝐸𝑞,𝑅(𝑅, Θ, Φ) 𝐸𝑞,Θ(𝑅, Θ, Φ) 𝐸𝑞,Φ(𝑅, Θ, Φ) ��������� = − 𝜔2 𝑞 𝑐2 exp(𝑖𝑘𝑞|𝑅|) |𝑅| ∭ ∞ −∞ 𝜌d𝜌d𝜙d𝑧 exp(𝑖𝑘𝑞rR) |𝑅| × ��������� 0 0 0 cos(Θ) cos(Φ) cos(Θ) sin(Φ) − sin(Θ) − sin(Φ) cos(Φ) 0 ��������� ��������� 𝑃(3) 𝑞,𝑥(r) 𝑃(3) 𝑞,𝑦(r) 𝑃(3) 𝑞,𝑧(r) ��������� (4) where q is replaced by aS or cS to calculate either the anti-Stokes or coherent Stokes radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Within the simulations, we segmented the focal area into (121×121×121≈) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='77 mio elements of a width of 50 nm equally spaced into the x, y and z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The far-field radiation sphere was discretized into (ΔΘ=1◦, ΔΦ=2◦) 32400 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' The coherent (anti-)Stokes radiation was qualified as either forward or backward directed if falling into the range Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='. 0-80◦ or Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='. 100-180◦, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content=' Funding Information We acknowledge financial support from the Centre National de la Recherche Scientifique (CNRS), Aix-Marseille University (A-M-AAP-ID-17-13-170228-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQf5wXA/content/2301.03516v1.pdf'} +page_content='22-RIGNEAULT), A*Midex (ANR- 11-IDEX-0001-02), Cancéropôle Provence-Alpes Côte d’Azur, French National Cancer institute (INCa), Région Sud, ANR grants (ANR-10-INSB-04-01, ANR-11-INSB-0006, ANR-16-CONV- 0001), INSERM PC201508 and 18CP128-00.' metadata={'source': 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+FOURIER–MUKAI TRANSFORMS FOR NON-COMMUTATIVE COMPLEX +TORI +NOBUKI OKUDA +Abstract. Let X be a complex torus of dimension g and ˆX be the dual torus. For any g(g − 1)/2- +tuple λ of complex numbers of absolute value 1, we define a non-commutative complex torus Xλ as +a sheaf of algebras on a real torus of dimension g. We prove that if all components of λ are roots of +unity, then the category of coherent sheaves on Xλ is abelian and derived-equivalent to the category +of coherent sheaves on ˆX twisted by an element of the Brauer group of ˆX determined by λ. +1. Introduction +Let X and ˆX be smooth projective varieties. An integral functor +Φ: Db(X) → Db( ˆX) +(1.1) +between derived categories of coherent sheaves is said to be a Fourier–Mukai functor if it is an +equivalence. A Fourier–Mukai functor Φ induces an isomorphism +HH(Φ): HH•(X) → HH•( ˆX) +(1.2) +of Hochschild cohomologies. For any u ∈ HH2(X), Toda [Tod09] gave a C[ε]/(ε2)-linear category +Db(X, u) of first order deformations of X along u and an equivalence +Φ† : Db(X, u) → Db( ˆX, HH(Φ)(u)) +(1.3) +extending Φ. +The integral functor FM with the Poincar´e line bundle as the integral kernel is the first example +of a Fourier–Mukai functor given by Mukai [Muk81] for an abelian variety X and the dual abelian +variety ˆX. Under the Hochschild–Kostant–Rosenberg isomorphism +HH2(X) ∼= H0(∧2TX) ⊕ H1(TX) ⊕ H2(OX), +(1.4) +the induced map HH(FM) sends H0(∧2TX) to H2(O ˆ +X). This suggests that non-commutative de- +formations of abelian varieties are Fourier–Mukai partners of gerby deformations of dual abelian +varieties. +While the notion of gerby deformations is well-established in terms of twisted sheaves, non- +commutative deformations are much harder to define in general. At the formal level, the Moyal +deformation quantizations give formal non-commutative deformations of complex tori, and derived +equivalences to formal gerby deformations are proved in [BBBP07]. The next problem is to construct +non-commutative tori and derived equivalences for non-formal parameters (or even as a family over +a complex manifold). +The first attempt to define non-commutative complex tori is given by Schwarz [Sch01], who intro- +duced the notion of complex structures on non-commutative tori, which are irrational rotation alge- +bras (see [Rie81], for example) regarded as non-commutative spaces in the sense of Connes[Con94]. +Categories of holomorphic vector bundles on them are studied in [PS03]. +In the paper [Blo10] and the preprint [Blo], Block discussed Fourier–Mukai functors between non- +commutative complex tori and gerby deformations of dual complex tori, using DG categories which +include objects corresponding to quasi-coherent sheaves. +In [Blo], he also announced to discuss +coherent sheaves in a paper in preparation. +We now explain the results of this paper. Let X = T/Γ be a complex torus, where T := (C∗)g +and Γ is a discrete subgroup of T isomorphic to Zg. The dual complex torus ˆX := Pic0 X can +naturally be identified with ˆΓ/ ˆT where ˆΓ := Hom(Γ, C∗) ∼= (C∗)g and ˆT := Hom(T, C∗) ∼= Zg. Let +λ ∈ H2( ˆT, U(1)) ∼= U(1)g(g−1)/2 be an element of the second group cohomology of ˆT with values in +1 + +U(1). We construct a non-commutative deformation of X with parameter λ. When λ takes values +in roots of unity, we give an equivalence of the derived category of coherent sheaves on the non- +commutative deformation of X with parameter λ and the derived category of coherent sheaves on +the gerby deformation of ˆX with the same parameter λ. +Our construction of non-commutative complex tori is different from those in [Sch01], [PS03] and +[Blo]. +Ours can be regarded as a patching of Archimedean analog of quantum analytic tori in +[Soi09]. To prove the equivalence of derived categories, we use the idea of equivariant Fourier–Mukai +transforms developed in [Sos12]. +This paper is organized as follows: In Section 2, we discuss q-Weyl algebras as toy models to +motivate constructions in later sections. In Section 3, we recall Fourier–Mukai transforms for complex +tori. +In Section 4, we define non-commutative complex tori by deforming sheaves of convergent +Laurent series rings on real tori. It can be regarded as an Archimedean analog of the construction +of quantum analytic tori in [Soi09]. In Section 5, we discuss a dual pair X → Y and ˆY → ˆX of +finite coverings of tori associated with a deformation parameter λ with values in roots of unity. In +Section 6, we describe non-commutative complex tori whose deformation parameters take values in +roots of unity in terms of a finite sheaf of algebras. In Section 7, we collect basic definitions on +twisted sheaves on complex manifolds. In Section 8, we introduce Fourier–Mukai transforms from +gerby complex tori to non-commutative complex tori at roots of unity and state Theorem 8.1, which +is the main result in this paper. In Section 9, we recall basic definitions and results on finite group +actions on abelian and derived categories, and discuss twistings by group cocycles. In Section 10, we +discuss group actions on categories appearing in our construction. In Section 11, we prove Theorem +8.1. +Notations and conventions. We fix the complex number field C as the ground field. All modules +(resp. actions) are right modules (resp. actions) unless otherwise specified. In contrast, all group ac- +tions on categories are left actions. The word ‘non-commutative’ is synonymous with ‘not necessarily +commutative’. +For a ringed space X = (Z, OX), we write the ringed space (Z, OX op := (OX)op) as X op. The +category of OX-modules will be denoted by Mod X . For an element a of a complex abelian Lie group +A, we write the right translation map A → A, x �→ xa as Ra. For two or three complex manifolds +Z1, Z2 or Z1, Z2, Z3, the projection to the first (resp. second) component Z1 (resp. Z2) will be denoted +by pZ1,Z2 or pZ1,Z2,Z3 (resp. qZ1,Z2 or qZ1,Z2,Z3). +Acknowledgment. The author is deeply grateful to his advisor, Kazushi Ueda, for a lot of guidance, +useful comments, and encouragement. +The author also thanks Masahiro Futaki for many useful +questions, one of which lead to the formula (6.9). Finally, the author has deep gratitude to his +parents for their various supports throughout his life. +2. q-Weyl algebras +The category of OX-modules is equivalent to the category of Γ-equivariant OT -modules, and the +category of Γ-equivariant C[T]-modules can be regarded as a toy model for it. The latter can be +identified with the category of modules over the crossed product algebra C[T] ⋊ Γ. +For a parameter λ = (λi,j)1≤i0)g, +(x1, x2, . . . , xg) �→ (|x1|, |x2|, . . . , |xg|). +(4.1) +For a product D = �g +i=1(ri, Ri) of open intervals, the ring ̟∗OT (D) consists of Laurent series in g +variables with radii of convergence (ri, Ri) for 1 ≤ i ≤ g. +Definition 4.1. A unitary deformation parameter is an element of Z2( ˆT, U(1)), i.e., a map λ: ˆT × +ˆT → U(1) satisfying +λ(t2, t3)λ(t1t2, t3)−1λ(t1, t2t3)λ(t1, t2)−1 = 1 +(4.2) +for all t1, t2, t3 ∈ ˆT. +Given a unitary deformation parameter λ, the star product ∗λ on ̟∗OT (D) is defined by + +� +t∈ ˆT +att + + ∗λ + +� +t∈ ˆT +btt + + = +� +t∈ ˆT + + +� +t1,t2∈ ˆT, t1t2=t +λ(t1, t2)at1bt2 + + t, +(4.3) +which is easily seen to be associative by using (4.2). The convergence of the right hand side follows +from +������ +� +t1,t2∈ ˆT, t1t2=t +λ(t1, t2)at1bt2 +������ +≤ +� +t1,t2∈ ˆT, t1t2=t +|at1||bt2| +(4.4) +and +� +t∈ ˆT + + +� +t1,t2∈ ˆT,t1t2=t +|at1||bt2| + + t = + +� +t∈ ˆT +|at|t + + + +� +t∈ ˆT +|bt|t + + , +(4.5) +4 + +which depends on the unitarity of λ. The resulting sheaf of associative algebras on |T| will be denoted +by OTλ which turns |T| into a non-commutative ringed space Tλ := (|T|, OTλ). +A cochain α ∈ Z1( ˆT, U(1)) bounding λ, λ′ ∈ Z2( ˆT, U(1)) is a map α: ˆT → U(1) satisfying +λ′(t1, t2) = λ(t1, t2)α(t1)α(t2)α(t1t2)−1. +(4.6) +It gives an isomorphism +� +t∈ ˆT +att �→ +� +t∈ ˆT +α(t)att +(4.7) +of the ring of sections, which ensures that the isomorphism class of the sheaf OTλ of associative +algebras depends only on the cohomology class [λ] ∈ H2( ˆT, U(1)). +The natural T-action on T induces a T-action on |T|, which lifts to a T-action on the ringed space +Tλ in such a way that the morphism ρa : OTλ → (R̟(a)−1)∗OTλ of sheaves of associative algebras for +a ∈ T is given by +� +t∈ ˆT +att �→ +� +t∈ ˆT +att(a)−1t. +(4.8) +The action of Γ on |T| is free since ̟(γ) = 1 for 1 ̸= γ ∈ Γ contradicts the freeness or the properness +of Γ-action on T. +Definition 4.2. The non-commutative complex torus associated with a complex torus X = T/Γ and +a unitary deformation parameter λ ∈ Z2( ˆT, U(1)) is the non-commutative ringed space Xλ := Tλ/Γ. +Recall that a sheaf M of OX-modules on a ringed space X = (X, OX) is said to be coherent if +(1) M is finitely generated, i.e., for any point x ∈ X, there exists an open neighborhood U of x +and an epimorphism O⊕m +X |U → M|U → 0 for some m ∈ N, and +(2) for any open set U and any m ∈ N, the kernel of any morphism O⊕m +X |U → M|U is finitely +generated. +The full subcategory of Mod X consisting of coherent modules will be denoted by coh X . +Lemma 4.3. The sheaf OT1 is coherent. +Proof. It is clear that OT1 is finitely generated. For any open subset U of |T|, a morphism α: O⊕m +T1 |U → +OT1|U is the same as a morphism ˜α: O⊕m +T +|̟−1(U) → OT|̟−1(U). For any x ∈ U, let V ⊂ U be an +open neighborhood of x obtained as the product of intervals. Then ̟−1(V ) is Stein and hence there +is an epimorphism ˜β from O⊕m′ +̟−1(V ) to the kernel of ˜α|̟−1(V ) for some integer m′. We ensure that ̟∗ +is exact (and hence the morphism β := ̟∗ ˜β is also an epimorphism) by applying [Tay02, Corollary +11.5.4] for every fiber of ̟. This shows that ker α is finitely generated, and Lemma 4.3 is proved. +□ +Non-commutative complex tori at λ = 1 are usual complex tori: +Propositon 4.4. The adjunction ̟∗ ⊣ ̟∗ induces an equivalence coh T ≃ coh T1. +Proof. Oka coherence theorem implies that an object of Mod T is coherent if and only if it is finitely +presented. Similarly, Lemma 4.3 implies that an object of Mod T1 is coherent if and only if it is +finitely presented. +The functors ̟∗ and ̟∗ induce mutually inverse functors on categories of finitely presented mod- +ules since +(1) the functor ̟∗ is exact (and hence preserves cokernels in particular), +(2) the functor ̟∗ preserves cokernels since it is a left adjoint, and +(3) ̟∗ and ̟∗ interchanges OT and OT1. +□ +Corollary 4.5. The adjunction (̟X)∗ ⊣ (̟X)∗ induces an equivalence coh X ≃ coh X1, where +̟X : X → |X| is the map induced by ̟. +The category coh OXλ is abelian if Problem 4.6 below has an affirmative answer: +5 + +Problem 4.6 (Oka coherence for non-commutative tori). Is OTλ coherent? +Yet another problem is a generalization to non-unitary deformation parameters, which would be +needed for the duality with general gerby deformations. +5. Deformation parameters at roots of unity +As one can see (e.g. by noting that ˆT-modules are equivalent to Z[T] ∼= Z[t±1 +1 , t± +2 , . . . , t± +g ]-modules, +and the trivial ˆT-modules Z has the Koszul resolution associated to t1 −1, t2 −1, . . . , tg −1) that the +cohomology Hi( ˆT, A) with coefficients in an abelian group A with the trivial ˆT-action is isomorphic to +Hom(∧i ˆT/(∧i ˆT)tors, A) ∼= A(g +i) (note that (∧i ˆT)tors is generated by torsion elements a∧a∧t1∧· · ·∧ti−2 +(a, t1, . . . , ti−2 ∈ ˆT) of order 2). Let Λ ∈ Hom(∧2 ˆT, U(1)) be the element corresponding to the class +[λ] ∈ H2( ˆT, U(1)) of a unitary deformation parameter λ ∈ Z2( ˆT, U(1)), and set +ˆH := +� +t ∈ ˆT +��� Λ(t ∧ t′) = 1 for all t′ ∈ ˆT +� +, +(5.1) +so that one has an exact sequence +1 → ˆH → ˆT → ˆK → 1, +(5.2) +and [λ] ∈ H2( ˆT, U(1)) descends to an element of H2( ˆK, U(1)), which can be represented by a bilinear +cochain in Z2( ˆK, U(1)). +For the rest of this paper and unless otherwise specified, we will assume that a unitary deformation +parameter λ ∈ Z2( ˆT, U(1)) is contained in Z2( ˆT, µN) for some positive integer N where µN := +� +ζ ∈ U(1) +�� ζN = 1 +� +. This implies that ˆK is a finite abelian group, and we will also fix a bilinear +map +λ: ˆK ⊗Z ˆK → µN +(5.3) +representing [λ]. +The dual group K := Hom( ˆK, C∗) can be identified with the kernel of the map T → H dual to +the inclusion ˆH → ˆT, so that one has an exact sequence +1 → K → T +πT +−→ H → 1. +(5.4) +Since Γ∩K is the trivial group, the free action of K on T descends to a free action of K on X := T/Γ, +so that one has an exact sequence +1 → K → X +π−→ Y → 1 +(5.5) +where Y is a complex torus. We also have an exact sequence +1 → ˆK → ˆY +ˆπ−→ ˆX → 1 +(5.6) +Since K goes to the identity under the homomorphism ̟: T → |T|, there exists ̟Y : Y → |X| +making the diagram +X +Y +|X| +� +π +�❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +̟X +�✤✤✤✤✤✤✤ +̟Y +(5.7) +commute. +6 + +6. Non-commutative deformations at roots of unity +We have an isomorphism +π∗OX ∼= +� +ˆk∈ ˆ +K +Lˆk +(6.1) +of sheaves of OY -algebras on ˆY := Pic0 Y . The summand Lˆk is locally generated by a monomial +function t ∈ ˆT representing ˆk. We define a star product ⋆λ on π∗OX by +⋆λ: +� +ˆk∈ ˆ +K +Lˆk × +� +ˆk∈ ˆ +K +Lˆk → +� +ˆk∈ ˆ +K +Lˆk +(6.2) +((φˆk)ˆk, (ψˆk)ˆk) �→ + + � +ˆk1ˆk2=ˆk +λ(ˆk1, ˆk2)mˆk1,ˆk2(φˆk1 ⊗ ψˆk2) + + +ˆk +. +(6.3) +We write the resulting sheaf (π∗OX, ⋆λ) of OY -algebras as OXλ, and the ringed space (Y, OXλ) as +Xλ. +Propositon 6.1. One has an isomorphism (̟Y )∗OXλ ∼= OXλ of sheaves of algebras. +Proof. A direct calculation shows that + +� +t1∈ ˆT +at1t1 + + ∗λ + +� +t2∈ ˆT +bt2t2 + + +(6.4) += + + � +t∈ ˆT/ ˆ +H + + +� +t1 mod ˆ +H=t +at1t1 + + + + ∗λ + + � +t′∈ ˆT/ ˆ +H + + +� +t2 mod ˆ +H=t′ +bt2t2 + + + + +(6.5) += +� +t∈ ˆT/ ˆ +H + + � +t′∈ ˆT/ ˆ +H + + +� +t1 mod ˆH=t + + +� +t2 mod ˆ +H=t′ +λ(t1, t2)at1bt2t1t2 + + + + + + +(6.6) += +� +t∈ ˆT/ ˆ +H + + � +t′∈ ˆT/ ˆ +H + + +� +t1 mod ˆH=t + + +� +t2 mod ˆ +H=t′ +λ(t, t′)at1bt2t1t2 + + + + + + +(6.7) += +� +t∈ ˆT/ ˆ +H + + � +t′∈ ˆT/ ˆ +H +λ(t, t′)mt,t′ + + +� +t1 mod ˆ +H=t +at1t1, +� +t2 mod ˆ +H=t′ +bt2t2 + + + + +(6.8) +coincide with ⋆λ. +□ +Propositon 6.2. The adjunction (̟Y )∗ ⊣ (̟Y )∗ induces an equivalence coh Xλ ≃ coh Xλ. +Proof. We write the full subcategory of Mod Xλ (resp. Mod Xλ) consisting of coherent OY -modules +(resp. OY1-modules) as (coh Y ) ˆ +K,λ (resp. (coh Y1) ˆ +K,λ). Since OXλ is a finite OY -algebra by defi- +nition and OXλ is a finite OY1-algebra by Proposition 6.1, the category coh Xλ (resp. coh Xλ) is +equivalent to (coh Y ) ˆ +K,λ (resp. (coh Y1) ˆ +K,λ). It follows from Corollary 4.5 and Proposition 6.1 that +OXλ ∈ ob(coh Y1) and the adjunction (̟Y )∗ ⊣ (̟Y )∗ induces an equivalence (coh Y ) ˆ +K,λ ≃ (coh Y1) ˆ +K,λ. +□ +In particular, OXλ is a coherent sheaf on Xλ and hence coh OXλ is an abelian category. +Remark 6.3. It follows from the definition of ˆK that there exists an isomorphism λ♯ : ˆK → K such +that λ(ˆk1, ˆk2) = ˆk2(λ♯(ˆk1)). If we write the inverse of λ♯ as λ♭ and define ω : K ⊗Z K → U(1) by +ω(k1, k2) = λ(λ♭(k1), λ♭(k2)), then the star product on OXλ can alternatively be described as +φ(x) ⋆λ ψ(x) = 1 +♯K +� +k1,k2∈K +ω(k1, k2)φ(xk1)ψ(xk2). +(6.9) +7 + +A coherent sheaf on Xλ consists of an OY -module M and a collection of morphisms {mˆk : M⊗Lˆk → +M}ˆk∈ ˆ +K such that the diagram +M ⊗ Lˆk1 ⊗ Lˆk2 +M ⊗ Lˆk2 +M ⊗ Lˆk1ˆk2 +M +� +mˆk1⊗id +� +λ(ˆk1,ˆk2) id ⊗mˆk1ˆk2 +� +mˆk2 +� +mˆk1ˆk2 +(6.10) +commutes. The dual OY -module M−1 := HomOY (M, OY ) equipped with the transposes of mt gives +a coherent sheaf on Xop +λ . +7. Twisted sheaves +Let M be a complex manifold and α ∈ H2(M, O∗ +M) be a second ´etale cohomology class of O∗ +M. +Take an ´etale covering U = (Ui)i and a representative (αi,j,k) of α on U. We write the projections +from Ui ×M Uj to the first (resp. second) component as Ii,j (resp. Ji,j). +Definition 7.1. An α-twisted sheaf on M is a collection ((Fi)i, (ρi,j)i,j) of OUi-modules Fi and +isomorphisms ρi,j : I∗ +i,jFi → J∗ +i,jFj satisfying ρ−1 +i,k ◦ ρj,k ◦ ρi,j = αi,j,k id . An α-twisted sheaf is coherent +if all Fi are coherent. +The category of α-twisted sheaves on M and the full subcategory consisting of α-twisted coherent +sheaves will be denoted by Mod Mα and coh Mα respectively. Note that for an OM-algebra A and the +resulting ringed space X := (M, A), we similarly can define the notion of α-twisted sheaves on X and +categories Mod X α, coh X α. They do not depend on the choice of U and (αi,j,k) up to equivalence. +One has the tensor product functor +⊗: Mod Mα × Mod Mα′ → Mod Mαα′, +(7.1) +� +((Fi)i, (ρi,j)i,j), ((F ′ +i)i, (ρ′ +i,j)i,j) +� +�→ ((Fi ⊗ F ′ +i)i, (ρi,j ⊗ ρ′ +i,j)i,j) +(7.2) +and the duality functor +(−)−1 : Mod Mα → Mod Mα−1, +(7.3) +((Fi)i, (ρi,j)i,j) �→ ((HomOUi(Fi, OUi))i, ((ρ−1 +i,j )∗)i,j). +(7.4) +If U consists of a principal G-bundle P on M for some discrete group G, then the isomorphism +P × Gp → P ×M P ×M · · · ×M P, +(y, g1, . . . , gp) �→ (y, yg1, yg1g2, . . . , yg1 · · · gp) +(7.5) +induces an isomorphism from the ˇCech complex C•(U, O∗ +M) to the standard complex C•(G, O∗ +P(P)) +for group cohomology. +The composite of the resulting map H2(G, O∗ +P(P)) → H2(M, O∗ +M) with +the map H2(G, C∗) → H2(G, O∗ +P(P)) will be denoted by ιP : H2(G, C∗) → H2(M, O∗ +M). For any +λ ∈ H2(G, C∗), an ιP(λ)-twisted sheaf will simply be called a λ-twisted sheaf. If G is a finite abelian +group, then ιP(λ) is a torsion element since any group cohomology of finite abelian group is torsion. +In other words, ιP(λ) is an element of the cohomological Brauer group Br(M) := H2(M, O∗ +M)tors. A +λ-twisted sheaf consists of an OP-module F and a λ-twisted G-linearization of F, i.e., a collection +(ρg)g∈G of morphisms ρg : F → R∗ +gF satisfying R∗ +g1ρg2 ◦ ρg1 = λ(g1, g2)ρg1g2. +A λ-twisted sheaf (F, (ρg)g∈G) will also be called a λ-twisted G-equivariant OP-module; it reduces +to a G-equivariant OP-module if λ = 1 (which in turn is equivalent to an OM-module). +8. Deformed Fourier–Mukai transforms +Let Q be the Poincar´e line bundle on Y × ˆY . For a complex manifold Z, a sheaf of associative +algebras (pY,Z)∗OXλ (resp. (pY,Z)∗Oop +Xλ) will denoted by OXλ×Z (resp. OXop +λ ×Z), and the resulting +ringed space will be denoted by Xλ ×Z (resp. Xop +λ ×Z). Symbols Y × ˆXλ (resp. Xλ × ˆXλ, Xop +λ × ˆXλ) +denote (Y × ˆX)1×λ (resp. (Xλ × ˆX)1×λ, (Xop +λ × ˆX)1×λ). +8 + +The deformed Poincar´e line bundle Pλ is an object of coh Xλ × ˆXλ−1 defined as the OY × ˆY -module +� +ˆk∈ ˆ +K +Q ⊗ (pY, ˆY )∗Lˆk ∼= +� +ˆk∈ ˆ +K +R∗ +ˆkQ +(8.1) +equipped with the OXλ× ˆY ∼= � +ˆk∈ ˆ +K(pY, ˆY )∗Lˆk-action +� +ˆk∈ ˆ +K +Q ⊗ (pY, ˆY )∗Lˆk × +� +ˆk∈ ˆ +K +(pY, ˆY )∗Lˆk → +� +ˆk∈ ˆ +K +Q ⊗ (pY, ˆY )∗Lˆk +(8.2) +((ψˆk ⊗ φˆk)ˆk, (φ′ +ˆk)ˆk) �→ + + +� +ˆk1,ˆk2∈ ˆ +K, ˆk1ˆk2=ˆk +ψˆk1 ⊗ (φˆk1 ⋆λ φ′ +ˆk2) + + +ˆk +, +(8.3) +and the λ−1-twisted ˆK-action (i.e. the λ-twisted left ˆK-action) +ρˆk : +� +ˆk′∈ ˆ +K +R∗ +ˆk′Q →R∗ +ˆk−1 +� +ˆk′∈ ˆ +K +R∗ +ˆk′Q +(8.4) +(φˆk′)ˆk′ �→(λ(ˆk, ˆk′ˆk−1)φˆk′ˆk−1)ˆk′. +(8.5) +The deformed Fourier–Mukai transform +FMλ: Db( ˆXλ) → Db(Xλ) +(8.6) +is the integral functor with the deformed Poincar´e line bundle as the integral kernel, i.e., the composite +of the pull-back +(qY, ˆY )∗: Db( ˆXλ) → Db(Y × ˆXλ), +(8.7) +the tensor product +(−) ⊗ Pλ : Db(Y × ˆXλ) → Db(Xλ × ˆX), +(8.8) +and the push-forward +R(pY, ˆ +X)∗: Db(Xλ × ˆX) → Db(Xλ). +(8.9) +Its right adjoint is the integral functor FM−1 +λ +with the g-shift of +P−1 +λ +:= HomOY × ˆ +Y (Pλ, OY × ˆY ) ∈ ob(coh Xop +λ × ˆXλ) +(8.10) +as the kernel, since +• the push-forward R(qY, ˆY )∗ is right adjoint to the pull-back (qY, ˆY )∗, +• the tensor product P−1 +λ +⊗ (−) is right adjoint to the tensor product (−) ⊗ Pλ, and +• the pull-back (pY, ˆ +X)∗[g] shifted by g is right adjoint to the push-forward R(pY, ˆ +X)∗ because +Db(Xλ) is Calabi–Yau of dimension g and Db(Xλ × ˆX) is Calabi–Yau of dimension 2g (it will +be proved in Section 10). +Theorem 8.1 below is the main result in this paper: +Theorem 8.1. The deformed Fourier–Mukai transform FMλ is an equivalence of derived categories. +Remark 8.2. Let OTλ×ˆΓ be the sheaf associative algebras (̟ × id)∗OT׈Γ, with a non-commutative +associative product defined by the formula similar to (4.3), but at and bt are functions on ˆΓ. For +any λ ∈ H2( ˆT, U(1)) not necessarily at roots of unity, it is natural to expect that coh ˆXλ is derived- +equivalent to coh Xλ. Although the latter is not known to be abelian, we can define the deformed +Poincar´e line bundle Pλ as an object of Mod +� +Xλ × ˆXλ−1� +, i.e. a OTλ׈Γ-module equipped with a +λ-twisted left ˆT-action and a right Γ-action. +Pλ is defined by a free OTλ׈Γ-module of rank 1 equipped with a λ-twisted left ˆT-action +ˆγ · φ(x, ˆx) = ˆγ(x) ∗λ φ(x, ˆxˆγ) +(8.11) +9 + +and the right Γ-action +φ(x, ˆx) · γ = φ(xγ−1, ˆx) ∗λ ˆx(γ)−1. +(8.12) +9. Finite group actions on abelian and derived categories +It is natural to examine group actions on DG-categories in relation to group actions on derived +categories and equivariant Fourier-Mukai transforms. However, coherent data for group actions on +DG-categories are more intricate than those for group actions on abelian categories. As such, we will +concentrate on group actions on abelian categories and the actions they induce on derived categories. +A weak action of a finite group G on a category C is a family (g∗)g∈G of autoequivalences g∗: C → C +such that the functor (g1)∗◦(g2)∗ is isomorphic to (g1g2)∗ for any g1, g2 ∈ G. An action is a weak action +equipped with a coherence data, i.e., a family (cg1,g2)g1,g2∈G of isomorphisms cg1,g2 : (g1)∗ ◦ (g2)∗ +∼−→ +(g1g2)∗ of functors such that the diagram +(g1)∗ ◦ (g2)∗ ◦ (g3)∗ +cg1,g2 +−−−→ (g1g2)∗ ◦ (g3)∗ +cg2,g3 +� +cg1g2,g3 +� +(g1)∗ ◦ (g2g3)∗ +cg1,g2g3 +−−−−→ +(g1g2g3)∗ +(9.1) +commutes for any g1, g2, g3 ∈ G (cf. e.g. [Del97]). An action is strict if the coherence data consists +of identities. +Let C be a category equipped with an action of a finite group G. The following definition is taken +from [Sos12]: +Definition 9.1 ([Sos12, Definition 3.1]). A linearization of A ∈ ob(C) is a family (ρg)g∈G of isomor- +phisms ρg : A +∼−→ g∗A such that the diagram +A +(g1)∗A +(g1g2)∗A +� +ρg1 +�❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +ρg1g2 +�✤✤✤✤✤✤✤✤ +cg1,g2◦(g1)∗(ρg2) +(9.2) +commutes for any g1, g2 ∈ G. An equivariant object is an object equipped with a linearization. A +morphism of equivariant objects from (A, (ρg)g∈G) to (A′, (ρ′ +g)g∈G) is a morphism φ: A → A′ such +that the diagram commute +A +φ +−−−→ +A′ +ρg +� +ρ′ +g +� +g∗A +g∗φ +−−−→ g∗A′ +(9.3) +commutes. +The category of G-equivariant objects in C will be denoted by CG. For the rest of this paper and +unless otherwise specified, we will assume that C is a C-linear category and weak actions consists of +C-linear functors. +Propositon 9.2 ([Sos12, Proposition 3.2]). If C is abelian, then so is CG. +We extend the above constructions to twisted group actions. Let φ be a second cocycle of G with +values in C∗. +10 + +Definition 9.3. A φ-twisted linearization of A ∈ ob(C) is a family (ρg)g∈G of isomorphisms ρg : A +∼−→ +g∗A such that the diagram +A +(g1)∗A +(g1g2)∗A +� +ρg1 +�❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +φ(g1,g2)ρg1g2 +�✤✤✤✤✤✤✤✤ +cg1,g2◦(g1)∗(ρg2) +(9.4) +commutes for any g1, g2 ∈ G. Morphisms of φ-twisted equivariant objects are defined in the same +way as in CG. +A φ-twisted linearization of A ∈ ob(C) is equivalent to a linearization of A for G-action {ρg}g∈G +equipped with a coherence data (φ(g1, g2)−1cg1,g2)g1,g2∈G. The category of φ-twisted equivariant ob- +jects will be denoted by CG,φ. The cocycle condition on φ ensures the equality of +ρg3 ◦ ρg2 ◦ ρg1 = ρg3 ◦ (φ(g1, g2)ρg1g2) = φ(g1, g2)φ(g1g2, g3)ρg1g2g3 +(9.5) +and +ρg3 ◦ ρg2 ◦ ρg1 = (φ(g2, g3)ρg2g3) ◦ ρg1 = φ(g2, g3)φ(g1, g2g3)ρg1g2g3, +(9.6) +where we have omitted (g1)∗ and so on. If a pair of cocycles φ and φ′ differ by the coboundary of +α ∈ C1(G, C∗), then there exists an equivalence CG,φ → CG,φ′ sending (A, (ρg)g∈G) to (A, (α(g)ρg)g∈G). +Explanations of relations between group cohomology of G in low degrees and (weak) G-actions are +found in [BO20]. +Corollary 9.4 below is obtained by applying Proposition 9.2 to the G-action equipped with a +coherence data (φ(g1, g2)−1cg1,g2)g1,g2∈G: +Corollary 9.4. If C is abelian, then so is CG,φ. +By applying the free-forgetful adjunction +Free ⊣ Forget +(9.7) +between +Free: C → CG,φ, +A �→ + +� +g′∈G +(g′)∗A, +� +ρg := +� +g′∈G +φ(g, g′) (id: (gg′)∗A → g∗(g′)∗A) +� +g∈G + + +(9.8) +and +Forget: CG,φ → C, +(A, (ρg)g∈G) �→ A +(9.9) +to the opposite categories and using the equivalence (CG,φ)op ≃ (Cop)G,φ−1, one obtains an adjunction +Forget ⊣ Free . +(9.10) +For any (A, (ρg)g∈G), (A′, (ρ′ +g)g∈G) ∈ CG,φ, the space HomC(A, A′) comes with a natural linear action +of G in such a way that the diagram +A +χ +−−−→ +A′ +ρg +� +ρ′ +g +� +g∗A +g∗(χ·g) +−−−−→ g∗A′ +(9.11) +11 + +commutes since +χ · g1 · g2 = ((g1∗)−1(ρ′ +g1 ◦ χ ◦ ρ−1 +g1 )) · g2 +(9.12) += (g2∗)−1(ρ′ +g2 ◦ ((g1∗)−1(ρ′ +g1 ◦ χ ◦ ρ−1 +g1 )) ◦ ρ−1 +g2 )) +(9.13) += (g2∗)−1(g1∗)−1(g1∗ρ′ +g2 ◦ (ρ′ +g1 ◦ χ ◦ ρ−1 +g1 ) ◦ g1∗ρ−1 +g2 ) +(9.14) += ((g1g2)∗)−1((φ(g1, g2)ρ′ +g1g2) ◦ χ ◦ (φ(g1, g2)ρg1g2)−1) +(9.15) += χ · g1g2. +(9.16) +It follows from the definition that +HomCG((A, (ρg)g∈G), (A′, (ρ′ +g)g∈G)) = HomC(A, A′)G. +(9.17) +A functor Φ: C → C′ between categories with G-actions is said to be G-equivariant if it is equipped +with a family (ag)g∈G of natural isomorphisms ag : Φ ◦ g∗ +∼−→ g∗ ◦ Φ of functors such that the diagram +Φ ◦ (g1)∗ ◦ (g2)∗ +(g1)∗ ◦ Φ ◦ (g2)∗ +(g1)∗ ◦ (g2)∗ ◦ Φ +Φ ◦ (g1g2)∗ +(g1g2)∗ ◦ Φ +� +ag1 +�✤✤✤✤✤✤✤✤ +cg1,g2 +� +ag2 +�♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ +cg1,g2 +� +ag1g2 +(9.18) +commutes. A G-equivariant functor Φ: C → C′ induces a functor ΦG,φ : CG,Φ → C′G,φ sending an +object (A, (ρg)g∈G) to (Φ(A), (ag ◦ Φ(ρg))g∈G) and a morphism f : (A, (ρg)g∈G) → (A′, (ρ′ +g)g∈G) to +Φ(f): Φ(A) → Φ(A′). It is straightforward to show that ΦG,φ send G-equivariant objects to G- +equivariant objects. +Propositon 9.5. If Φ is right (resp. left) exact, then so is ΦG,φ. +Proof. Since Free and Forget are mutually both left and right adjoint to each other, they are exact, +so that a sequence A → B → C in CG,φ is exact if and only if Forget(A) → Forget(B) → Forget(C) +is exact in C. +□ +Now we discuss the derived category of CG,φ. +Propositon 9.6. An object (I, (ρg)g∈G) ∈ ob(CG,φ) is injective if and only if so is I ∈ ob(C). The +category CG,φ has enough injectives if and only if so is C. +Proof. If I is injective, then the functor +A �→ HomCG,φ(A, (I, (ρg))) = HomC(Forget(A), I)G +(9.19) +is exact, since Forget is exact, I is injective, and taking the G-invariant part is exact. +Conversely, if (I, (ρg)) is injective, then the functor +A �→ HomC(A, I) ∼= HomCG,φ(Free(A), (I, (ρg))) +(9.20) +is exact. +Let (A, (ρg)g) be an object in CG,φ. If C has enough injectives, then a monomorphism A → I into +an injective object I ∈ ob(C) gives a monomorphism A → Free I into Free I, which is injective in +CG,φ. +Conversely, if CG,φ has enough injectives, then for any A ∈ ob(C), a monomorphism Free A → I +into an injective I ∈ ob(CG,φ) gives a monomorphism A → Forget I into an injective Forget I. +□ +Assume that C has enough injectives. For any pair A, B ∈ ob(D+(CG,φ)) of objects, the space +Hom(Forget(A), Forget(B)) has a natural G-action in such a way that +Hom(A, B) ∼= Hom(Forget(A), Forget(B))G. +(9.21) +Propositon 9.7. If Db(C) is Calabi–Yau of dimension n, then so is Db(CG,φ). +12 + +Proof. There exists a natural isomorphism +HomDb(C)(A, B) ∼= HomDb(C)(B, A[n])∗ +(9.22) +for A, B ∈ ob(Db(CG,φ)), since Db(C) is Calabi–Yau of dimension n. +It is G-equivariant by the +naturality, so it induces an isomorphism on G-invariant part. +This means that Db(CG,φ) is also +Calabi–Yau of dimension n by (9.21). +□ +A G-equivariant left exact functor Φ: C → C′ induces functors RΦ: D+(C) → D+(C′) and +RΦG,φ : D+(CG,φ) → D+(C′G,φ). If RΦ is fully faithful, then so is RΦG,φ by (9.21). +10. Group actions on coherent sheaves +An action of a finite group G on a complex manifold Z induces a strict G-action (R∗ +g)g∈G on coh Z. +In the case of the ˆK-action on ˆY , one obtains (coh ˆY ) ˆ +K,λ ≃ coh ˆXλ. +Another example of a finite group action on the category of coherent sheaves comes from a coherent +injection from a finite abelian group G to the Picard group Pic0 Z of a complex manifold Z, i.e., a +family {Lg}g∈G of line bundles and a family (mg1,g2)g1,g2∈G of isomorphisms mg1,g2 : Lg1 ⊗Lg2 +∼−→ Lg1g2 +such that the diagrams +Lg1 ⊗ Lg2 ⊗ Lg3 +mg1,g2⊗id +−−−−−−→ Lg1g2 ⊗ Lg3 +id ⊗mg2,g3 +� +mg1g2,g3 +� +Lg1 ⊗ Lg2g3 +mg1,g2g3 +−−−−−→ +Lg1g2g3 +(10.1) +Lg1 ⊗ Lg2 +Lg1g2 +Lg2 ⊗ Lg1 +� +mg1,g2 +� +�t +t +t +t +t +t +t +t +t +t +t +mg2,g1 +(10.2) +commutes, inducing a G-action +� +(−) ⊗ L−1 +g ) +� +g∈G on coh Z. Here, the vertical arrow in (10.2) comes +from the canonical symmetric monoidal structure in coh Z. If Z is compact and connected, then one +has Aut L = C∗ for any line bundle L, and an example of a coherence data (mg1,g2)g1,g2∈G comes from +a choice of a collection (ϕg)g∈G of linear isomorphisms ϕ: (Lg)z +∼−→ C from the fibers (Lg)z of Lg at +an arbitrarily chosen and fixed base point z ∈ Z to the complex line. +A φ-twisted G-linearization (ρg)g∈G of M is equivalent to a family (mg)g∈G of morphisms +mg = (g∗)−1ρg : M ⊗ Lg → M +(10.3) +such that diagram +M ⊗ Lg1 ⊗ Lg2 +M ⊗ Lg2 +M +� +mg1 +�❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +φ(g1,g2)mg1g2 +�✤✤✤✤✤✤✤ +mg2 +(10.4) +commutes. The category (coh Z)G,φ is equivalent to coh Aφ, where Aφ is the sheaf of OZ-algebras ob- +tained as the OZ-module � +g∈G Lg equipped with the multiplication given by � +g1,g2∈G φ(g1, g2)mg1,g2. +If φ = 1, Aφ is commutative by the commutativity of (10.2). In particular, coh Xλ is equivalent to +(coh Y ) ˆ +K,λ. By Proposition 9.7, Db(Xλ) is Calabi–Yau of dimension g and Db(Xλ× ˆX) is Calabi–Yau +of dimension 2g, which were needed to prove that FM−1 +λ +is right adjoint to FMλ. +13 + +11. Proof of Theorem 8.1 +Functors FMλ and FMQ are the right derived functors of functors +FMab +λ : coh ˆXλ → coh Xλ +(11.1) +M �→ (pY, ˆ +X)∗((qY, ˆY )∗M ⊗OY × ˆY Pλ) +(11.2) +and +FMab +Q : coh ˆY → coh Y +(11.3) +M �→ (pY, ˆY )∗((qY, ˆY )∗M ⊗OY × ˆ +Y Q) +(11.4) +of abelian categories. +Since +FMab +Q (R∗ +ˆyM) = (pY, ˆY )∗((qY, ˆY )∗R∗ +ˆyM ⊗ Q) +(11.5) +∼= (pY, ˆY )∗R∗ +(1,ˆy−1)((qY, ˆY )∗R∗ +ˆyM ⊗ Q) +(11.6) +∼= (pY, ˆY )∗((qY, ˆY )∗M ⊗ R∗ +(1,ˆy−1)Q) +(11.7) +∼= (pY, ˆY )∗((qY, ˆY )∗M ⊗ Q ⊗ (pY, ˆY )∗L−1 +ˆy ) +(11.8) +∼= (pY, ˆY )∗((qY, ˆY )∗M ⊗ Q) ⊗ L−1 +ˆy +(11.9) += FMab +Q (M) ⊗ L−1 +ˆy , +(11.10) +for any ˆy ∈ ˆK and M ∈ ob(coh ˆY ), FMab +Q commutes with the weak ˆK-action. The commutativity +of the diagram (9.18) in this case is a straightforward diagram chasing. This turns FMab +Q into a +ˆK-equivariant functor, inducing a functor +(FMab +Q ) +ˆ +K,λ: coh ˆXλ → coh Xλ. +(11.11) +Lemma 11.1. The functor (FMab +Q ) ˆ +K,λ is isomorphic to FMab +λ . +Proof. The squares on the left and the right of the diagram +(coh ˆY ) ˆ +K,λ +((qY, ˆ +Y )∗) ˆ +K,λ +� +� +(coh(Y × ˆY )) ˆ +K,λ (π∗(−⊗Q)) ˆ +K,λ +� +� +(coh(Y × ˆX)) ˆ +K,λ ((pY, ˆ +X)∗) ˆ +K,λ +� +� +(coh Y ) ˆ +K,λ +� +coh ˆXλ +(qY, ˆ +Y )∗ +� coh(Y × ˆXλ) +−⊗Pλ +� coh(Xλ × ˆX) +(pY, ˆ +X)∗ +� coh ˆXλ +commute by definition, and the natural isomorphism +⊕ˆy′∈ ˆ +Kρ−1 +ˆy′ ⊗ id: +� +ˆy′∈ ˆ +K +R∗ +(1,ˆy′)(M ⊗OY × ˆY Q) → +� +ˆy′∈ ˆ +K +M ⊗OY × ˆY R∗ +(1,ˆy′)Q, +(11.12) +whose ˆK-equivariance can be checked by a straightforward computation, gives the commutativity of +the square in the middle. +□ +Therefore R(FMab +Q ) ˆ +K,λ and FMλ are isomorphic. Hence FMλ is fully faithful as explained at the +end of Section 9. Similarly, the right adjoint FM−1 +λ +of FMλ, whose kernel is given by the g-shift of +(8.10), is also fully faithful, and Theorem 8.1 is proved. +14 + +References +[BBBP07] O. Ben-Bassat, J. Block, and T. Pantev, Non-commutative tori and Fourier-Mukai duality, Compos. Math. +143 (2007), no. 2, 423–475. MR 2309993 1, 4 +[Blo] +Jonathan Block, Duality and equivalence of module categories in noncommutative geometry II: Mukai du- +ality for holomorphic noncommutative tori, arXiv:math/0604296. 1, 2 +[Blo10] +, Duality and equivalence of module categories in noncommutative geometry, A celebration of the +mathematical legacy of Raoul Bott, CRM Proc. Lecture Notes, vol. 50, Amer. Math. Soc., Providence, RI, +2010, pp. 311–339. MR 2648899 1 +[BO20] +Thorsten Beckmann and Georg Oberdieck, On equivariant derived categories, arxiv:2006.13626v2, 2020. 11 +[Con94] +Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779 1 +[Del97] +P. Deligne, Action du groupe des tresses sur une cat´egorie, Invent. Math. 128 (1997), no. 1, 159–175. +MR 1437497 10 +[Muk81] +Shigeru Mukai, Duality between D(X) and D( ˆX) with its application to Picard sheaves, Nagoya Math. J. +81 (1981), 153–175. MR 607081 1, 4 +[PS03] +A. Polishchuk and A. Schwarz, Categories of holomorphic vector bundles on noncommutative two-tori, +Comm. Math. Phys. 236 (2003), no. 1, 135–159. MR 1977884 1, 2 +[Rie81] +Marc A. Rieffel, C∗-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), no. 2, 415–429. +MR 623572 1 +[Sch01] +Albert Schwarz, Theta functions on noncommutative tori, Lett. Math. Phys. 58 (2001), no. 1, 81–90. +MR 1865115 1, 2 +[Soi09] +Y. Soibelman, On non-commutative analytic spaces over non-Archimedean fields, Homological mirror sym- +metry, Lecture Notes in Phys., vol. 757, Springer, Berlin, 2009, pp. 221–247. 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MR 2477894 1 +Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, +Tokyo, 153-8914, Japan. +Email address: hiokuc8h18@gmail.com +15 + diff --git a/4NE2T4oBgHgl3EQfOQYe/content/tmp_files/load_file.txt b/4NE2T4oBgHgl3EQfOQYe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d53702f8f277451f7d4a374c16c57821db075c2b --- /dev/null +++ b/4NE2T4oBgHgl3EQfOQYe/content/tmp_files/load_file.txt @@ -0,0 +1,588 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf,len=587 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content='03745v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content='AG] 10 Jan 2023 FOURIER–MUKAI TRANSFORMS FOR NON-COMMUTATIVE COMPLEX TORI NOBUKI OKUDA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' Let X be a complex torus of dimension g and ˆX be the dual torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' For any g(g − 1)/2- tuple λ of complex numbers of absolute value 1, we define a non-commutative complex torus Xλ as a sheaf of algebras on a real torus of dimension g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' We prove that if all components of λ are roots of unity, then the category of coherent sheaves on Xλ is abelian and derived-equivalent to the category of coherent sheaves on ˆX twisted by an element of the Brauer group of ˆX determined by λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' Introduction Let X and ˆX be smooth projective varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' An integral functor Φ: Db(X) → Db( ˆX) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content='1) between derived categories of coherent sheaves is said to be a Fourier–Mukai functor if it is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' A Fourier–Mukai functor Φ induces an isomorphism HH(Φ): HH•(X) → HH•( ˆX) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content='2) of Hochschild cohomologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' For any u ∈ HH2(X), Toda [Tod09] gave a C[ε]/(ε2)-linear category Db(X, u) of first order deformations of X along u and an equivalence Φ† : Db(X, u) → Db( ˆX, HH(Φ)(u)) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content='3) extending Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' The integral functor FM with the Poincar´e line bundle as the integral kernel is the first example of a Fourier–Mukai functor given by Mukai [Muk81] for an abelian variety X and the dual abelian variety ˆX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' Under the Hochschild–Kostant–Rosenberg isomorphism HH2(X) ∼= H0(∧2TX) ⊕ H1(TX) ⊕ H2(OX), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content='4) the induced map HH(FM) sends H0(∧2TX) to H2(O ˆ X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' This suggests that non-commutative de- formations of abelian varieties are Fourier–Mukai partners of gerby deformations of dual abelian varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' While the notion of gerby deformations is well-established in terms of twisted sheaves, non- commutative deformations are much harder to define in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' At the formal level, the Moyal deformation quantizations give formal non-commutative deformations of complex tori, and derived equivalences to formal gerby deformations are proved in [BBBP07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' The next problem is to construct non-commutative tori and derived equivalences for non-formal parameters (or even as a family over a complex manifold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' The first attempt to define non-commutative complex tori is given by Schwarz [Sch01], who intro- duced the notion of complex structures on non-commutative tori, which are irrational rotation alge- bras (see [Rie81], for example) regarded as non-commutative spaces in the sense of Connes[Con94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' Categories of holomorphic vector bundles on them are studied in [PS03].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' In the paper [Blo10] and the preprint [Blo], Block discussed Fourier–Mukai functors between non- commutative complex tori and gerby deformations of dual complex tori, using DG categories which include objects corresponding to quasi-coherent sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' In [Blo], he also announced to discuss coherent sheaves in a paper in preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' We now explain the results of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' Let X = T/Γ be a complex torus, where T := (C∗)g and Γ is a discrete subgroup of T isomorphic to Zg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' The dual complex torus ˆX := Pic0 X can naturally be identified with ˆΓ/ ˆT where ˆΓ := Hom(Γ, C∗) ∼= (C∗)g and ˆT := Hom(T, C∗) ∼= Zg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' Let λ ∈ H2( ˆT, U(1)) ∼= U(1)g(g−1)/2 be an element of the second group cohomology of ˆT with values in 1 U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' We construct a non-commutative deformation of X with parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' When λ takes values in roots of unity, we give an equivalence of the derived category of coherent sheaves on the non- commutative deformation of X with parameter λ and the derived category of coherent sheaves on the gerby deformation of ˆX with the same parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' Our construction of non-commutative complex tori is different from those in [Sch01], [PS03] and [Blo].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' Ours can be regarded as a patching of Archimedean analog of quantum analytic tori in [Soi09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' To prove the equivalence of derived categories, we use the idea of equivariant Fourier–Mukai transforms developed in [Sos12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' This paper is organized as follows: In Section 2, we discuss q-Weyl algebras as toy models to motivate constructions in later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' In Section 3, we recall Fourier–Mukai transforms for complex tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' In Section 4, we define non-commutative complex tori by deforming sheaves of convergent Laurent series rings on real tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' It can be regarded as an Archimedean analog of the construction of quantum analytic tori in [Soi09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' In Section 5, we discuss a dual pair X → Y and ˆY → ˆX of finite coverings of tori associated with a deformation parameter λ with values in roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' In Section 6, we describe non-commutative complex tori whose deformation parameters take values in roots of unity in terms of a finite sheaf of algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' In Section 7, we collect basic definitions on twisted sheaves on complex manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' In Section 8, we introduce Fourier–Mukai transforms from gerby complex tori to non-commutative complex tori at roots of unity and state Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content='1, which is the main result in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' In Section 9, we recall basic definitions and results on finite group actions on abelian and derived categories, and discuss twistings by group cocycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' In Section 10, we discuss group actions on categories appearing in our construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' In Section 11, we prove Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' Notations and conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' We fix the complex number field C as the ground field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' All modules (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' actions) are right modules (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' actions) unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' In contrast, all group ac- tions on categories are left actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' The word ‘non-commutative’ is synonymous with ‘not necessarily commutative’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' For a ringed space X = (Z, OX), we write the ringed space (Z, OX op := (OX)op) as X op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' The category of OX-modules will be denoted by Mod X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' For an element a of a complex abelian Lie group A, we write the right translation map A → A, x �→ xa as Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' For two or three complex manifolds Z1, Z2 or Z1, Z2, Z3, the projection to the first (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' second) component Z1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' Z2) will be denoted by pZ1,Z2 or pZ1,Z2,Z3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' qZ1,Z2 or qZ1,Z2,Z3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' The author is deeply grateful to his advisor, Kazushi Ueda, for a lot of guidance, useful comments, and encouragement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' The author also thanks Masahiro Futaki for many useful questions, one of which lead to the formula (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' Finally, the author has deep gratitude to his parents for their various supports throughout his life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' q-Weyl algebras The category of OX-modules is equivalent to the category of Γ-equivariant OT -modules, and the category of Γ-equivariant C[T]-modules can be regarded as a toy model for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' The latter can be identified with the category of modules over the crossed product algebra C[T] ⋊ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE2T4oBgHgl3EQfOQYe/content/2301.03745v1.pdf'} +page_content=' For a parameter λ = (λi,j)1≤i 0), we +have +h = +� +h1 +h2 +... +hN +� += +� +CB +CAB +... +CAN−1B +� +(16) +Then according to Eq. (15)- (16) and with the knowledge of the observation +matrix Y and the control input matrix U, one can estimate the impulse response +in a least-squares fashion +� +h1 +h2 +... +hN +� += Y +� +���� +U0 +U1 +... +UN−1 +0 +U0 +... +UN−2 +... +... +... +... +0 +0 +. . . +U0 +� +���� +† +(17) +9 + +where the operator † represents the Moore-Penrose pseudo-inverse. Note that +the noise is not optimally filtered by the least-squares inverse as presented in Eq. +(17). To address the issue, the conventional OKID can be designed based on an +optimal observer system whereby optimal system parameters can be identified. +For simplicity, we refer readers to [39] (Pages 340-343) for detailed explanation +and implementation. +Next, with the obtained impulse response, the Hankel matrix H and the +next-step Hankel matrix H +′ can be written as follows: +H = +� +���� +h1 +h2 +... +hN +0 +h1 +... +hN−1 +... +... +... +... +0 +0 +. . . +h1 +� +���� , H +′ = +� +���� +h2 +h3 +... +hN+1 +0 +h2 +... +hN +... +... +... +... +0 +0 +. . . +h2 +� +���� +(18) +The Hankel matrix H could be truncated with Singular Value Decomposition +(SVD): +H = UΣVT = [�U, U tr] +��Σ +0 +0 +Σtr +� � +�V +T +VT +tr +� +≈ � +U �Σ �V +T +(19) +Let +O = [C, CA, CA2, ..., CAN−1]T +(20) +be the observability matrix, and +C = [B, AB, A2B, ..., AN−1B] +(21) +be the controllability matrix. Then, by observing Eq. (16) and Eq. (18), we +have +H = OC, +H +′ = OAC +(22) +Furthermore, considering Eq. (19), we can assume that O = �U �Σγ and C = +�Σ1−γ �V +T , where γ is an arbitrary real value. +Conventional OKID algorithm. For the conventional OKID algorithm, ERA +is thereafter used to identify the matrix A and B, with γ set to a constant +1 +2 for a special balanced realization. That is, one can assume O = �U �Σ +1 +2 and +C = �Σ +1 +2 �V +T , whereby a state space model with balanced Grammians is realized +(i.e., the same degree of controllability and observability) that agrees with the +control input and the observation data. As such, with γ = 1 +2 and by Eq. (20) - +Eq. (22), the matrices A and B can be identified by the conventional OKID +as follows [39]: +� +A = �Σ− 1 +2 �U +T H +′ �V �Σ− 1 +2 +(23a) +� +B = CNS×NU = +� +�Σ +1 +2 �V +T � +NS×NU +(23b) +where the operator +�.� +NS×NU represents the first NS rows and the first NU +columns of the matrix in the bracket; NS is the dimension of Koopman embed- +ding space and NU is the dimension of control inputs. +10 + +In this paper, to better identify the Koopman-based process dynamics, we +propose a Koopman-inspired algorithm to find an optimal γ rather than assum- +ing γ = 1 +2 as in the conventional OKID. Consider a general form with γ unfixed +�U �Σγ = O = [C, CA, CA2, ..., CAN−1]T += IN×N ⊗ C · [I, A, A2, ..., AN−1]T +(24a) +�Σ1−γ �V +T = C = [B, AB, A2B, ..., AN−1B] += [I, A, A2, ..., AN−1](IN×N ⊗ B) +(24b) +where IN×N is the identity matrix with the dimension N × N, and ⊗ denotes +the Kronecker product which is +IN×N ⊗ C = +� +�� +C +... +C +� +�� , +IN×N ⊗ B = +� +�� +B +... +B +� +�� +(25) +Then +(IN×N ⊗ C)† �U �Σγ = [I, A, A2, ..., AN−1]T +(26a) +�Σ1−γ �V +T (IN×N ⊗ B)† = [I, A, A2, ..., AN−1] +(26b) +By observing Eq. (26a) and Eq. (26b), we have +�Σγ �U +T ((IN×N ⊗ C)†)T = �Σ1−γ �V +T (IN×N ⊗ B)† +⇒ �Σ2γ−1 �U +T � +(IN×N ⊗ C)†�T = �V +T (IN×N ⊗ B)† +(27) +Treating Eq. (27) as a soft constraint for the parameter γ, one can formulate a +quadratic optimization problem to solve the optimal parameter γopt: +γopt = arg min +γ +∥�Σ2γ−1 �U +T � +(IN×N ⊗ C)†�T − �V +T (IN×N ⊗ B)∥F +subject to: +0 ≤ γ ≤ 1 +(28) +where ∥.∥F represents the Frobenius norm of a matrix. The inequality 0 ≤ γ ≤ 1 +is added to constrain problem complexity. The novel OKID-based algorithm for +parameter estimation is summarized below. The flowchart of the algorithm is +also presented in Fig. 3. +The Proposed Online Koompan-Inspired Enhanced OKID Algorithm +Algorithm Initialization. Initialize γopt = γopt,0, the smoothing factor η, +and the time step TOP T between two updates of γopt. The selection of these +parameters will be discussed in Remarks after the presentation of the algorithm. +At each time step of identification and the secondary control, i.e., for k = +1, 2, ..., conduct Step 1 -Step 5. +Step 1: Data preparation. Collect the last N data samples from microPMUs +to obtain the data matrices of phasor angle Θ, voltage deviation ∆V and +angular frequency deviation ∆Ω from the nominal values. Collect control input +11 + +data U from the secondary controller. +For example, the phasor angle Θ is +stacked in a form of +Θ = +� +� +| +| +| +Θ1 +... +ΘN +| +| +| +� +� +(29) +∆V , ∆Ω and U are formed in the same way. The approximated operation +points of voltage phasor angles and magnitudes Θ∗ +L and V ∗ +L are defined as the +first data sample from a window of collected PMU data, prepared in a matrix +form as follows: +Θ∗ +L = +� +� +| +| +| +Θ1 +... +Θ1 +| +| +| +� +� , +V ∗ +L = +� +� +| +| +| +V1 +... +V1 +| +| +| +� +� +(30) +Prepare the data matrices for y and z as follows: Y = [Θ − Θ∗ +L, V − V ∗ +L ]T , +and Z = [∆V, sin (Θ) − sin (Θ∗ +L), cos (Θ) − cos (Θ∗ +L), ∆Ω]T . +Step 2: Hankel matrix preparation and SVD. Estimate the impulse re- +sponse and prepare the Hankel matrices according to Eq. (17)-Eq. (18). Con- +duct the SVD on the obtained Hankel matrix H ≈ �U �Σ �V +T . +Step 3: Estimation of C. Ignoring the error term in Eq. (13b), we have +Y = CZ. Thus, one can estimate the observation matrix C at each time step +k in a least-squares fashion by multiplying the pseudo-inverse on both sides of +the equation, which is +� +Ck = Y Z† +(31) +Step 4: Optimization for γopt. Check if the run time of optimization between +the last update of γopt is larger than TOP T . If no, γopt,k = γopt,k−1, go to Step +5; otherwise, solve the optimization problem in Eq. (28) for γ− +opt. To do so, by +Eq. (23b), replace B with +� +�Σ1−γ �V +T � +NS×NU +and replace C with � +Ck from Step +3 in Eq. (28). Then, adaptively update γopt by +γopt,k =ηγ− +opt + (1 − η)γopt,k−1, +for +k = TOP T , 2TOP T , 3TOP T , ... +(32) +where γ− +opt is the optimal value of the realization parameter γ according to Eq. +(28). That is, once γ− +opt is updated, we update γopt with the weighted sum of the +old γopt at last time step and the updated value γ− +opt. η is the weight to smooth +online learning. The role of η is to smooth the online learning of γ. As the small +piece of online data used for identification is characterized by stochasticity, +the smoothing factor η can mitigate aggressive change to make the learning +process more reliable. This is so because the estimation is equivalent to the +Robbins–Monro form [40], which is γopt,k = ηγ− +opt + (1 − η)γopt,k−1 = γopt,k−1 + +η(γ− +opt−γopt,k−1). The larger the value of γ is, the smoother the learning process +tends to be, whereas the adaptiveness of learning is compromised. +Step 5: Estimation of A and B. By Eq. (23a)-(23b) +� +Ak = +� +η �Σ−γopt,k �U +T H +′ �V �Σγopt,k−1 + (1 − η) � +Ak−1 +if k ≥ 1 +�Σ−γopt,k �U +T H +′ �V �Σγopt,k−1 +if k = 0 +(33) +12 + +� +Bk = +� +� +� +� +� +η +� +�Σ1−γopt,k �V +T � +NS×NU ++ (1 − η) � +Bk−1 +if k ≥ 1 +� +�Σ1−γopt,k �V +T � +NS×NU +if k = 0 +(34) +After implementing the identification algorithm, the identified Koopman state +space model at the time step k is obtained as: +zk+1 = � +Akzk + � +Bkuk +(35a) +yk = � +Ckzk +(35b) +Compared to the traditional EDMDc used in power systems [21], the pro- +posed Koopman-inspired OKID can use the observation data y as in Eq. (13b) +to help learn the Koopman state space model in Eq. (13a), while the traditional +EDMDc only estimates the Koopman state space in Eq. (13a) in a least-squares +fashion without the incorporation of observation data. The fusion of the infor- +mation from the observation data provides extra opportunities to enhance the +modeling efficacy. +Remarks +• γopt: in this paper, γopt,0 = 1 +2. Thus the enhanced OKID is initially equiv- +alent to the conventional one while it gradually learns the optimized value +for γopt with the online OKID and the periodically enabled optimization +in Eq. (28). +• The smoothing factor η: it is used to weigh the past estimations and +the latest one, and set to +1 +N in this paper with the assumption that all +estimations have the same weight independent on the time of occurrence. +A larger η means the estimation put more weight on the newest data, and +vice versa. +• The time step TOP T for updating γopt: it is set to 0.6s, which is longer +than the run time of the proposed Koopman-inspired enhanced OKID +and the time step of secondary control (30ms) as detailed in Section 4. +A small TOP T is favorable as a fast update of γ to compensate for the +uncertainty of the Koopman process model (13a), while it should be longer +than the run time of the optimization (28) to ensure the feasibility of online +implementation. +3.4. The Linear Control Based on the Koopman-Inspired Enhanced OKID +After obtaining the identified model (35a) - (35b), a discrete-time linear +quadratic regulator (LQR) is applied at each time step of secondary control, +aiming to reduce the voltage and frequency deviations by minimizing the cost +J(u) = +∞ +� +k=0 +zT +k Qzk + uT +k Ruk, +subject to +zk+1 = � +Azk + � +Buk +(36) +13 + +Figure 3: Algorithm flowchart of the proposed Koopman-inspired enhanced OKID +where Q and R are cost matrices defined as: +Q = +� +��� +qV +qsin θ +qcos θ +qω +� +��� , +R = +�rP +rQ +� +(37) +where qV , qsin θ, qcos θ and qω are cost submatrices for the Koopman observ- +ables presented in (14). rP and rQ are cost submatrices for the control signals +∆P ∗ and ∆Q∗. They are basically selected empirically in this paper based on +which factor is treated to be more important. The optimal control input can be +obtained by: +uk = +� +� +� +ULB +uk < ULB +−Kzk +ULB ⩽ uk ⩽ UUB +UUB +uk > UUB +with +K = ( � +BT S � +B + R)−1 � +BT S � +A +(38) +where K is the control gain matrix. S is the solution of Riccati equation [41]. +UUB and ULB are the upper and lower saturation limits that can bound the +uncertainty introduced by control inputs. The bounds are user-defined values, +which are determined empirically in the paper. Usually, large bounds can lead +to faster response whereas the uncertainty introduced through control input +channels could be increased to an unmanageable level that degrades the dynamic +control performance or even stability. On the other hand, the bounds cannot be +set to too small values, otherwise, the response could be slow and the capability +of the controller cannot be fully taken use of. +The Koopman-inspired enhanced OKID illustrated in Section 3.3 and the +LQR illustrated in Section 3.4 can be respectively applied to the identification +block and the control algorithm block of secondary control in Fig. 1. Specifi- +cally, Fig. 4 presents the proposed online identification and control structure. +The stability of such Koopman-inspired identification and control is guaranteed, +which is proved in what follows. +3.5. +Stability Analysis +MG dynamics can be expressed in a Koopman-based structure and can be +approximated with the online Koopman-inspired identification in Section 3.3. +The approximation error is bounded but often not quantifiable as it depends on +14 + +At each time +Steps 1-2 +Step 3 +Step 4 +Step 5 +step k: +No +Keep Old opt +Data Matrix +Initialization +Estimate Ck +If k = ToPT, 2ToPT, : +Preparation & SVD +OKID (update Ak and Bk) +Yes. +Update Yopt +EndFigure 4: Online structure of the proposed Koopman-inspired enhanced OKID and control +the appropriateness of Koopman observables and the online parameter identi- +fication algorithm. In what follows, we aim to prove stability properties in a +general sense. +3.5.1. Proof of BIBO Stability +We prove that the proposed Koopman-inspired OKID-based control is BIBO +(bounded-input-bounded-output) stable. Denoted by ˆxk+1 the one-step-ahead +prediction of the state vector x at the time step k with the OKID-based es- +timation. +Denoted by ˆKk the estimated Koopman operator at time step k. +According to Eq. (12), we have +g(ˆxk+1) = Φ(ˆxk+1, 0) = ˆKkΦ(xk, uk) = ˆKk +Nϕ +� +j=1 +ϕj(xk, uk)vj = +Nϕ +� +j=1 +(ρj,kϕj(xk, uk)vj) +(39) +where ρj,k is the eigenvalue corresponding to the jth eigenfunction ϕj for the +estimated Koopman operator ˆKk. Recall that Φ(x, u) = g(x) + l(u) discussed +in Section III.A, where l(u) = [l1(u), l2(u), . . . lp(u)]T and l(0) = 0. Then +g(xk+1) = g(ˆxk+1) + δk = ˆKkΦ(xk, uk) + δk += ˆKk(g(xk) + l(uk)) + δk += ˆKk( ˆKk−1Φ(xk−1, uk−1) + δk−1 + l(uk)) + δk += ˆKk( ˆKk−1(g(xk−1) + l(uk−1)) + δk−1 + l(uk)) + δk += ˆKk( ˆKk−1( ˆKk−2Φ(xk−2, uk−2) + δk−2 + l(uk−2)) + l(uk−1)) + δk−1 + l(uk)) + δk += · · · = +k +� +h=0 +ˆKk−hΦ(x0, u0) + +k +� +h=1 +k +� +i=h +ˆKk−i+h(δh−1 + l(uh)) + δk += +Nϕ +� +j=1 +( +k +� +h=0 +ρj,h)ϕj(x0, u0)vj + +k +� +h=1 +k +� +i=h +ˆKk−i+h(δh−1 + l(uh)) + δk +15 + +Microgrid Network +Control Command +Microgrid Node/Bus +(e.g., four-bus, thirteen-bus microgrids) +Measurement Data Sensing +Microgrid Distribution Line +Action:(control inputs)u=[△P",Q*jr +Primary +Control +Secondary Controller: +DER 1 +Proposed Koopman-Inspired Enhanced OKID and LQR Control +Control +MicroPMU Data +Action Generator (LQR) +Online +Minimize the Cost Function +Identification +J(u) =z Qz +utRui +MicroPMUl +Control +Measurement: +ak = jox, VaJT +Cyber +Koopman Embedding Mapping g : +Data +[Section 3.4] +Inputs +Network +DER 2 +(Action) +Zk =g(ak[Section 3.2] +Data +Primary +Koopman-inspired Enhanced OKID +Control +[Section 3.3] +3 +Primary +States +DER +Control +MicroPMU Data +zk +Estimated System Model: +Estimated Parameters +Zk+1 = Akzk + Buk +Ak +Bk +Yk = Ckzk +4where δk is the Koopman modeling error which has been defined in Eq. (13a), +and vj is the jth Koopman mode associated with the Koopman eigenfunction +ϕj. Apparently, +0 ⩽ ∥ +Nϕ +� +j=0 +( +k +� +j=0 +ρj,h)ϕj(x0, u0)vj∥2 ⩽ lim +k→∞(maxj,h|ρj,h|)k+1 +Nϕ +� +j=1 +∥ϕj(x0, u0)vj∥2 +(40) +With LQR in the Koopman invariant subspace, assume the MG secondary +controller can optimally make the magnitudes of all system eigenvalues smaller +than 1 (if the system is stabilizable). That is ˆKkϕj = ρj,kϕj with |ρj,k| < 1. +Due to the online rolling-based estimation in the proposed method, we can +assume the global error ∥ �k +h=1 Πk +i=h ˆKk(δh−1 + l(uh))∥2 is bounded by ζg, and +the modeling error is ∥δk∥2 bounded by ϵm. According to (40), we have +lim +k→∞ ∥g(xk+1)∥2 ⩽ lim +k→∞(maxj,h|ρj,h|)k+1 +Nϕ +� +j=1 +∥ϕj(x0, u0)vj∥2 + ζg + lim +k→∞ ∥δk∥2 ⩽ ζg + ϵm +(41) +Based on (41), g(x) converges till reaching the area Ξ = +� +g(x)|∥g(x)∥2 ⩽ +ζg + ϵm +� +. Thus, the system is BIBO stable. Besides, the Koopman-based LQR +can guarantee asymptotic stability subject to the disturbance in control input +channels under mild conditions. See Section 3.5.2. +3.5.2. Stability Margins of Koopman-Enabled LQR +The discrete-time LQR used in this paper has analytical disc stability mar- +gins [42], within which asymptotic stability subject to the disturbance in control +input channels is guaranteed. Specifically, consider the identified Koopman state +space model described as below: +g(xk+1) = Ag(xk) + Buk + BMuk = Ag(xk) + BKg(xk) + BMKg(xk) += Ag(xk) + B(I + M)Kg(xk) +(42) +where M = diag([m1, m2, . . . m2NDER]) is an introduced diagonal matrix to +represent model uncertainty in control input channels. +In other words, the +introduced matrix parameter M can be used to quantify the uncertainty from +control input channels, whereby one can provide the stability analysis based +on the disc margin for each channel (which will be provided below). K is the +control gain matrix such that uk = Kg(xk) in line with the LQR. +Consider M and g(xk) to be complex-valued to reflect both gain and phase +disturbances. Define a Lyapunov function V (x) = g(x)∗Sg(x) (where S is the +solution of Riccati equation). Based on the Lyapunov function and following +the steps in [42], we provide the disk stability margin for the ith control input +channel in (43) without further explanation (also see Fig. 5). Interested readers +can refer to [42] for the derivation of the disc margin. +1 + mi = +� +αi + jβi : +� +αi − (1 + ri +µ ) +�2 ++ β2 +i +< (1 + ri +µ )2 + ρ − ri +µ +− 1 +� +, +where +i = 1, 2, ..., 2NDER +(43) +where ρ = σmin[Q]/(σmax[K])2 and µ = σmax[BT SB]. σmax[.] and σmin[.] +represent the matrix operation to obtain the maximum and minimum singular +16 + +Figure 5: Disk stability margin for the discrete-time LQR +values, respectively. ri is the ith diagonal element of the cost matrix R. Fig. +5 shows the disc margin, within which the system is asymptotically stable. +Specifically, according to Eq. +(43) and Fig. +5 , the sufficient conditions of +asymptotic convergence against the model uncertainty is: 1 + GL,i < αi < +1 + GU,i for the gain margin and PML,i < arctan βi +αi < PMU,i for the phase +margin, with +GL,i = ri +µ − +� +(1 + ri +µ )2 + ρ − ri +µ +− 1, +GU,i = ri +µ + +� +(1 + ri +µ )2 + ρ − ri +µ +− 1 (44) +and +PML,i = − arccos(1/(1 + ri +µ )) = − arccos +µ +µ + ri +PMU,i = arccos(1/(1 + ri +µ )) = arccos +µ +µ + ri , +for +i = 1, 2, ..., 2NDER. +(45) +4. Case Studies +This section presents case studies based on two MG test systems, namely +a four-bus MG as shown in Fig. 6 and a thirteen-bus MG as shown in Fig. +7, to verify the effectiveness of the proposed Koopman-inspired identification +and control. +The two test systems were established in MATLAB Simulink +2021b. The DERs in the test systems are primary-controlled in different control +modes (grid-forming converters, grid-following converters, and an isochronous- +controlled diesel generator as given in Fig. 6 and Fig. 7) with the inner control +loops modeled in detail. Therefore, the interaction of primary and secondary +control is preserved in simulation to test the effectiveness of secondary control +in realistic setups. The implementation of the converter voltage and current +control inner-loops can be found [9, 24]. +Besides, randomized measurement noises, control time delays, and ambient +perturbations were incorporated into the test systems to mimic practical oper- +ation. The simulation parameters of the two test systems are summarized in +17 + +βi +Disc Stability Margin (1+m;=α;+jβ): +When the model uncertainty is within the +disc (grey area), the system is Lyapunov +asymptotically stable. +Gain Margin +1 +αi +11+ +Phase +1 +Disc Radius : +Margin +μTable 1 and Table 2, respectively. The readers can find more information about +the test systems at https://github.com/nash13123/MG-Test-System.git. +4.1. Identification and Control in the 4-Bus MG Test System +The small 4-bus MG test system was used to test the proposed Koopman- +inspired enhanced OKID with control under load variations and the MG transi- +tion from the grid-connected mode to islanded mode. The DERs at Bus 1 and 3 +are droop-based grid-forming, and the DERs at Bus 2 and 4 are inverse-droop- +based grid-following. At 0.7s, the MG was disconnected from the main grid by +turning off the switch SW, which causes sudden voltage drops and consequent +dynamics. After detecting the sudden change, the secondary control was en- +abled and kept online from 0.8s, i.e., approximately 0.1s lag to mimic a time +delay of islanding event detection in practical applications. +Modeling accuracy of the Koopman-inspired OKID. First, we eval- +uate the modeling accuracy with the one-step-ahead prediction error of the +Figure 6: The MG 4-bus test system +Figure 7: The MG 13-bus test system +18 + +@Utility Grid +R14 +Local +L14 +Local +Busl +DER4 +DER1 +Bus1 +Bus4 +sw +Battery +Rf1 +Vo1 +Bus4 +Re1 +Vo4 +DC/ +R +4 +AC/ +Rfel +n +AC +DC +dc +R +Droop Control +Inverse Droop-Based +(Grid Forming) +Cf1 + Grid-Following Control +P13 +DER2 +Local +Local +DER3 +Battery +Bus2 +Bus3 +Bus2 +Bus3 +Vo2 +R +V。 +Rf3 +R +c3 +AC/ +DC/ +AC +DC +dc +R +Droop Control +Inverse Droop-Based +(Grid Forming) +Grid-Following ControlSecondary Uncontrollable Distributed Resources +PCC: Point of Common Coupling +Utility Grid +MPPT: Maximum Power Point Tracking +Secondary Controllable Distributed Resources +PV: Photovoltaic +Transformer +BESS: Battery Energy Storage System +Load 1 +T1 +DER: Distributed Energy Resource +3 +Grid-Feeding PV +Switch SW1 +Diesel +Farm 1 (MPPT) +PCC +2 +23 +9 +Grid-Following BESS 1 +Load 6 +Grid-Forming DER 2 +SW2 +13 +4 +Load 2 +Load 4 +5 +10 +2 +Grid-Forming DER 1 +Grid-Feeding PV +119 +Grid-Following BESS 2 +6 +Farm 2 (MPPT) +120 +Load 5 +Load 3 +7Table 1: Parameters of the 4-Bus MG Test System +Parameters +Value +Power base Sbase +30kVA +Voltage Base Vbase +480V +Primary control time step Tsp +0.1ms +Secondary control time step Ts +30ms +Sliding window length for estimation N +9 (270ms) +Local Voltage proportional gain KP +0.5 +Local Voltage integral gain KS +523 +Local current proportional gain KP +0.3 +Local current integral gain KS +635 +Frequency droop parameters for DERs 1,2: σω +2.14 × 10−3rad/(W · s) +Voltage droop parameters for DERs 1: σV +1.0 × 10−3V/V ar +Voltage droop parameters for DERs 2: σV +6.3 × 10−3V/V ar +Frequency droop parameters for DERs 3,4: σω +2.83 × 10−3rad/(W · s) +Voltage droop parameters for DERs 3: σV +1.5 × 10−3V/V ar +Voltage droop parameters for DERs 4: σV +9.4 × 10−3V/V ar +PMU measurement noise +N(0, 0.00562) +Control Time delay +N(0.05, 0.0022)s +Ambient perturbation level added to the reference +of DER output voltage and angle: +N(0, 0.012) +Filter resistance Rf1,2,3,4(Ω) +0.1 +Filter inductance Lf1,2,3,4, Lc1,2(mH) +1.35 +Filter capacitance Cf1,2,3,4(µF) +50 +Filter capacitor resistance Rfc1,2,3,4(Ω) +1 +Line resistance Rc1,2(Ω) +0.08 +Line resistance Rc3,4(Ω) +0.09 +Line inductance Lc1,2(mH) +0.35 +Line inductance Lc3,4(mH) +0.45 +Line Resistance Rl1,2,3,4(Ω) +0.15, 0.35, 0.23, 0.17 +Line inductance Ll1,2,3,4(mH) +0.42, 0.33, 0.55, 2.40 +Load PL1,2,3 (active power in kW) +20, 16, 12 +Load QL1,2,3 (reactive powe in kVar) +9, 9, 6 +LQR control parameter qV +1 × 103I +LQR control parameter qsin , qcos +0 +LQR control parameter qω +1 × 10−6I +LQR control parameter rP , rQ +1 × 10−6I +Control input lower bounds ULB +-1.0 kVA +Control input upper bounds UUB +1.0 kVA +Time period for the optimization (28) +TOP T +0.6 s +Time constant of the power low-pass filter +0.02857s +* N(a, b) is the normal distribution with mean of a and variance of b. Control +parameters are designed based on Per Unit. +19 + +Table 2: Parameters of the 13-Bus MG Test System +Parameters +Value +Power base Sbase +150kVA +Voltage Base Vbase +4.16kV +Sliding window length for estimation N +14(420ms) +Droop parameters for all DERs: σω +3.14 × 10−4rad/(W · s) +Droop parameters for all DERs: σV +1.5 × 10−3V/V ar +Ambient perturbation level +N(0, 0.022) +LQR control parameter qV +1 × 103I +LQR control parameter qsin , qcos +0 +LQR control parameter qω +0.01I +LQR control parameter rP , rQ +1 × 10−6I +* Other control parameters are the same to the values in Table 1. +Figure 8: Comparison of the prediction error +voltage magnitude, which is defined as +e(pred) +k+1 += +1 +dim(∆V )∥∆Vk+1 − ∆ ˆVk+1∥ +(46) +where dim[.] represents the dimension of the vector in the bracket, and ∆ ˆVk+1 +represents the predicted voltage magnitude at time step k + 1 by the identified +model of interest. In Fig. 8, we compared the prediction error of two different +ways of modeling: (i) the proposed Koopman-inspired enhanced OKID with +the basis z = [∆V , sin θ − sin(θ∗ +L), cos θ − cos(θ∗ +L), ∆ω]T ; (ii) the conventional +OKID (i.e., linearize the system model in Eq. (11) and apply OKID with γopt +fixed at 1 +2). It was found that the proposed Koopman-inspired enhanced OKID +leads to smaller prediction error than the conventional OKID. These results +show that the salient features of the proposed Koopman-inspired OKID, i.e., the +Koopman nonlinear basis and the adaptive γopt, can ensure a good modeling +accuracy regardless of nonlinearity and uncertainty during large disturbances. +Control results comparison. Fig. 9 compares the voltage and frequency +trajectories with different secondary control methods. As Fig. 9(a)-(c) show, +the bus voltage suddenly drops with incurred transients after the disturbance at +0.7s, which triggers the secondary control to restore the voltage and frequency +to their nominal values (1p.u and 60Hz). Fig. 9(a) shows the control results +20 + +0.5 +(n' +Koopman-InspiredEnhancedOKID(proposed) +0.4 +Conventional OKID +Error +? +0.3 +UO +0.2 +0. +0 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +Time (seconds)(a) The proposed Koopman-inspired enhanced OKID with LQR control +(b) The secondary PI control +(c) The conventional OKID with LQR control (γ = 0.5) +(d) The classical EDMDc (least-squares-based Koopman operator control with +the Koopman observables z) +Figure 9: Voltage and frequency trajectories of the 4-bus MG test system with different +secondary control methods +21 + + (p.u.) +1.05 +0.95 +DER 1 +DER 2 +Voltage +0.9 +DER 3 +DER 4 +0.85 +1 +2 +3 +4 +Time (seconds)61 +59 +DER 1 +DER 2 +58 +DER 3 +DER 4 +57 +1 +2 +3 +4 +Time(seconds)(p.u.) +1.05 +0.95 +DER 1 +DER 2 +Voltage +0.9 +DER 3 +DER 4 +0.85 +1 +3 +4 +Time (seconds)61 +DER 1 +DER 2 +60 +DER 3 +DER 4 +59 +58 +57 +1 +2 +3 +4 +Time(seconds) (p.u.) +1.05 +0.95 +DER 1 +DER 2 +Voltage +0.9 +DER 3 +DER 4 +0.85 +1 +2 +3 +4 +Time (seconds)61 +59 +DER 1 +DER 2 +58 +DER 3 +DER 4 +57 +1 +2 +3 +4 +Time(seconds)Voltage Magnitude (p.u.) +1.05 +0.95 +DER +DER 2 +0.9 +DER 3 +DER 4 +0 +1 +2 +3 +4 +5 +Time (seconds)61 +60 +Frequency +59 +DER +DER +2 +58 +DER +3 +DER +4 +57 +2 +3 +4 +Time (seconds)with the proposed Koopman-inspired enhanced OKID with LQR control; both +voltage and frequency are corrected approximately to the nominal values. For +comparison, Fig. 9(b) presents the voltage and frequency trajectories using sec- +ondary PI control that is tuned with the best effort. The PI control with respect +to the voltage magnitude and the frequency shows a slower response for voltage +restoration compared to the proposed control, and has non-zero steady-state +errors. +It also suffers from a larger frequency deviation as it cannot handle +the voltage-frequency dependence properly. Fig. 9(c) shows the results of con- +ventional OKID with LQR control. In contrast to the proposed method, the +conventional OKID with LQR cannot realize the same fast voltage restoration. +Fig. 9(d) shows the voltage and frequency trajectories of the classical ED- +MDc (i.e., LQR with pseudo-inverse least-squares identification based on the +Koopman observables z). By comparing Fig. 9(d) with Fig. 9(a), we found +that the LQR with the least-squares-based identification cannot perform as well +as the LQR with the proposed Koopman-inspired Koopman-inspired identifica- +tion. These results indicate the effectiveness of the proposed Koopman-inspired +enhanced OKID that possibly results from the two ingredients: (i) the nonlin- +ear basis functions of the Koopman observables proposed in Eq. (14); (ii) the +OKID with adaptive γopt. Both ingredients help better describe the MG sys- +temwide dynamics under big disturbances, realizing more effective control for +both voltage and frequency. +Fig. 10 presents the optimized parameters γopt during control, and Fig. 11 +shows the run time of the proposed Koopman-inspired enhanced OKID at each +Figure 10: Estimated γopt of the proposed Koopman-inspired enhanced OKID +Figure 11: Run time of the proposed Koopman-inspired enhanced OKID +22 + +0.56 +0.54 +opt +0.52 +0.5 +0.48 +0.46 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +Time (seconds)600 +(su) +400 +Time +Run +200 +20ms +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +Time (seconds)time step of secondary control. The run time of the proposed method is about +20ms in case that γopt is not updated, less than the time step of secondary +control (30ms). The run time of the proposed method is around 250-500ms in +case that γopt is updated, which is still less than the time period TOP T = 0.6s +between two updates of γopt. These indicate the feasibility to implement the +proposed Koopman-inspired identification and control online. +4.2. Identification and Control in the 13-Bus MG Test System +To show the performance of the Koopman-inspired enhanced OKID with +LQR control in larger systems for generality, we consider the 13-bus MG test +system presented in Fig. 7, which is adapted from the IEEE 13-node test feeder +[43]. +The DERs at Bus 6 and 9 are droop-based grid-forming. +The BESSs +at Bus 1 and 11 are inverse-droop-based grid-following. +The solar farms at +Bus 3 and 5 are grid-following under MPPT, which are not controllable for +secondary control. The MG system is under transition from the grid-connected +to the islanded modes, and under generation/load variations. At 0.4s, the MG +is disconnected from the main grid by turning off the switch SW1, causing the +sudden drop of voltage with incurred transient. After detecting the islanding +transient, the secondary control is triggered and kept online from 0.5s, i.e., 0.1s +lag to mimic a time delay of islanding detection in practical application. Next, +an active power perturbation of the two solar farms (around 80kW for each) +occurs at 1.0s due to a drop of the solar irradiation from 1000 to 200 W/m2. +Then, a load perturbation happens at the Bus 4 at 1.05s: the consumed active +power increases by 150kW and the consumed reactive power increased by 50kVar +by turning on the switch SW2. +Fig. 12 compares the voltage and frequency trajectories with different sec- +ondary control methods. Fig. 12(a) shows that the proposed Koopman-inspired +OKID with LQR control can correct the voltage and frequency approximately to +the nominal values. For comparison, the voltage and frequency trajectories with +the secondary PI control are shown in Fig. 12(b), which illustrates that the PI +control with the best effort of tuning still fails to realize the stable and accurate +voltage and frequency restoration. Fig. 12(c) shows the results of the conven- +tional OKID with LQR control, which suffers from larger voltage and frequency +oscillations after 1.0s. Fig. 12(d) shows the voltage and frequency trajectories +of classical EDMDc (i.e., LQR with pseudo-inverse least-squares identification +based on the Koopman observables z). +By comparing Fig. +12(d) with Fig. +12(a), we found that the classical EDMDc cannot perform as well as the LQR +with the proposed Koopman-inspired Koopman-inspired identification. These +results further demonstrate the advantages of the proposed Koopman-inspired +OKID with LQR control. Because of the nonlinear Koopman embeddings and +the adaptive γopt, the proposed method can effectively restore the voltage and +frequency to their nominal values despite nonlinearity and uncertainty due to +large disturbances. +23 + +(a) The proposed Koopman-inspired enhanced OKID with LQR control +(b) The secondary PI control +(c) The conventional OKID with LQR control (γ = 0.5) +(d) The classical EDMDc (least-squares-based Koopman operator control with +the Koopman observables z) +Figure 12: Voltage and frequency trajectories of the 13-bus MG test system with different +secondary control methods +24 + +(p.u.) +1.1 +Magnitude +0.9 +DER 1 +Voltagel +0.8 +DER 2 +0.7 +DER 3 +DER 4 +0.6 +0.5 +1 +1.5 +Time (seconds)60.5 +(zH) +60 +DER 1 +59.5 +DER 2 +DER 3 +DER 4 +59 +0.5 +1.5 +Time (seconds)(p.u.) +1.2 +Magnitude +0.8 +0.6 +DER 1 +DER 2 +DER 3 +0.2 +DER 4 +0.5 +1.5 +Time (seconds)60.5 +(zH) +60E +59.5 +59 +DER 1 +DER 2 +58.5 +DER 3 +DER 4 +58 +0.5 +1.5 +Time (seconds)(p.u.) +1.1 +Magnitude +0.9 +DER 1 +Voltagel +0.8 +DER 2 +0.7 +DER 3 +DER 4 +0.6 +0.5 +1.5 +Time (seconds)60.5 +(ZH) +59.5 +DER 1 +DER 2 +DER 3 +DER 4 +59 +0.5 +1.5 +Time(seconds)MA +0.9 +0.8 +DER +DER 2 +0.7 +DER 3 +DER 4 +0.6 +0 +0.5 +1 +1.5 +Time (seconds)60.5 +(ZH) +60 +Frequency ( +59.5 +DER +DER +2 +DER 3 +DER +4 +59 +0 +0.5 +1 +1.5 +Time (seconds)5. Conclusions +This paper proposed a data-driven Koopman-inspired identification and con- +trol method for MG secondary voltage and frequency control. The proposed +method requires no knowledge of network information and primary controllers. +It requires no warm-up training yet with guaranteed BIBO stability and even +asymptotic stability under some mild conditions. In this method, a Koopman +operator-inspired enhanced OKID (observer Kalman filter identification) algo- +rithm is proposed, whereby the Koopman state space model is estimated online +and used for control to handle microgrid nonlinearity and uncertainty adap- +tively. Case studies in the 4-bus and 13-bus MG test systems (with different +converter control modes) demonstrate the effectiveness and robustness of the +proposed Koopman-inspired identification and control method subject to mode +transitions, varying operating conditions, measurement noises and time delays. +References +[1] K. P. Schneider, N. Radhakrishnan, Y. 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Kerst- +ing, Analytic considerations and design basis for the ieee distribution test +feeders, IEEE Transactions on Power Systems 33 (3) (2018) 3181–3188. +doi:10.1109/TPWRS.2017.2760011. +29 + diff --git a/59AzT4oBgHgl3EQff_xu/content/tmp_files/load_file.txt b/59AzT4oBgHgl3EQff_xu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..19f6125d3842677140a4597b8dfe7e7ba8c7249c --- /dev/null +++ b/59AzT4oBgHgl3EQff_xu/content/tmp_files/load_file.txt @@ -0,0 +1,1271 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf,len=1270 +page_content='A Novel Koopman-Inspired Method for the Secondary Control of Microgrids with Grid-Forming and Grid-Following Sources Xun Gong, Xiaozhe Wang∗ Department of Electrical and Computer Engineering, McGill University, 3480 Rue University, Montreal,H3A 0E9, Quebec, Canada Abstract This paper proposes an online data-driven Koopman-inspired identification and control method for microgrid secondary voltage and frequency control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Unlike typical data-driven methods, the proposed method requires no warm-up train- ing yet with guaranteed bounded-input-bounded-output (BIBO) stability and even asymptotic stability under some mild conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The proposed method estimates the Koopman state space model adaptively so as to perform effective secondary voltage and frequency control that can handle microgrid nonlinearity and uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Case studies in the 4-bus and 13-bus microgrid test systems (with grid-forming and grid-following sources) demonstrate the effectiveness and robustness of the proposed identification and control method subject to the change of operating conditions and large disturbances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', microgrid mode transitions, generation/load variations) even with measurement noises and time delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Keywords: data-driven control, adaptive Koopman-inspired identification, microgrid secondary control, grid-forming, grid-following, Koopman operator control, observer Kalman filter identification 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Introduction The microgrids (MGs) are small local grids that can disconnect from the bulk grid to operate independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The MGs facilitate the integration of sus- tainable distributed energy resources (DERs) like wind, solar as well as energy storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Nonetheless, the DERs are interfaced with microgrids by power con- verters, making MGs low-inertia or even inertia-less [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' In addition, MGs are characterized by frequency-voltage dependence due to low X/R ratios as ∗Corresponding author 1This work was supported by the Fonds de Recherche du Quebec-Nature et technologies under Grant FRQ-NT PR-298827 and NSERC ALLRP 571554 - 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Preprint submitted to Applied Energy January 5, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='01461v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='SY] 4 Jan 2023 opposed to conventional power systems [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Therefore, the frequency and voltage of MGs tend to experience coupled large deviations subject to volatile operation conditions of generation and load, transitions between islanded and grid-connected modes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The hierarchical control is commonly adopted to maintain the MG’s voltage and frequency stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The hierarchical control includes primary control at the individual DER level, and secondary/tertiary control at the systemwide level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Even though the droop-based primary control at individual DERs can coordinate the power of DERs in a decentralized manner and improve the local stability, the frequency and voltage deviation at the system level may not be eliminated by merely using primary control [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The system stability may even be compromised when the droop gains are improperly designed to high values [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Hence, the secondary control is essential to achieve stable voltage and frequency restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The scope of the paper lies in the secondary control, aiming to restore fre- quency and voltage for islanded MGs under larger disturbances or MGs in mode transitions(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', from grid-connected to islanded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The secondary control of MGs can be classified as model-based and model-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' There have been many research papers on model-based control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' For example, the small-signal models have been used in [3, 6] to regulate droop gains and improve the systemwide small-signal stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' To handle large disturbance, multi-agent distributed co- operative control with feedback linearization was proposed in [7, 8] to deal with the nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Despite advancement, all the aforementioned methods rely heavily on accurate physical models that may not always be available to MG operators due to time-varying topologies and operating conditions, as well as high uncertainty introduced by volatile renewables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' To relax the pre-knowledge of accurate models, researchers have designed various model-free control methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' A common model-free method is Propor- tional and Integral (PI) control [4, 9, 10], which nevertheless may lack online adaptiveness to compensate for uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Besides, the MG may suffer from high starting overshoot, high sensitivity to controller gains, and sluggish re- sponse to disturbances if the PI control is not properly tuned [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Another category of MG secondary control method is the averaging/consensus-based secondary droop control [11–13] that targets on accurate power sharing in quasi steady-state rather than voltage and frequency stability under large distur- bances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' To improve systemwide voltage and frequency stability under both small and big disturbances, machine learning based methods were proposed [9, 14–18] for secondary voltage and frequency control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' However, the universal learning machines such as artificial neural network (ANN) and reinforcement learning (RL) may lack physical interpretability and thus reliability of repre- senting the system’s dynamics in diverse topologies and operating conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Obtaining adequate offline training data that can sufficiently represent the sys- tem dynamics is challenging too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Moreover, individual DERs can be either controllable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', energy storage systems (ESSs), renewable energy with ESSs), or non-controllable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', renew- able generation operating under maximum power point tracking (MPPT)) at the 2 secondary level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' They possess diverse modes of primary control (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', conven- tional isochronous grid-forming, power-based grid-forming and grid-following).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The resulting model complexity may affect the performance of the secondary control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Yet all the aforementioned works (both model-based and model-free) on secondary voltage and frequency control assume that all DERs in islanded MGs work under the grid-forming or voltage control mode [5, 7–10, 15–17, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' However, in existing MGs, the mix of grid-forming and grid-following control with diverse control structures and parameters introduces uncertainty that chal- lenges MG secondary control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Particularly, when large disturbances occur, the interaction among diverse grid-forming and grid-following converters and the dynamics of the affiliated phasor-locked loops (PLLs) may deteriorate the sys- tem stability and control performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' In this paper, we propose a new data-driven secondary voltage and frequency control method for MGs with both grid-forming and grid-following DERs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The method is able to handle MG nonlinearity and uncertainty (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', MG mode transitions from grid-connected to islanded, generation and load variations) in an adaptive data-driven fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The proposed method requires no offline training and uses only a small window of phasor angle and voltage data from synchrophasors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', micoPMUs) at the DER output ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' In the proposed method, Koopman operator theory [20] is leveraged to convert the nonlinear dynamical system into a linear one under Koopman embedding mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' As such, the system can be identified and controlled with mature and powerful linear system techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Particularly, we tailor the OKID (Observer Kalman- filter IDentification)-based algorithm so that the Koopman-based linear dynam- ical system can be identified optimally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Then, the discrete-time linear quadratic regulator (LQR) is applied to the identified Koopman-based linear dynamical system with well-characterized stability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' It is noteworthy that M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Korda et al [21] utilized Koopman operator control for power system transient stability and control, while the method required offline training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Besides, the identification based on the brute-force least-squares estimation, could lead to unsatisfactory identification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Gong et al [22] presented a combined ap- plication of the Koopman operator and identification method for MG secondary control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' However, the method assumes that the droop parameters of DERs are known by the secondary controller, whereby the control matrix in the Koopman state space can be directly obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' In this paper, we lift the assumption that the local control mechanism and parameters are fully unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' In short, the advantages of the proposed method are summarized as below: (i) The proposed Koopman-inspired enhanced OKID method can help identify the system dynamics accurately and adaptively due to the capacity of dealing with nonlinearity and uncertainty under large disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (ii) The proposed Koopman-inspired identification and control method is purely data-driven using only a small window of synchrophasor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' It requires no knowledge of network information and primary controllers, and no offline train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (iii) The MG system with the proposed Koopman-inspired identification and control is guaranteed to be bounded-input-bounded-output (BIBO) stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' On 3 Figure 1: Microgrid control architecture top of the BIBO stability, the sufficient condition under which the MG system is asymptotically stable is also developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (iv) The proposed control method is robust to measurement noises and time delays as tested in numerical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The remainder of the paper is organized as follows: Section II describes the MG hierarchical control and the interfaces between secondary and primary control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Section III details the proposed Koopman-inspired identification and control method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Section IV presents case studies for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Section V concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Microgrid System Description A MG can be controlled hierarchically with the secondary and primary con- trol as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The primary controllers enable fast response of the individual DERs to guarantee local stability, while the secondary controller globally dictates the primary controllers of controllable DERs according to the data collected from microPMUs, whereby the systemwide interaction dynamics of MGs can be handled and the voltage and frequency can be restored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The local primary control modes can be different, such as grid-forming or grid-following [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' In an islanded MG or future power system without syn- chronous generators, at least one DER is required to work under the grid-forming mode to actively form the grid voltage and frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' then the rest of DERs can remain operating under the grid-following modes [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' As discussed in [24, 25], grid-forming DERs define the voltage magnitude and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' In contrast, grid-following DERs follow the measured frequency and voltage magnitude in the grid via PLL, which represents the prevalent type of control strategy for grid-connected PV and wind converters in existing power grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' As droop is commonly used in MGs, we consider the droop-based grid-forming [26, 27] and 4 Control Command Microgrid Node/Bus Measurement Data Sensing Microgrid Distribution Line Microgrid Network Secondary Control Primary Control Control Center DER 1 Adaptive Online MicroPMUData Identification Real-Time MicroPMU Control Cvber Data Algorithms Network DER 2 Primary Control Primary Control MicroPMUDatainverse-droop-based grid-following control [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Specifically, consider a DER at the bus i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 2(a), the droop for grid-forming converter control is defined as: Droop : � ωi − ω∗ i Vi − V ∗ i � = � −σω(Pi − P ∗ i ) −σV (Qi − Q∗ i ) � (1) where ωi and Vi denote the frequency and voltage magnitude for the grid- forming control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' ω∗ i and V ∗ i are the rated frequency and voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The parameters σω and σV are frequency and voltage droop gains, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' P ∗ i and Q∗ i are the reference power in the droop, which can be the steady-state power without the secondary control or an augmented reference power after the secondary control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Pi and Qi are the active and reactive power, which are measured with a low-pass filter embedded in the power measurement block in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The filter is in the form of [29, 30]: Pi = 1 Tfs + 1P (IN) i , Qi = 1 Tfs + 1Q(IN) i (2) where Tf is the time constant of the first-order low-pass filter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' P (IN) i and Q(IN) i represent the active and reactive power before filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Generally, the filter is required to attenuate high-frequency dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', harmonics) and preserve low-frequency dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', sub-synchronous components which can be fur- ther managed by secondary control).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' With the secondary control, the reference power P ∗ i and Q∗ i of the droop in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 2 was updated in discrete time as: P ∗(+) i = P ∗ i + ∆P ∗ i , Q∗(+) i = Q∗ i + ∆Q∗ i (3) To distinguish P ∗ i and Q∗ i before and after secondary control, the superscription (+) is added in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (3) to denote the values of P ∗ i and Q∗ i after considering the secondary control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Similarly, if the DER at the bus i is grid-following as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 2(b), the inverse droop control is defined as: Inverse Droop : � ¯Pi ¯Qi � = � − 1 σω (ωi − ω∗ i ) − 1 σV (Vi − V ∗ i ) � (4) where ¯Pi and ¯Qi are the power generated by the inverse droop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The eventual real power Pi and reactive power Qi sent to the grid-following control as references are: Pi = ¯Pi + P ∗ i , Qi = ¯Qi + Q∗ i (5) where P ∗ i and Q∗ i are reference power guided by the secondary control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' In a similar form to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (3), P ∗ i and Q∗ i was updated in discrete time as: P ∗(+) i = P ∗ i + ∆P ∗ i and Q∗(+) i = Q∗ i + ∆Q∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Consequently, the droop for grid-forming control and the inverse droop for the grid-following control can be represented as: Droop: �ωi − ω∗ i Vi − V ∗ i � = � −σω(Pi − P ∗(+) i ) −σV (Qi − Q∗(+) i ) � = �−σω(Pi − P ∗ i ) −σV (Qi − Q∗ i ) � + �σω σV � ui, with ui = �∆P ∗ i ∆Q∗ i � (6a) 5 (a) (b) Figure 2: Different primary control modes: (a) droop-based grid-forming control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (b) inverse- droop-based grid-following control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Inverse Droop: � Pi − P ∗(+) i Qi − Q∗(+) i � = �− 1 σω (ωi − ω∗ i ) − 1 σV (Vi − V ∗ i ) � ⇒ �ωi − ω∗ i Vi − V ∗ i � = �−σω(Pi − P ∗ i ) −σV (Qi − Q∗ i ) � + �σω σV � ui (6b) According to (6a)-(6b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' both the droop and the inverse droop take the same form: � ˙θi ˙Vi � = � −σω(Pi − P ∗ i ) − σV τV (Qi − Q∗ i ) � + �σω σV τV � ui (7) where ˙θi = ωi − ω∗ i (8) Pi = n � j=1 ViVj(Gij cos(θi − θj) + Bij sin(θi − θj)) (9) Qi = n � j=1 ViVj(Gij cos(θi − θj) − Bij sin(θi − θj)) (10) τV is the equivalent time constant of voltage magnitude dynamics due to the grid-forming or the grid-following control loops,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' which can be treated as a first- order inertia system when properly tuned;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' u denotes the external control inputs due to secondary control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' θ is the voltage phasor angle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' j denotes the bus number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Gij and Bij represent the equivalent conductance and susceptance between bus i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 2, the droop generates the voltage reference for the grid- forming control system, and the inverse droop generates the power references for the grid-following control system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Note that the grid-forming control based on a PLL is adopted in this paper to mitigate negative impacts on systemwide stability [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The dynamics of the grid-forming and grid-following loops are not presented in detail but will be considered in all simulations presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Interested readers are referred to [24, 27] for the detailed modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Uncertainty from Grid-Forming Control: As the local converter control is much faster than the secondary control, the voltage reference fed to grid-forming control in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 2(a) is approximately equal to ω and V assuming that the grid- forming control loop is well tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Nonetheless, when large disturbances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 6 Primary Power io Control I Measurement Droop Local 3 Vo Grid-Forming Bus w = *-O(P- P*) DER V V = V* - v(Q - Q*) Control p*Q* I Updated by Secondary ControlPrimary Control Voltage Magnitude Inverse droop w* Measurement 30 w (estimated by PLL) V* 0 A- 7 40 io Local P Grid-Following Bus P=P+P* Control DER Q Q=Q +Q* (with PLL) p* 0* Updated by Secondary ControlMG transitions from the grid-connected mode to the islanded mode,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' volatile generation and load) occur that cause large power perturbations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' the nonlin- earity driven by the system power flows (8)-(10) and by the control interaction between the droop module and the grid-forming control loops can emerge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' thus leading to modeling uncertainty in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Uncertainty from Grid-Following Control: Likewise, the control interaction between the inverse-droop module and the grid-following control loops may emerge when there are system disturbances causing big perturbations to the angle and voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' In addition, when there are large disturbances or measure- ment noises that make the grid voltage measurement distorted, the uncertainty due to the PLL can also directly introduce the modeling error to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (7) [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' To describe the uncertainty from either the grid-forming or the grid-following control, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (7) can be modified as � ˙θi ˙Vi � = � −σω(Pi − P ∗ i ) − σV τV (Qi − Q∗) � + �σω σV τV � ui + �fω(P, Q, θ, V ) fV (P, Q, θ, V ) � (11) where fω(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=') and fV (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=') are unknown nonlinear functions to describe the residual dynamics for the voltage phasor angle and magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The aforementioned nonlinearity and uncertainty pose challenges to the con- ventional secondary control of MGs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', model-based ones and PI) especially under large disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' To address these challenges, we propose a Koopman- inspired method that can help identify the system accurately and adaptively using data despite nonlinearity and uncertainty such that effective control can be designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Koopman-Inspired Identification and Control 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Koopman Operator Theory Koopman operator theory [20] shows that a nonlinear dynamical system can be transformed into an infinite-dimensional linear system under a Koopman em- bedding mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The Koopman-enabled linear model is valid for global non- linearity with the infinite-dimensional representation as opposed to traditional locally linearized small-signal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' However, in practice, one can consider finite-dimensional Koopman invariant subspaces where dominant dynamics can be described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Particularly, given a nonlinear dynamical system with external control xk+1 = F(xk, uk), where x ∈ M and u ∈ U with M and U being the manifolds of state and control input, we consider the Koopman embedding map- ping Φ from the two manifolds to a new Hilbert space Φ : M×U → H, which lies within the span of the eigenfunctions ϕj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' That is, Φ(x, u) = �Nϕ j=1 ϕj(x, u)vj, where Φ(x, u) = [Φ1(x, u), Φ2(x, u), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' , Φi(x, u), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' , Φp(x, u)]T is a set of Koopman observables, vj are the vector-valued coefficients called Koopman modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The Koopman operator K, acting on the span of ϕj, advances the embeddings Φ(x, u) linearly in the Hilbert space H as [20]: Φ(xk+1, uk+1) = KΦ(xk, uk) = K Nϕ � j=1 ϕj(x, u)vj = Nϕ � j=1 (ρjϕj(xk, uk)vj) (12) 7 where ρj are the eigenvalues satisfying Kϕj(x, u) = ρjϕj(x, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' To be con- sistent with the linear form of control inputs in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (11), we assume that Φi(x, u) = gi(x)+li(u) where gi(x) is a nonlinear observable function and li(u) is linear with li(0) = 0 [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' In addition, we assume Φi(xk+1, 0) = KΦi(xk, uk) for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Then, gi(xk+1) + li(0) = Kgi(xk) + Kli(uk) ⇒ gi(xk+1) = Kgi(xk) + Kli(uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' This assumption means that the Koopman operator is only attempt- ing to propagate the observable functions at the current state xk and inputs uk to the future observable functions on the state xk+1 but not on future in- puts uk+1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', [∆P ∗, ∆Q∗]T are not state-dependent) [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Let us define z := g(x) = [g1(x), g2(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' , gi(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' gp(x)]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Then we have an approxima- tion of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (11) in a form of extended dynamic mode decomposition with control (EDMDc) [32] as below (Process model) zk+1 = Azk + Buk + δk (13a) (Observation model) yk = Czk + ek (13b) where yk are the outputs of the Koopman state space model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' We define yk = [dθk, dVk]T = [θk − θ∗ L, Vk − V ∗ L]T in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (13b) as the PMU-measured phasor angle and voltage magnitude deviations from the local operation points [θ∗ L, V ∗ L]T that are the first data sample from a window of collected PMU data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' A and B are the state transition matrix and control matrix, satisfying that Azk = Kg(xk) and Buk = Kl(uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' δk is the Koopman modeling error associ- ated with the EDMDc approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' ek is the observation model error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Given proper Koopman observables z, the Koopman state space model (13a)-(13b) can describe large signal-driven nonlinear dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' That being said, under the Koopman embedding z = g(x), the nonlinear dynamical system (11) can be represented by the linear dynamical system (13a)-(13b) that is valid under both small and large perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' There are three consecutive tasks to use this model for control: determination of Koopman observables, online identifi- cation of the Koopman state space model, and implementation of linear control (illustrated in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='4, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Koopman Observables for MG Secondary Control The selection of Koopman observables is important for realizing accurate modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The observables can be selected either empirically [21, 22, 33] or with the help of machine learning techniques [34–36], while it remains an open question to obtain the best possible observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' In this paper, we selected the Koopman observables based on our experience and domain knowledge of power systems and microgrids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (7)-Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (10), sinusoidal-driven interaction dynamics may emerge when subject to large perturbations and low inertia (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', the general solution for the droop-control differential equations contains trigonometric patterns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Inspired by this, we include the functions sin θ and cos θ into the Koopman embedding to describe such underlying dynamics, which were shown effective to describe interaction transients of power grids [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Thus, let us define the MG original states xk = [θk, Vk]T and the Koopman real- valued observables zk = g(xk) as: zk = g(xk) = [∆Vk, sin θk − sin(θ∗ L,k), cos θk − cos(θ∗ L,k), ∆ωk]T (14) 8 where ∆V and ∆ω are voltage and angular frequency deviations from the nominal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' θ∗ L,k represents the approximate underlying operation point of voltage phasor angle at time step k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The Koopman observables z constitute the Koopman state space in the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (13a)-Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (13b), where the parameter matrices A, B and C are to be determined by an advanced system identification method online as described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Online Identification: A Koopman-Inspired Enhanced OKID Algorithm Considering the Koopman-based linear dynamical system model (13a)-(13b), we propose an observer Kalman filter identification (OKID)–based optimization algorithm to optimally identify the MG Koopman state space model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', the matrix parameters A,B and C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The OKID Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Belonging to the category of closed-loop subspace methods, the conventional OKID algorithm is commonly used to identify linear systems [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' It is free of the bias problem that most typical closed-loop subspace methods have [37], and has been applied in many areas such as aircraft control and autonomous underwater vehicles [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' In OKID, the impulse response of the system is estimated in a least-squares fashion with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Then, a state space model of the system is obtained with the eigensystem realization algorithm (ERA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Specifically, let Y and U represent the matrix stacking the time series data of the outputs y and the control inputs u in a matrix form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Let Yi and Ui represent the observation outputs and the control inputs at the ith time step in the data matrix, and consider the length of the sliding window is N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' By observing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (13a)-Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (13b) and assuming zero initial conditions, yk can be expressed with iterations in a form of yk = Czk = C(Azk−1 + Buk−1) = C(A(Azk−2+Buk−2)+Buk−1) = CAk−1Bu0+CAk−2Bu1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='CBuk−1 = �k−1 i=0 CAk−i−1Bui, whereby we obtain Y = � CB · · CAN−1B � � ���� U0 U1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' UN−1 0 U0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' UN−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' U0 � ���� (15) Let h denote the impulse response of the Koopman state space model (13a)- (13b) in the sliding window of size N (from k = 1 to k = N) with zero initial conditions (x0 = 0) and impulse inputs (u0 = 1 and uk = 0 when k > 0), we have h = � h1 h2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' hN � = � CB CAB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' CAN−1B � (16) Then according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (15)- (16) and with the knowledge of the observation matrix Y and the control input matrix U, one can estimate the impulse response in a least-squares fashion � h1 h2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' hN � = Y � ���� U0 U1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' UN−1 0 U0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' UN−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' U0 � ���� † (17) 9 where the operator † represents the Moore-Penrose pseudo-inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Note that the noise is not optimally filtered by the least-squares inverse as presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' To address the issue, the conventional OKID can be designed based on an optimal observer system whereby optimal system parameters can be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' For simplicity, we refer readers to [39] (Pages 340-343) for detailed explanation and implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Next, with the obtained impulse response, the Hankel matrix H and the next-step Hankel matrix H ′ can be written as follows: H = � ���� h1 h2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' hN 0 h1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' hN−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' h1 � ���� , H ′ = � ���� h2 h3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' hN+1 0 h2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' hN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' h2 � ���� (18) The Hankel matrix H could be truncated with Singular Value Decomposition (SVD): H = UΣVT = [�U, U tr] ��Σ 0 0 Σtr � � �V T VT tr � ≈ � U �Σ �V T (19) Let O = [C, CA, CA2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', CAN−1]T (20) be the observability matrix, and C = [B, AB, A2B, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', AN−1B] (21) be the controllability matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Then, by observing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (16) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (18), we have H = OC, H ′ = OAC (22) Furthermore, considering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (19), we can assume that O = �U �Σγ and C = �Σ1−γ �V T , where γ is an arbitrary real value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Conventional OKID algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' For the conventional OKID algorithm, ERA is thereafter used to identify the matrix A and B, with γ set to a constant 1 2 for a special balanced realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' That is, one can assume O = �U �Σ 1 2 and C = �Σ 1 2 �V T , whereby a state space model with balanced Grammians is realized (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', the same degree of controllability and observability) that agrees with the control input and the observation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' As such, with γ = 1 2 and by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (20) - Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (22), the matrices A and B can be identified by the conventional OKID as follows [39]: � A = �Σ− 1 2 �U T H ′ �V �Σ− 1 2 (23a) � B = CNS×NU = � �Σ 1 2 �V T � NS×NU (23b) where the operator �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='� NS×NU represents the first NS rows and the first NU columns of the matrix in the bracket;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' NS is the dimension of Koopman embed- ding space and NU is the dimension of control inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 10 In this paper, to better identify the Koopman-based process dynamics, we propose a Koopman-inspired algorithm to find an optimal γ rather than assum- ing γ = 1 2 as in the conventional OKID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Consider a general form with γ unfixed �U �Σγ = O = [C, CA, CA2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', CAN−1]T = IN×N ⊗ C · [I, A, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', AN−1]T (24a) �Σ1−γ �V T = C = [B, AB, A2B, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', AN−1B] = [I, A, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', AN−1](IN×N ⊗ B) (24b) where IN×N is the identity matrix with the dimension N × N, and ⊗ denotes the Kronecker product which is IN×N ⊗ C = � �� C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' C � �� , IN×N ⊗ B = � �� B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' B � �� (25) Then (IN×N ⊗ C)† �U �Σγ = [I, A, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', AN−1]T (26a) �Σ1−γ �V T (IN×N ⊗ B)† = [I, A, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', AN−1] (26b) By observing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (26a) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (26b), we have �Σγ �U T ((IN×N ⊗ C)†)T = �Σ1−γ �V T (IN×N ⊗ B)† ⇒ �Σ2γ−1 �U T � (IN×N ⊗ C)†�T = �V T (IN×N ⊗ B)† (27) Treating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (27) as a soft constraint for the parameter γ, one can formulate a quadratic optimization problem to solve the optimal parameter γopt: γopt = arg min γ ∥�Σ2γ−1 �U T � (IN×N ⊗ C)†�T − �V T (IN×N ⊗ B)∥F subject to: 0 ≤ γ ≤ 1 (28) where ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='∥F represents the Frobenius norm of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The inequality 0 ≤ γ ≤ 1 is added to constrain problem complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The novel OKID-based algorithm for parameter estimation is summarized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The flowchart of the algorithm is also presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The Proposed Online Koompan-Inspired Enhanced OKID Algorithm Algorithm Initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Initialize γopt = γopt,0, the smoothing factor η, and the time step TOP T between two updates of γopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The selection of these parameters will be discussed in Remarks after the presentation of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' At each time step of identification and the secondary control, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', conduct Step 1 -Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Step 1: Data preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Collect the last N data samples from microPMUs to obtain the data matrices of phasor angle Θ, voltage deviation ∆V and angular frequency deviation ∆Ω from the nominal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Collect control input 11 data U from the secondary controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' For example, the phasor angle Θ is stacked in a form of Θ = � � | | | Θ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' ΘN | | | � � (29) ∆V , ∆Ω and U are formed in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The approximated operation points of voltage phasor angles and magnitudes Θ∗ L and V ∗ L are defined as the first data sample from a window of collected PMU data, prepared in a matrix form as follows: Θ∗ L = � � | | | Θ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Θ1 | | | � � , V ∗ L = � � | | | V1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' V1 | | | � � (30) Prepare the data matrices for y and z as follows: Y = [Θ − Θ∗ L, V − V ∗ L ]T , and Z = [∆V, sin (Θ) − sin (Θ∗ L), cos (Θ) − cos (Θ∗ L), ∆Ω]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Step 2: Hankel matrix preparation and SVD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Estimate the impulse re- sponse and prepare the Hankel matrices according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (17)-Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Con- duct the SVD on the obtained Hankel matrix H ≈ �U �Σ �V T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Step 3: Estimation of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Ignoring the error term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (13b), we have Y = CZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Thus, one can estimate the observation matrix C at each time step k in a least-squares fashion by multiplying the pseudo-inverse on both sides of the equation, which is � Ck = Y Z† (31) Step 4: Optimization for γopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Check if the run time of optimization between the last update of γopt is larger than TOP T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' If no, γopt,k = γopt,k−1, go to Step 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' otherwise, solve the optimization problem in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (28) for γ− opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' To do so, by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (23b), replace B with � �Σ1−γ �V T � NS×NU and replace C with � Ck from Step 3 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Then, adaptively update γopt by γopt,k =ηγ− opt + (1 − η)γopt,k−1, for k = TOP T , 2TOP T , 3TOP T , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (32) where γ− opt is the optimal value of the realization parameter γ according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' That is, once γ− opt is updated, we update γopt with the weighted sum of the old γopt at last time step and the updated value γ− opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' η is the weight to smooth online learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The role of η is to smooth the online learning of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' As the small piece of online data used for identification is characterized by stochasticity, the smoothing factor η can mitigate aggressive change to make the learning process more reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' This is so because the estimation is equivalent to the Robbins–Monro form [40], which is γopt,k = ηγ− opt + (1 − η)γopt,k−1 = γopt,k−1 + η(γ− opt−γopt,k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The larger the value of γ is, the smoother the learning process tends to be, whereas the adaptiveness of learning is compromised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Step 5: Estimation of A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (23a)-(23b) � Ak = � η �Σ−γopt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='k �U T H ′ �V �Σγopt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='k−1 + (1 − η) � Ak−1 if k ≥ 1 �Σ−γopt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='k �U T H ′ �V �Σγopt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='k−1 if k = 0 (33) 12 � Bk = � � � � � η � �Σ1−γopt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='k �V T � NS×NU + (1 − η) � Bk−1 if k ≥ 1 � �Σ1−γopt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='k �V T � NS×NU if k = 0 (34) After implementing the identification algorithm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' the identified Koopman state space model at the time step k is obtained as: zk+1 = � Akzk + � Bkuk (35a) yk = � Ckzk (35b) Compared to the traditional EDMDc used in power systems [21],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' the pro- posed Koopman-inspired OKID can use the observation data y as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (13b) to help learn the Koopman state space model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (13a), while the traditional EDMDc only estimates the Koopman state space in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (13a) in a least-squares fashion without the incorporation of observation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The fusion of the infor- mation from the observation data provides extra opportunities to enhance the modeling efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Remarks γopt: in this paper, γopt,0 = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Thus the enhanced OKID is initially equiv- alent to the conventional one while it gradually learns the optimized value for γopt with the online OKID and the periodically enabled optimization in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The smoothing factor η: it is used to weigh the past estimations and the latest one, and set to 1 N in this paper with the assumption that all estimations have the same weight independent on the time of occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' A larger η means the estimation put more weight on the newest data, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The time step TOP T for updating γopt: it is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='6s, which is longer than the run time of the proposed Koopman-inspired enhanced OKID and the time step of secondary control (30ms) as detailed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' A small TOP T is favorable as a fast update of γ to compensate for the uncertainty of the Koopman process model (13a), while it should be longer than the run time of the optimization (28) to ensure the feasibility of online implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The Linear Control Based on the Koopman-Inspired Enhanced OKID After obtaining the identified model (35a) - (35b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' a discrete-time linear quadratic regulator (LQR) is applied at each time step of secondary control,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' aiming to reduce the voltage and frequency deviations by minimizing the cost J(u) = ∞ � k=0 zT k Qzk + uT k Ruk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' subject to zk+1 = � Azk + � Buk (36) 13 Figure 3: Algorithm flowchart of the proposed Koopman-inspired enhanced OKID where Q and R are cost matrices defined as: Q = � ��� qV qsin θ qcos θ qω � ��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' R = �rP rQ � (37) where qV ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' qsin θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' qcos θ and qω are cost submatrices for the Koopman observ- ables presented in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' rP and rQ are cost submatrices for the control signals ∆P ∗ and ∆Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' They are basically selected empirically in this paper based on which factor is treated to be more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The optimal control input can be obtained by: uk = � � � ULB uk < ULB −Kzk ULB ⩽ uk ⩽ UUB UUB uk > UUB with K = ( � BT S � B + R)−1 � BT S � A (38) where K is the control gain matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' S is the solution of Riccati equation [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' UUB and ULB are the upper and lower saturation limits that can bound the uncertainty introduced by control inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The bounds are user-defined values, which are determined empirically in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Usually, large bounds can lead to faster response whereas the uncertainty introduced through control input channels could be increased to an unmanageable level that degrades the dynamic control performance or even stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' On the other hand, the bounds cannot be set to too small values, otherwise, the response could be slow and the capability of the controller cannot be fully taken use of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The Koopman-inspired enhanced OKID illustrated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='3 and the LQR illustrated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='4 can be respectively applied to the identification block and the control algorithm block of secondary control in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Specifi- cally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 4 presents the proposed online identification and control structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The stability of such Koopman-inspired identification and control is guaranteed, which is proved in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Stability Analysis MG dynamics can be expressed in a Koopman-based structure and can be approximated with the online Koopman-inspired identification in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The approximation error is bounded but often not quantifiable as it depends on 14 At each time Steps 1-2 Step 3 Step 4 Step 5 step k: No Keep Old opt Data Matrix Initialization Estimate Ck If k = ToPT, 2ToPT, : Preparation & SVD OKID (update Ak and Bk) Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Update Yopt EndFigure 4: Online structure of the proposed Koopman-inspired enhanced OKID and control the appropriateness of Koopman observables and the online parameter identi- fication algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' In what follows, we aim to prove stability properties in a general sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Proof of BIBO Stability We prove that the proposed Koopman-inspired OKID-based control is BIBO (bounded-input-bounded-output) stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Denoted by ˆxk+1 the one-step-ahead prediction of the state vector x at the time step k with the OKID-based es- timation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Denoted by ˆKk the estimated Koopman operator at time step k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (12), we have g(ˆxk+1) = Φ(ˆxk+1, 0) = ˆKkΦ(xk, uk) = ˆKk Nϕ � j=1 ϕj(xk, uk)vj = Nϕ � j=1 (ρj,kϕj(xk, uk)vj) (39) where ρj,k is the eigenvalue corresponding to the jth eigenfunction ϕj for the estimated Koopman operator ˆKk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Recall that Φ(x, u) = g(x) + l(u) discussed in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='A, where l(u) = [l1(u), l2(u), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' lp(u)]T and l(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Then g(xk+1) = g(ˆxk+1) + δk = ˆKkΦ(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' uk) + δk = ˆKk(g(xk) + l(uk)) + δk = ˆKk( ˆKk−1Φ(xk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' uk−1) + δk−1 + l(uk)) + δk = ˆKk( ˆKk−1(g(xk−1) + l(uk−1)) + δk−1 + l(uk)) + δk = ˆKk( ˆKk−1( ˆKk−2Φ(xk−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' uk−2) + δk−2 + l(uk−2)) + l(uk−1)) + δk−1 + l(uk)) + δk = · · · = k � h=0 ˆKk−hΦ(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' u0) + k � h=1 k � i=h ˆKk−i+h(δh−1 + l(uh)) + δk = Nϕ � j=1 ( k � h=0 ρj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='h)ϕj(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' u0)vj + k � h=1 k � i=h ˆKk−i+h(δh−1 + l(uh)) + δk 15 Microgrid Network Control Command Microgrid Node/Bus (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', four-bus, thirteen-bus microgrids) Measurement Data Sensing Microgrid Distribution Line Action:(control inputs)u=[△P",Q*jr Primary Control Secondary Controller: DER 1 Proposed Koopman-Inspired Enhanced OKID and LQR Control Control MicroPMU Data Action Generator (LQR) Online Minimize the Cost Function Identification J(u) =z Qz +utRui MicroPMUl Control Measurement: ak = jox, VaJT Cyber Koopman Embedding Mapping g : Data [Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='4] Inputs Network DER 2 (Action) Zk =g(ak[Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='2] Data Primary Koopman-inspired Enhanced OKID Control [Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='3] 3 Primary States DER Control MicroPMU Data zk Estimated System Model: Estimated Parameters Zk+1 = Akzk + Buk Ak Bk Yk = Ckzk 4where δk is the Koopman modeling error which has been defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (13a), and vj is the jth Koopman mode associated with the Koopman eigenfunction ϕj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Apparently, 0 ⩽ ∥ Nϕ � j=0 ( k � j=0 ρj,h)ϕj(x0, u0)vj∥2 ⩽ lim k→∞(maxj,h|ρj,h|)k+1 Nϕ � j=1 ∥ϕj(x0, u0)vj∥2 (40) With LQR in the Koopman invariant subspace, assume the MG secondary controller can optimally make the magnitudes of all system eigenvalues smaller than 1 (if the system is stabilizable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' That is ˆKkϕj = ρj,kϕj with |ρj,k| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Due to the online rolling-based estimation in the proposed method, we can assume the global error ∥ �k h=1 Πk i=h ˆKk(δh−1 + l(uh))∥2 is bounded by ζg, and the modeling error is ∥δk∥2 bounded by ϵm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' According to (40), we have lim k→∞ ∥g(xk+1)∥2 ⩽ lim k→∞(maxj,h|ρj,h|)k+1 Nϕ � j=1 ∥ϕj(x0, u0)vj∥2 + ζg + lim k→∞ ∥δk∥2 ⩽ ζg + ϵm (41) Based on (41), g(x) converges till reaching the area Ξ = � g(x)|∥g(x)∥2 ⩽ ζg + ϵm � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Thus, the system is BIBO stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Besides, the Koopman-based LQR can guarantee asymptotic stability subject to the disturbance in control input channels under mild conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' See Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Stability Margins of Koopman-Enabled LQR The discrete-time LQR used in this paper has analytical disc stability mar- gins [42], within which asymptotic stability subject to the disturbance in control input channels is guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Specifically, consider the identified Koopman state space model described as below: g(xk+1) = Ag(xk) + Buk + BMuk = Ag(xk) + BKg(xk) + BMKg(xk) = Ag(xk) + B(I + M)Kg(xk) (42) where M = diag([m1, m2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' m2NDER]) is an introduced diagonal matrix to represent model uncertainty in control input channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' In other words, the introduced matrix parameter M can be used to quantify the uncertainty from control input channels, whereby one can provide the stability analysis based on the disc margin for each channel (which will be provided below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' K is the control gain matrix such that uk = Kg(xk) in line with the LQR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Consider M and g(xk) to be complex-valued to reflect both gain and phase disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Define a Lyapunov function V (x) = g(x)∗Sg(x) (where S is the solution of Riccati equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Based on the Lyapunov function and following the steps in [42], we provide the disk stability margin for the ith control input channel in (43) without further explanation (also see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Interested readers can refer to [42] for the derivation of the disc margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 1 + mi = � αi + jβi : � αi − (1 + ri µ ) �2 + β2 i < (1 + ri µ )2 + ρ − ri µ − 1 � , where i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', 2NDER (43) where ρ = σmin[Q]/(σmax[K])2 and µ = σmax[BT SB].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' σmax[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='] and σmin[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='] represent the matrix operation to obtain the maximum and minimum singular 16 Figure 5: Disk stability margin for the discrete-time LQR values, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' ri is the ith diagonal element of the cost matrix R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 5 shows the disc margin, within which the system is asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Specifically, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (43) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 5 , the sufficient conditions of asymptotic convergence against the model uncertainty is: 1 + GL,i < αi < 1 + GU,i for the gain margin and PML,i < arctan βi αi < PMU,i for the phase margin, with GL,i = ri µ − � (1 + ri µ )2 + ρ − ri µ − 1, GU,i = ri µ + � (1 + ri µ )2 + ρ − ri µ − 1 (44) and PML,i = − arccos(1/(1 + ri µ )) = − arccos µ µ + ri PMU,i = arccos(1/(1 + ri µ )) = arccos µ µ + ri , for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', 2NDER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (45) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Case Studies This section presents case studies based on two MG test systems, namely a four-bus MG as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 6 and a thirteen-bus MG as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 7, to verify the effectiveness of the proposed Koopman-inspired identification and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The two test systems were established in MATLAB Simulink 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The DERs in the test systems are primary-controlled in different control modes (grid-forming converters, grid-following converters, and an isochronous- controlled diesel generator as given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 6 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 7) with the inner control loops modeled in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Therefore, the interaction of primary and secondary control is preserved in simulation to test the effectiveness of secondary control in realistic setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The implementation of the converter voltage and current control inner-loops can be found [9, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Besides, randomized measurement noises, control time delays, and ambient perturbations were incorporated into the test systems to mimic practical oper- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The simulation parameters of the two test systems are summarized in 17 βi Disc Stability Margin (1+m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='=α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='+jβ): When the model uncertainty is within the disc (grey area), the system is Lyapunov asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Gain Margin 1 αi 11+ Phase 1 Disc Radius : Margin μTable 1 and Table 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The readers can find more information about the test systems at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='com/nash13123/MG-Test-System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='git.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Identification and Control in the 4-Bus MG Test System The small 4-bus MG test system was used to test the proposed Koopman- inspired enhanced OKID with control under load variations and the MG transi- tion from the grid-connected mode to islanded mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The DERs at Bus 1 and 3 are droop-based grid-forming, and the DERs at Bus 2 and 4 are inverse-droop- based grid-following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' At 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='7s, the MG was disconnected from the main grid by turning off the switch SW, which causes sudden voltage drops and consequent dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' After detecting the sudden change, the secondary control was en- abled and kept online from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='8s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='1s lag to mimic a time delay of islanding event detection in practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Modeling accuracy of the Koopman-inspired OKID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' First, we eval- uate the modeling accuracy with the one-step-ahead prediction error of the Figure 6: The MG 4-bus test system Figure 7: The MG 13-bus test system 18 @Utility Grid R14 Local L14 Local Busl DER4 DER1 Bus1 Bus4 sw Battery Rf1 Vo1 Bus4 Re1 Vo4 DC/ R 4 AC/ Rfel n AC DC dc R Droop Control Inverse Droop-Based (Grid Forming) Cf1 Grid-Following Control P13 DER2 Local Local DER3 Battery Bus2 Bus3 Bus2 Bus3 Vo2 R V。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Rf3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='AC/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='DC/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='AC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='DC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='dc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Droop Control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Inverse Droop-Based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='(Grid Forming) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Grid-Following ControlSecondary Uncontrollable Distributed Resources ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='PCC: Point of Common Coupling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Utility Grid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='MPPT: Maximum Power Point Tracking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Secondary Controllable Distributed Resources ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='PV: Photovoltaic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Transformer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='BESS: Battery Energy Storage System ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Load 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='T1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='DER: Distributed Energy Resource ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Grid-Feeding PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Switch SW1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Diesel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Farm 1 (MPPT) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='PCC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Grid-Following BESS 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Load 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Grid-Forming DER 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='SW2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Load 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Load 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Grid-Forming DER 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Grid-Feeding PV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='119 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Grid-Following BESS 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Farm 2 (MPPT) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Load 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Load 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='7Table 1: Parameters of the 4-Bus MG Test System ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Power base Sbase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='30kVA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Voltage Base Vbase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='480V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='Primary control time step Tsp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='1ms Secondary control time step Ts 30ms Sliding window length for estimation N 9 (270ms) Local Voltage proportional gain KP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 Local Voltage integral gain KS 523 Local current proportional gain KP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='3 Local current integral gain KS 635 Frequency droop parameters for DERs 1,2: σω 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='14 × 10−3rad/(W · s) Voltage droop parameters for DERs 1: σV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='0 × 10−3V/V ar Voltage droop parameters for DERs 2: σV 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='3 × 10−3V/V ar Frequency droop parameters for DERs 3,4: σω 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='83 × 10−3rad/(W · s) Voltage droop parameters for DERs 3: σV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 × 10−3V/V ar Voltage droop parameters for DERs 4: σV 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='4 × 10−3V/V ar PMU measurement noise N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='00562) Control Time delay N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='0022)s Ambient perturbation level added to the reference of DER output voltage and angle: N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='012) Filter resistance Rf1,2,3,4(Ω) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='1 Filter inductance Lf1,2,3,4, Lc1,2(mH) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='35 Filter capacitance Cf1,2,3,4(µF) 50 Filter capacitor resistance Rfc1,2,3,4(Ω) 1 Line resistance Rc1,2(Ω) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='08 Line resistance Rc3,4(Ω) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='09 Line inductance Lc1,2(mH) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='35 Line inductance Lc3,4(mH) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='45 Line Resistance Rl1,2,3,4(Ω) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='23, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='17 Line inductance Ll1,2,3,4(mH) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='42, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='33, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='55, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='40 Load PL1,2,3 (active power in kW) 20, 16, 12 Load QL1,2,3 (reactive powe in kVar) 9, 9, 6 LQR control parameter qV 1 × 103I LQR control parameter qsin , qcos 0 LQR control parameter qω 1 × 10−6I LQR control parameter rP , rQ 1 × 10−6I Control input lower bounds ULB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='0 kVA Control input upper bounds UUB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='0 kVA Time period for the optimization (28) TOP T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='6 s Time constant of the power low-pass filter 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='02857s N(a, b) is the normal distribution with mean of a and variance of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Control parameters are designed based on Per Unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 19 Table 2: Parameters of the 13-Bus MG Test System Parameters Value Power base Sbase 150kVA Voltage Base Vbase 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='16kV Sliding window length for estimation N 14(420ms) Droop parameters for all DERs: σω 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='14 × 10−4rad/(W · s) Droop parameters for all DERs: σV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 × 10−3V/V ar Ambient perturbation level N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='022) LQR control parameter qV 1 × 103I LQR control parameter qsin , qcos 0 LQR control parameter qω 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='01I LQR control parameter rP , rQ 1 × 10−6I Other control parameters are the same to the values in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Figure 8: Comparison of the prediction error voltage magnitude, which is defined as e(pred) k+1 = 1 dim(∆V )∥∆Vk+1 − ∆ ˆVk+1∥ (46) where dim[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='] represents the dimension of the vector in the bracket, and ∆ ˆVk+1 represents the predicted voltage magnitude at time step k + 1 by the identified model of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 8, we compared the prediction error of two different ways of modeling: (i) the proposed Koopman-inspired enhanced OKID with the basis z = [∆V , sin θ − sin(θ∗ L), cos θ − cos(θ∗ L), ∆ω]T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (ii) the conventional OKID (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', linearize the system model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (11) and apply OKID with γopt fixed at 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' It was found that the proposed Koopman-inspired enhanced OKID leads to smaller prediction error than the conventional OKID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' These results show that the salient features of the proposed Koopman-inspired OKID, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', the Koopman nonlinear basis and the adaptive γopt, can ensure a good modeling accuracy regardless of nonlinearity and uncertainty during large disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Control results comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 9 compares the voltage and frequency trajectories with different secondary control methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 9(a)-(c) show, the bus voltage suddenly drops with incurred transients after the disturbance at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='7s, which triggers the secondary control to restore the voltage and frequency to their nominal values (1p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='u and 60Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 9(a) shows the control results 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content="5 (n' Koopman-InspiredEnhancedOKID(proposed) 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='4 Conventional OKID Error ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='3 UO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 0 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 5 Time (seconds)(a) The proposed Koopman-inspired enhanced OKID with LQR control (b) The secondary PI control (c) The conventional OKID with LQR control (γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5) (d) The classical EDMDc (least-squares-based Koopman operator control with the Koopman observables z) Figure 9: Voltage and frequency trajectories of the 4-bus MG test system with different secondary control methods 21 (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='95 DER 1 DER 2 Voltage 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='9 DER 3 DER 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='85 1 2 3 4 Time (seconds)61 59 DER 1 DER 2 58 DER 3 DER 4 57 1 2 3 4 Time(seconds)(p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='95 DER 1 DER 2 Voltage 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='9 DER 3 DER 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='85 1 3 4 Time (seconds)61 DER 1 DER 2 60 DER 3 DER 4 59 58 57 1 2 3 4 Time(seconds) (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='95 DER 1 DER 2 Voltage 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='9 DER 3 DER 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='85 1 2 3 4 Time (seconds)61 59 DER 1 DER 2 58 DER 3 DER 4 57 1 2 3 4 Time(seconds)Voltage Magnitude (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='95 DER DER 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='9 DER 3 DER 4 0 1 2 3 4 5 Time (seconds)61 60 Frequency 59 DER DER 2 58 DER 3 DER 4 57 2 3 4 Time (seconds)with the proposed Koopman-inspired enhanced OKID with LQR control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' both voltage and frequency are corrected approximately to the nominal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' For comparison, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 9(b) presents the voltage and frequency trajectories using sec- ondary PI control that is tuned with the best effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The PI control with respect to the voltage magnitude and the frequency shows a slower response for voltage restoration compared to the proposed control, and has non-zero steady-state errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' It also suffers from a larger frequency deviation as it cannot handle the voltage-frequency dependence properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 9(c) shows the results of con- ventional OKID with LQR control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' In contrast to the proposed method, the conventional OKID with LQR cannot realize the same fast voltage restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 9(d) shows the voltage and frequency trajectories of the classical ED- MDc (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', LQR with pseudo-inverse least-squares identification based on the Koopman observables z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' By comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 9(d) with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 9(a), we found that the LQR with the least-squares-based identification cannot perform as well as the LQR with the proposed Koopman-inspired Koopman-inspired identifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' These results indicate the effectiveness of the proposed Koopman-inspired enhanced OKID that possibly results from the two ingredients: (i) the nonlin- ear basis functions of the Koopman observables proposed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (14);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' (ii) the OKID with adaptive γopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Both ingredients help better describe the MG sys- temwide dynamics under big disturbances, realizing more effective control for both voltage and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 10 presents the optimized parameters γopt during control, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 11 shows the run time of the proposed Koopman-inspired enhanced OKID at each Figure 10: Estimated γopt of the proposed Koopman-inspired enhanced OKID Figure 11: Run time of the proposed Koopman-inspired enhanced OKID 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='54 opt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 4 Time (seconds)600 (su) 400 Time Run 200 20ms 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 4 Time (seconds)time step of secondary control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The run time of the proposed method is about 20ms in case that γopt is not updated, less than the time step of secondary control (30ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The run time of the proposed method is around 250-500ms in case that γopt is updated, which is still less than the time period TOP T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='6s between two updates of γopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' These indicate the feasibility to implement the proposed Koopman-inspired identification and control online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Identification and Control in the 13-Bus MG Test System To show the performance of the Koopman-inspired enhanced OKID with LQR control in larger systems for generality, we consider the 13-bus MG test system presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 7, which is adapted from the IEEE 13-node test feeder [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The DERs at Bus 6 and 9 are droop-based grid-forming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The BESSs at Bus 1 and 11 are inverse-droop-based grid-following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The solar farms at Bus 3 and 5 are grid-following under MPPT, which are not controllable for secondary control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The MG system is under transition from the grid-connected to the islanded modes, and under generation/load variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' At 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='4s, the MG is disconnected from the main grid by turning off the switch SW1, causing the sudden drop of voltage with incurred transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' After detecting the islanding transient, the secondary control is triggered and kept online from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='1s lag to mimic a time delay of islanding detection in practical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Next, an active power perturbation of the two solar farms (around 80kW for each) occurs at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='0s due to a drop of the solar irradiation from 1000 to 200 W/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Then, a load perturbation happens at the Bus 4 at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='05s: the consumed active power increases by 150kW and the consumed reactive power increased by 50kVar by turning on the switch SW2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 12 compares the voltage and frequency trajectories with different sec- ondary control methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 12(a) shows that the proposed Koopman-inspired OKID with LQR control can correct the voltage and frequency approximately to the nominal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' For comparison, the voltage and frequency trajectories with the secondary PI control are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 12(b), which illustrates that the PI control with the best effort of tuning still fails to realize the stable and accurate voltage and frequency restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 12(c) shows the results of the conven- tional OKID with LQR control, which suffers from larger voltage and frequency oscillations after 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='0s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 12(d) shows the voltage and frequency trajectories of classical EDMDc (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=', LQR with pseudo-inverse least-squares identification based on the Koopman observables z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' By comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 12(d) with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 12(a), we found that the classical EDMDc cannot perform as well as the LQR with the proposed Koopman-inspired Koopman-inspired identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' These results further demonstrate the advantages of the proposed Koopman-inspired OKID with LQR control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Because of the nonlinear Koopman embeddings and the adaptive γopt, the proposed method can effectively restore the voltage and frequency to their nominal values despite nonlinearity and uncertainty due to large disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 23 (a) The proposed Koopman-inspired enhanced OKID with LQR control (b) The secondary PI control (c) The conventional OKID with LQR control (γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5) (d) The classical EDMDc (least-squares-based Koopman operator control with the Koopman observables z) Figure 12: Voltage and frequency trajectories of the 13-bus MG test system with different secondary control methods 24 (p.' 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+page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='8 DER DER 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='7 DER 3 DER 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 Time (seconds)60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 (ZH) 60 Frequency ( 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 DER DER 2 DER 3 DER 4 59 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='5 Time (seconds)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Conclusions This paper proposed a data-driven Koopman-inspired identification and con- trol method for MG secondary voltage and frequency control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' The proposed method requires no knowledge of network information and primary controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' It requires no warm-up training yet with guaranteed BIBO stability and even asymptotic stability under some mild conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' In this method, a Koopman operator-inspired enhanced OKID (observer Kalman filter identification) algo- rithm is proposed, whereby the Koopman state space model is estimated online and used for control to handle microgrid nonlinearity and uncertainty adap- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Case studies in the 4-bus and 13-bus MG test systems (with different converter control modes) demonstrate the effectiveness and robustness of the proposed Koopman-inspired identification and control method subject to mode transitions, varying operating conditions, measurement noises and time delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' References [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Schneider, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Radhakrishnan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Tang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' K.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' Kerst- ing, Analytic considerations and design basis for the ieee distribution test feeders, IEEE Transactions on Power Systems 33 (3) (2018) 3181–3188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='1109/TPWRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content='2760011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} +page_content=' 29' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AzT4oBgHgl3EQff_xu/content/2301.01461v1.pdf'} diff --git a/59E2T4oBgHgl3EQfOwac/content/tmp_files/2301.03752v1.pdf.txt b/59E2T4oBgHgl3EQfOwac/content/tmp_files/2301.03752v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5606e97dfd7cd55ddc340449b41b567d3abf2020 --- /dev/null +++ b/59E2T4oBgHgl3EQfOwac/content/tmp_files/2301.03752v1.pdf.txt @@ -0,0 +1,2396 @@ +Draft version January 11, 2023 +Typeset using LATEX default style in AASTeX631 +Magnetic Fields, Star Formation Rates and Gas Densities at Sub-kpc Scales in a Pilot Sample of +Nearby Galaxies +Souvik Manna1 and Subhashis Roy1 +1National Center for Radio Astrophysics, TIFR, +Pune University Campus, Ganeshkhind, Pune 411007, India +ABSTRACT +We have estimated the magnetic field strengths of a sample of seven galaxies using their non-thermal +synchrotron radio emission at metre wavelengths, and assuming energy equipartition between magnetic +fields and cosmic ray particles. We tested for deviation of magnetic fields from energy equipartition with +cosmic ray particles, and found that deviations of ∼25% are typical for the sample galaxies. Spatially +resolved star formation rates (SFR) were estimated for the seven galaxies along with five galaxies +studied previously. For the combined sample of twelve galaxies, the equipartition magnetic fields (Beq) +are correlated with the SFR surface densities (ΣSFR) at sub-kpc scales with Beq ∝ Σ0.31±0.06 +SFR +, consistent +with model predictions. We estimated gas densities (ρgas) for a sub-sample of seven galaxies using +archival observations of the carbon monoxide (CO) rotational transitions and the atomic hydrogen +(Hi) 21 cm line and studied the spatially-resolved correlation between the magnetic fields and ρgas. +Magnetic fields and gas densities are found to be correlated at sub-kpc scale as Beq ∝ ρ0.40±0.09 +gas +. This +is broadly consistent with models, which typically predict B ∝ ρ0.5 +gas. +Keywords: Radio continuum emission — Interstellar medium — Star formation — Magnetic fields +1. INTRODUCTION +Magnetic fields are believed to influence several physical processes in a galaxy at almost every scale (e.g. Elmegreen +1981; Niklas & Beck 1997; Groves et al. 2003; Price & Bate 2008; Adebahr et al. 2013). Magnetic fields have been +found to consist of two main components: a small-scale turbulent magnetic field up to a few hundred parsecs (e.g. +Batchelor 1950; Groves et al. 2003) and a large-scale “ordered” or “regular” magnetic field component at scales of a few +kpcs (e.g. Moss & Shukurov 1996; Shukurov et al. 2006; Kulsrud & Zweibel 2008). Magnetic fields in galaxies can be +measured using their effects on different radiation processes like Zeeman splitting of emission lines, polarized emission +from dust, the polarization of starlight, Faraday rotation of polarized radio emission, and intensity of synchrotron +emission which we use in this work. Measurement of the line-of-sight component of the magnetic field via the Zeeman +effect in galaxies other than the Milky Way has been possible for only a few systems (Kazes et al. 1991; Sarma et al. +2005; Robishaw et al. 2008); a significant expansion of such studies is very difficult with current-generation telescopes. +Magnetic fields in galaxies can be measured and studied using synchrotron emission at radio frequencies, at scales +larger than the resolution of the radio observation. For example, a Very Large Array (VLA) polarization study of +NGC 4736 at 8.46 and 4.86 GHz found that the magnetic field in the galaxy was ordered in a spiral shape (Chy˙zy +& Buta 2008). An X-shaped structure of the magnetic field in the galactic halo region was observed by stacking the +Karl G. Jansky VLA polarized emission maps of 16 nearly edge-on spiral galaxies, obtained as part of the CHANG-ES +survey (Krause et al. 2020); such structures had also been observed in individual spiral galaxies (e.g. Krause et al. +2006; Krause 2009; Heesen et al. 2009). However, polarized radio emission from external individual galaxies is difficult +to study at low radio frequencies due to Faraday depolarization (e.g. Sokoloff et al. 1998). +Corresponding author: Souvik Manna +souvik@ncra.tifr.res.in +arXiv:2301.03752v1 [astro-ph.GA] 10 Jan 2023 + +2 +Manna and Roy +The average magnetic field strength can also be estimated from the total intensity of synchrotron radio emission, +assuming energy equipartition between magnetic fields and cosmic ray particles (e.g. Miley 1980; Beck & Krause 2005). +Equipartition magnetic fields have been studied in several nearby galaxies, but primarily at frequencies >1 GHz (e.g. +Chy˙zy et al. 2000; Soida et al. 2001; Heesen et al. 2009; Fletcher et al. 2011; Adebahr et al. 2013). Vargas et al. (2018) +studied a sample of three nearly edge-on galaxies from the CHANG-ES survey to separate the thermal Bremsstrahlung +from the non-thermal synchrotron emission at 1.5 and 6 GHz. At these frequencies, the thermal component is large and +hence the correction for the thermal emission can be as large as ∼ 20%, making the derived magnetic field strengths +prone to errors. Conversely, the steep spectral index of synchrotron emission implies that it will dominate the total +emission at frequencies < 1 GHz, with ∼ 95% contribution (Basu et al. 2012b; Roy & Manna 2021). Thus, magnetic +field strengths derived using observations at <1 GHz are very robust to any correction for thermal emission. +Magnetic fields are believed to play an important role at various stages of the star-formation process - from the +fragmentation of clouds at the few kpc scales to the final collapse of gas into stars (e.g. Elmegreen 1981; Crutcher +1999; Price & Bate 2008; Van Loo et al. 2015). To understand the influence of magnetic fields and star-formation +activities on different physical processes in the ISM at different physical scales, several studies on radio-infrared +correlations have been carried out in the past (e.g. Murphy et al. 2006a,b, 2008; Tabatabaei et al. 2013). Magnetic +fields (B) and star formation rate surface densities (SFRSD) are expected to be correlated (Niklas & Beck 1997). Semi- +analytical model also predicts a strong correlation between B and SFRSDs (ΣSFR) as B ∝ Σ1/3 +SFR at sub-kpc scales +to explain the local radio-FIR correlation (Schleicher & Beck 2013, 2016). Observational studies of the correlation +between B and star formation rates (SFR) have been done primarily in samples of nearby dwarf galaxies. For example, +Chy˙zy et al. (2011) studied 12 local group dwarf galaxies to find that the galaxy-averaged magnetic field and the SFR +follow B ∼ SFR0.30±0.04, consistent with the prediction of B ∝ Σ1/3 +SFR. However, Jurusik et al. (2014) found the same +power-law index in a sample of Magellanic type dwarf galaxies to be 0.25±0.02, somewhat lower than the expectation. +Recently, a study of the dwarf galaxy IC 10 by Basu et al. (2017) provides the only study of the correlation between +spatially-resolved magnetic fields and SFRSDs; these authors found that the SFRSD is related to the magnetic field +as B ∝ Σ0.35±0.03 +SFR +. Therefore, it is important to test such predictions by carrying out systematic spatially-resolved +studies of magnetic fields in galaxies and their connection to the star formation rate in nearby large galaxies. +The energy density of magnetic fields and gas in galaxies are expected to be in equipartition, which implies B +∝ √ρgas (e.g. Chandrasekhar & Fermi 1953; Groves et al. 2003). The observed Radio-FIR correlation can be explained +based on such equipartition between the energy density of magnetic fields and gas (Niklas & Beck 1997). Several other +numerical magnetohydrodynamic (MHD) simulations of the ISM have predicted the coupling constant (k) between +magnetic fields and gas (B ∝ ρk +gas) to be in the range of ≈0.4−0.6 (Fiedler & Mouschovias 1993; Kim et al. 2001; +Thompson et al. 2006). Niklas & Beck (1997) studied the correlation between galaxy-integrated equipartition magnetic +fields and gas densities for a sample of 43 galaxies to find a power-law index of 0.48 ± 0.05; the observed correlation is +consistent with B ∝ √ρgas. Although the correlation between gas surface densities and SRFSDs has been extensively +studied in the nearby Universe (Kennicutt-Schmidt law; e.g. Kennicutt 1998a; Onodera et al. 2010; Roychowdhury +et al. 2015), systematic studies of spatially-resolved correlations between magnetic fields, SFRs and gas densities in +nearby galaxies are yet to be carried out. It is thus important to carry out a systematic investigation of both the +B-ρ and the B-SFR correlations, at high-spatial resolutions (≈ sub-kpc scales), using direct estimates of the magnetic +fields, gas densities, and star-formation rates, in a sample of nearby galaxies. In this paper, we present a pilot study +of the connection between spatially resolved magnetic fields, SFRSDs and gas densities in a sample of nearby galaxies. +We have selected a sample of 46 galaxies (Sample 0; Table 2) from the Spitzer Local Volume Legacy (LVL) sample +of 258 galaxies within 11 Mpc (Dale et al. 2009). As a pilot project, seven (Sample 1; Table 2) of these 46 galaxies +have been observed with the Giant Metrewave Radio Telescope (GMRT) at 0.33 GHz (Roy & Manna 2021). Six of +our seven sample galaxies are spirals and the other one is a dwarf irregular Magellanic-type galaxy. +In this paper, we present spatially resolved equipartition magnetic field (Beq) maps of the seven galaxies in Sample +1 (Table 2). We also incorporate the magnetic field maps of five galaxies studied by Basu et al. (2012a) from previous +GMRT observations in our study. We derived SFRSD maps of all 12 galaxies (Sample 2; Table 2) using extinction- +free diagnostics and used these maps to study the relation between SFRSDs and Beq at sub-kpc scales in our pilot +study. We used available archival CO and Hi 21 cm data to measure the gas densities (ρgas) of seven (Sample 3; +Table 2) of the combined sample of 12 galaxies and studied the correlation between ρgas and Beq in these galaxies. +We also studied the magnetic field-gas connection through an indirect measurement of their coupling coefficient using +radio−FIR correlations of the galaxies in Sample 1. + +A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities +3 +Table 1. Details of the seven sample galaxies. Note that the images at 0.33 GHz were obtained from observations with the +GMRT reported in Roy & Manna (2021) while those at 1.4 GHz were obtained from archival VLA data. The distances to +the galaxies were taken from Dale et al. (2009). Galaxies with an asterisk are those for which spatially-resolved CO data are +available. +Name +Class +Distance +Inclination +Position +uv +Angular +Spatial +RMS +RMS +VLA +(Mpc) +angle +angle +range +resolution +resolution +(0.33 GHz) +(1.4 GHz) +Project ID +(deg) +(deg) +(kλ) +(arcsec2) +(pc) +(µJy/beam) +(µJy/beam) +(1.4 GHz) +NGC 2683 +Sb +7.7 +83 +43 +0.19 - 15 +19 × 13 +670 +200 +40 +AI23 +NGC 3627∗ +SAB +10. +65 +170 +0.26 - 25 +16 × 11 +760 +800 +370 +AS541, AP462 +NGC 4096 +SABc +8.3 +76 +20 +0.14 - 17 +14 × 12 +730 +100 +25 +16A-013 +NGC 4449 +Irregular +4.2 +0 +0 +0.15 - 15 +26 × 15 +360 +300 +180 +AB167 +NGC 4490 +SBm +8.0 +60 +126 +0.13 - 14 +19 × 18 +560 +230 +100 +AA181 +NGC 4826∗ +SAab +7.5 +60 +120 +0.22 - 20 +15 × 14 +650 +280 +70 +AS541 +NGC 5194∗ +Sbc +8.0 +20 +10 +0.15 - 10 +23 × 18 +740 +310 +30 +AB505, AN57 +Table 2. +List of different samples studied in this paper. +Sample Name +Galaxies +Sample 0 +Full sample containing 46 galaxies from Spitzer LVL survey +Sample 1 +Pilot sample containing 7 galaxies from Sample 0; galaxies listed in Table 1 +Sample 2 +Sample 1 + 5 galaxies (NGC 1097, NGC 4736, NGC 5055, NGC 5236 and NGC 6946) +from Basu et al. (2012b) = 12 galaxies; used to probe the Beq-SFRSD correlations +Sample 3 +A subset of 7 galaxies (NGC 3627, NGC 4826, NGC 5194, NGC 4736, NGC 5055, NGC 5236 and +NGC 6946) from Sample 2 which have archival CO data; used to study the Beq-gas density correlations +The paper is organized as follows. The analysis of the data is discussed in Section 2. In Section 3, we present +the results of our analysis, including the correlation between magnetic fields, SFRSDs and gas densities of the seven +galaxies in Sample 1. In Section 4, we have extended our study to include a sample of five galaxies of Basu et al. +(2012a) . We discuss the results in Section 5. A summary of this paper is presented in Section 6. +2. DATA ANALYSIS +As can be seen in Table 1, six of the seven galaxies in Sample 1 are spirals of varying inclination angles. The seventh +galaxy NGC 4449 is a dwarf irregular galaxy. Basic information about the seven sample galaxies, including their +types, distances, inclination angles, position angles, angular resolutions, spatial resolutions, and RMS noise obtained +on the GMRT and VLA images are also listed in Table 1. The distances, inclination angles, and position angles of +the galaxies were taken from Dale et al. (2009). Radio observations and the data reduction procedures are discussed +in detail in Roy & Manna (2021). Briefly, we used GMRT 0.33 GHz observations (covering 0.309−0.342 GHz) and +archival VLA observations at 1.4 and ∼6 GHz to derive non-thermal emission maps for each galaxy. We used Hα and +24µm observations of the seven galaxies to model free-free emission from them and subsequently, we subtracted the +modelled free-free emission from the observed radio emission to get the non-thermal radio maps at 0.33, 1.4 and ∼6 +GHz (Roy & Manna 2021). To generate the non-thermal spectral index maps, we used the non-thermal radio maps +at 0.33 and ∼6 GHz for NGC 2683, NGC 3627, NGC 4096, and NGC 4449. For the rest of the galaxies (NGC 4490, +NGC 4826, and NGC 5194), we used non-thermal images at 0.33 and 1.4 GHz to generate the non-thermal spectral +index maps (Roy & Manna 2021). In the following subsections, we present the analysis of other ancillary data and +relevant measurements. +2.1. Magnetic Field Strengths +The average magnetic field strengths can be estimated from the observed synchrotron flux densities, assuming energy +equipartition between cosmic ray particles and magnetic fields (“Classical Equipartition Formula”, e.g. Pacholczyk +1970; Miley 1980; Longair 2011). The equipartition condition is achieved when the total energy in magnetic fields and +cosmic ray particles is minimum. + +4 +Manna and Roy +The classical equipartition formalism has shortcomings that lead to an overestimation of the magnetic field strength +(B) at regions of steep spectral indices and underestimation of B at flat spectral index regions. To overcome these +shortcomings of the classical equipartition formula, Beck & Krause (2005) proposed a revised formula to estimate the +average magnetic field strength. The formula is expressed as +Beq = [4π(K0 + 1)E1−2αnt +p +f(αnt) +C4(i) +Iνναnt +l +] +1 +αnt+3 +(1) +K0, Ep, Iν, and αnt are the number density ratio of cosmic ray protons to electrons, the proton rest mass energy, +the intensity of the synchrotron emission at frequency ν, and the spectral index of synchrotron emission, respectively. +f(αnt) is a function of αnt given as f(αnt) = (2αnt + 1)[2(αnt − 1)c2(αnt)cαnt +1 +] (Beck & Krause 2005). C4(i) is a +constant that depends on the inclination angle (i) of the galaxy and is expressed as C4(i) = [cos(i)](γ+1)/2, where +γ = (2αnt +1). l is the path length of the synchrotron emission. The path length was assumed to be 1 kpc for a galaxy +with an inclination angle of 0 degree (face-on). For galaxies with low- and moderate- inclination angles (< 75◦), +the assumed path length was corrected for the inclinations of the galaxies as l/cos(i). For the two nearly edge-on +galaxies in Sample 1, NGC 2683 and NGC 4096, we have assumed an oblate spheroidal shape of the synchrotron +emission, such that the diameter on the plane of the galaxy is equal to its major axis. The path lengths (l) were then +appropriately calculated, with the path length being maximum (equal to the galaxy’s major axis) at the optical centre +of the galaxy and gradually declining to the edge of the galaxy. We note that Beq has only a weak dependence on l as +Beq(r) = l(r) +−1 +αnt+3 and hence is less sensitive to the exact choice of l. Values of K0 and Ep were assumed to be 100 +and 938.28 MeV, respectively, the same as used by Beck & Krause (2005). Finally, we used non-thermal radio maps at +0.33 GHz (Iν) and spectral index maps (αnt) made using 0.33 and 1.4 or ∼6 GHz radio observations (Roy & Manna +2021) to produce magnetic field maps of the sample galaxies using Equation 1. +The revised equipartition formula diverges for spectral index values ≤ 0.5 because such flat spectra indicate energy +loss of electrons through ionizations or Coulomb interactions (Sarazin 1999). The central bulge and arm regions have a +mostly flatter spectrum due to the association of star-forming regions and the estimates of equipartition magnetic fields +in such regions might be affected by systematic uncertainties. This issue affects the derived magnetic field strengths +for 8%, 12%, 3%, 70%, 17%, 7%, and 6% of the projected total surface area of NGC 2683, NGC 3627, NGC 4096, +NGC 4449, NGC 4490, NGC 4826, and NGC 5194, respectively. We note that a large fraction of the derived magnetic +field values are affected for NGC 4449 due to its non-thermal spectral indices being predominantly flat. This could +bias the Beq for NGC 4449. +2.1.1. Uncertainties on Magnetic Field Maps +The procedure we used to estimate the uncertainties on our magnetic field maps is similar to that of Basu & Roy +(2013). We used a Monte Carlo method that generated 104 random flux density values for each pixel in a galaxy map +at 0.33 GHz and either 1.4 GHz or 6 GHz. These flux density values have Gaussian probability distributions with rms +values equal to the measured rms of each of the 0.33 and 1.4/6 GHz maps. For each of the 104 intensity maps, we +computed a magnetic field map using the procedure described in the beginning of Section 2.1. The rms of these 104 +magnetic field maps provided us with the magnetic field uncertainty maps for each of the seven galaxies in sample 1. +2.2. Star Formation Rates +Rest frame Hα and ultraviolet (UV) observations are the best tracers of recent SFRs as the radiation from these +predominantly originate in newly formed massive stars. However, the observations are affected by extinction caused by +interstellar dust in both the host galaxy as well as the Milky Way. SFRs estimated from Hα and UV observations are +therefore corrected for the extinction. Dust-corrected SFRs can be estimated by combining far-ultraviolet (FUV) and +Hα data with infrared (IR) data to exploit the complementary strengths at different wavelengths (e.g. Kennicutt & +Evans 2012a; Buat 1992; Meurer et al. 1995, 1999; Cortese et al. 2008; Leroy et al. 2012). In addition to the FUV+IR +and Hα+IR tracers, the low-frequency radio emission from galaxies, which is predominantly optically thin synchrotron +emission, can be used to estimate their dust-unobscured SFRs via the radio-FIR correlation (e.g. Yun et al. 2001). +We estimated the spatially-resolved star formation rates of our Sample 1 galaxies using FUV+24µm, Hα+24µm, +and 1.4 GHz data, which are discussed, respectively, in the following Sections, 2.2.1, 2.2.2, and 2.2.3. We used data +of these different frequencies as tracers in order to (1) get a fair comparison between different SFR diagnostics and +(2) for studying star-formation history at different timescales. All SFRs in this paper assume a Kroupa IMF (Kroupa +2001). + +A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities +5 +2.2.1. SFRs using FUV and 24µm Observations +To estimate SFRSD maps of the seven galaxies in Sample 1 (Table 2) using FUV+24µm emission, we used SPITZER +24 µm IR data (Dale et al. 2009) and GALEX FUV data (11HUGS survey; Kennicutt et al. 2008). We first convolved +both the 24 µm and the FUV maps of all galaxies to the same resolutions as our magnetic field maps. The FUV data +were corrected for extinction due to dust in the Milky Way (see Section 2.2.4). The FUV images were in units of +counts/sec/pixel and were converted to flux-density units of MJy Sr−1. We also converted the 24µm images to units of +MJy Sr−1 and used the following calibration from Leroy et al. (2012) to derive SFRSD maps for the sample galaxies: +ΣSFR[M⊙yr−1kpc−2] = 0.081 IFUV[MJy sr−1] + 0.032 I24µm[MJy sr−1] +(2) +The uncertainties of the coefficients are ∼10-30%. Note that the uncertainty in SFR estimates arises from issues +such as the error in sampling the stellar IMF of different star-forming regions, determining the contribution of different +emission which are not associated with recent star formation, etc. (e.g Kennicutt & Evans 2012b; Leroy et al. 2012). +2.2.2. SFRs using Hα and 24µm Observations +To estimate SFRSD maps using Hα+24µm as a tracer, we used 24 µm emission along with Hα emission from +11HUGS (Kennicutt et al. 2008), for all but NGC 5194, for which we used data from the SINGS survey (Kennicutt +et al. 2003). All the maps were convolved and regridded to the resolution and pixel size of the magnetic field maps. For +the Hα maps from 11HUGS and SINGS, the flux density units were converted to erg/s/cm−2. We used the following +calibration from Leroy et al. (2012) to estimate SFRSDs of the galaxies in Sample 1. +ΣSFR[M⊙yr−1kpc−2] = 634.0 IHα[erg s−1 sr−1] + 0.0025 I24µm[MJy sr−1] +(3) +2.2.3. SFRs using 1.4 GHz Observations +Our 1.4 GHz non-thermal maps of the galaxies (Sample 1) (Roy & Manna 2021) and an SFR calibration from +Murphy et al. (2011) were used to derive SFRSD maps (Equation 4). The calibration is based on the observed radio- +FIR correlation in a sample of nearby star-forming galaxies (Bell 2003) and has a scatter of 0.26 dex. We used this +galaxy-integrated calibration (Equation 4) to derive the formula for spatially-resolved radio-ΣSFR calibration. +SFR1.4GHz +M⊙yr−1 += 6.35 × 10−29 +L1.4GHz +erg Hz−1s−1 +(4) +The spatially-resolved calibration is consistent with the calibration of Heesen et al. (2014). We used the above +relation to estimate the SFRSD maps of the sample galaxies from the measured 1.4 GHz surface brightness. +2.2.4. Galactic Extinction Correction for FUV Emission +We corrected for the extinction of FUV emission due to dust in the Milky Way using the E(B-V) values along the +line of sight to the sample galaxies from Bianchi et al. (2017). The extinction coefficients (AFUV) of the GALEX FUV +bands were measured using Table 1 from Bianchi et al. (2017) and intrinsic fluxes (Fintrinsic) were estimated from the +following formula: +AFUV = −2.5 × log[Fobserved +Fintrinsic +] +(5) +The extinction percentage of the FUV emission is listed in Table 3. +2.3. Gas Densities +Atomic hydrogen (Hi) and molecular hydrogen (H2) predominantly contribute to the total gas mass of galaxies. +H2 is best traced using rotational transitions in CO (e.g. Bolatto et al. 2013). Spatially-resolved observations of CO +transitions exist for only three of our seven galaxies in Sample 1 (Table 2). We have used CO J=2-1 line data of NGC +3627 and NGC 5194 from the HERA CO-Line Extragalactic Survey (HERACLES; Leroy et al. 2009) and CO J=1-0 +data of NGC 4826 from the BIMA Survey of Nearby Galaxies (BIMA SONG; Regan et al. 2001). The HERACLES +and BIMA survey have a spatial resolution of 13′′ and 6′′, respectively. The velocity resolution of the HERACLES and +BIMA spectral cubes are ∼5 and 6 km/s, respectively. We restrict our study of the connection between gas densities +and magnetic fields to only these three galaxies for which spatially-resolved CO data are available. + +6 +Manna and Roy +Table 3. FUV extinction values of the Sample 1 galaxies due to the Milky Way foreground dust. The extinctions were computed +using E(B-V) values along the line of sight to the sample galaxies from Bianchi et al. (2017). +Name +Percentage extinction +NGC 2683 +22 +NGC 3627 +23 +NGC 4096 +21 +NGC 4449 +15 +NGC 4490 +15 +NGC 4826 +13 +NGC 5194 +27 +The HI Nearby Galaxy Survey (THINGS; Walter et al. 2008) used VLA observations to obtain very high spectral +(≤ 5.2 km/s) and spatial (∼ 6 +′′) resolution maps of nearby galaxies at 21cm. We used the publicly available 21cm +moment maps from this THINGS survey to estimate the distribution of Hi in the three galaxies for which CO data +are available. All CO and Hi 21 cm maps were convolved and regridded to a common resolution and pixel size of the +non-thermal radio maps. Gas densities were estimated (for NGC 3627, NGC 4826 and NGC 5194) following Basu & +Roy (2013) assuming CO to H2 conversion factor of 2 ×1020 (K km s−1)−1 (e.g. Bolatto et al. 2013). A line ratio of 0.8 +was assumed to convert COJ=2-1 to COJ=1-0 (e.g. Leroy et al. 2009). We accounted for the contribution of helium to +the gas density using ρgas=1.36 × (ρHi + ρH2). Line of sight depths were assumed to be 300 and 400 pc for molecular +and atomic gas, respectively (Basu & Roy 2013). +3. RESULTS +3.1. Magnetic Fields in the Galaxies +We have estimated spatially resolved revised equipartition magnetic field maps for seven galaxies in Sample 1, using +the procedures of Section 2.1; these maps are shown in Figures 1 & 2. Flux density contours of 1.4 GHz observations +are overlaid on magnetic field maps. The resolution of these maps corresponds to spatial scales of ∼ 0.4−0.8 kpc +(see Table 1). The bottom right panel of Figure 2 shows the radial variation of the magnetic field with galactocentric +radius of all the seven galaxies where both the axes are normalized by their maximum values. Here, we have averaged +the magnetic field strengths over an annular elliptical region of width equal to the beam size of the corresponding +map. Position and inclination angle (Table 1) of each galaxy were used while selecting the elliptical regions. We find +magnetic fields to be stronger at the central region and at the star formation sites (arm regions) with field strengths +up to 50 µG. Field strengths fall by ∼50% at the edges of the magnetic field maps. The Milky Way also shows such a +trend in the variation of magnetic field strengths (Beck et al. 1996). We note that our analysis was limited to distances +where the signal-to-noise ratio in spectral index maps is > 5; the magnetic field strengths at these distances are thus +likely to be reliable. +We note that, compared to the magnetic field strengths obtained using the classical equipartition expression, these +values are higher by ∼ 1.3−1.5 for a non-thermal spectral index of -0.6, and they match for a spectral index of -0.75 +(Beck & Krause 2005). +Figure 7 shows the uncertainties in the magnetic field values for Sample 1 derived using the Monte Carlo method +described in Section 2.1.1. Statistical uncertainties on mean magnetic fields for these seven galaxies are provided in +Table 4. +3.2. Star Formation Rates in the Galaxies +We have estimated the global, galaxy-averaged SFRs of Sample 1 galaxies using 1.4 GHz, FUV+24µm, and Hα+24µm +emission using calibrations discussed in Section 2.2. Globally integrated star formation rates of the sample galaxies +are given in Table 5. No systematic offset was found in the SFR values estimated using these tracers. The differences +in the SFR values for our galaxies are much less than the calibration uncertainty except for NGC 4490. For NGC +4490, SFR calculated from 1.4 GHz emission is higher than the same from FUV+24µm emission by a factor of 2.2. +As discussed in Section 2.2, we have estimated SFRSD maps of the seven galaxies (Sample 1) using FUV+24µm, +Hα+24µm and 1.4GHz emission. We show SFRSD maps of the seven galaxies in the Appendix (Figures 8-9), where +SFRSDs estimated using 1.4 GHz and FUV+24µm emission are shown in contours and colors, respectively. In the + +A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities +7 +Figure 1. The equipartition magnetic field maps of NGC 2683, NGC 3627, NGC 4449 and NGC 4096 (clockwise from top +left) (Sample 1). Non-thermal radio contours at 1.4 GHz are overlaid on magnetic field maps. The magnetic field strengths are +shown in color with non-thermal emission at 1.4 GHz shown as overlaid contours. Contour levels are presented below each panel +in the figure. The circle in the bottom-left corner of the panels indicates the angular resolution of the maps. The uncertainties +on mean magnetic fields are 0.06µG, 0.17µG, 0.04µG and 0.18µG for the above galaxies, respectively. + +COLOR:NGC26832683.B.final.TH.SUB.1 +CONT:NGC2683IPOL1490.572MHz2683.L.Ths.TH.SUB.1 +10 +20 +30 +40 +33 29 +28 +27 +Declination (J2000) +26 +25 +24 +23 +22 +21 +085305 +00 +52 55 +50 +45 +40 +35 +30 +25 +20 +RightAscension(J2000) +Colorscalerange=5.0040.00uG +Contpeakflux=4.8560E-03JY/BEAM +Levs = 1.600E-04 * (-2, -1, 1, 2, 4, 8, 16, 32, +64, 128, 256, 512)COLOR:NGC36273627.B.final.TH.SUB.1 +CONT:N3627LIPOL1430.389MHz3627.L.Ths.TH.SUB.1 +10 +20 +30 +40 +1302 +01 +Declination (J2000) +00 +1259 +58 +57 +1120 25 +20 +15 +10 +05 +RightAscension(J2000) +Colorscale ranqe=5.0040.00uG +Contpeakflux=1.4880E-02JY/BEAM +Levs = 1.480E-03 * (-2, -1, 1, 2, 4, 8, 16, 32 +64,128,256,512COLOR:NGC40964096.B.final.TH.SUB.1 +CONT:NGC4096IPOL1432.873MHz4096.L.Ths.TH.SUB.1 +5 +10 +15 +20 +25 +4734 +32 +Declination (J2000) +30 +28 +26 +24 +可 +120630 +15 +00 +05 45 +30 +RightAscension(J2000) +Colorscalerange=5.0025.00uG +Contpeakflux=1.9020E-03JY/BEAM +Levs = 1.000E-04 * (-2, -1, 1, 2, 4, 8, 16, 32 +64,128,256,512)COLOR:NGC44494449.B.final.TH.SUB.1 +CONT:N49IPOL1489.900MHz4449.L.Ths.TH.SUB.1 +10 +20 +30 +4412 +10 +08 +Declination (J2000) +06 +04 +02 +00 +4358 +12 28 45 +30 +15 +00 +27 45 +30 +RightAscension(J2000) +Colorscalerange=5.0035.00uG +Contpeakflux=2.6366E-01JY/BEAM +Levs = 7.200E-04 *(-2, -1, 1,2, 4, 8, 16, 32. +64, 128, 256,512)8 +Manna and Roy +0.2 +0.4 +0.6 +0.8 +1.0 +Radial distance (Normalised) +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +Magnetic field (Normalised) +NGC 2683 +NGC 3627 +NGC 4096 +NGC 4449 +NGC 4490 +NGC 4826 +NGC 5194 +Figure 2. The equipartition magnetic field maps of NGC 4490 (top left), NGC 4826 (top right) and NGC 5194 (bottom left). +The magnetic field strengths are shown in color with non-thermal emission at 1.4 GHz shown as overlaid contours. Contour +levels are presented below each panel in the figure. The circle in the bottom-left corner of the panels indicates the angular +resolution of the maps. The uncertainties on mean magnetic fields are 0.06µG, 0.11µG and 0.02µG, respectively. The bottom +right panel presents the radial variation of magnetic field strengths with galactocentric distance for all seven galaxies in Sample +1. + +COLOR:NGC44904490.B.final.TH.SUB.1 +CONT:N4490IPOL1435.114MHz4490.L.ThS.TH.SUB.1 +10 +20 +30 +40 +4144 +42 +1 +Declination (J2000) +40 +38 +36 +34 +123100 +30 45 +30 +15 +RightAscension(J2000) +Colorscalerange=5.0045.00uG +Contpeakflux=3.3720E-02JY/BEAM +Levs = 4.000E-04 * (-2, -1, 1, 2, 4, 8, 16, 32, +64,128, 256, 512)COLOR:NGC48264826.B.final.TH.SUB.1 +CONT:N4826LIPOL1425.677MHz4826.L.Ths.TH.SUB.1 +10 +20 +30 +21 45 +44 +43 +Declination (J2000) +42 +41 +40 +39 +38 +37 +Q +12 57 00 +56 55 +50 +45 +40 +35 +30 +25 +RightAscension(J2000) +Colorscalerange=5.0035.00uG +Contpeakflux=2.5190E-02JY/BEAM +Levs = 2.800E-04 * (-2, -1, 1, 2, 4, 8, 16, 32, +64, 128,256,512)COLOR:NGC51945194.B.final.OHGSPX.1 +CONT:M51IPOL1664.900MHz5194.L.Ths.TH.SUB.1 +10 +20 +30 +40 +4718 +16 +14 +Declination (J2000) +12 +10 +08 +06 +13 30 30 +15 +00 +29 45 +30 +15 +RightAscension(J2000) +Colorscale range=5.0040.00 uG +Contpeakflux=4.6656E-02JY/BEAM +Levs = 1.200E-04 *(-2, -1, 1,2, 4, 8, 16, 32 +64,128,256,512)A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities +9 +Table 4. Statistical uncertainties on mean magnetic fields for galaxies in Sample 1. +Name +Statistical uncertainty +on mean magnetic fields +(µG) +NGC 2683 +0.06 +NGC 3627 +0.17 +NGC 4096 +0.04 +NGC 4449 +0.18 +NGC 4490 +0.06 +NGC 4826 +0.11 +NGC 5194 +0.02 +Table 5. Galaxy-averaged star formation rates of the galaxies in Sample 1, using 1.4 GHz, FUV+24µm, and Hα+24µm data. +The uncertainties on the SFR values are ≈ 30%. +Name +SFR from 1.4 GHz (M⊙yr−1) +SFR from FUV+24µm (M⊙yr−1) +SFR from Hα+24µm (M⊙yr−1) +NGC 2683 +0.28 +0.25 +0.33 +NGC 3627 +1.56 +2.00 +1.84 +NGC 4096 +0.42 +0.35 +0.38 +NGC 4449 +0.37 +0.38 +0.32 +NGC 4490 +4.63 +2.13 +2.30 +NGC 4826 +0.63 +0.73 +0.78 +NGC 5194 +4.16 +3.88 +3.65 +Appendix (Figures 10-11), we also present the SFRSD maps estimated using Hα+24µm and 1.4 GHz emission in colors +and contours, respectively. The SFRSD maps of each galaxy in Figures 8−11 are shown in the same color scale and +contours. To determine the radial variation of SFRSDs, we have averaged the SFRSD maps of our sample galaxies +over tilted rings centred on the optical centre of each galaxy using their inclinations and position angles. The width of +the tilted rings was taken to be equal to the beam size of the corresponding image. Figure 3 shows the radial variation +of the average SFRSD, derived using FUV+24µm and Hα+24µm emission, with galactocentric distance where both +the axes are normalized to their maximum values. We also derived the radial variation of SFRSDs for the galaxies +using 1.4 GHz emission and it is consistent within 1σ statistical uncertainties, with those derived using FUV+24µm +and Hα+24µm data. Azimuthally averaged SFRSDs of all the seven galaxies decrease gradually towards the outer +region and drop by a factor of 6 to 8 at the edge. +3.3. Details of the Individual Galaxies of Sample 1 +(i) NGC 2683: In this galaxy, Krause et al. (2020) found very weak linear polarisation using C-band and L-band +VLA observations. Based on the optical image, we could separate the central region from the disk. The average +magnetic field in the central region is found to be ≈31 µG and the outer region of the disk has an average value of +≈19 µG (see Figure 1 and Table 6). +Wiegert et al. (2015) used WISE 22 µm data to estimate a galaxy-averaged SFR of ≈0.09 M⊙yr−1 for NGC 2683. +From our analysis, integrated SFR was measured to be ∼0.24 M⊙yr−1 and ∼0.28 M⊙yr−1 using FUV+24µm and 1.4 +GHz radio emission, respectively. However, we note that Wiegert et al. (2015) used a distance of 6.27 Mpc for this +galaxy, but we have used a distance of 7.7 Mpc. The SFR is estimated to be 0.16 M⊙yr−1 using FUV+24µm emission, +assuming the same distance as used by Wiegert et al. (2015). Taking the calibration uncertainties and the assumed +distance into account, our estimated SFR is hence consistent with that of Wiegert et al. (2015). We note that the +contours on the background sources (Figure 8) are not real SFRSDs, as these are likely to be background AGNs. +(ii) NGC 3627: NGC 3627 was observed at 8.46 GHz and 4.85 GHz using the VLA in its D-configuration (Soida +et al. 2001). These authors estimated the magnetic field strengths using the classical equipartition formula (Longair +2011) and found an average equipartition magnetic field strength of 11±2 µG, assuming a constant non-thermal spectral + +10 +Manna and Roy +0.2 +0.4 +0.6 +0.8 +1.0 +Radial distance (Normalised) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +SFRSD from FUV+24μm (Normalised) +NGC 2683 +NGC 3627 +NGC 4096 +NGC 4449 +NGC 4490 +NGC 4826 +NGC 5194 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Radial distance (Normalised) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +SFRSD using Hα+24μm (Normalised) +NGC 2683 +NGC 3627 +NGC 4096 +NGC 4449 +NGC 4490 +NGC 4826 +NGC 5194 +Figure 3. +The variation of SFRSDs (normalized), estimated using FUV+24µm (left panel) and Hα+24µm (right panel) +emission as a function of galactocentric distance (normalized) for all seven galaxies in Sample 1. +index of 0.9 and a disk thickness of 2 kpc. Soida et al. (2001) also studied the polarized emission at these frequencies +to find a regular magnetic field of 4±1 µG. They suggested two distinct magnetic field components of NGC 3627: one +for the spiral arms and another for the inter-arm regions. We have separately studied equipartition magnetic fields +in the arm and interarm regions of the galaxy. We find that the central region and the edges of the extended bar +have magnetic field strengths of ≈ 34 µG (see Figure 1). The arm region has a field strength of ≈28 µG (see Table +6). However, the magnetic field strength in the interarm regions has values ≈21 µG. We note that our estimates +of the equipartition magnetic field strengths in the galaxy are higher than those found by Soida et al. (2001); this +difference likely arises from the fact that Soida et al. (2001) estimated the magnetic field strengths using the classical +equipartition formula, which is known to significantly underestimate the magnetic field in the star-forming regions. +We measured a galaxy-averaged SFR of ≈2.0 M⊙yr−1 and ≈1.56 M⊙yr−1 from FUV+24µm and 1.4 GHz emission, +respectively. +Our measurements of spatially resolved SFRs in different regions are consistent, within calibration +uncertainties, with the SFR estimates of Watanabe et al. (2011). +(iii) NGC 4096: Our estimate of the equipartition magnetic field in NGC 4096 varies from ≈21 µG at the centre +to ≈12 µG at the edge (table 6). The magnetic field strength in both the central region and northern periphery is +quite similar, with typical field strengths of ≈ 20 µG; this is presumably due to its high inclination. The outer part of +the galaxy has an average field strength of ≈14 µG. NGC 4096 was observed (Irwin et al. 2012; Wiegert et al. 2015) +with its B-field and further studied by Krause et al. (2020) who found very little polarized emission from the galaxy. +Wiegert et al. (2015) used the 22 µm−SFR calibration to measure a galaxy-averaged SFR of 0.27±0.02 M⊙yr−1. +Our measurement of the galaxy-averaged SFR is ≈0.35 M⊙yr−1 and ≈0.43 M⊙yr−1 using FUV+24µm and 1.4 GHz +emission, respectively. Considering the calibration uncertainties, our estimates are consistent with that of Wiegert +et al. (2015). +(iv) NGC 4449: This is an optically bright irregular starburst galaxy. Chy˙zy et al. (2000) used VLA 4.86 and +8.46 GHz observations to find a galaxy-averaged equipartition magnetic field of ≈14 µG. These authors also used +polarization emission to estimate a regular field of ≈8 µG. The equipartition magnetic field map of NGC 4449 from +our study is shown in Figure 1. As noted in Section 2.1, about 70 % of the total projected area of this galaxy has +spectral index values of less than 0.55. We have replaced the pixel values with αnt < 0.55 with αnt = 0.55 while +computing the magnetic field for NGC 4449 (see Section 2.1). The average magnetic field strength is ≈17 µG in this +galaxy, which is comparable to the findings of Chy˙zy et al. (2000). +Our measurements of the galaxy-averaged SFR are ≈0.38 M⊙yr−1 and ≈0.37 M⊙yr−1 using FUV+24µm and 1.4 +GHz emission, respectively, which are consistent with the SFR of 0.47 M⊙yr−1 estimated by Chy˙zy et al. (2011). +(v) NGC 4490: Nikiel-Wroczy´nski et al. (2016) observed NGC 4490 at 0.61 GHz using the GMRT, and at 4.86 & +8.44 GHz using VLA + Effelsberg. The authors used these observations to find a mean equipartition magnetic field + +A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities +11 +Table 6. Magnetic field strengths in different regions of the galaxies in Sample 1. For the irregular galaxy NGC 4449, we could +only measure the galaxy-integrated magnetic field. We have separated the two nearly face-on galaxies (NGC 3627 and NGC +5194) into arm and inter-arm regions. For the rest of the galaxies, we could not separate the arm and inter-arm region due to +their higher inclinations. +Galaxy +Galaxy-average +Beq in +Beq in +Beq in +Beq in +name +Beq +central region +disk region +arm region +inter-arm region +(µG) +(µG) +(µG) +(µG) +(µG) +NGC 2683 +24±6 +31±3 +19±5 +– +– +NGC 3627 +25±4 +34±8 +– +28±5 +21±4 +NGC 4096 +16±4 +21±5 +14±3 +– +– +NGC 4449 +17±6 +– +– +– +– +NGC 4490 +23±10 +40±6 +17±7 +– +– +NGC 4826 +23±9 +38±8 +20±5 +– +– +NGC 5194 +16±6 +34±6 +– +25±5 +18±4 +of 21.9±2.9 µG, with typical field strengths in the range of 18 µG to 40 µG. We have found a typical equipartition +magnetic field strength of ≈40 µG in the central region, which decreases to ≈17 µG in the outer region (see Figure 2); +these values are consistent with the estimates of Nikiel-Wroczy´nski et al. (2016). We find a relatively lower magnetic +field strength of ≈15 µG in both the interacting region and the companion galaxy NGC 4485. Therefore, a gradual +decrease in the average magnetic field strength occurs from the center to the outer region. +Clemens et al. (1999) used radio observations to find a galaxy-averaged SFR of 4.7 M⊙yr−1. We found a similar SFR +(≈4.63 M⊙yr−1) using 1.4 GHz radio emission but a factor of ∼2 lower SFR (2.13 M⊙yr−1) using the FUV+24µm +emission (Table 5). Extinction corrections for NGC 4490 are believed to be higher than those typically assumed and +this may lead to an underestimation of the SFR while using the FUV+24µm diagnostics (Clemens et al. 1999). +(vi) NGC 4826: No spatially resolved maps of magnetic fields and SFRSDs are available in the literature. We +measure the central and outer regions of the galaxy to have an average equipartition magnetic field strength of ≈38 +µG and ≈20 µG, respectively (see Figure 2 and Table 6). We find galaxy-averaged SFR of ≈0.73 M⊙yr−1 and ≈0.63 +M⊙yr−1 using FUV+24µm and 1.4 GHz data, respectively. +(vii) NGC 5194: Fletcher et al. (2011) used VLA C-band observations of the galaxy and assumed a constant +thermal and non-thermal spectral index of 0.1 and 1.1 to find an average equipartition magnetic field strength of 20 +µG using the revised formula by Beck & Krause (2005). They found a magnetic field of 20−25 µG in the spiral arms, +higher than the 15−20 µG typical in the interarm regions. Using VLA observations at S-band (2−4 GHz) frequencies, +Kierdorf et al. (2020) found the field strength of turbulent and regular components of the magnetic field in the arm +regions of 18−24 µG and 8−16 µG, respectively. We find an equipartition magnetic field strength of ≈25 µG in the +arm region and ≈18 µG in the interarm region (see Table 6). The peripheral region has a magnetic field of ≈12 µG, +while the overlapping region between NGC 5194 and NGC 5195 has an average Beq of ≈16 µG. Considering our use +of Equation 1 (Beck & Krause 2005), measurements are roughly consistent with the earlier study of Fletcher et al. +(2011) and Kierdorf et al. (2020). +Spatially resolved SFRs were measured in several star-forming regions of NGC 5194 using Hα+24µm and Hα+Paα +emission (Kennicutt et al. 2007). +SFRSDs in different regions were found to be in the range of 0.10 to 0.46 +M⊙yr−1kpc−2. Our estimates using the two tracers are consistent with the estimates of Kennicutt et al. (2007) (See +Figures 9 & 11). Furthermore, we find that the galaxy-integrated SFR derived using FUV+24µm (≈3.88 M⊙yr−1) +and 1.4 GHz data (≈4.16 M⊙yr−1) are consistent with each other, within 1-sigma statistical uncertainty. +3.4. Is the Minimum Energy Condition Valid for the Sample Galaxies? +We have estimated magnetic fields for the galaxies in Sample 1 assuming the “minimum energy condition” or +“equipartition condition”, i.e. by assuming that the energy density in the magnetic field is approximately equal to +the energy density in cosmic ray particles. Therefore, it is important to verify the validity of this assumption in our +sample galaxies. The tightness of the spatially-resolved radio−FIR correlation can be used to estimate the deviation +of the energy densities from the minimum energy condition (Hummel 1986; Basu & Roy 2013). According to the +simplified model of Hummel (1986), when the minimum energy condition is satisfied, the distribution of Int/IFIR will +be similar to the distribution of B1+αnt. The model assumes the following to be constant across galaxies: (a) the ratio +of the number densities of relativistic electrons and dust-heating stars, (b) the volume ratio of radio and FIR emitting + +12 +Manna and Roy +regions, and (c) the ratio of efficiency factors for both the radio and FIR emission. In this model, the cumulative +distribution function (CDF) of the quantity Int/IFIR and B1+αnt +eq +is expected to follow each other if Beq is close to B. +To verify the validity of the minimum energy condition in our sample galaxies, we have followed the procedure as +in Hummel (1986) and Basu & Roy (2013). The CDF of Int/IFIR and B1+αnt +eq +were estimated using our radio maps +of the sample galaxies at both 0.33 and 1.4 GHz. We used an ensemble of spatially-resolved values of αnt, Int (both +at 0.33 and 1.4 GHz), IFIR (70 µm) and magnetic fields (Beq), which are averaged over the beam size from all the +galaxies in Sample 1 (Table 2) to generate these distributions. The CDFs of all quantities were normalized by their +median values. The top panels in Figure 4 show the median-normalized CDFs of Int/IFIR and B1+αnt +eq +at both 0.33 +and 1.4 GHz. +We find that the CDFs of Int/IFIR and B1+αnt +eq +at both 0.33 and 1.4 GHz broadly follow each other but with slight +deviations at high and low ends (see top panels in Figure 4). This implies that the minimum energy condition is +broadly valid and is consistent with earlier findings. For example, Hummel (1986) found the distribution of the two +quantities is similar in a sample of Sbc galaxies while Basu & Roy (2013) reached similar conclusions in a study of 5 +nearby large spiral galaxies, but with slight deviations observed in the CDFs of Int/IFIR and B1+αnt +eq +in the interarm +regions of the galaxies. +The observed deviation in the CDFs of Int/IFIR and B1+αnt +eq +for our sample galaxies imply a corresponding deviation +from the minimum-energy condition. In order to quantify this deviation, we performed a Monte Carlo simulation orig- +inally proposed by Hummel (1986). In this simulation, random numbers (X) were drawn from a Gaussian distribution +with standard deviation σ. Thereafter, we multiplied 10X with the observed equipartition magnetic fields to introduce +deviations from the minimum-energy condition. We thus constructed the CDF of B1+αnt +eq +using the deviated magnetic +field values. The CDF of B1+αnt +eq +were then compared to the observed CDF of Int/IFIR via a Kolmogorov-Smirnov +(KS) test. This procedure was repeated for a range of σ from 0 to 0.2. We find that the p-values for the KS test +comparing the distributions are maximized when σ = 0.1. Indeed, B1+αnt +eq +derived after deviating the magnetic field +using σ = 0.1 and Int/IFIR are consistent with being derived from the same distribution, with a KS test p-value of +0.41 and 0.55, when using Int at 0.33 and 1.4 GHz, respectively. The bottom panels in Figure 4 show the CDFs of the +two quantities for σ = 0.1 at 0.33 and 1.4 GHz; it is clear that the CDFs follow each other. This implies the actual +magnetic field values may deviate from the equipartition values by ∼ 25% in our galaxies in Sample 1. We note that +any violation of the assumptions made by Hummel (1986) may also lead to the observed deviation in the CDFs. +3.5. Correlation Between Magnetic Fields and SFRSDs +We have studied the correlation between the spatially-resolved equipartition magnetic field and SFRSDs for the +galaxies in Sample 1 (Table 2) at scales of ≈360−760 pc (Table 1). For the seven sample galaxies, we used the SFRSD +maps estimated using the FUV+24µm emission. The correlations between magnetic fields and SFRSDs for the seven +galaxies are shown in Figure 5. Each point represents the logarithms of equipartition magnetic fields and SFRSD values +that are averaged over the beam size of the corresponding maps. da Silva et al. (2014) found that SFR calibrations +could be biased and strongly affected by stochasticity at small spatial scales where the star formation rate is low (≤ +10−2.5 M⊙yr−1); we have therefore excluded regions of low star formation rates from the correlation study. +We find that the equipartition magnetic field and the SFSRD are correlated in all seven sample galaxies. We use +orthogonal distance regression in Scipy (Virtanen et al. 2020) to fit a power law of the form B = B0 (ΣSFR)η to the +magnetic field − SFRSD data points; the spatially-resolved uncertainty maps of equipartition magnetic fields and rms +noise on the SFRSD maps were used to estimate the uncertainties on each data point during the fitting procedure. The +best-fit parameters of the power-law are given in Table 7. We have also estimated the scatter (rms of the data points +along the y-axis) of the correlations which are presented in Table 7 and are shown in dashed lines in the corresponding +plots (Figure 5). We find that six of the seven galaxies have slopes (η) in the range of ≈ 0.27 − 0.40 but that the slope +is relatively lower for NGC 4449 with η ≈ 0.18. Averaging over the slope of all galaxies in Sample 1, we find a mean +slope of 0.32 ± 0.06. +3.6. Correlation Between Magnetic Fields and Gas Densities +We have studied the correlation between spatially-resolved equipartition magnetic fields and gas densities for three +of the galaxies in Sample 1, NGC 3627, NGC 4826, and NGC 5194, for which spatially resolved CO observations +were available (see Section 2.3). Similar to the study of correlations between Beq and SFRSDs, we have studied the +correlations between Beq and gas density values, both averaged over the beam size of the corresponding maps. The + +A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities +13 +Figure 4. The top panels show the cumulative distribution function (CDF) of Int,radio/I70µm (in red) and B1+αnt +eq +(in blue), +where Int is the non-thermal emission at 0.33 GHz (top left) and 1.4 GHz (top right) (Sample 1). The variables are normalized +by their median values. The bottom panels show the same but now with the magnetic field perturbed from its measured value +using σ=0.1 (see Section 3.4); the CDFs of the Int,radio/I70µm and B1+αnt +eq +are now consistent with being derived from the same +distribution. +Table 7. Best-fit parameters and the scatter of the correlation between magnetic fields and SFRSDs for the seven galaxies in +Sample 1. The data were fitted with a power law of the form B=B0(ΣSFR)η. +Name +Slope (η) +Intercept (B0) (log(µG)) +Intercept (B0) (µG) +Scatter +NGC 2683 +0.34 ± 0.04 +2.10 ± 0.07 +125 ± 1.2 +0.05 +NGC 3627 +0.31 ± 0.03 +1.71 ± 0.03 +51 ± 1.1 +0.04 +NGC 4096 +0.33 ± 0.04 +1.80 ± 0.08 +63 ± 1.2 +0.05 +NGC 4449 +0.18 ± 0.03 +1.64 ± 0.04 +43 ± 1.1 +0.03 +NGC 4490 +0.27 ± 0.02 +1.90 ± 0.03 +79 ± 1.1 +0.06 +NGC 4826 +0.38 ± 0.02 +1.80 ± 0.02 +63 ± 1.0 +0.05 +NGC 5194 +0.40 ± 0.01 +2.00 ± 0.02 +100 ± 1.0 +0.07 + +1.0 +0.8 +X +0.6 +=） +P +0.4 +0.2 +lo.33GHz/170μm +0.0 +0 +2 +4 +6 +X1.0 +0.8 +X +0.6 +=） +v +P +0.4 +0.2 +eq +l1.4GHz/l70μm +0.0 +0 +1 +2 +3 +4 +5 +X1.0 +0.8 +X +0.6 +=） +V +P +0.4 +0.2 +lo.33GHz/170μm +0.0 +0 +2 +4 +6 +X1.0 +0.8 +X +0.6 +=） +v +P +0.4 +0.2 +eq +l1.4GHz/l70μm +0.0 +0 +1 +2 +3 +4 +5 +X14 +Manna and Roy +2.25 +2.00 +1.75 +1.50 +1.2 +1.3 +1.4 +1.5 +1.6 +log(Magnetic Field in G) +NGC 2683 +1.0 +0.8 +0.6 +0.4 +1.3 +1.4 +1.5 +1.6 +1.7 +NGC 3627 +2.0 +1.8 +1.6 +1.1 +1.2 +1.3 +1.4 +NGC 4096 +1.6 +1.4 +1.2 +1.0 +1.25 +1.30 +1.35 +1.40 +1.45 +1.50 +log(Magnetic Field in G) +NGC 4449 +2.0 +1.5 +1.2 +1.4 +1.6 +NGC 4490 +2.0 +1.5 +1.0 +1.2 +1.4 +1.6 +NGC 4826 +2.0 +1.5 +1.0 +1.0 +1.2 +1.4 +1.6 +log(Magnetic Field in G) +NGC 5194 +2.0 +1.5 +1.0 +1.0 +1.1 +1.2 +1.3 +NGC 1097 +2.5 +2.0 +1.5 +1.0 +1.0 +1.1 +1.2 +1.3 +1.4 +1.5 +NGC 4736 +1.00 +0.75 +0.50 +0.25 +log(SFRSD in M + yr +1 kpc +2) +1.05 +1.10 +1.15 +1.20 +log(Magnetic Field in G) +NGC 5055 +2.0 +1.5 +1.0 +log(SFRSD in M + yr +1 kpc +2) +1.0 +1.2 +1.4 +1.6 +NGC 5236 +2.0 +1.5 +1.0 +log(SFRSD in M + yr +1 kpc +2) +1.1 +1.2 +1.3 +1.4 +NGC 6946 +Figure 5. The correlation between magnetic fields and SFRSD for the combined sample of 12 galaxies (Sample 2, Table 2). +For the seven galaxies in Sanple 1, the SFRSD estimates shown in the plots were derived using FUV + 24µm (Section 3.5). The +SFRSD estimates for the five galaxies from Basu et al. (2012a) (Sample 2) were derived using Hα + 24µm (Section 4) The red +line shows a linear fit to the data points. The black dashed lines show the ±1σ vertical scatter. + +A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities +15 +23.2 +23.0 +1.3 +1.4 +1.5 +1.6 +log(Magnetic Field in G) +NGC 3627 +23.0 +22.5 +1.2 +1.4 +1.6 +NGC 4826 +23.0 +22.5 +1.0 +1.2 +1.4 +1.6 +NGC 5194 +23.5 +23.0 +1.0 +1.1 +1.2 +1.3 +1.4 +1.5 +log(Magnetic Field in G) +NGC 4736 +23.6 +23.4 +23.2 +log(Gas density in gm/cm3) +1.05 +1.10 +1.15 +1.20 +NGC 5055 +23.5 +23.0 +22.5 +log(Gas density in gm/cm3) +1.1 +1.2 +1.3 +1.4 +1.5 +NGC 5236 +23.0 +22.5 +log(Gas density in gm/cm3) +1.1 +1.2 +1.3 +log(Magnetic Field in G) +NGC 6946 +Figure 6. +The correlations between magnetic fields (µG) and gas densities (gm/cm−3) for seven galaxies of Sample 3 (Table 2). +The red line shows a linear fit to the data points. The black dashed lines show the ±1σ vertical scatter. +correlations between magnetic fields and gas densities of NGC 3627, NGC 4826, and NGC 5194 are shown in Figure +6. We have again used orthogonal distance regression in Scipy (Virtanen et al. 2020) to fit a power-law to the Beq and +gas density data points. The scatters of the three correlations are shown in dashed lines in all the figures. +The measured best-fit power-law indices are 0.40±0.02, 0.49±0.03 and 0.53±0.02 (Table 9) for NGC 3627, NGC +4826 and NGC 5194, respectively. The mean of the power-law indices is 0.47±0.05. +4. EXTENDING THE SAMPLE WITH 5 GALAXIES FROM EXISTING GMRT OBSERVATIONS +As mentioned earlier, a study of Beq and radio-FIR correlations for a sample of five large nearly face-on galaxies was +carried out by Basu et al. (2012a,b); Basu & Roy (2013), using low-radio frequency observations at 0.33 and 1.4 GHz + +16 +Manna and Roy +Table 8. Best-fit parameters and the scatter of the correlation between magnetic fields and SFRSDs for the five galaxies in +Basu et al. (2012a) (Sample 2). The data were fitted with a power law of the form B=B0(ΣSFR)η. +Name +Slope (η) +Intercept (B0) (log(µG)) +Intercept (B0) (µG) +Scatter +NGC 1097 +0.27 ± 0.01 +1.61 ± 0.01 +41 ± 1.0 +0.02 +NGC 4736 +0.32 ± 0.02 +1.78 ± 0.05 +60 ± 1.1 +0.04 +NGC 5055 +0.27 ± 0.04 +1.26 ± 0.02 +18 ± 1.0 +0.02 +NGC 5236 +0.38 ± 0.07 +1.91 ± 0.02 +81 ± 1.0 +0.08 +NGC 6946 +0.25 ± 0.03 +1.62 ± 0.05 +42 ± 1.1 +0.05 +Table 9. Best-fit parameters and the scatter of the correlation between spatially-resolved magnetic fields and gas densities for +the seven galaxies in Sample 3. Galaxies with an asterisk are from the sample of Basu et al. (2012a). +Name +Exponent +Scatter +NGC 3627 +0.40 ± 0.02 +0.03 +NGC 4826 +0.49 ± 0.03 +0.05 +NGC 5194 +0.53 ± 0.02 +0.06 +NGC 4736∗ +0.44 ± 0.03 +0.03 +NGC 5055∗ +0.25 ± 0.02 +0.02 +NGC 5236∗ +0.40 ± 0.03 +0.04 +NGC 6946∗ +0.31 ± 0.03 +0.04 +at sub-kpc linear resolutions. In this paper, we expand our study of spatially-resolved correlations between magnetic +fields, gas densities, and SFRSDs by including these five galaxies. +We refer readers to Basu et al. (2012a) for a detailed discussion of their sample, GMRT observations, data reduction +procedures, and estimation of non-thermal spectral indices. It is to be noted that the modelling of the thermal free-free +emission from these galaxies is performed in the same way as was done for our seven galaxies in Sample 1. +We have estimated the SFRSD maps of these five galaxies using Hα data along with 24 µm IR data. We obtained +Hα maps of four of the galaxies, NGC 1097, NGC 4736, NGC 5055, and NGC 6946 from the ancillary data at the +SINGS website1 and obtained the Hα map of NGC 5236 from 11HUGS (Kennicutt et al. 2008). We used Hα and MIPS +24 µm data in combination to derive the SFRSD maps of these galaxies using the calibration from Leroy et al. (2012) +(Equation 3, Section 2.2). To estimate the equipartition magnetic field strengths of these five galaxies, we have used +the non-thermal radio maps at 0.33 and 1.4 GHz from Basu et al. (2012a). The correlations between equipartition +magnetic fields and SFRSDs are shown in Figure 5 where, similar to the previous correlation studies, each point +represents the logarithms of magnetic fields and SFRSD values that are averaged over the beam size. Similar to the +previous correlations (Section 3.5), we used orthogonal distance regression in Scipy to fit a power law to the data. We +have provided the best-fit parameters of the power-law fit in Table 8. The scatters of all five correlations (presented +in Table 8) are shown in dashed lines in all the figures. We find a mean exponent of 0.30±0.05 for the five galaxies +where the exponent of individual galaxies varies from ≈0.25 to ≈0.38. +We have computed maps of cold gas densities of four out of the five galaxies; NGC 4736, NGC 5055, NGC 5236 +and NGC 6946, using the atomic and molecular gas surface density maps from Basu & Roy (2013). The assumed +parameters are taken to be the same as described in Section 2.3. For the remaining galaxy, NGC 1097, we could not +measure gas densities as there are no archival CO data available for the galaxy. Following the procedures of Section 3.5, +we have also studied the spatially-resolved correlation between equipartition magnetic fields and gas densities for the +four sample galaxies, which are shown in Figure 6. The best-fit parameters are presented in Table 9. The exponents +of the individual galaxies vary between ≈0.25 to ≈0.44 where the mean exponent is found to be 0.35±0.07. +5. DISCUSSION +Understanding the relationship between the physical condition of the interstellar medium (ISM) and the star +formation process is crucial to understand galaxy evolution. +Gas and magnetic fields are key constituents of the +1 https://irsa.ipac.caltech.edu/data/SPITZER/SINGS/ + +A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities +17 +ISM and therefore it is important to study the interrelations between gas, magnetic fields, and SFRs. Though the +Kennicutt−Schmidt relation, i.e. the relation between gas densities and SFRs, has been extensively studied at high +spatial resolutions in various types of nearby galaxies (e.g. Onodera et al. 2010; Roychowdhury et al. 2015; Filho et al. +2016; Miettinen et al. 2017), similar high-resolution observations of how the magnetic fields are related to SFRs and +gas densities are yet to be systematically investigated. Such observations are critical to understand the validity of +several models that predict strong correlations between the magnetic fields and gas densities (e.g. Chandrasekhar & +Fermi 1953; Fiedler & Mouschovias 1993; Cho & Vishniac 2000; Groves et al. 2003) as well as magnetic fields and +SFRSDs (e.g. Niklas & Beck 1997; Schleicher & Beck 2013, 2016). Here, we have studied these correlations in a sample +of twelve galaxies (Sample 3) at sub-kpc scales (see Sections 3.5, 3.6 & 4). To our knowledge, this is the first spatially +resolved study of the above correlations in nearby large galaxies. In this section, we place these findings in the light +of predictions made by various models and in the process attempt to provide physical insights into the interrelation +between magnetic fields, gas densities, and star formation rates at sub-kpc scales. +5.1. Magnetic Fields and SFRSDs +Several Magneto-Hydrodynamical simulations find that galactic magnetic fields are amplified by gas turbulence in +very short timescales (i.e. ∼100 Myr) (e.g. Brandenburg & Subramanian 2005; Beresnyak 2012; Schober et al. 2012; +Schleicher & Beck 2013; Bovino et al. 2013). The primary driver of gas-turbulence in the ISM of galaxies is supernova +explosion (Bacchini et al. 2020), the rate of which is in turn directly coupled to the SFR in the galaxy. Therefore, +it is expected that the star formation rates and the magnetic fields in a galaxy will be correlated. Indeed, using +semi-analytical models, Schleicher & Beck (2013, 2016) found that in order to explain the radio-FIR correlation at +sub-kpc scales, magnetic fields and SFRSDs, again at sub-kpc scales, must be related as B ∝ Σ1/3 +SFR. +Studies in the literature on the correlation between magnetic fields and SFRSDs have focused on dwarf galaxies and +those studies were carried out using galaxy-integrated magnetic fields and SFRSDs. As mentioned in Section 1, to our +knowledge, there is only one published work of the spatially-resolved study of the correlation between magnetic fields +and SFRSDs (Basu et al. 2017). +For the 12 galaxies in Sample 2 (Table 2), we find that the mean value of the power-law index of the correlation +between Beq and SFRSDs is 0.31±0.06, i.e Beq ∝ Σ0.31±0.06 +SFR +(2), consistent (at < 1σ error) with the model of Schleicher +& Beck (2013, 2016). Thus, it appears that the semi-analytical models that are based on the amplification of magnetic +fields due to supernova-driven gas turbulence work remarkably well for the pilot sample, in predicting the correlation +between magnetic fields and SFRSDs down to sub-kpc scales. +We note that the power-law index for the correlation between Beq and SFRSDs for NGC 4449 was found to be +0.18 ± 0.03, significantly lower than for the remaining galaxies (Table 7) as well as lower than the model prediction +of B ∝ Σ1/3 +SFR Schleicher & Beck (2013) (at > 5σ significance). For the case of NGC 4449, the relatively flat spectral +index values (αnt ≤ 0.55) in ≈ 70% of the galaxy meant that the magnetic field values could not be estimated reliably +for a large part of the galaxy (see Section 2.1 and 3.5). This could lead to biases in the correlation and therefore, the +low value of the power-law index for NGC 4449 should be taken with caution. +5.1.1. Intercept of the Correlation +According to the model proposed by Schleicher & Beck (2013), the intercept of the B-ΣSFR correlation depends on +several ISM parameters such as gas density (ρ0), the fraction of turbulent kinetic energy converted into magnetic energy +(fsat), the injection rate of turbulent supernova energy (C) and the intercept of Kennicutt-Schmidt (KS) relation (C1) +(Equation 5). +B ∼ +� +fsat8π ρ1/6 +0 +( C +C1 +)1/3 Σ1/3 +SFR. +(6) +Schleicher & Beck (2013) predicted the intercept of the B-ΣSFR correlation to be ∼ 26 µG assuming ρ0 = 10−24 +g cm−3 and fsat ∼ 5 percent. +We have found an average intercept at 65±25 µG of the Beq-ΣSFR correlation of +the 12 galaxies in sample 2 (see Table 7 & 8). Although the mean value is a factor of ≈2.5 higher than the value +predicted by Schleicher & Beck (2013), this value is consistent with the predicted value, within the scatter (at ≈1.6σ). +Future follow-up studies, such as using our full survey (Sample 0 which consists of 46 galaxies), are required to draw +statistically robust conclusions about the value of the intercept. +2 The uncertainty quoted is the scatter of the measured value of η across the galaxies in Sample 2. + +18 +Manna and Roy +If the value of fsat is indeed higher, this would imply a higher than assumed value of one or more of ρ0, C, and fsat. +The intercept is broadly insensitive to the assumed value of ρ0 (Equation 6) and therefore, in order to explain a factor +of ≈ 2.5 higher value of the intercept, the actual value ρ0 has to be higher than the assumed value of 10−24 g cm−3 +by a factor of ≈ 240; such high gas densities are unphysical and are not observed in typical regions of a galaxy. The +other possibility that the assumed value of the injection rate of turbulent supernova energy (C) is higher by a factor +of ≈ 16 is also contrary to expectation; Basu et al. (2017) found that under reasonable conditions the value of C can +be higher by at most a factor of 1.4. Therefore, fsat must be higher than 0.05 to explain a significantly higher value +of the intercept. An understanding of how galaxies can achieve such efficient amplification of magnetic fields with fsat +much greater than 5% requires detailed MHD simulations. We note that Basu et al. (2017) found that the value of +the intercept for B-ΣSFR for the dwarf galaxy IC 10 is 51 µG, similar to our findings of a higher than predicted value +of the intercept. +5.2. Magnetic Fields and Gas +Magnetic fields and gas are expected to be correlated as B ∝ √ρgas (e.g. Chandrasekhar & Fermi 1953; Groves et al. +2003). We find that equipartition magnetic fields are correlated with gas densities for the seven galaxies (Sample 3) +with an average power-law index, k=0.40±0.09 (see Section 3.6 & 4)3. This value of k is consistent with the numerical +simulations that predict k ≈ 0.4−0.6 and also consistent with the theories that predict B ∝ ρ0.5 +gas. The power-law index +of the correlation between Beq and gas densities is found to be 0.25±0.02 and 0.31±0.03 for NGC 5055 and NGC +6946 respectively, significantly lower than the model predictions and as compared to the other galaxies in Sample 3. +A lower value of k could mean that either the efficiency of the amplification of the magnetic field is less or that the +magnetic field strengths derived assuming the “minimum energy condition” are underestimated (Dumas et al. 2011). +Strong synchrotron or inverse Compton losses of cosmic-ray electrons could suppress the radio synchrotron emission +which would then cause the equipartition magnetic fields to be underestimated. +5.2.1. Magnetic Fields, Gas Densities and the Radio-FIR Correlations +Energy equipartition between the magnetic field (B) and the gas density (ρgas), and between magnetic fields and +cosmic ray particles implies that the non-thermal emission is related to the gas density as Int ∝ ρk(3+αnt) +gas +where k +is the power-law index relating magnetic fields and gas densities (Beq ∝ ρk +gas) (Niklas & Beck 1997). Further, the +Kennicutt-Schmidt law and the radio-FIR correlation imply that Int is related to gas densities as (1) Int ∝ ρm(n+1) +gas +for +optically thin dust to UV photons and (2) Int ∝ ρmn +gas for optically thick dust to UV photons, where m is the power-law +index of the radio-FIR correlation and n is the power-law index of the Kennicutt-Schmidt law. Therefore, we can +obtain the following relation between the power-law index of all four correlations (Dumas et al. 2011): +k = (n + 1)m +3 + αnt +; Optically thin dust +(7) +k = +nm +3 + αnt +; Optically thick dust +(8) +We can use the above equations to indirectly estimate the power-law index, k, of the correlation between magnetic +fields and gas densities. For the three galaxies, NGC 3627, NGC 4826, and NGC 5194 (Roy & Manna 2021), we have +estimated gas densities using CO and Hi observations. Now we can compare the direct measurement of k with an +indirect estimate of k using Equations 7 and 8; this will provide additional information on the validity of both the +minimum energy conditions that were assumed between magnetic fields and the gas densities as well as the magnetic +fields and cosmic ray particles. +For the galaxies from Basu et al. (2012a), this study was already presented and +discussed in Basu et al. (2012b). +We have estimated k for all the seven sample galaxies from Roy & Manna (2021) (Sample 1), using the assumption of +optically thin dust to UV photons, using (i) the slope of radio-FIR correlation (m) as derived in Roy & Manna (2021), +(ii) the measured galaxy-averaged spectral index (αnt) from Roy & Manna (2021), and (iii) a Kennicutt-Schmidt +power-law index of 1.4±0.15 (Kennicutt 1998b). +Table 10 provides the relevant values as well as estimated values of k derived using the measured value of m using +radio emission at both 0.33 and 1.4 GHz. For two of the galaxies, NGC 3627 & NGC 5194, the value of k estimated +3 The uncertainty quoted is the scatter of the measured value of k across the galaxies in Sample 3. + +A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities +19 +Table 10. Power law index (k) of the relation between magnetic fields and gas densities (B ∝ ρk) of galaxies in Sample 1, +indirectly estimated using the slope of radio-FIR correlation (m) and the slope of the Kennicutt−Schmidt law. See Section 5.2.1 +for a discussion on these. +Name +m +m +αnt +k (Optically thin) +k (Optically thin) +0.33 GHz +1.4 GHz +0.33 GHz +1.4 GHz +NGC 2683 +0.54±0.06 +0.91±0.07 +-0.84±0.08 +0.33 ± 0.04 +0.57 ± 0.06 +NGC 3627 +0.55±0.03 +0.85±0.13 +-1.10±0.07 +0.32 ± 0.03 +0.50 ± 0.08 +NGC 4096 +0.74±0.05 +0.90±0.04 +-0.78±0.06 +0.47 ± 0.04 +0.57 ± 0.05 +NGC 4449 +0.77±0.05 +0.65±0.04 +-0.48±0.06 +0.53 ± 0.05 +0.45 ± 0.04 +NGC 4490 +0.68±0.02 +0.75±0.02 +-0.59±0.07 +0.45 ± 0.03 +0.50 ± 0.04 +NGC 4826 +1.39±0.1 +1.47±0.08 +-0.49±0.06 +0.95 ± 0.09 +1.00 ± 0.09 +NGC 5194(arm) +0.50±0.05 +0.65±0.04 +-0.63±0.05 +0.33± 0.04 +0.43 ± 0.04 +NGC 5194(interarm) +0.73±0.11 +1.03±0.05 +-0.85±0.10 +0.46± 0.08 +0.64 ± 0.05 +using Equation 7 is comparable to the direct measurement of k. This broadly validates the assumption of energy +equipartition between magnetic fields and cosmic ray particles in these two galaxies. +For the optically thin case, the mean of indirectly-estimated k values of the sample of seven galaxies are 0.59 ± 0.16 +and 0.53 ± 0.19 at 1.4 and 0.33 GHz, respectively. However, this includes the galaxy NGC 4826, which shows an +anomalously high value of k=1.0 and 0.95 derived at 1.4 and 0.33 GHz, respectively. Excluding this galaxy from the +mean calculation, we find that k=0.52±0.04 and 0.47±0.09 at 1.4 and 0.33 GHz, respectively. Remarkably, for all the +galaxies except NGC 4826, the k value at 1.4 GHz, for the optically thin case, is consistent with 0.5 within error bars. +Thus, the indirectly estimated values of k are consistent with equipartition between magnetic fields and gas energy +densities (Chandrasekhar & Fermi 1953; Fiedler & Mouschovias 1993; Cho & Vishniac 2000; Groves et al. 2003). This +is similar to the findings of Niklas & Beck (1997) for their sample of 43 galaxies and Basu et al. (2012b) for their +sample of four galaxies. +The value of k derived for NGC 4826, for the optically thin case, is a consequence of the anomalously high value +of the power-law index of the radio-FIR correlation (≈1.39 and ≈1.47 for 0.33 and 1.4 GHz respectively, Table 10) +which is different from the other six galaxies in the sample. NGC 4826 has been classified as a Seyfert 2 galaxy in +the past (Malkan et al. 2017) and therefore the emission from the core contributes to the observed power-law index of +the radio-FIR correlation (Roy & Manna 2021). It is likely that the significant contribution of the AGN to the radio +emission makes the estimate of k for NGC 4826 unreliable. +6. SUMMARY +1. We made spatially resolved maps of equipartition magnetic fields in seven galaxies (Sample 1): NGC 2683, +NGC 3627, NGC 4096, NGC 4449, NGC 4490, NGC 4826, and NGC 5194 and find that the magnetic fields are +strongest near the central region and go down by a factor of ∼2 at the edge of the magnetic field maps. +2. We have used the tightness of the spatially-resolved radio-FIR correlations to verify the validity of the equipar- +tition condition between magnetic fields and cosmic ray particles for the sample galaxies. We find that the +magnetic field values may deviate from the equipartition values by ∼25%. +3. We have estimated spatially resolved maps of SFRSDs of the galaxies in Sample 1 using FUV+24µm, Hα+24µm, +and 1.4 GHz data. Azimuthally averaged SFRSDs drop by a factor of 6 to 8 at the edge of the galaxies, where +SFRSD values are 5 times the rms of the maps. +4. We also included five additional galaxies: NGC 1097, NGC 4736, NGC 5055, NGC 5236, and NGC 6946 from +previous GMRT observations of Basu et al. (2012a) and estimated their equipartition magnetic field, SFRSD +and gas density maps. +5. We studied the spatial correlation between magnetic fields and star formation rates at < 1 kpc resolution for +the 12 galaxies (Sample 2) and find that magnetic field strengths and SFRSDs are correlated with an average +power-law index of 0.31±0.06. This result is in remarkable agreement (at < 1σ error) with semi-analytical model +predictions of B ∝ Σ1/3 +SFR (Schleicher & Beck 2013, 2016). + +20 +Manna and Roy +6. We measure an average intercept of ≈ 65 µG from the B-ΣSFR correlations of our galaxies in Sample 2. This is +higher than the predictions of Schleicher & Beck (2013) by a factor of ≈ 2.5, and, if confirmed with a larger sample, +would imply a significantly higher efficiency of magnetic field amplification than what is typically assumed. +7. We used spatially resolved gas density maps for seven (Sample 3) of the 12 galaxies, for which archival CO +data was available, to find that magnetic fields are correlated with gas densities as B ∝ ρ0.40±0.09 +gas +. This result +is consistent with numerical simulations that predict k ≈ 0.4−0.6 and broadly consistent (within ≈1 sigma +uncertainty) with theories that predict B ∝ ρ0.5 +gas. +8. We have indirectly estimated the power-law index (k) of the correlation between the magnetic fields and the +gas densities using the slope of the radio-FIR correlation, the slope of the Kennicutt-Schmidt law, and the non- +thermal spectral index. The mean value of k, for optically thin dust, was found to be 0.52±0.04 and 0.47±0.09 +at 1.4 and 0.33 GHz respectively for the six galaxies in Sample 1, with NGC 4826 excluded due to its high value +of k. This is consistent with the equipartition between magnetic fields and gas. The anomalously high values +of k (1.0 and 0.95 at 1.4 and 0.33 GHz respectively) for NGC 4826 are possibly due to the contribution of the +central AGN to the radio emission. +We have started to follow up these pilot study results with a survey of a much larger sample of galaxies (Sample +0, Table 2). For this, we have already observed another 24 galaxies using the upgraded GMRT (uGMRT), a Square +Kilometer Array (SKA) pathfinder facility. Sensitivities of the images from these uGMRT observations are significantly +better (≈ 3 times) than those of the observations presented here and the result will be part of a future publication. +In addition, SKA precursors such as the MeerKAT will also provide very deep images of the diffuse radio-continuum +emission around nearby galaxies. Eventually, the dramatic increase in sensitivity and ∼arc-sec resolution of the SKA +has the potential to significantly advance our understanding of magnetic fields in nearby galaxies. For example, the +SKA is expected to provide sensitive images of polarised synchrotron emission from nearby galaxies at a few GHz +frequencies which would provide information on the large-scale ordered fields on the plane of the sky (e.g. Johnston- +Hollitt et al. 2015). Further, polarised emission from nearby galaxies at <∼1 GHz, where significant depolarisations +take place, could be modelled through Faraday tomography (e.g. Heald et al. 2015). +A combination of the two +approaches could eventually allow us to infer the three-dimensional structure of the magnetic fields in nearby galaxies. +SKA observations will also provide detailed images of star formation with resolutions of tens of parsecs. These will +help to identify any dependence of SFR and IMF on galaxy type, evolution and environment within the local volume +(Beswick et al. 2015). +We would like to thank Aditya Chowdhury for his help at various stages of this research. We thank Yogesh Wadadekar, +Preeti Kharb, and Dipanjan Mitra for reading the manuscript and providing useful comments. Aritra Basu provided +their earlier published images and also suggested checking the B vs SFRSD relation for our sample galaxies. We +thank him for the above. We also thank the anonymous referee whose comments helped significantly improve the +presentation of the paper. We thank the staff of GMRT that allowed these observations to be made. GMRT is run by +National Centre for Radio Astrophysics of the Tata Institute of fundamental research. We acknowledge the support +of the Department of Atomic Energy, Government of India, under project no. 12-R&D-TFR-5.02-0700. +1 +2 +3 +4 +5 +6 +7 +REFERENCES +Adebahr, B., Krause, M., Klein, U., et al. 2013, A&A, 555, +A23, doi: 10.1051/0004-6361/201220226 +Bacchini, C., Fraternali, F., Iorio, G., et al. 2020, A&A, +641, A70, doi: 10.1051/0004-6361/202038223 +Basu, A., Mitra, D., Wadadekar, Y., & Ishwara-Chandra, +C. 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J. 2001, ApJ, 554, +803, doi: 10.1086/323145 + +A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities +23 +200 +0 +200 +Relative RA (arcseconds) +300 +200 +100 +0 +100 +200 +300 +Relative DEC (arcseconds) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Error on B ( G) +200 +100 +0 +100 +200 +Relative RA (arcseconds) +200 +100 +0 +100 +200 +Relative DEC (arcseconds) +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +Error on B ( G) +400 +200 +0 +200 +400 +Relative RA (arcseconds) +400 +200 +0 +200 +400 +Relative DEC (arcseconds) +0.2 +0.4 +0.6 +0.8 +Error on B ( G) +400 +200 +0 +200 +400 +Relative RA (arcseconds) +400 +200 +0 +200 +400 +Relative DEC (arcseconds) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Error on B ( G) +200 +0 +200 +Relative RA (arcseconds) +200 +0 +200 +Relative DEC (arcseconds) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Error on B ( G) +200 +0 +200 +Relative RA (arcseconds) +300 +200 +100 +0 +100 +200 +300 +Relative DEC (arcseconds) +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +Error on B ( G) +250 +0 +250 +Relative RA (arcseconds) +400 +200 +0 +200 +400 +Relative DEC (arcseconds) +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +Error on B ( G) +Figure 7. The magnetic field uncertainty maps (in µG) of NGC 2683 (top left), NGC 3627 (top centre), NGC 4096 (top right), +NGC 4449 (middle left), NGC 4490 (middle centre), NGC 4826 (middle right) and NGC 5194 (bottom) (Sample 1), shown in +colour scale. Blanked regions (in white colour) in the centre of each galaxy correspond to regions with spectral index values ≤ +0.55. +APPENDIX +A. MAGNETIC FIELD UNCERTAINTY MAPS +We present here (Figure 7) magnetic field uncertainty maps of the galaxies in Sample 1, generated using the procedure +described in Section 2.1.1. +B. STAR FORMATION RATE SURFACE DENSITY MAPS +We show SFRSD maps of the seven galaxies (Sample 1) in Figures 8 and 9, where SFRSDs estimated using 1.4 GHz +and FUV+24µm emission are shown in contours and colors, respectively. In Figures 10 and 11, we have also shown +the SFRSD maps estimated using Hα+24µm and 1.4GHz data in colors and contours, respectively. The SFRSD maps +of each galaxy in Figures 8, 9, 10, and 11 have been shown in the same color scale and contours. + +24 +Manna and Roy +Figure 8. SFRSD (M⊙yr−1kpc−2) maps of NGC 2683, NGC 3627, NGC 4449 and NGC 4096(clockwise from top left) (Sample +1). SFRSDs estimated using 1.4 GHz radio and FUV+24µm emission are shown in contours and colors, respectively. Contour +levels are listed below each panel of the figure. The circle in the bottom-left corner of the images indicates the angular resolution +of the maps. + +COLOR:NGC26832683sfrhyb.OHGSPX.1 +CONT:NGC2683IPOL1490.572MHz2683.sfr.L.TH.SUB.1 +0 +10 +20 +30 +33 29 +28 +27 +Declination (J2000) +26 +25 +24 +23 +22 +21 +085300 +5255 +50 +45 +40 +35 +30 +25 +20 +Right Ascension (J2000) +Colorscale ranqe=-0.1039.85MilliSolarmass/yr/kpc^2 +Contpeakflux=5.4958E-02Solarmass/yr/kpc^2 +Levs = 2.327E-03 * (-1,1,2, 4,8, 16,25, 50, +80,100,130,160)COLOR:NGC36273627sfrhyb.OHGSPX.1 +CONT:N3627LIPOL1430.389MHz3627.sfr.L.TH.SUB.1 +200 +400 +600 +13 02 +01 +00 +Declination (J2000) +1259 +58 +57 +56 +11 20 25 +20 +15 +10 +05 +RightAscension(J2000) +Colorscalerange=-0.0604.9MilliSolarmass/yr/kpc^2 +Contpeakflux=2.4733E-01Solarmass/yr/kpc^2 +Levs = 2.000E-02 * (-1, 1,2, 4, 8, 16, 25, 50, +80,100,150,200,400,800)COLOR:NGC40964096sfrhyb.OHGSPX.1 +CONT:NGC4096IPOL1432.873MHz4096.sfr.L.TH.SUB.1 +0 +10 +20 +30 +40 +4731 +30 +Declination (J2000) +29 +28 +0 +27 +26 +25 +120620 +15 +10 +05 +00 +0555 +50 +45 +40 +Right Ascension (J2000) +Colorscale range=-0.1546.02MilliSolarmass/yr/kpc^2 +Contpeakflux=3.1588E-02Solarmass/yr/kpc^2 +Levs = 1.511E-03 * (-1, 1, 2, 4, 8, 16, 25, 50, +80,100,150,200,400,800)COLOR:NGC44494449sfrFUV.OHGSPX.1 +CONT:N49IPOL1489.984MHz4449sfrL.TH.SUB.2 +100 +200 +4412 +10 +08 +Declination (J2000) +06 +04 +02 +00 +12 28 45 +30 +15 +00 +27 45 +RightAscension(J2000) +Colorscalerange=-0.1251.0MilliSolarmass/yr/kpc^2 +Contpeakflux=7.4433E+00Solarmass/yr/kpc^2 +Levs = 5.421E-03 * (-1, 1,2, 4, 8, 16, 25, 50, +80,100,150,200,400,800)A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities +25 +Figure 9. SFRSD (M⊙yr−1kpc−2) maps of NGC 4490, NGC 4826 and NGC 5194 (clockwise from top left) (Sample 1). SFRSDs +estimated using 1.4 GHz radio and FUV+24µm emission are shown in contours and colors, respectively. Contour levels are +listed below each panel of the figure. The circle in the bottom-left corner of the images indicates the angular resolution of the +maps. + +COLOR:NGC44904490sfrFUV.OHGSPX.1 +CONT:N4490IPOL1435.114MHz4490sfrL.TH.SUB.2 +0 +50 +100 +150 +200 +41 44 +42 +Declination (J2000) +40 +38 +36 +34 +0 +12 31 00 +30 45 +30 +15 +RightAscension(J2000) +Colorscale range=-0.1204.4MilliSolarmass/yr/kpc^2 +Contpeakflux=2.9951E-01Solarmass/yr/kpc^2 +Levs = 7.000E-03 * (-1, 1, 2, 4, 8, 16, 25, 50, +80,100,150,200,400,800)COLOR:NGC48264826sfrhyb.OHGSPX.1 +CONT:N4826LIPOL1425.677MHz4826.sfr.L.TH.SUB.1 +0 +100 +200 +300 +214300 +42 30 +00 +Declination (J2000) +41 30 +00 +40 30 +00 +39 30 +00 +12 56 50 +45 +40 +35 +RightAscension (J2000) +Color scalerange=-0.1354.2Milli Solarmass/yr/kpc^2 +Contpeakflux=3.4554E-01Solarmass/yr/kpc^2 +Levs=3.758E-03 *(-1, 1,2, 4,8, 16,25, 50, +80,100,150,200,400,800)COLOR:NGC51945194sfrhyb.OHGSPX.1 +CONT: M51 iPOL 1664.900 MHz 5194.sfr.L.TH.SUB.1 +100 +200 +4718 +16 +14 +Declination (J2000) +12 +10 +08 +06 +13 30 15 +00 +29 45 +30 +RightAscension (J2000) +Color scale range= -0.0293.2'Milli Solar mass/yr/kpc2 +Contpeakflux=3.1707E-01Solarmass/yr/kpc^2 +Levs = 1.215E-03 * (-1, 1, 2, 4, 8, 16, 25, 50, +80,100,150,200,400,800)26 +Manna and Roy +Figure 10. +SFRSD (M⊙yr−1kpc−2) maps of NGC 2683, NGC 3627, NGC 4449 and NGC 4096 (clockwise from top left) +(Sample 1). SFRSDs estimated using 1.4 GHz radio and Hα+24µm emission are shown in contours and colors, respectively. +Contour levels are listed below each panel of the figure. The circle in the bottom-left corner of the images indicates the angular +resolution of the maps. + +COLOR:N2683652683sfrha.OHGEO.1 +CONT:NGC2683 IPOL 1490.572MHz 2683 sfr L.TH.SUB.1 +0 +10 +20 +30 +3329 +28 +27 +Declination (J2000) +26 +25 +24 +23 +22 +085300 +52 55 +50 +45 +40 +35 +30 +25 +RightAscension(J2000) +Colorscalerange=-0.1039.85MilliSolarmass/yr/kpc^2 +Contpeakflux=5.4958E-02Solarmass/yr/kpc^2 +Levs=2.327E-03 *(-1,1,2, 4,8,16,25,50, +80,100,130,160)COLOR:nqc36273627sfrha.OHGEO.1 +CONT:N3627LIPOL1430.389MHz3627sfrL.TH.SUB.1 +200 +400 +600 +13 02 +01 +Declination (J2000) +00 +1259 +58 +57 +56 +11 20 25 +20 +15 +10 +05 +RightAscension(J2000) +Colorscalerange=0.0604.9MilliSolarmass/yr/kpc^2 +Contpeak flux= 2.4733E-01Solarmass/yr/kpc^2 +Levs = 2.000E-02 *(-1, 1,2,4, 16,25, 50, 80, +100,150,200,400,800)COLOR:N4096654096sfrha.OHGEO.1 +CONT:NGC4096IPOL1432.873MHz4096sfrL.TH.SUB.1 +0 +10 +20 +30 +40 +47 32 +31 +30 +Declination (J2000) +29 +28 +27 +26 +25 +12.0620 +15 +10 +05 +0555 +50 +45 +40 +RightAscension(J2000) +Colorscalerange=-0.1546.02MilliSolarmass/yr/kpc^2 +Contpeak flux=3.1588E-02Solarmass/yr/kpc^2 +Levs = 1.511E-03 *(-1, 1,2, 4,8, 16, 25, 50, +80,100,150,200,400,800)COLOR:N4449654449sfrha.OHGEO.1 +CONT:N49IPOL1489.984MHz4449sfrL.TH.SUB.2 +100 +200 +4412 +10 +08 +? +Declination (J2000) +06 +04 +02 +00 +12 28 30 +15 +00 +27 45 +RightAscension(J2000) +Colorscalerange=-0.1251.0MilliSolarmass/yr/kpc^2 +Contpeakflux=7.4433E+00Solarmass/yr/kpc^2 +Levs = 5.421E-03 * (-1, 1, 2, 4, 8, 16, 25, 50, +80,100,150,200,400,800)A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities +27 +Figure 11. SFRSD (M⊙yr−1kpc−2) maps of NGC 4490, NGC 4826 and NGC 5194 (clockwise from top left) (Sample 1). +SFRSDs estimated using 1.4 GHz radio and Hα+24µm emission are shown in contours and colors, respectively. Contour levels +are listed below each panel of the figure. The circle in the bottom-left corner of the images indicates the angular resolution of +the maps. + +COLOR:N4485/904490sfrha.OHGEO.1 +CONT:N4490IPOL1435.114MHz4490sfrL.TH.SUB.2 +50 +100 +150 +200 +41 44 +42 +Declination (J2000) +40 +38 +36 +34 +0 +12 31 00 +30 45 +30 +15 +RightAscension(J2000) +Colorscale range=-0.1204.4MilliSolarmass/yr/kpc^2 +Contpeakflux=2.9951E-01Solarmass/yr/kpc^2 +Levs=7.000E-03 *(-1,1,2,4,8,16,25,50, +80,100,150,200,400,800)COLOR:N4826654826sfrha.OHGEO.1 +CONT:N4826LIPOL1425.677MHz4826sfrL.TH.SUB.1 +100 +200 +300 +21 43 30 +00 +42 30 +Declination (J2000) +00 +41 30 +00 +40 30 +00 +39 30 +00 +38 30 +125655 +50 +45 +40 +35 +30 +RightAscension(J2000) +Colorscalerange=-0.1354.2MilliSolarmass/yr/kpc^2 +Contpeakflux=3.4554E-01Solarmass/yr/kpc^2 +Levs = 3.758E-03 * (-1, 1, 2, 4, 8, 16, 25, 50, +80,100,150,200,400,800)COLOR:NGC5194+5194sfrha.OHGEO.1 +CONT:M51IPOL1664.900MHz5194sfrL.TH.SUB.1 +100 +200 +4718 +16 +D +14 +Declination (J2000) +12 +10 +08 +06 +Q +13 30 30 +15 +00 +29 45 +30 +15 +Right Ascension (J2000) +Color scale range=0.0293.2Mili Solar mass/yr/kpc^2 +Contpeakflux=3.1707E-01Solarmass/yr/kpc^2 +Levs=1.215E-03 *(-1,1,2, 4,8, 16,25,50, +80,100,150,200,400,800) \ No newline at end of file diff --git a/59E2T4oBgHgl3EQfOwac/content/tmp_files/load_file.txt b/59E2T4oBgHgl3EQfOwac/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cef3c2d7b2f5c0d2d51ede9d48939d0ae68cb070 --- /dev/null +++ b/59E2T4oBgHgl3EQfOwac/content/tmp_files/load_file.txt @@ -0,0 +1,2345 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf,len=2344 +page_content='Draft version January 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2023 Typeset using LATEX default style in AASTeX631 Magnetic Fields,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Star Formation Rates and Gas Densities at Sub-kpc Scales in a Pilot Sample of Nearby Galaxies Souvik Manna1 and Subhashis Roy1 1National Center for Radio Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' TIFR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Pune University Campus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Ganeshkhind,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Pune 411007,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' India ABSTRACT We have estimated the magnetic field strengths of a sample of seven galaxies using their non-thermal synchrotron radio emission at metre wavelengths,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' and assuming energy equipartition between magnetic fields and cosmic ray particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We tested for deviation of magnetic fields from energy equipartition with cosmic ray particles, and found that deviations of ∼25% are typical for the sample galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Spatially resolved star formation rates (SFR) were estimated for the seven galaxies along with five galaxies studied previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For the combined sample of twelve galaxies, the equipartition magnetic fields (Beq) are correlated with the SFR surface densities (ΣSFR) at sub-kpc scales with Beq ∝ Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='31±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='06 SFR , consistent with model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We estimated gas densities (ρgas) for a sub-sample of seven galaxies using archival observations of the carbon monoxide (CO) rotational transitions and the atomic hydrogen (Hi) 21 cm line and studied the spatially-resolved correlation between the magnetic fields and ρgas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Magnetic fields and gas densities are found to be correlated at sub-kpc scale as Beq ∝ ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='40±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='09 gas .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' This is broadly consistent with models, which typically predict B ∝ ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Keywords: Radio continuum emission — Interstellar medium — Star formation — Magnetic fields 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' INTRODUCTION Magnetic fields are believed to influence several physical processes in a galaxy at almost every scale (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Elmegreen 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Niklas & Beck 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Price & Bate 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Adebahr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Magnetic fields have been found to consist of two main components: a small-scale turbulent magnetic field up to a few hundred parsecs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Batchelor 1950;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2003) and a large-scale “ordered” or “regular” magnetic field component at scales of a few kpcs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Moss & Shukurov 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Shukurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Kulsrud & Zweibel 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Magnetic fields in galaxies can be measured using their effects on different radiation processes like Zeeman splitting of emission lines, polarized emission from dust, the polarization of starlight, Faraday rotation of polarized radio emission, and intensity of synchrotron emission which we use in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Measurement of the line-of-sight component of the magnetic field via the Zeeman effect in galaxies other than the Milky Way has been possible for only a few systems (Kazes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Sarma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Robishaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' a significant expansion of such studies is very difficult with current-generation telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Magnetic fields in galaxies can be measured and studied using synchrotron emission at radio frequencies, at scales larger than the resolution of the radio observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For example, a Very Large Array (VLA) polarization study of NGC 4736 at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='46 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='86 GHz found that the magnetic field in the galaxy was ordered in a spiral shape (Chy˙zy & Buta 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' An X-shaped structure of the magnetic field in the galactic halo region was observed by stacking the Karl G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Jansky VLA polarized emission maps of 16 nearly edge-on spiral galaxies, obtained as part of the CHANG-ES survey (Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' such structures had also been observed in individual spiral galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Krause 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Heesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' However, polarized radio emission from external individual galaxies is difficult to study at low radio frequencies due to Faraday depolarization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Sokoloff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Corresponding author: Souvik Manna souvik@ncra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='tifr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='in arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03752v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='GA] 10 Jan 2023 2 Manna and Roy The average magnetic field strength can also be estimated from the total intensity of synchrotron radio emission, assuming energy equipartition between magnetic fields and cosmic ray particles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Miley 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Beck & Krause 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Equipartition magnetic fields have been studied in several nearby galaxies, but primarily at frequencies >1 GHz (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Chy˙zy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Soida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Heesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Adebahr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Vargas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2018) studied a sample of three nearly edge-on galaxies from the CHANG-ES survey to separate the thermal Bremsstrahlung from the non-thermal synchrotron emission at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 and 6 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' At these frequencies, the thermal component is large and hence the correction for the thermal emission can be as large as ∼ 20%, making the derived magnetic field strengths prone to errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Conversely, the steep spectral index of synchrotron emission implies that it will dominate the total emission at frequencies < 1 GHz, with ∼ 95% contribution (Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2012b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Roy & Manna 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Thus, magnetic field strengths derived using observations at <1 GHz are very robust to any correction for thermal emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Magnetic fields are believed to play an important role at various stages of the star-formation process - from the fragmentation of clouds at the few kpc scales to the final collapse of gas into stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Elmegreen 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Crutcher 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Price & Bate 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Van Loo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' To understand the influence of magnetic fields and star-formation activities on different physical processes in the ISM at different physical scales, several studies on radio-infrared correlations have been carried out in the past (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2006a,b, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Tabatabaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Magnetic fields (B) and star formation rate surface densities (SFRSD) are expected to be correlated (Niklas & Beck 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Semi- analytical model also predicts a strong correlation between B and SFRSDs (ΣSFR) as B ∝ Σ1/3 SFR at sub-kpc scales to explain the local radio-FIR correlation (Schleicher & Beck 2013, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Observational studies of the correlation between B and star formation rates (SFR) have been done primarily in samples of nearby dwarf galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For example, Chy˙zy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2011) studied 12 local group dwarf galaxies to find that the galaxy-averaged magnetic field and the SFR follow B ∼ SFR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='30±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04, consistent with the prediction of B ∝ Σ1/3 SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' However, Jurusik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2014) found the same power-law index in a sample of Magellanic type dwarf galaxies to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02, somewhat lower than the expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Recently, a study of the dwarf galaxy IC 10 by Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2017) provides the only study of the correlation between spatially-resolved magnetic fields and SFRSDs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' these authors found that the SFRSD is related to the magnetic field as B ∝ Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='35±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03 SFR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Therefore, it is important to test such predictions by carrying out systematic spatially-resolved studies of magnetic fields in galaxies and their connection to the star formation rate in nearby large galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The energy density of magnetic fields and gas in galaxies are expected to be in equipartition, which implies B ∝ √ρgas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Chandrasekhar & Fermi 1953;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The observed Radio-FIR correlation can be explained based on such equipartition between the energy density of magnetic fields and gas (Niklas & Beck 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Several other numerical magnetohydrodynamic (MHD) simulations of the ISM have predicted the coupling constant (k) between magnetic fields and gas (B ∝ ρk gas) to be in the range of ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 (Fiedler & Mouschovias 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Niklas & Beck (1997) studied the correlation between galaxy-integrated equipartition magnetic fields and gas densities for a sample of 43 galaxies to find a power-law index of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' the observed correlation is consistent with B ∝ √ρgas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Although the correlation between gas surface densities and SRFSDs has been extensively studied in the nearby Universe (Kennicutt-Schmidt law;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Kennicutt 1998a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Onodera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Roychowdhury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2015), systematic studies of spatially-resolved correlations between magnetic fields, SFRs and gas densities in nearby galaxies are yet to be carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' It is thus important to carry out a systematic investigation of both the B-ρ and the B-SFR correlations, at high-spatial resolutions (≈ sub-kpc scales), using direct estimates of the magnetic fields, gas densities, and star-formation rates, in a sample of nearby galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' In this paper, we present a pilot study of the connection between spatially resolved magnetic fields, SFRSDs and gas densities in a sample of nearby galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We have selected a sample of 46 galaxies (Sample 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Table 2) from the Spitzer Local Volume Legacy (LVL) sample of 258 galaxies within 11 Mpc (Dale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' As a pilot project, seven (Sample 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Table 2) of these 46 galaxies have been observed with the Giant Metrewave Radio Telescope (GMRT) at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 GHz (Roy & Manna 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Six of our seven sample galaxies are spirals and the other one is a dwarf irregular Magellanic-type galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' In this paper, we present spatially resolved equipartition magnetic field (Beq) maps of the seven galaxies in Sample 1 (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We also incorporate the magnetic field maps of five galaxies studied by Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2012a) from previous GMRT observations in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We derived SFRSD maps of all 12 galaxies (Sample 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Table 2) using extinction- free diagnostics and used these maps to study the relation between SFRSDs and Beq at sub-kpc scales in our pilot study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We used available archival CO and Hi 21 cm data to measure the gas densities (ρgas) of seven (Sample 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Table 2) of the combined sample of 12 galaxies and studied the correlation between ρgas and Beq in these galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We also studied the magnetic field-gas connection through an indirect measurement of their coupling coefficient using radio−FIR correlations of the galaxies in Sample 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Details of the seven sample galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Note that the images at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 GHz were obtained from observations with the GMRT reported in Roy & Manna (2021) while those at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz were obtained from archival VLA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The distances to the galaxies were taken from Dale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Galaxies with an asterisk are those for which spatially-resolved CO data are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Name Class Distance Inclination Position uv Angular Spatial RMS RMS VLA (Mpc) angle angle range resolution resolution (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 GHz) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz) Project ID (deg) (deg) (kλ) (arcsec2) (pc) (µJy/beam) (µJy/beam) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz) NGC 2683 Sb 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='7 83 43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='19 - 15 19 × 13 670 200 40 AI23 NGC 3627∗ SAB 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 65 170 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='26 - 25 16 × 11 760 800 370 AS541, AP462 NGC 4096 SABc 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='3 76 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='14 - 17 14 × 12 730 100 25 16A-013 NGC 4449 Irregular 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='15 - 15 26 × 15 360 300 180 AB167 NGC 4490 SBm 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 60 126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='13 - 14 19 × 18 560 230 100 AA181 NGC 4826∗ SAab 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 60 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='22 - 20 15 × 14 650 280 70 AS541 NGC 5194∗ Sbc 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 20 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='15 - 10 23 × 18 740 310 30 AB505, AN57 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' List of different samples studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Sample Name Galaxies Sample 0 Full sample containing 46 galaxies from Spitzer LVL survey Sample 1 Pilot sample containing 7 galaxies from Sample 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' galaxies listed in Table 1 Sample 2 Sample 1 + 5 galaxies (NGC 1097, NGC 4736, NGC 5055, NGC 5236 and NGC 6946) from Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2012b) = 12 galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' used to probe the Beq-SFRSD correlations Sample 3 A subset of 7 galaxies (NGC 3627, NGC 4826, NGC 5194, NGC 4736, NGC 5055, NGC 5236 and NGC 6946) from Sample 2 which have archival CO data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' used to study the Beq-gas density correlations The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The analysis of the data is discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' In Section 3, we present the results of our analysis, including the correlation between magnetic fields, SFRSDs and gas densities of the seven galaxies in Sample 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' In Section 4, we have extended our study to include a sample of five galaxies of Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2012a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We discuss the results in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' A summary of this paper is presented in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' DATA ANALYSIS As can be seen in Table 1, six of the seven galaxies in Sample 1 are spirals of varying inclination angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The seventh galaxy NGC 4449 is a dwarf irregular galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Basic information about the seven sample galaxies, including their types, distances, inclination angles, position angles, angular resolutions, spatial resolutions, and RMS noise obtained on the GMRT and VLA images are also listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The distances, inclination angles, and position angles of the galaxies were taken from Dale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Radio observations and the data reduction procedures are discussed in detail in Roy & Manna (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Briefly, we used GMRT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 GHz observations (covering 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='309−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='342 GHz) and archival VLA observations at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 and ∼6 GHz to derive non-thermal emission maps for each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We used Hα and 24µm observations of the seven galaxies to model free-free emission from them and subsequently, we subtracted the modelled free-free emission from the observed radio emission to get the non-thermal radio maps at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 and ∼6 GHz (Roy & Manna 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' To generate the non-thermal spectral index maps, we used the non-thermal radio maps at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 and ∼6 GHz for NGC 2683, NGC 3627, NGC 4096, and NGC 4449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For the rest of the galaxies (NGC 4490, NGC 4826, and NGC 5194), we used non-thermal images at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz to generate the non-thermal spectral index maps (Roy & Manna 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' In the following subsections, we present the analysis of other ancillary data and relevant measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Magnetic Field Strengths The average magnetic field strengths can be estimated from the observed synchrotron flux densities, assuming energy equipartition between cosmic ray particles and magnetic fields (“Classical Equipartition Formula”, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Pacholczyk 1970;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Miley 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Longair 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The equipartition condition is achieved when the total energy in magnetic fields and cosmic ray particles is minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 4 Manna and Roy The classical equipartition formalism has shortcomings that lead to an overestimation of the magnetic field strength (B) at regions of steep spectral indices and underestimation of B at flat spectral index regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' To overcome these shortcomings of the classical equipartition formula, Beck & Krause (2005) proposed a revised formula to estimate the average magnetic field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The formula is expressed as Beq = [4π(K0 + 1)E1−2αnt p f(αnt) C4(i) Iνναnt l ] 1 αnt+3 (1) K0, Ep, Iν, and αnt are the number density ratio of cosmic ray protons to electrons, the proton rest mass energy, the intensity of the synchrotron emission at frequency ν, and the spectral index of synchrotron emission, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' f(αnt) is a function of αnt given as f(αnt) = (2αnt + 1)[2(αnt − 1)c2(αnt)cαnt 1 ] (Beck & Krause 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' C4(i) is a constant that depends on the inclination angle (i) of the galaxy and is expressed as C4(i) = [cos(i)](γ+1)/2, where γ = (2αnt +1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' l is the path length of the synchrotron emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The path length was assumed to be 1 kpc for a galaxy with an inclination angle of 0 degree (face-on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For galaxies with low- and moderate- inclination angles (< 75◦), the assumed path length was corrected for the inclinations of the galaxies as l/cos(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For the two nearly edge-on galaxies in Sample 1, NGC 2683 and NGC 4096, we have assumed an oblate spheroidal shape of the synchrotron emission, such that the diameter on the plane of the galaxy is equal to its major axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The path lengths (l) were then appropriately calculated, with the path length being maximum (equal to the galaxy’s major axis) at the optical centre of the galaxy and gradually declining to the edge of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We note that Beq has only a weak dependence on l as Beq(r) = l(r) −1 αnt+3 and hence is less sensitive to the exact choice of l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Values of K0 and Ep were assumed to be 100 and 938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='28 MeV, respectively, the same as used by Beck & Krause (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Finally, we used non-thermal radio maps at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 GHz (Iν) and spectral index maps (αnt) made using 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 or ∼6 GHz radio observations (Roy & Manna 2021) to produce magnetic field maps of the sample galaxies using Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The revised equipartition formula diverges for spectral index values ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 because such flat spectra indicate energy loss of electrons through ionizations or Coulomb interactions (Sarazin 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The central bulge and arm regions have a mostly flatter spectrum due to the association of star-forming regions and the estimates of equipartition magnetic fields in such regions might be affected by systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' This issue affects the derived magnetic field strengths for 8%, 12%, 3%, 70%, 17%, 7%, and 6% of the projected total surface area of NGC 2683, NGC 3627, NGC 4096, NGC 4449, NGC 4490, NGC 4826, and NGC 5194, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We note that a large fraction of the derived magnetic field values are affected for NGC 4449 due to its non-thermal spectral indices being predominantly flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' This could bias the Beq for NGC 4449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Uncertainties on Magnetic Field Maps The procedure we used to estimate the uncertainties on our magnetic field maps is similar to that of Basu & Roy (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We used a Monte Carlo method that generated 104 random flux density values for each pixel in a galaxy map at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 GHz and either 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz or 6 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' These flux density values have Gaussian probability distributions with rms values equal to the measured rms of each of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4/6 GHz maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For each of the 104 intensity maps, we computed a magnetic field map using the procedure described in the beginning of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The rms of these 104 magnetic field maps provided us with the magnetic field uncertainty maps for each of the seven galaxies in sample 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Star Formation Rates Rest frame Hα and ultraviolet (UV) observations are the best tracers of recent SFRs as the radiation from these predominantly originate in newly formed massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' However, the observations are affected by extinction caused by interstellar dust in both the host galaxy as well as the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' SFRs estimated from Hα and UV observations are therefore corrected for the extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Dust-corrected SFRs can be estimated by combining far-ultraviolet (FUV) and Hα data with infrared (IR) data to exploit the complementary strengths at different wavelengths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Kennicutt & Evans 2012a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Buat 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Meurer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 1995, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Cortese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' In addition to the FUV+IR and Hα+IR tracers, the low-frequency radio emission from galaxies, which is predominantly optically thin synchrotron emission, can be used to estimate their dust-unobscured SFRs via the radio-FIR correlation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Yun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We estimated the spatially-resolved star formation rates of our Sample 1 galaxies using FUV+24µm, Hα+24µm, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz data, which are discussed, respectively, in the following Sections, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We used data of these different frequencies as tracers in order to (1) get a fair comparison between different SFR diagnostics and (2) for studying star-formation history at different timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' All SFRs in this paper assume a Kroupa IMF (Kroupa 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' SFRs using FUV and 24µm Observations To estimate SFRSD maps of the seven galaxies in Sample 1 (Table 2) using FUV+24µm emission, we used SPITZER 24 µm IR data (Dale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2009) and GALEX FUV data (11HUGS survey;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Kennicutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We first convolved both the 24 µm and the FUV maps of all galaxies to the same resolutions as our magnetic field maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The FUV data were corrected for extinction due to dust in the Milky Way (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The FUV images were in units of counts/sec/pixel and were converted to flux-density units of MJy Sr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We also converted the 24µm images to units of MJy Sr−1 and used the following calibration from Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2012) to derive SFRSD maps for the sample galaxies: ΣSFR[M⊙yr−1kpc−2] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='081 IFUV[MJy sr−1] + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='032 I24µm[MJy sr−1] (2) The uncertainties of the coefficients are ∼10-30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Note that the uncertainty in SFR estimates arises from issues such as the error in sampling the stellar IMF of different star-forming regions, determining the contribution of different emission which are not associated with recent star formation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g Kennicutt & Evans 2012b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' SFRs using Hα and 24µm Observations To estimate SFRSD maps using Hα+24µm as a tracer, we used 24 µm emission along with Hα emission from 11HUGS (Kennicutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2008), for all but NGC 5194, for which we used data from the SINGS survey (Kennicutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' All the maps were convolved and regridded to the resolution and pixel size of the magnetic field maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For the Hα maps from 11HUGS and SINGS, the flux density units were converted to erg/s/cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We used the following calibration from Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2012) to estimate SFRSDs of the galaxies in Sample 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' ΣSFR[M⊙yr−1kpc−2] = 634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 IHα[erg s−1 sr−1] + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0025 I24µm[MJy sr−1] (3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' SFRs using 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz Observations Our 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz non-thermal maps of the galaxies (Sample 1) (Roy & Manna 2021) and an SFR calibration from Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2011) were used to derive SFRSD maps (Equation 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The calibration is based on the observed radio- FIR correlation in a sample of nearby star-forming galaxies (Bell 2003) and has a scatter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='26 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We used this galaxy-integrated calibration (Equation 4) to derive the formula for spatially-resolved radio-ΣSFR calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' SFR1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4GHz M⊙yr−1 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='35 × 10−29 L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4GHz erg Hz−1s−1 (4) The spatially-resolved calibration is consistent with the calibration of Heesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We used the above relation to estimate the SFRSD maps of the sample galaxies from the measured 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz surface brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Galactic Extinction Correction for FUV Emission We corrected for the extinction of FUV emission due to dust in the Milky Way using the E(B-V) values along the line of sight to the sample galaxies from Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The extinction coefficients (AFUV) of the GALEX FUV bands were measured using Table 1 from Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2017) and intrinsic fluxes (Fintrinsic) were estimated from the following formula: AFUV = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 × log[Fobserved Fintrinsic ] (5) The extinction percentage of the FUV emission is listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Gas Densities Atomic hydrogen (Hi) and molecular hydrogen (H2) predominantly contribute to the total gas mass of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' H2 is best traced using rotational transitions in CO (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Bolatto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Spatially-resolved observations of CO transitions exist for only three of our seven galaxies in Sample 1 (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We have used CO J=2-1 line data of NGC 3627 and NGC 5194 from the HERA CO-Line Extragalactic Survey (HERACLES;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2009) and CO J=1-0 data of NGC 4826 from the BIMA Survey of Nearby Galaxies (BIMA SONG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Regan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The HERACLES and BIMA survey have a spatial resolution of 13′′ and 6′′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The velocity resolution of the HERACLES and BIMA spectral cubes are ∼5 and 6 km/s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We restrict our study of the connection between gas densities and magnetic fields to only these three galaxies for which spatially-resolved CO data are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 6 Manna and Roy Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' FUV extinction values of the Sample 1 galaxies due to the Milky Way foreground dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The extinctions were computed using E(B-V) values along the line of sight to the sample galaxies from Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Name Percentage extinction NGC 2683 22 NGC 3627 23 NGC 4096 21 NGC 4449 15 NGC 4490 15 NGC 4826 13 NGC 5194 27 The HI Nearby Galaxy Survey (THINGS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Walter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2008) used VLA observations to obtain very high spectral (≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 km/s) and spatial (∼ 6 ′′) resolution maps of nearby galaxies at 21cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We used the publicly available 21cm moment maps from this THINGS survey to estimate the distribution of Hi in the three galaxies for which CO data are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' All CO and Hi 21 cm maps were convolved and regridded to a common resolution and pixel size of the non-thermal radio maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Gas densities were estimated (for NGC 3627, NGC 4826 and NGC 5194) following Basu & Roy (2013) assuming CO to H2 conversion factor of 2 ×1020 (K km s−1)−1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Bolatto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' A line ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='8 was assumed to convert COJ=2-1 to COJ=1-0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We accounted for the contribution of helium to the gas density using ρgas=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='36 × (ρHi + ρH2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Line of sight depths were assumed to be 300 and 400 pc for molecular and atomic gas, respectively (Basu & Roy 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Magnetic Fields in the Galaxies We have estimated spatially resolved revised equipartition magnetic field maps for seven galaxies in Sample 1, using the procedures of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' these maps are shown in Figures 1 & 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Flux density contours of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz observations are overlaid on magnetic field maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The resolution of these maps corresponds to spatial scales of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='8 kpc (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The bottom right panel of Figure 2 shows the radial variation of the magnetic field with galactocentric radius of all the seven galaxies where both the axes are normalized by their maximum values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Here, we have averaged the magnetic field strengths over an annular elliptical region of width equal to the beam size of the corresponding map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Position and inclination angle (Table 1) of each galaxy were used while selecting the elliptical regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We find magnetic fields to be stronger at the central region and at the star formation sites (arm regions) with field strengths up to 50 µG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Field strengths fall by ∼50% at the edges of the magnetic field maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The Milky Way also shows such a trend in the variation of magnetic field strengths (Beck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We note that our analysis was limited to distances where the signal-to-noise ratio in spectral index maps is > 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' the magnetic field strengths at these distances are thus likely to be reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We note that, compared to the magnetic field strengths obtained using the classical equipartition expression, these values are higher by ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='3−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 for a non-thermal spectral index of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6, and they match for a spectral index of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='75 (Beck & Krause 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Figure 7 shows the uncertainties in the magnetic field values for Sample 1 derived using the Monte Carlo method described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Statistical uncertainties on mean magnetic fields for these seven galaxies are provided in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Star Formation Rates in the Galaxies We have estimated the global, galaxy-averaged SFRs of Sample 1 galaxies using 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz, FUV+24µm, and Hα+24µm emission using calibrations discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Globally integrated star formation rates of the sample galaxies are given in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' No systematic offset was found in the SFR values estimated using these tracers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The differences in the SFR values for our galaxies are much less than the calibration uncertainty except for NGC 4490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For NGC 4490, SFR calculated from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz emission is higher than the same from FUV+24µm emission by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' As discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2, we have estimated SFRSD maps of the seven galaxies (Sample 1) using FUV+24µm, Hα+24µm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4GHz emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We show SFRSD maps of the seven galaxies in the Appendix (Figures 8-9), where SFRSDs estimated using 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz and FUV+24µm emission are shown in contours and colors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' In the A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities 7 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The equipartition magnetic field maps of NGC 2683, NGC 3627, NGC 4449 and NGC 4096 (clockwise from top left) (Sample 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Non-thermal radio contours at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz are overlaid on magnetic field maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The magnetic field strengths are shown in color with non-thermal emission at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz shown as overlaid contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Contour levels are presented below each panel in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The circle in the bottom-left corner of the panels indicates the angular resolution of the maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The uncertainties on mean magnetic fields are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='06µG, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='17µG, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04µG and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='18µG for the above galaxies, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' COLOR:NGC26832683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='final.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='SUB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 CONT:NGC2683IPOL1490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='572MHz2683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='Ths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='SUB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 10 20 30 40 33 29 28 27 Declination (J2000) 26 25 24 23 22 21 085305 00 52 55 50 45 40 35 30 25 20 RightAscension(J2000) Colorscalerange=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='00uG Contpeakflux=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='8560E-03JY/BEAM Levs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='600E-04 * (-2, -1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512)COLOR:NGC36273627.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='final.' metadata={'source': 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64,128,256,512COLOR:NGC40964096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='final.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='SUB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 CONT:NGC4096IPOL1432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='873MHz4096.' metadata={'source': 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128, 256,512)8 Manna and Roy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 Radial distance (Normalised) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 Magnetic field (Normalised) NGC 2683 NGC 3627 NGC 4096 NGC 4449 NGC 4490 NGC 4826 NGC 5194 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The equipartition magnetic field maps of NGC 4490 (top left), NGC 4826 (top right) and NGC 5194 (bottom left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The magnetic field strengths are shown in color with non-thermal emission at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz shown as overlaid contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Contour levels are presented below each panel in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The circle in the bottom-left corner of the panels indicates the angular resolution of the maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The uncertainties on mean magnetic fields are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='06µG, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='11µG and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02µG, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The bottom right panel presents the radial variation of magnetic field strengths with galactocentric distance for all seven galaxies in Sample 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='ThS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='SUB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 10 20 30 40 4144 42 1 Declination (J2000) 40 38 36 34 123100 30 45 30 15 RightAscension(J2000) Colorscalerange=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0045.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='final.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='OHGSPX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 CONT:M51IPOL1664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='900MHz5194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='Ths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='SUB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 10 20 30 40 4718 16 14 Declination (J2000) 12 10 08 06 13 30 30 15 00 29 45 30 15 RightAscension(J2000) Colorscale range=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='00 uG Contpeakflux=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6656E-02JY/BEAM Levs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='200E-04 *(-2, -1, 1,2, 4, 8, 16, 32 64,128,256,512)A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities 9 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Statistical uncertainties on mean magnetic fields for galaxies in Sample 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Name Statistical uncertainty on mean magnetic fields (µG) NGC 2683 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='06 NGC 3627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='17 NGC 4096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04 NGC 4449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='18 NGC 4490 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='06 NGC 4826 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='11 NGC 5194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Galaxy-averaged star formation rates of the galaxies in Sample 1, using 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz, FUV+24µm, and Hα+24µm data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The uncertainties on the SFR values are ≈ 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Name SFR from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz (M⊙yr−1) SFR from FUV+24µm (M⊙yr−1) SFR from Hα+24µm (M⊙yr−1) NGC 2683 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 NGC 3627 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='56 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='84 NGC 4096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='38 NGC 4449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='32 NGC 4490 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='30 NGC 4826 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='78 NGC 5194 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='88 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='65 Appendix (Figures 10-11), we also present the SFRSD maps estimated using Hα+24µm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz emission in colors and contours, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The SFRSD maps of each galaxy in Figures 8−11 are shown in the same color scale and contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' To determine the radial variation of SFRSDs, we have averaged the SFRSD maps of our sample galaxies over tilted rings centred on the optical centre of each galaxy using their inclinations and position angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The width of the tilted rings was taken to be equal to the beam size of the corresponding image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Figure 3 shows the radial variation of the average SFRSD, derived using FUV+24µm and Hα+24µm emission, with galactocentric distance where both the axes are normalized to their maximum values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We also derived the radial variation of SFRSDs for the galaxies using 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz emission and it is consistent within 1σ statistical uncertainties, with those derived using FUV+24µm and Hα+24µm data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Azimuthally averaged SFRSDs of all the seven galaxies decrease gradually towards the outer region and drop by a factor of 6 to 8 at the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Details of the Individual Galaxies of Sample 1 (i) NGC 2683: In this galaxy, Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2020) found very weak linear polarisation using C-band and L-band VLA observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Based on the optical image, we could separate the central region from the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The average magnetic field in the central region is found to be ≈31 µG and the outer region of the disk has an average value of ≈19 µG (see Figure 1 and Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Wiegert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2015) used WISE 22 µm data to estimate a galaxy-averaged SFR of ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='09 M⊙yr−1 for NGC 2683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' From our analysis, integrated SFR was measured to be ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='24 M⊙yr−1 and ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='28 M⊙yr−1 using FUV+24µm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz radio emission, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' However, we note that Wiegert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2015) used a distance of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='27 Mpc for this galaxy, but we have used a distance of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='7 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The SFR is estimated to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='16 M⊙yr−1 using FUV+24µm emission, assuming the same distance as used by Wiegert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Taking the calibration uncertainties and the assumed distance into account, our estimated SFR is hence consistent with that of Wiegert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We note that the contours on the background sources (Figure 8) are not real SFRSDs, as these are likely to be background AGNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (ii) NGC 3627: NGC 3627 was observed at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='46 GHz and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='85 GHz using the VLA in its D-configuration (Soida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' These authors estimated the magnetic field strengths using the classical equipartition formula (Longair 2011) and found an average equipartition magnetic field strength of 11±2 µG, assuming a constant non-thermal spectral 10 Manna and Roy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 Radial distance (Normalised) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 SFRSD from FUV+24μm (Normalised) NGC 2683 NGC 3627 NGC 4096 NGC 4449 NGC 4490 NGC 4826 NGC 5194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 Radial distance (Normalised) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 SFRSD using Hα+24μm (Normalised) NGC 2683 NGC 3627 NGC 4096 NGC 4449 NGC 4490 NGC 4826 NGC 5194 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The variation of SFRSDs (normalized), estimated using FUV+24µm (left panel) and Hα+24µm (right panel) emission as a function of galactocentric distance (normalized) for all seven galaxies in Sample 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' index of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='9 and a disk thickness of 2 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Soida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2001) also studied the polarized emission at these frequencies to find a regular magnetic field of 4±1 µG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' They suggested two distinct magnetic field components of NGC 3627: one for the spiral arms and another for the inter-arm regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We have separately studied equipartition magnetic fields in the arm and interarm regions of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We find that the central region and the edges of the extended bar have magnetic field strengths of ≈ 34 µG (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The arm region has a field strength of ≈28 µG (see Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' However, the magnetic field strength in the interarm regions has values ≈21 µG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We note that our estimates of the equipartition magnetic field strengths in the galaxy are higher than those found by Soida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' this difference likely arises from the fact that Soida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2001) estimated the magnetic field strengths using the classical equipartition formula, which is known to significantly underestimate the magnetic field in the star-forming regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We measured a galaxy-averaged SFR of ≈2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 M⊙yr−1 and ≈1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='56 M⊙yr−1 from FUV+24µm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz emission, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Our measurements of spatially resolved SFRs in different regions are consistent, within calibration uncertainties, with the SFR estimates of Watanabe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (iii) NGC 4096: Our estimate of the equipartition magnetic field in NGC 4096 varies from ≈21 µG at the centre to ≈12 µG at the edge (table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The magnetic field strength in both the central region and northern periphery is quite similar, with typical field strengths of ≈ 20 µG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' this is presumably due to its high inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The outer part of the galaxy has an average field strength of ≈14 µG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' NGC 4096 was observed (Irwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Wiegert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2015) with its B-field and further studied by Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2020) who found very little polarized emission from the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Wiegert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2015) used the 22 µm−SFR calibration to measure a galaxy-averaged SFR of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='27±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02 M⊙yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Our measurement of the galaxy-averaged SFR is ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='35 M⊙yr−1 and ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='43 M⊙yr−1 using FUV+24µm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz emission, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Considering the calibration uncertainties, our estimates are consistent with that of Wiegert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (iv) NGC 4449: This is an optically bright irregular starburst galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Chy˙zy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2000) used VLA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='86 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='46 GHz observations to find a galaxy-averaged equipartition magnetic field of ≈14 µG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' These authors also used polarization emission to estimate a regular field of ≈8 µG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The equipartition magnetic field map of NGC 4449 from our study is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' As noted in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1, about 70 % of the total projected area of this galaxy has spectral index values of less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We have replaced the pixel values with αnt < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='55 with αnt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='55 while computing the magnetic field for NGC 4449 (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The average magnetic field strength is ≈17 µG in this galaxy, which is comparable to the findings of Chy˙zy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Our measurements of the galaxy-averaged SFR are ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='38 M⊙yr−1 and ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='37 M⊙yr−1 using FUV+24µm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz emission, respectively, which are consistent with the SFR of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='47 M⊙yr−1 estimated by Chy˙zy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (v) NGC 4490: Nikiel-Wroczy´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2016) observed NGC 4490 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='61 GHz using the GMRT, and at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='86 & 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='44 GHz using VLA + Effelsberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The authors used these observations to find a mean equipartition magnetic field A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities 11 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Magnetic field strengths in different regions of the galaxies in Sample 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For the irregular galaxy NGC 4449, we could only measure the galaxy-integrated magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We have separated the two nearly face-on galaxies (NGC 3627 and NGC 5194) into arm and inter-arm regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For the rest of the galaxies, we could not separate the arm and inter-arm region due to their higher inclinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Galaxy Galaxy-average Beq in Beq in Beq in Beq in name Beq central region disk region arm region inter-arm region (µG) (µG) (µG) (µG) (µG) NGC 2683 24±6 31±3 19±5 – – NGC 3627 25±4 34±8 – 28±5 21±4 NGC 4096 16±4 21±5 14±3 – – NGC 4449 17±6 – – – – NGC 4490 23±10 40±6 17±7 – – NGC 4826 23±9 38±8 20±5 – – NGC 5194 16±6 34±6 – 25±5 18±4 of 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='9±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='9 µG, with typical field strengths in the range of 18 µG to 40 µG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We have found a typical equipartition magnetic field strength of ≈40 µG in the central region, which decreases to ≈17 µG in the outer region (see Figure 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' these values are consistent with the estimates of Nikiel-Wroczy´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We find a relatively lower magnetic field strength of ≈15 µG in both the interacting region and the companion galaxy NGC 4485.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Therefore, a gradual decrease in the average magnetic field strength occurs from the center to the outer region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Clemens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (1999) used radio observations to find a galaxy-averaged SFR of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='7 M⊙yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We found a similar SFR (≈4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='63 M⊙yr−1) using 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz radio emission but a factor of ∼2 lower SFR (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='13 M⊙yr−1) using the FUV+24µm emission (Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Extinction corrections for NGC 4490 are believed to be higher than those typically assumed and this may lead to an underestimation of the SFR while using the FUV+24µm diagnostics (Clemens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (vi) NGC 4826: No spatially resolved maps of magnetic fields and SFRSDs are available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We measure the central and outer regions of the galaxy to have an average equipartition magnetic field strength of ≈38 µG and ≈20 µG, respectively (see Figure 2 and Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We find galaxy-averaged SFR of ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='73 M⊙yr−1 and ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='63 M⊙yr−1 using FUV+24µm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (vii) NGC 5194: Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2011) used VLA C-band observations of the galaxy and assumed a constant thermal and non-thermal spectral index of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 to find an average equipartition magnetic field strength of 20 µG using the revised formula by Beck & Krause (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' They found a magnetic field of 20−25 µG in the spiral arms, higher than the 15−20 µG typical in the interarm regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Using VLA observations at S-band (2−4 GHz) frequencies, Kierdorf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2020) found the field strength of turbulent and regular components of the magnetic field in the arm regions of 18−24 µG and 8−16 µG, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We find an equipartition magnetic field strength of ≈25 µG in the arm region and ≈18 µG in the interarm region (see Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The peripheral region has a magnetic field of ≈12 µG, while the overlapping region between NGC 5194 and NGC 5195 has an average Beq of ≈16 µG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Considering our use of Equation 1 (Beck & Krause 2005), measurements are roughly consistent with the earlier study of Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2011) and Kierdorf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Spatially resolved SFRs were measured in several star-forming regions of NGC 5194 using Hα+24µm and Hα+Paα emission (Kennicutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' SFRSDs in different regions were found to be in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='10 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='46 M⊙yr−1kpc−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Our estimates using the two tracers are consistent with the estimates of Kennicutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2007) (See Figures 9 & 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Furthermore, we find that the galaxy-integrated SFR derived using FUV+24µm (≈3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='88 M⊙yr−1) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz data (≈4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='16 M⊙yr−1) are consistent with each other, within 1-sigma statistical uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Is the Minimum Energy Condition Valid for the Sample Galaxies?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We have estimated magnetic fields for the galaxies in Sample 1 assuming the “minimum energy condition” or “equipartition condition”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' by assuming that the energy density in the magnetic field is approximately equal to the energy density in cosmic ray particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Therefore, it is important to verify the validity of this assumption in our sample galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The tightness of the spatially-resolved radio−FIR correlation can be used to estimate the deviation of the energy densities from the minimum energy condition (Hummel 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Basu & Roy 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' According to the simplified model of Hummel (1986), when the minimum energy condition is satisfied, the distribution of Int/IFIR will be similar to the distribution of B1+αnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The model assumes the following to be constant across galaxies: (a) the ratio of the number densities of relativistic electrons and dust-heating stars, (b) the volume ratio of radio and FIR emitting 12 Manna and Roy regions, and (c) the ratio of efficiency factors for both the radio and FIR emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' In this model, the cumulative distribution function (CDF) of the quantity Int/IFIR and B1+αnt eq is expected to follow each other if Beq is close to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' To verify the validity of the minimum energy condition in our sample galaxies, we have followed the procedure as in Hummel (1986) and Basu & Roy (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The CDF of Int/IFIR and B1+αnt eq were estimated using our radio maps of the sample galaxies at both 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We used an ensemble of spatially-resolved values of αnt, Int (both at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz), IFIR (70 µm) and magnetic fields (Beq), which are averaged over the beam size from all the galaxies in Sample 1 (Table 2) to generate these distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The CDFs of all quantities were normalized by their median values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The top panels in Figure 4 show the median-normalized CDFs of Int/IFIR and B1+αnt eq at both 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We find that the CDFs of Int/IFIR and B1+αnt eq at both 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz broadly follow each other but with slight deviations at high and low ends (see top panels in Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' This implies that the minimum energy condition is broadly valid and is consistent with earlier findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For example, Hummel (1986) found the distribution of the two quantities is similar in a sample of Sbc galaxies while Basu & Roy (2013) reached similar conclusions in a study of 5 nearby large spiral galaxies, but with slight deviations observed in the CDFs of Int/IFIR and B1+αnt eq in the interarm regions of the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The observed deviation in the CDFs of Int/IFIR and B1+αnt eq for our sample galaxies imply a corresponding deviation from the minimum-energy condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' In order to quantify this deviation, we performed a Monte Carlo simulation orig- inally proposed by Hummel (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' In this simulation, random numbers (X) were drawn from a Gaussian distribution with standard deviation σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Thereafter, we multiplied 10X with the observed equipartition magnetic fields to introduce deviations from the minimum-energy condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We thus constructed the CDF of B1+αnt eq using the deviated magnetic field values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The CDF of B1+αnt eq were then compared to the observed CDF of Int/IFIR via a Kolmogorov-Smirnov (KS) test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' This procedure was repeated for a range of σ from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We find that the p-values for the KS test comparing the distributions are maximized when σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Indeed, B1+αnt eq derived after deviating the magnetic field using σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 and Int/IFIR are consistent with being derived from the same distribution, with a KS test p-value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='41 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='55, when using Int at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The bottom panels in Figure 4 show the CDFs of the two quantities for σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' it is clear that the CDFs follow each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' This implies the actual magnetic field values may deviate from the equipartition values by ∼ 25% in our galaxies in Sample 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We note that any violation of the assumptions made by Hummel (1986) may also lead to the observed deviation in the CDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Correlation Between Magnetic Fields and SFRSDs We have studied the correlation between the spatially-resolved equipartition magnetic field and SFRSDs for the galaxies in Sample 1 (Table 2) at scales of ≈360−760 pc (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For the seven sample galaxies, we used the SFRSD maps estimated using the FUV+24µm emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The correlations between magnetic fields and SFRSDs for the seven galaxies are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Each point represents the logarithms of equipartition magnetic fields and SFRSD values that are averaged over the beam size of the corresponding maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' da Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2014) found that SFR calibrations could be biased and strongly affected by stochasticity at small spatial scales where the star formation rate is low (≤ 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 M⊙yr−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' we have therefore excluded regions of low star formation rates from the correlation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We find that the equipartition magnetic field and the SFSRD are correlated in all seven sample galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We use orthogonal distance regression in Scipy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2020) to fit a power law of the form B = B0 (ΣSFR)η to the magnetic field − SFRSD data points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' the spatially-resolved uncertainty maps of equipartition magnetic fields and rms noise on the SFRSD maps were used to estimate the uncertainties on each data point during the fitting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The best-fit parameters of the power-law are given in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We have also estimated the scatter (rms of the data points along the y-axis) of the correlations which are presented in Table 7 and are shown in dashed lines in the corresponding plots (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We find that six of the seven galaxies have slopes (η) in the range of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='27 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='40 but that the slope is relatively lower for NGC 4449 with η ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Averaging over the slope of all galaxies in Sample 1, we find a mean slope of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Correlation Between Magnetic Fields and Gas Densities We have studied the correlation between spatially-resolved equipartition magnetic fields and gas densities for three of the galaxies in Sample 1, NGC 3627, NGC 4826, and NGC 5194, for which spatially resolved CO observations were available (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Similar to the study of correlations between Beq and SFRSDs, we have studied the correlations between Beq and gas density values, both averaged over the beam size of the corresponding maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities 13 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The top panels show the cumulative distribution function (CDF) of Int,radio/I70µm (in red) and B1+αnt eq (in blue), where Int is the non-thermal emission at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 GHz (top left) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz (top right) (Sample 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The variables are normalized by their median values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The bottom panels show the same but now with the magnetic field perturbed from its measured value using σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' the CDFs of the Int,radio/I70µm and B1+αnt eq are now consistent with being derived from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Best-fit parameters and the scatter of the correlation between magnetic fields and SFRSDs for the seven galaxies in Sample 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The data were fitted with a power law of the form B=B0(ΣSFR)η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Name Slope (η) Intercept (B0) (log(µG)) Intercept (B0) (µG) Scatter NGC 2683 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='07 125 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05 NGC 3627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03 51 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04 NGC 4096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='08 63 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05 NGC 4449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04 43 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03 NGC 4490 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03 79 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='06 NGC 4826 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02 63 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05 NGC 5194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02 100 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='8 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 =） P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 lo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33GHz/170μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 0 2 4 6 X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='8 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 =） v P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 eq l1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4GHz/l70μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 0 1 2 3 4 5 X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='8 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 =） V P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 lo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33GHz/170μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 0 2 4 6 X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='8 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 =） v P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 eq l1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4GHz/l70μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 0 1 2 3 4 5 X14 Manna and Roy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 log(Magnetic Field in G) NGC 2683 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='7 NGC 3627 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 NGC 4096 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='50 log(Magnetic Field in G) NGC 4449 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 NGC 4490 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 NGC 4826 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 log(Magnetic Field in G) NGC 5194 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='3 NGC 1097 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 NGC 4736 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='25 log(SFRSD in M yr 1 kpc 2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='20 log(Magnetic Field in G) NGC 5055 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 log(SFRSD in M yr 1 kpc 2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 NGC 5236 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 log(SFRSD in M yr 1 kpc 2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 NGC 6946 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The correlation between magnetic fields and SFRSD for the combined sample of 12 galaxies (Sample 2, Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For the seven galaxies in Sanple 1, the SFRSD estimates shown in the plots were derived using FUV + 24µm (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The SFRSD estimates for the five galaxies from Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2012a) (Sample 2) were derived using Hα + 24µm (Section 4) The red line shows a linear fit to the data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The black dashed lines show the ±1σ vertical scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities 15 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 log(Magnetic Field in G) NGC 3627 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 NGC 4826 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 NGC 5194 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 log(Magnetic Field in G) NGC 4736 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 log(Gas density in gm/cm3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='20 NGC 5055 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 log(Gas density in gm/cm3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 NGC 5236 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 log(Gas density in gm/cm3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='3 log(Magnetic Field in G) NGC 6946 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The correlations between magnetic fields (µG) and gas densities (gm/cm−3) for seven galaxies of Sample 3 (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The red line shows a linear fit to the data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The black dashed lines show the ±1σ vertical scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' correlations between magnetic fields and gas densities of NGC 3627, NGC 4826, and NGC 5194 are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We have again used orthogonal distance regression in Scipy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2020) to fit a power-law to the Beq and gas density data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The scatters of the three correlations are shown in dashed lines in all the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The measured best-fit power-law indices are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='40±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='49±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='53±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02 (Table 9) for NGC 3627, NGC 4826 and NGC 5194, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The mean of the power-law indices is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' EXTENDING THE SAMPLE WITH 5 GALAXIES FROM EXISTING GMRT OBSERVATIONS As mentioned earlier, a study of Beq and radio-FIR correlations for a sample of five large nearly face-on galaxies was carried out by Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2012a,b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Basu & Roy (2013), using low-radio frequency observations at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz 16 Manna and Roy Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Best-fit parameters and the scatter of the correlation between magnetic fields and SFRSDs for the five galaxies in Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2012a) (Sample 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The data were fitted with a power law of the form B=B0(ΣSFR)η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Name Slope (η) Intercept (B0) (log(µG)) Intercept (B0) (µG) Scatter NGC 1097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='01 41 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02 NGC 4736 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05 60 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04 NGC 5055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02 18 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02 NGC 5236 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02 81 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='08 NGC 6946 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05 42 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05 Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Best-fit parameters and the scatter of the correlation between spatially-resolved magnetic fields and gas densities for the seven galaxies in Sample 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Galaxies with an asterisk are from the sample of Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2012a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Name Exponent Scatter NGC 3627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03 NGC 4826 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05 NGC 5194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='06 NGC 4736∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03 NGC 5055∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02 NGC 5236∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04 NGC 6946∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04 at sub-kpc linear resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' In this paper, we expand our study of spatially-resolved correlations between magnetic fields, gas densities, and SFRSDs by including these five galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We refer readers to Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2012a) for a detailed discussion of their sample, GMRT observations, data reduction procedures, and estimation of non-thermal spectral indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' It is to be noted that the modelling of the thermal free-free emission from these galaxies is performed in the same way as was done for our seven galaxies in Sample 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We have estimated the SFRSD maps of these five galaxies using Hα data along with 24 µm IR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We obtained Hα maps of four of the galaxies, NGC 1097, NGC 4736, NGC 5055, and NGC 6946 from the ancillary data at the SINGS website1 and obtained the Hα map of NGC 5236 from 11HUGS (Kennicutt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We used Hα and MIPS 24 µm data in combination to derive the SFRSD maps of these galaxies using the calibration from Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2012) (Equation 3, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' To estimate the equipartition magnetic field strengths of these five galaxies, we have used the non-thermal radio maps at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz from Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2012a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The correlations between equipartition magnetic fields and SFRSDs are shown in Figure 5 where, similar to the previous correlation studies, each point represents the logarithms of magnetic fields and SFRSD values that are averaged over the beam size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Similar to the previous correlations (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5), we used orthogonal distance regression in Scipy to fit a power law to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We have provided the best-fit parameters of the power-law fit in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The scatters of all five correlations (presented in Table 8) are shown in dashed lines in all the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We find a mean exponent of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='30±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05 for the five galaxies where the exponent of individual galaxies varies from ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='25 to ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We have computed maps of cold gas densities of four out of the five galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' NGC 4736, NGC 5055, NGC 5236 and NGC 6946, using the atomic and molecular gas surface density maps from Basu & Roy (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The assumed parameters are taken to be the same as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For the remaining galaxy, NGC 1097, we could not measure gas densities as there are no archival CO data available for the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Following the procedures of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5, we have also studied the spatially-resolved correlation between equipartition magnetic fields and gas densities for the four sample galaxies, which are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The best-fit parameters are presented in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The exponents of the individual galaxies vary between ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='25 to ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='44 where the mean exponent is found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='35±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' DISCUSSION Understanding the relationship between the physical condition of the interstellar medium (ISM) and the star formation process is crucial to understand galaxy evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Gas and magnetic fields are key constituents of the 1 https://irsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='edu/data/SPITZER/SINGS/ A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities 17 ISM and therefore it is important to study the interrelations between gas, magnetic fields, and SFRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Though the Kennicutt−Schmidt relation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' the relation between gas densities and SFRs, has been extensively studied at high spatial resolutions in various types of nearby galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Onodera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Roychowdhury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Filho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Miettinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2017), similar high-resolution observations of how the magnetic fields are related to SFRs and gas densities are yet to be systematically investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Such observations are critical to understand the validity of several models that predict strong correlations between the magnetic fields and gas densities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Chandrasekhar & Fermi 1953;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Fiedler & Mouschovias 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Cho & Vishniac 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2003) as well as magnetic fields and SFRSDs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Niklas & Beck 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Schleicher & Beck 2013, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Here, we have studied these correlations in a sample of twelve galaxies (Sample 3) at sub-kpc scales (see Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 & 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' To our knowledge, this is the first spatially resolved study of the above correlations in nearby large galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' In this section, we place these findings in the light of predictions made by various models and in the process attempt to provide physical insights into the interrelation between magnetic fields, gas densities, and star formation rates at sub-kpc scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Magnetic Fields and SFRSDs Several Magneto-Hydrodynamical simulations find that galactic magnetic fields are amplified by gas turbulence in very short timescales (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' ∼100 Myr) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Brandenburg & Subramanian 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Beresnyak 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Schober et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Schleicher & Beck 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Bovino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The primary driver of gas-turbulence in the ISM of galaxies is supernova explosion (Bacchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2020), the rate of which is in turn directly coupled to the SFR in the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Therefore, it is expected that the star formation rates and the magnetic fields in a galaxy will be correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Indeed, using semi-analytical models, Schleicher & Beck (2013, 2016) found that in order to explain the radio-FIR correlation at sub-kpc scales, magnetic fields and SFRSDs, again at sub-kpc scales, must be related as B ∝ Σ1/3 SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Studies in the literature on the correlation between magnetic fields and SFRSDs have focused on dwarf galaxies and those studies were carried out using galaxy-integrated magnetic fields and SFRSDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' As mentioned in Section 1, to our knowledge, there is only one published work of the spatially-resolved study of the correlation between magnetic fields and SFRSDs (Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For the 12 galaxies in Sample 2 (Table 2), we find that the mean value of the power-law index of the correlation between Beq and SFRSDs is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='31±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='06, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='e Beq ∝ Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='31±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='06 SFR (2), consistent (at < 1σ error) with the model of Schleicher & Beck (2013, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Thus, it appears that the semi-analytical models that are based on the amplification of magnetic fields due to supernova-driven gas turbulence work remarkably well for the pilot sample, in predicting the correlation between magnetic fields and SFRSDs down to sub-kpc scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We note that the power-law index for the correlation between Beq and SFRSDs for NGC 4449 was found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03, significantly lower than for the remaining galaxies (Table 7) as well as lower than the model prediction of B ∝ Σ1/3 SFR Schleicher & Beck (2013) (at > 5σ significance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For the case of NGC 4449, the relatively flat spectral index values (αnt ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='55) in ≈ 70% of the galaxy meant that the magnetic field values could not be estimated reliably for a large part of the galaxy (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' This could lead to biases in the correlation and therefore, the low value of the power-law index for NGC 4449 should be taken with caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Intercept of the Correlation According to the model proposed by Schleicher & Beck (2013), the intercept of the B-ΣSFR correlation depends on several ISM parameters such as gas density (ρ0), the fraction of turbulent kinetic energy converted into magnetic energy (fsat), the injection rate of turbulent supernova energy (C) and the intercept of Kennicutt-Schmidt (KS) relation (C1) (Equation 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' B ∼ � fsat8π ρ1/6 0 ( C C1 )1/3 Σ1/3 SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (6) Schleicher & Beck (2013) predicted the intercept of the B-ΣSFR correlation to be ∼ 26 µG assuming ρ0 = 10−24 g cm−3 and fsat ∼ 5 percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We have found an average intercept at 65±25 µG of the Beq-ΣSFR correlation of the 12 galaxies in sample 2 (see Table 7 & 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Although the mean value is a factor of ≈2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 higher than the value predicted by Schleicher & Beck (2013), this value is consistent with the predicted value, within the scatter (at ≈1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Future follow-up studies, such as using our full survey (Sample 0 which consists of 46 galaxies), are required to draw statistically robust conclusions about the value of the intercept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2 The uncertainty quoted is the scatter of the measured value of η across the galaxies in Sample 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 18 Manna and Roy If the value of fsat is indeed higher, this would imply a higher than assumed value of one or more of ρ0, C, and fsat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The intercept is broadly insensitive to the assumed value of ρ0 (Equation 6) and therefore, in order to explain a factor of ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 higher value of the intercept, the actual value ρ0 has to be higher than the assumed value of 10−24 g cm−3 by a factor of ≈ 240;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' such high gas densities are unphysical and are not observed in typical regions of a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The other possibility that the assumed value of the injection rate of turbulent supernova energy (C) is higher by a factor of ≈ 16 is also contrary to expectation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2017) found that under reasonable conditions the value of C can be higher by at most a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Therefore, fsat must be higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05 to explain a significantly higher value of the intercept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' An understanding of how galaxies can achieve such efficient amplification of magnetic fields with fsat much greater than 5% requires detailed MHD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We note that Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2017) found that the value of the intercept for B-ΣSFR for the dwarf galaxy IC 10 is 51 µG, similar to our findings of a higher than predicted value of the intercept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Magnetic Fields and Gas Magnetic fields and gas are expected to be correlated as B ∝ √ρgas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Chandrasekhar & Fermi 1953;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We find that equipartition magnetic fields are correlated with gas densities for the seven galaxies (Sample 3) with an average power-law index, k=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='40±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='09 (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 & 4)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' This value of k is consistent with the numerical simulations that predict k ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 and also consistent with the theories that predict B ∝ ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The power-law index of the correlation between Beq and gas densities is found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='31±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03 for NGC 5055 and NGC 6946 respectively, significantly lower than the model predictions and as compared to the other galaxies in Sample 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' A lower value of k could mean that either the efficiency of the amplification of the magnetic field is less or that the magnetic field strengths derived assuming the “minimum energy condition” are underestimated (Dumas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Strong synchrotron or inverse Compton losses of cosmic-ray electrons could suppress the radio synchrotron emission which would then cause the equipartition magnetic fields to be underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Magnetic Fields, Gas Densities and the Radio-FIR Correlations Energy equipartition between the magnetic field (B) and the gas density (ρgas), and between magnetic fields and cosmic ray particles implies that the non-thermal emission is related to the gas density as Int ∝ ρk(3+αnt) gas where k is the power-law index relating magnetic fields and gas densities (Beq ∝ ρk gas) (Niklas & Beck 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Further, the Kennicutt-Schmidt law and the radio-FIR correlation imply that Int is related to gas densities as (1) Int ∝ ρm(n+1) gas for optically thin dust to UV photons and (2) Int ∝ ρmn gas for optically thick dust to UV photons, where m is the power-law index of the radio-FIR correlation and n is the power-law index of the Kennicutt-Schmidt law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Therefore, we can obtain the following relation between the power-law index of all four correlations (Dumas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2011): k = (n + 1)m 3 + αnt ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Optically thin dust (7) k = nm 3 + αnt ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Optically thick dust (8) We can use the above equations to indirectly estimate the power-law index, k, of the correlation between magnetic fields and gas densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For the three galaxies, NGC 3627, NGC 4826, and NGC 5194 (Roy & Manna 2021), we have estimated gas densities using CO and Hi observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Now we can compare the direct measurement of k with an indirect estimate of k using Equations 7 and 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' this will provide additional information on the validity of both the minimum energy conditions that were assumed between magnetic fields and the gas densities as well as the magnetic fields and cosmic ray particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For the galaxies from Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2012a), this study was already presented and discussed in Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2012b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We have estimated k for all the seven sample galaxies from Roy & Manna (2021) (Sample 1), using the assumption of optically thin dust to UV photons, using (i) the slope of radio-FIR correlation (m) as derived in Roy & Manna (2021), (ii) the measured galaxy-averaged spectral index (αnt) from Roy & Manna (2021), and (iii) a Kennicutt-Schmidt power-law index of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='15 (Kennicutt 1998b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Table 10 provides the relevant values as well as estimated values of k derived using the measured value of m using radio emission at both 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For two of the galaxies, NGC 3627 & NGC 5194, the value of k estimated 3 The uncertainty quoted is the scatter of the measured value of k across the galaxies in Sample 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities 19 Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Power law index (k) of the relation between magnetic fields and gas densities (B ∝ ρk) of galaxies in Sample 1, indirectly estimated using the slope of radio-FIR correlation (m) and the slope of the Kennicutt−Schmidt law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' See Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 for a discussion on these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Name m m αnt k (Optically thin) k (Optically thin) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 GHz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 GHz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz NGC 2683 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='54±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='91±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='84±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='06 NGC 3627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='55±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='85±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='08 NGC 4096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='74±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='90±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05 NGC 4449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='77±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='65±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='48±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04 NGC 4490 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='68±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='59±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04 NGC 4826 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='49±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='09 NGC 5194(arm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='50±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='65±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='63±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04 NGC 5194(interarm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='73±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='03±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='85±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='46± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='05 using Equation 7 is comparable to the direct measurement of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' This broadly validates the assumption of energy equipartition between magnetic fields and cosmic ray particles in these two galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For the optically thin case, the mean of indirectly-estimated k values of the sample of seven galaxies are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='16 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='19 at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' However, this includes the galaxy NGC 4826, which shows an anomalously high value of k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='95 derived at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Excluding this galaxy from the mean calculation, we find that k=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='09 at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Remarkably, for all the galaxies except NGC 4826, the k value at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz, for the optically thin case, is consistent with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 within error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Thus, the indirectly estimated values of k are consistent with equipartition between magnetic fields and gas energy densities (Chandrasekhar & Fermi 1953;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Fiedler & Mouschovias 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Cho & Vishniac 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Groves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' This is similar to the findings of Niklas & Beck (1997) for their sample of 43 galaxies and Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2012b) for their sample of four galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The value of k derived for NGC 4826, for the optically thin case, is a consequence of the anomalously high value of the power-law index of the radio-FIR correlation (≈1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='39 and ≈1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='47 for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz respectively, Table 10) which is different from the other six galaxies in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' NGC 4826 has been classified as a Seyfert 2 galaxy in the past (Malkan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2017) and therefore the emission from the core contributes to the observed power-law index of the radio-FIR correlation (Roy & Manna 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' It is likely that the significant contribution of the AGN to the radio emission makes the estimate of k for NGC 4826 unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' SUMMARY 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We made spatially resolved maps of equipartition magnetic fields in seven galaxies (Sample 1): NGC 2683, NGC 3627, NGC 4096, NGC 4449, NGC 4490, NGC 4826, and NGC 5194 and find that the magnetic fields are strongest near the central region and go down by a factor of ∼2 at the edge of the magnetic field maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We have used the tightness of the spatially-resolved radio-FIR correlations to verify the validity of the equipar- tition condition between magnetic fields and cosmic ray particles for the sample galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We find that the magnetic field values may deviate from the equipartition values by ∼25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We have estimated spatially resolved maps of SFRSDs of the galaxies in Sample 1 using FUV+24µm, Hα+24µm, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Azimuthally averaged SFRSDs drop by a factor of 6 to 8 at the edge of the galaxies, where SFRSD values are 5 times the rms of the maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We also included five additional galaxies: NGC 1097, NGC 4736, NGC 5055, NGC 5236, and NGC 6946 from previous GMRT observations of Basu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' (2012a) and estimated their equipartition magnetic field, SFRSD and gas density maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We studied the spatial correlation between magnetic fields and star formation rates at < 1 kpc resolution for the 12 galaxies (Sample 2) and find that magnetic field strengths and SFRSDs are correlated with an average power-law index of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='31±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' This result is in remarkable agreement (at < 1σ error) with semi-analytical model predictions of B ∝ Σ1/3 SFR (Schleicher & Beck 2013, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 20 Manna and Roy 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We measure an average intercept of ≈ 65 µG from the B-ΣSFR correlations of our galaxies in Sample 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' This is higher than the predictions of Schleicher & Beck (2013) by a factor of ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5, and, if confirmed with a larger sample, would imply a significantly higher efficiency of magnetic field amplification than what is typically assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We used spatially resolved gas density maps for seven (Sample 3) of the 12 galaxies, for which archival CO data was available, to find that magnetic fields are correlated with gas densities as B ∝ ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='40±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='09 gas .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' This result is consistent with numerical simulations that predict k ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='6 and broadly consistent (within ≈1 sigma uncertainty) with theories that predict B ∝ ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='5 gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We have indirectly estimated the power-law index (k) of the correlation between the magnetic fields and the gas densities using the slope of the radio-FIR correlation, the slope of the Kennicutt-Schmidt law, and the non- thermal spectral index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The mean value of k, for optically thin dust, was found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='04 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='09 at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 GHz respectively for the six galaxies in Sample 1, with NGC 4826 excluded due to its high value of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' This is consistent with the equipartition between magnetic fields and gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The anomalously high values of k (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='95 at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='33 GHz respectively) for NGC 4826 are possibly due to the contribution of the central AGN to the radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We have started to follow up these pilot study results with a survey of a much larger sample of galaxies (Sample 0, Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For this, we have already observed another 24 galaxies using the upgraded GMRT (uGMRT), a Square Kilometer Array (SKA) pathfinder facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Sensitivities of the images from these uGMRT observations are significantly better (≈ 3 times) than those of the observations presented here and the result will be part of a future publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' In addition, SKA precursors such as the MeerKAT will also provide very deep images of the diffuse radio-continuum emission around nearby galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Eventually, the dramatic increase in sensitivity and ∼arc-sec resolution of the SKA has the potential to significantly advance our understanding of magnetic fields in nearby galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' For example, the SKA is expected to provide sensitive images of polarised synchrotron emission from nearby galaxies at a few GHz frequencies which would provide information on the large-scale ordered fields on the plane of the sky (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Johnston- Hollitt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Further, polarised emission from nearby galaxies at <∼1 GHz, where significant depolarisations take place, could be modelled through Faraday tomography (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Heald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' A combination of the two approaches could eventually allow us to infer the three-dimensional structure of the magnetic fields in nearby galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' SKA observations will also provide detailed images of star formation with resolutions of tens of parsecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' These will help to identify any dependence of SFR and IMF on galaxy type, evolution and environment within the local volume (Beswick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We would like to thank Aditya Chowdhury for his help at various stages of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We thank Yogesh Wadadekar, Preeti Kharb, and Dipanjan Mitra for reading the manuscript and providing useful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Aritra Basu provided their earlier published images and also suggested checking the B vs SFRSD relation for our sample galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We thank him for the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We also thank the anonymous referee whose comments helped significantly improve the presentation of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' We thank the staff of GMRT that allowed these observations to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' GMRT is run 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 2001, ApJ, 554, 803, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1086/323145 A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities 23 200 0 200 Relative RA (arcseconds) 300 200 100 0 100 200 300 Relative DEC (arcseconds) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='50 Error on B ( G) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The magnetic field uncertainty maps (in µG) of NGC 2683 (top left), NGC 3627 (top centre), NGC 4096 (top right), NGC 4449 (middle left), NGC 4490 (middle centre), NGC 4826 (middle right) and NGC 5194 (bottom) (Sample 1), shown in colour scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Blanked regions (in white colour) in the centre of each galaxy correspond to regions with spectral index values ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' MAGNETIC FIELD UNCERTAINTY MAPS We present here (Figure 7) magnetic field uncertainty maps of the galaxies in Sample 1, generated using the procedure described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' STAR FORMATION RATE SURFACE DENSITY MAPS We show SFRSD maps of the seven galaxies (Sample 1) in Figures 8 and 9, where SFRSDs estimated using 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz and FUV+24µm emission are shown in contours and colors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' In Figures 10 and 11, we have also shown the SFRSD maps estimated using Hα+24µm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4GHz data in colors and contours, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The SFRSD maps of each galaxy in Figures 8, 9, 10, and 11 have been shown in the same color scale and contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' 24 Manna and Roy Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' SFRSD (M⊙yr−1kpc−2) maps of NGC 2683, NGC 3627, NGC 4449 and NGC 4096(clockwise from top left) (Sample 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' SFRSDs estimated using 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz radio and FUV+24µm emission are shown in contours and colors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Contour levels are listed below each panel of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The circle in the bottom-left corner of the images indicates the angular resolution of the maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' COLOR:NGC26832683sfrhyb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='OHGSPX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 CONT:NGC2683IPOL1490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='572MHz2683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='sfr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='SUB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 0 10 20 30 33 29 28 27 Declination (J2000) 26 25 24 23 22 21 085300 5255 50 45 40 35 30 25 20 Right Ascension (J2000) Colorscale ranqe=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='85MilliSolarmass/yr/kpc^2 Contpeakflux=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4958E-02Solarmass/yr/kpc^2 Levs = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='327E-03 * (-1,1,2, 4,8, 16,25, 50, 80,100,130,160)COLOR:NGC36273627sfrhyb.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 200 400 600 13 02 01 00 Declination (J2000) 1259 58 57 56 11 20 25 20 15 10 05 RightAscension(J2000) Colorscalerange=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='9MilliSolarmass/yr/kpc^2 Contpeakflux=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4733E-01Solarmass/yr/kpc^2 Levs = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='000E-02 * (-1, 1,2, 4, 8, 16, 25, 50, 80,100,150,200,400,800)COLOR:NGC40964096sfrhyb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='OHGSPX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 CONT:NGC4096IPOL1432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='873MHz4096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='sfr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='SUB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 0 10 20 30 40 4731 30 Declination (J2000) 29 28 0 27 26 25 120620 15 10 05 00 0555 50 45 40 Right Ascension (J2000) Colorscale range=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='02MilliSolarmass/yr/kpc^2 Contpeakflux=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1588E-02Solarmass/yr/kpc^2 Levs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='511E-03 * (-1, 1, 2, 4, 8, 16, 25, 50, 80,100,150,200,400,800)COLOR:NGC44494449sfrFUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='OHGSPX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 CONT:N49IPOL1489.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='984MHz4449sfrL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='SUB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 100 200 4412 10 08 Declination (J2000) 06 04 02 00 12 28 45 30 15 00 27 45 RightAscension(J2000) Colorscalerange=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0MilliSolarmass/yr/kpc^2 Contpeakflux=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4433E+00Solarmass/yr/kpc^2 Levs = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='421E-03 * (-1, 1,2, 4, 8, 16, 25, 50, 80,100,150,200,400,800)A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities 25 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' SFRSD (M⊙yr−1kpc−2) maps of NGC 4490, NGC 4826 and NGC 5194 (clockwise from top left) (Sample 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' SFRSDs estimated using 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz radio and FUV+24µm emission are shown in contours and colors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Contour levels are listed below each panel of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The circle in the bottom-left corner of the images indicates the angular resolution of the maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' COLOR:NGC44904490sfrFUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='OHGSPX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 CONT:N4490IPOL1435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='114MHz4490sfrL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='SUB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='2 0 50 100 150 200 41 44 42 Declination (J2000) 40 38 36 34 0 12 31 00 30 45 30 15 RightAscension(J2000) Colorscale range=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4MilliSolarmass/yr/kpc^2 Contpeakflux=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='9951E-01Solarmass/yr/kpc^2 Levs = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='000E-03 * (-1, 1, 2, 4, 8, 16, 25, 50, 80,100,150,200,400,800)COLOR:NGC48264826sfrhyb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='OHGSPX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 CONT:N4826LIPOL1425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='677MHz4826.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='sfr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='SUB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 0 100 200 300 214300 42 30 00 Declination (J2000) 41 30 00 40 30 00 39 30 00 12 56 50 45 40 35 RightAscension (J2000) Color scalerange=-0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='900 MHz 5194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='sfr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='SUB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1 100 200 4718 16 14 Declination (J2000) 12 10 08 06 13 30 15 00 29 45 30 RightAscension (J2000) Color scale range= -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='0293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content="2'Milli Solar mass/yr/kpc2 Contpeakflux=3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='1707E-01Solarmass/yr/kpc^2 Levs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='215E-03 * (-1, 1, 2, 4, 8, 16, 25, 50, 80,100,150,200,400,800)26 Manna and Roy Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' SFRSD (M⊙yr−1kpc−2) maps of NGC 2683, NGC 3627, NGC 4449 and NGC 4096 (clockwise from top left) (Sample 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' SFRSDs estimated using 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz radio and Hα+24µm emission are shown in contours and colors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Contour levels are listed below each panel of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' The circle in the bottom-left corner of the images indicates the angular resolution of the maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' COLOR:N2683652683sfrha.' metadata={'source': 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4, 8, 16, 25, 50, 80,100,150,200,400,800)A Pilot Study of Magnetic Fields, Star Formation Rates and Gas densities 27 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' SFRSD (M⊙yr−1kpc−2) maps of NGC 4490, NGC 4826 and NGC 5194 (clockwise from top left) (Sample 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' SFRSDs estimated using 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content='4 GHz radio and Hα+24µm emission are shown in contours and colors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E2T4oBgHgl3EQfOwac/content/2301.03752v1.pdf'} +page_content=' Contour levels are listed below each panel of the figure.' metadata={'source': 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sha256:f106d8145848555566972b2dc548a78eb30d6c9a90996e1a95cc410f0a9eb4b2 +size 3866669 diff --git a/5NFKT4oBgHgl3EQfSS3g/content/tmp_files/2301.11775v1.pdf.txt b/5NFKT4oBgHgl3EQfSS3g/content/tmp_files/2301.11775v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3315fb5d464d20f79485795016e3e382c0507466 --- /dev/null +++ b/5NFKT4oBgHgl3EQfSS3g/content/tmp_files/2301.11775v1.pdf.txt @@ -0,0 +1,2988 @@ +arXiv:2301.11775v1 [math.AP] 11 Jan 2023 +REGULARITY IN THE TWO-PHASE BERNOULLI PROBLEM FOR THE +p-LAPLACE OPERATOR +MASOUD BAYRAMI AND MORTEZA FOTOUHI +Abstract. We show that any minimizer of the well-known ACF functional (for the +p-Laplacian) is a viscosity solution. This allows us to establish a uniform flatness +decay at the two-phase free boundary points to improve the flatness, that boils +down to C1,η regularity of the flat part of the free boundary. This result, in turn, is +used to prove the Lipschitz regularity of minimizers by a dichotomy argument. +1. Introduction and main result +We study the problem of minimizing the following two-phase functional +JTP(v, D) := +� +D +|∇v|p + (p − 1)λp ++χ{v>0} + (p − 1)λp +−χ{v<0} dx, +v ∈ K, +where D is a bounded and smooth domain in Rn, χA is the characteristic function +of the set A, 1 < p < ∞, and λ± > 0 are given constants. The class of admissible +functions K, consists of all functions v ∈ g + W1,p +0 (D), where g ∈ W1,p(D). +Any minimizer u satisfies, in a certain weak sense, the following system of +equations +(1) + +∆pu := div(|∇u|p−2∇u) = 0, +in +Ω+ +u ∪ Ω− +u, +|∇u+|p − |∇u−|p = λp ++ − λp +−, +on +�∂Ω+ +u ∩ ∂Ω− +u +� ∩ D, +|∇u+| ≥ λ+, |∇u−| ≥ λ−, +on +�∂Ω+ +u ∩ ∂Ω− +u +� ∩ D, +|∇u+| = λ+, +on +�∂Ω+ +u \ ∂Ω− +u +� ∩ D, +|∇u−| = λ−, +on +�∂Ω− +u \ ∂Ω+ +u +� ∩ D, +where Ω± +u = {x ∈ D : ±u(x) > 0}, u± := max{±u, 0}, and ∆pu = div(|∇u|p−2∇u) is the +p-Laplace operator; see Lemma 3.1. +These types of problems are known as Bernoulli-type free boundary problems +which appear in various models of fluid mechanics or heat conduction (see e.g. +[2, 4, 5, 7, 21, 18]). For the admissible functions in K+ := {v ∈ K : v ≥ 0}, the +analogous one-phase functional and the corresponding overdetermined problem +called the one-phase Bernoulli problem, was first studied in [1] for the case p = 2, +and then in [6] for the two-phase problem. Also, the case of uniformly elliptic +quasilinear equations in the one-phase case has been treated in [3]. The difficulty +of the problem (1) is that the governing operator, ∆pu = div(|∇u|p−2∇u), is not +Date: January 30, 2023. +1991 Mathematics Subject Classification. 35R35, 35J92. +Key words and phrases. Free boundary regularity, Two-phase Bernoulli problem, p-Laplacian. +M. Bayrami and M. Fotouhi was supported by Iran National Science Foundation (INSF) under +project No. 4001885. +1 + +2 +M. BAYRAMI AND M. FOTOUHI +uniformly elliptic. Obviously, close to regular free boundary points one expects +that |∇u| > 0 implying uniform ellipticity of the p-Laplacian. However, without +such a regularity assumption, it is difficult to prove non-degeneracy up to the free +boundary. In [10], the authors circumvent this issue by simultaneously showing +the non-degeneracy of the gradient and the regularity of the free boundary. +Here below we list terminologies and definitions that are frequently used in this +paper: +• A function u : D → R is said to be a minimizer of JTP in D if and only if +JTP(u, D) ≤ JTP(v, D), +for all v ∈ K. +• F(u) := �∂Ω+ +u ∪ ∂Ω− +u +� ∩ D, denotes the free boundary of the minimizer u. +• The set ΓTP := ∂Ω+ +u ∩ ∂Ω− +u ∩ D is the two-phase points of the free boundary +F(u). +• The boundary of positive and negative phases, i.e. ∂Ω± +u ∩ D can be decom- +posed as +∂Ω± +u ∩ D = Γ± +OP ∪ ΓTP, +where Γ+ +OP := �∂Ω+ +u \ ∂Ω− +u +�∩D and Γ− +OP := �∂Ω− +u \ ∂Ω+ +u +�∩D are the one-phase +parts of F(u). +• We will say that x0 ∈ ΓTP is an interior two-phase point and will denote it +by x0 ∈ Γint +TP, if +|Br(x0) ∩ {u = 0}| = 0, +for some +r > 0. +• We will say that x0 ∈ ΓTP is a branching point and will denote it by x0 ∈ Γbr +TP, +if +|Br(x0) ∩ {u = 0}| > 0, +for every +r > 0. +• We denote by Hα,e the following one-dimensional function +Hα,e(x) = α (x · e)+ − β (x · e)− , +with a unit vector e ∈ Sn−1 and the constants α and β satisfying the condi- +tions +(2) +α ≥ λ+, +β ≥ λ−, +αp − βp = λp ++ − λp +−. +Hα,e is the so-called two-plane solution to (1). +Our goal is to study the regularity of the free boundary F(u) = �∂Ω+ +u ∪ ∂Ω− +u +�∩D, +for minimizers of JTP in D, around the two-phase points. More precisely, we prove +that in a suitable neighborhood of the two-phase points, the sets Ω+ +u and Ω− +u are +two C1,η-regular domains touching along the closed set of two-phase points ΓTP. +For the special case p = 2, this result has been recently obtained in [12], by invoking +the linearization technique and we will closely follow this technique in order to +generalize this result to any 1 < p < ∞. +As is usual for problems of this type, prior to applying any method to determine +the regularity of the free boundary, the Lipschitz continuity of the minimizers +across the free boundary is required. Our partial result for the regularity of the +free boundary, however, gives us the Lipschitz regularity of the solution as well. +We first show C1,η-regularity of the free boundary with a flatness assumption in +the following theorem. + +3 +Theorem 1.1 (Flatness implies C1,η). Let u : D → R be a minimizer of JTP in D. For +any positive constants Λ0 and Λ1, there exists a constant ¯ǫ = ¯ǫ(n, p, Λ0, Λ1) such that if +(3) +∥u − Hα,e∥L∞(B1) ≤ ¯ǫ, +for some e ∈ Sn−1 and max(Λ0, λ+) ≤ α ≤ Λ1, then ∂Ω± +u ∩ Br0 are C1,η graphs for some +r0 > 0 and for any η ∈ (0, 1 +3). +We need to remark that the critical flatness to obtain the regularity does not +depend on λ±. Indeed, as long as we are close enough to a two-plane solution with +coefficient α ∈ [Λ0, Λ1], we obtain the regularity of the free boundary. This result +in turn is crucial to obtain the Lipschitz regularity of minimizers in the following +theorem. +Theorem 1.2 (Lipschitz regularity). Let u : D → R be a minimizer of JTP in D. Then +u is locally Lipschitz continuous; u ∈ C0,1 +loc(D). +2. Basic properties of minimizers +In this section, we gather some basic properties of minimizers of JTP. +Theorem 2.1 (Existence). If the admissible set K is nonempty, then there exists a mini- +mizer u of JTP over K. Moreover, every minimizer satisfies + +∆pu = 0, +in +Ω+ +u ∪ Ω− +u, +∆pu± ≥ 0, +in +D, +∥u∥L∞(D) ≤ ∥g∥L∞(D). +Proof. The existence of a bounded minimizer u of the functional JTP can be easily +established using the semi-continuity of the p-Dirichlet energy and the weak con- +vergence in W1,p, and can be obtained in the standard way. See e.g. [6, 23] for +the details. Also, notice that by comparison of u and u + tϕ, where ϕ is a suitable +smooth that supp ϕ ⊂ Ω+ +u ∪ Ω− +u, it is easy to find that ∆pu = 0 in Ω+ +u ∪ Ω− +u in the +sense of distributions. +To prove that u+ are p-subharmonic, we first note that since ∆pu = 0 in Ω+ +u, we +may choose ǫk → 0 such that {u = ǫk} to be a C1 manifold by the Sard’s Theorem, +resulting in −∇u +|∇u| to be the outer normal vector on ∂{u > ǫk}. Now take 0 ≤ ϕ ∈ C∞ +c (D), +the integration by parts implies that +� +D +|∇u+|p−2∇u+ · ∇ϕ dx = +� +{u>0} +|∇u+|p−2∇u+ · ∇ϕ dx += lim +ǫk→0 +� +{u>ǫk} +|∇u+|p−2∇u+ · ∇ϕ dx += lim +ǫk→0 +� +{u=ǫk} +|∇u+|p−2 +� +∇u+ · −∇u +|∇u| +� +ϕ dx +− +� +{u>ǫk} +∆pu ϕ dx += − lim +ǫk→0 +� +{u=ǫk} +|∇u+|p−1ϕ dx ≤ 0. +The proof of ∆pu− ≥ 0 is the same. Finally, the last estimate +∥u∥L∞(D) ≤ ∥g∥L∞(D) + +4 +M. BAYRAMI AND M. FOTOUHI +is the consequence of the p-subharmonicity of u± in D. +□ +In the following proposition we show the non-degeneracy property for the +minimizers. It reveals the fact that each of the two phases Ω+ +u and Ω− +u are optimal +with respect to one-sided inwards perturbations. The proof is the same as the proof +of non-degeneracy for one-phase problems; see [10, Lemma 4.2]. We postpone the +proof to Appendix C. +Proposition 2.2 (Non-degeneracy). Let D ⊂ Rn be an open set, and u be a minimizer +of JTP. Then, u is non-degenerate; i.e. there is a constant C = C(n, λ±, p) > 0 such that +⧸ +� +Br(x0) +�u±�p dx ≥ Crp, +for every x0 ∈ Ω± +u ∩ D and every 0 < r < dist (x0, ∂D). +The next proposition concerns the Lipschitz regularity of the minimizers around +the one-phase points. +Proposition 2.3 (Lipschitz regularity at one-phase points). Let u : D → R be a +minimizer of JTP in D. There there is constant C = C(n, p, ±λ) such that if x0 ∈ Γ+ +OP (or +x0 ∈ Γ− +OP) is one-phase point and Br(x0) ∩ Ω− +u = ∅ (Br(x0) ∩ Ω+ +u = ∅), then +∥∇u∥L∞(B r +2 (x0)) ≤ C. +We remark that the condition Br(x0) ∩ Ω− +u = ∅ always holds for some r > 0 by +the definition of one-phase points. +Proof. We know that ux0,r(x) = u(x0+rx) +r +is a minimizer of the following one phase +functional in B1, i.e. minimizer of +JOP(v, B1) := +� +B1 +|∇v|p dx + (p − 1)λp ++|{v > 0} ∩ B1|, +over the class of nonnegative functions. Then the boundedness of the gradient +∥∇u∥L∞(B r +2 (x0)) = ∥∇ux0,r∥L∞(B 1 +2 ) ≤ C(n, p, λ+), +follows from [10, Theorem 3.3]. We shall remark that the Lipschitz constant for +one-phase problems does not depend on the boundary values of the minimizer as +long as we stay uniformly far from the boundary. +□ +Next, we mention the following continuity result for minimizers. +Proposition 2.4 (BMO estimates for the gradient). Let u be a minimizer of JTP and +D′ ⋐ D. Then, +(i) for 1 < p < 2, we have that |∇u| +p−2 +2 ∇u ∈ BMO(D′), and consequently u ∈ Cσ(D′) +for any σ ∈ (0, 1); +(ii) for 2 < p < ∞, we have that ∇u ∈ BMO(D′) and thus u is locally log-Lipschitz +continuous. +In particular, ∇u ∈ Lq(D′) for any 1 < q < ∞ and for any 1 < p < ∞. +Proof. The proof is the same as the proof of Lemma 3.1 in [16]. +□ + +5 +The BMO estimate for the gradient of minimizers is sufficient to obtain the fol- +lowing compactness result. Since we have not yet proved the Lipschitz continuity, +this result will be extremely valuable for our argument in the next section. We +postpone the proof to Appendix A. +Proposition 2.5. Let uj be a bounded minimizer of JTP in B2 with the points xj ∈ B1 such +that uj(xj) = 0. Also, set vj(x) = +uj(xj+rjx) +Sj +, for any x ∈ BR, with 0 < R < 1 +rj , where rj → 0, +as j → +∞ and Sj > 0. Then, vj is the minimizer (according to its own boundary values) +of the following scaled functional +(4) +ˆJTP(v) := +� +BR +|∇v|p + (p − 1)σp +jλp ++χ{v>0} + (p − 1)σp +jλp +−χ{v<0} dx, +where σj := +rj +Sj . Moreover, if |vj| ≤ M in BR, for any fixed 0 < R < +1 +rj and for some +M = M(R) > 0, then up to a subsequence, the followings hold: +(i) For any q > 1, and some α ∈ (0, 1) (if q > n, one can take α = 1 − n +q), vj converges +to some function v0 as j → +∞ in Cα(BR) and weakly in W1,q(BR); +(ii) vj → v0 strongly in W1,p(BR); +(iii) If moreover, σj := +rj +Sj → σ, as j → +∞, then v0 is a minimizer of +ˆJTP(v) := +� +BR +|∇v|p + (p − 1)σpλp ++χ{v>0} + (p − 1)σpλp +−χ{v<0} dx. +In particular, if σ = 0, then v0 is p-harmonic in BR. +The following lemma states that u+ and u− have coherent growth at two-phase +points. This is essential to show that the minimizers are the viscosity solution of +(1). +Lemma 2.6. Let u be a bounded minimizer of JTP. Let x0 ∈ ΓTP and r0 > 0 be small +such that Br0(x0) ⊂ D. Assume that supBr(x0) u− ≤ C0r (resp. supBr(x0) u+ ≤ C0r) for +all r ∈ (0, r0]. Then there exist constant C1 > 0 such that supBr(x0) u+ ≤ C1r (resp. +supBr(x0)u− ≤ C1r) for all r ∈ (0, r0]. +Proof. We will just demonstrate one of the claims; the other can be demonstrated +similarly. By the assumption of the lemma +(5) +sup +Br(x0) +u− ≤ C0r, +∀r ∈ (0, r0]. +We claim that there is ˜C1 > 0 such that +(6) +S(k + 1) ≤ max +� ˜C1 +2k+1 , 1 +2S(k) +� +, +where S(k) := ∥u∥L∞(B2−k(x0)), for any k ∈ N that 2−k ≤ r0. To prove this, we argue +by contradiction and suppose that (6) fails. Then there is a sequence of integers kj, +with j = 1, 2, · · · such that +(7) +S(kj + 1) > max +� +j +2kj+1 , 1 +2S(kj) +� +. + +6 +M. BAYRAMI AND M. FOTOUHI +Observe that since u is a bounded minimizer, then (7) implies that kj → +∞ as +j → +∞. Also, notice that (7) implies that +(8) +σj := +2−kj +S(kj + 1) ≤ 2 +j → 0 +as +j → +∞. +Now, we introduce the scaled functions vj(x) := u(x0+2−kjx) +S(kj+1) , for any x ∈ B1. Then, +from (5) and (8), it follows that vj(0) = 0 and +(9) +v− +j (x) = u−(x0 + 2−kjx) +S(kj + 1) +≤ +2−kjC0 +S(kj + 1) ≤ 2C0 +j +→ 0, +as +j → +∞. +Furthermore, it is simple to show that (7) implies that +(10) +sup +B1 +|vj| ≤ 2, +and +sup +B 1 +2 +|vj| = 1. +Also, Proposition 2.5 entails that vj is a minimizer of the scaled functional (4) for +R = 3 +4 and we can extract a converging subsequence such that vj → v0 uniformly +in B 3 +4 that v0 is p-harmonic. The uniform convergence of vj to v0 along with (9), +(10), give that +∆pv0(x) = 0, +v0(x) ≥ 0 if x ∈ B 3 +4 , +v(0) = 0, +sup +B 1 +2 +v0 = 1, +which is in contradiction with the strong minimum principle. Thus (6) obtains. +Now we show how (6) implies the lemma. Assume that k0 is the smallest integer +k that 2−k ≤ r0. Let ¯C1 = max( ˜C1, 2k0S(k0)). It is not difficult to observe from (6) that +S(k) ≤ ¯C12−k. For an arbitrary r ∈ (0, r0] choose k ≥ k0 such that 2−(k+1) < r ≤ 2−k, +then +∥u∥L∞(Br(x0)) ≤ ∥u∥L∞(B2−k(x0)) = S(k) ≤ ¯C12−k ≤ 2 ¯C1r. +Thus the statement in the lemma holds for C1 = 2 ¯C1. +□ +The following theorem roughly says that, in a very weak sense,the free boundary +conditions (1) hold. +Proposition 2.7. Suppose that u is a minimizer of JTP in D and D′ ⊂ D be such that +|D′ ∩ {u = 0}| = 0. Then, we have the following free boundary condition in the very weak +sense +lim +ǫ→0+ +� +∂{u>ǫ}∩D′ +� +|∇u+|p − λp ++ +� +η · ν + lim +δ→0+ +� +∂{u<−δ}∩D′ +� +|∇u−|p − λp +− +� +η · ν = 0, +for any η ∈ W1,p +0 (D′, Rn), and where ν is the outward normal. +Proof. The proof can be established precisely as in [6, Theorem 2.4]. +□ +Corollary 2.8. Suppose u(x) = α (x · e)+ − β (x · e)− is a global minimizer of JTP for some +unite vector e ∈ Sn−1 and the positive constants α and β. Then α and β satisfy conditions +(2). +Proof. The equality αp − βp = λp ++ − λp +− is obvious by invoking Proposition 2.7. +Besides, conditions +α ≥ λ+, +and +β ≥ λ−, + +7 +can be obtained by a smooth variation of the free boundary {u = 0} = {x · e = 0}. +Indeed, by considering competitors of the form ut(x) = u+(x) − u−(x + tξ(x)) for +vector fields ξ ∈ C∞ +c (Rn; Rn) with ξ · e ≤ 0 so that it moves negative phase only +inwards, that is, {ut < 0} ⊂ {u < 0}, and taking the derivative of JTP(ut, BR) at t > 0 +and letting t → 0 (where R is sufficiently large such that supp ξ ⊂ BR), we get +� +{u=0}∩BR +(ξ · e) +� +|∇u−|p − λp +− +� +≤ 0, +which gives β ≥ λ−. The estimate on α is analogous. +□ +3. Free boundary conditions in the viscosity sense +Let u : D → R be a local minimizer of JTP. In this section, we will show that u +satisfies the free boundary conditions (1) in a weak (viscosity) sense. +Definition 3.1. Let D be an open set. We say that a function Q : D → R touches a +function w : D → R from below (resp. from above) at a point x0 ∈ D if Q(x0) = w(x0) and +Q(x) − w(x) ≤ 0 +(resp. Q(x) − w(x) ≥ 0), +for every x in a neighborhood of x0. We will say that Q touches w strictly from below (resp. +above) if the above inequalities are strict for x � x0. +A function Q is an (admissible) comparison function in D if +(a) Q ∈ C1({Q > 0} ∩ D) ∩ C1({Q < 0} ∩ D); +(b) Q ∈ C2({Q > 0} ∩ D) ∩ C2({Q < 0} ∩ D); +(c) ∂{Q > 0} and ∂{Q < 0} are smooth manifolds in D. +We should remark that if ∇Q � 0 on ∂{Q > 0} ∪ ∂{Q < 0}, the condition (c) above holds. +Lemma 3.1. Let u be a local minimizer of JTP in the open set D ⊂ Rn. Then the following +optimality conditions on the free boundary F(u) hold. +(A) Suppose that Q is a comparison function that touches u from below at x0. +(A.1) If x0 ∈ Γ+ +OP, then |∇Q+(x0)| ≤ λ+; +(A.2) if x0 ∈ Γ− +OP, then Q+ ≡ 0 in a neighborhood of x0 and |∇Q−(x0)| ≥ λ−; +(A.3) if x0 ∈ ΓTP, then |∇Q−(x0)| ≥ λ− and +|∇Q+(x0)|p − |∇Q−(x0)|p ≤ λp ++ − λp +−. +(B) Suppose that Q is a comparison function that touches u from above at x0. +(B.1) If x0 ∈ Γ+ +OP, then Q− ≡ 0 in a neighborhood of x0 and |∇Q+(x0)| ≥ λ+; +(B.2) if x0 ∈ Γ− +OP, then |∇Q−(x0)| ≤ λ−; +(B.3) if x0 ∈ ΓTP, then |∇Q+(x0)| ≥ λ+ and +|∇Q+(x0)|p − |∇Q−(x0)|p ≥ λp ++ − λp +−. +Proof. First, we will prove the gradient bounds in (A.1) and (B.1). The case x0 ∈ Γ− +OP, +and the proofs of (A.2) and (B.2) can be obtained similarly. +Let x0 ∈ Γ+ +OP be a one-phase point and let Q touches u from below at x0. Then, Q+ +touches u from below at x0, too. Consider ux0,rk(x) = u(x0+rkx) +rk +and Q+ +x0,rk(x) = Q+(x0+rkx) +rk +as the blow-up sequences of u and Q+ at x0. By virtue of Proposition 2.3, the +functions ux0,rk are uniformly Lipschitz for sufficiently rk small and up to extracting +a subsequence, we can assume that ux0,rk converges uniformly to a blow-up limit +v. The limit v is a minimizer of one-phase functional JOP and so ∆pv = 0 in {v > 0}. + +8 +M. BAYRAMI AND M. FOTOUHI +On the other hand, since Q+ is differentiable at x0 in Ω+ +Q, we get that Q+ +x0,rk +converges to the function +(11) +HQ+(x) = (x · e′)+ +with +e′ = ∇Q+(x0). +If ∇Q+(x0) = 0, (A.1) is trivially valid. We assume that e′ � 0, and since HQ+ touches +v from below at x = 0, we get +v(x) = α(x · e′)+ + o(|x|), +α ≥ 1, +for some α; see [19, Lemma B.1]. We get that any blow-ups of v will be v0(x) = +α(x ·e′)+ which is also a minimizer of JOP. Thus α|e′| = λ+ due to the free boundary +condition for one-phase minimizers (Proposition 2.7 or see [10, Theorem 2.1]) and +so +|∇Q+(x0)| ≤ λ+. +Similarly, when Q touches u from above at x0, then also Q+ touches u from above +at x0, and the claim Q− ≡ 0 in (B.1) is trivially true. Again, consider ux0,rk, which up +to extracting a subsequence, converges uniformly to a blow-up limit v and Q+ +x0,rk, +as the blow-up sequences of Q+ at x0, which converges to the function (11). Now, +we argue similar to the proof of (A.1) to get +|∇Q+(x0)| ≥ λ+. +Now, we prove (A.3). Suppose x0 ∈ ΓTP and assume that Q touches u from +below at x0. Then u− ≤ Q− and u−(x) ≤ C0|x − x0| for C0 = 2|∇Q−(x0)| if |x − x0| is +sufficiently small. Now we employ Lemma 2.6 to deduce that |u(x)| ≤ C1|x − x0| in +a neighborhood of x0. +Let ux0,rk and Qx0,rk be the blow-up sequences of u and Q at x0. Then, by using +Proposition 2.5, up to extracting a subsequence, we can assume that ux0,rk converges +uniformly to some function v which is also a minimizer of JTP. Moreover, it satisfies +(12) +|v(x)| ≤ C1|x|. +On the other hand, since Q+ and Q− are differentiable at x0 (respectively in Ω+ +Q +and Ω− +Q), we get that Qx0,rk converges to the function +HQ(x) = (x · ˜e+)+ − �x · ˜e−�− , +where ˜e± = ∇Q±(x0). Since HQ touches v from below at x = 0, we have ([19, Lemma +B.1]) +v+(x) = α(x · ˜e−)+ + o(|x|), +|˜e+| ≤ α|˜e−|, +v−(x) = β(x · ˜e−)− + o(|x|), +β ≤ 1, +for some α, β ≥ 0. Note that by virtue of the non-degeneracy, Proposition 2.2, +v− � 0 and so ˜e− � 0. If v0 is a blowup of v (recall (12) and Proposition 2.5), it +will be v0(x) = α(x · ˜e−)+ − β(x · ˜e−)− which is also a minimizer of JTP. Now apply +Corollary 2.8, we get +(αp − βp)|˜e−|p = λp ++ − λp +−, +β|˜e−| ≥ λ−. +Hence, +|∇Q+(x0)|p − |∇Q−(x0)|p ≤ αp − βp = λp ++ − λp +−, +as well as |∇Q−(x0)| ≥ λ−. The proof of (B.3) is analogous. +□ + +9 +If u : D → R is a continuous function such that the claims (A) and (B) hold for +every comparison function Q, then we say that u satisfies the boundary condition +(1) on the free boundary in viscosity sense. +We need the following straightforward consequence of the definition of viscosity +solution. It emphasizes what happens when a function is touching only one of the +two phases. +Lemma 3.2. Let u : D → R be a continuous function that satisfies (1). +(i) Assume that Q is a comparison function touching u+ from above at x0 ∈ ∂Ω+ +u +(resp. −u− from below at x0 ∈ ∂Ω− +u), then +|∇Q+(x0)| ≥ λ+ +�resp. |∇Q−(x0)| ≥ λ− +� . +(ii) Assume that Q is a comparison function touching u+ from below at x0 ∈ Γ+ +OP +(resp. −u− from above at x0 ∈ Γ− +OP), then +|∇Q+(x0)| ≤ λ+ +�resp. |∇Q−(x0)| ≤ λ− +� . +Proof. Statement (i) will be obtained directly from (B). The proof of (ii) follows the +same lines of arguments as the proof of Lemma 3.1. +□ +4. Flatness decay at two-phase points +In this section, we will follow the method of improvement of flatness. In fact, +we will prove that at two-phase points x0 ∈ ΓTP, there is a constant ǫ0 > 0 such that +if u is ǫ0-flat in Br(x0) with respect to H = Hα,e, then it has excess flatness in smaller +scales with respect to another ˜H = H ˜α,˜e. +Theorem 4.1. For every 1 < p < ∞, 0 < L0, L1 and γ ∈ (0, 1 +2), there exist ǫ0 > 0, C > 0 +and ρ > 0 such that if the function u : B1 → R satisfies: +(a) the origin is on the two-phase free boundary, 0 ∈ ΓTP; +(b) u is p-harmonic in Ω+ +u ∪ Ω− +u; +(c) u satisfies the free boundary condition (1) in viscosity sense; +(d) u is ǫ0-flat in B1, that is, +(13) +∥u − Hα,en∥L∞(B1) ≤ ǫ0, +for some +max(λ+, L0) ≤ α ≤ L1, +then, there are e ∈ Sn−1 and ˜α ≥ max(λ+, L0), such that +(14) +|e − en| + | ˜α − α| ≤ C∥u − Hα,en∥L∞(B1), +and +(15) +∥uρ − H ˜α,e∥L∞(B1) ≤ ργ∥u − Hα,en∥L∞(B1), +where uρ(x) denotes u0,ρ(x) = u(ρx) +ρ . +Theorem 4.1 is an easy consequence of the two upcoming lemmas. In the first +one, we deal with the situation where the two-plane is, roughly, Hλ+,e for some +e ∈ Sn−1. Note that this is the case where one might expect the presence of branching +points and it is indeed in this setting that we will obtain the two membrane +problems as ”linearization” (see e.g. [12, Subsection 1.3] for a presentation of the +linearization method in studying the regularity of free boundaries). In the second +lemma, we deal with the case when the closest half-plane solution has a gradient +much larger than λ+. In this case, the origin will be an interior two-phase point. +In fact, in one-phase problems, it is possible to obtain universal interior bounds, in + +10 +M. BAYRAMI AND M. FOTOUHI +the sense that, if u is a solution in a ball B1 and 0 ∈ F(u), then |∇u| is bounded in B 1 +2 +by a universal constant, no matter what the boundary data are. However, in two- +phase problems, this is generally not possible. For instance, in the one-dimensional +minimization scenario, increasing the boundary data leads to the appearance of a +solution with a large gradient near the origin, see [9, Section 1.1]. +Lemma 4.2 (Improvement of flatness: branching points). For every 1 < p < ∞, +0 < L0, L1, γ ∈ (0, 1 +2), and M > 0, there exist ǫ1 = ǫ1(p, γ, n, L0, L1, M), C1 = +C1(p, γ, n, L0, L1, M) and ρ = ρ(p, γ, n, L0, L1, M) such that if function u : B1 → R +satisfies (a) − (b) − (c) of Theorem 4.1 and furthermore +∥u − Hα,en∥L∞(B1) ≤ ǫ1, +with +L0 ≤ λ+ ≤ α ≤ λ+ + M∥u − Hα,en∥L∞(B1), +then there exist e ∈ Sn−1 and ˜α ≥ λ+, for which (14) and (15) hold. +Lemma 4.3 (Improvement of flatness: non-branching points). For every 1 < p < ∞, +0 < L0, L1 and γ ∈ (0, 1), there exist ǫ2 = ǫ2(p, γ, n, L0, L1), M = M(p, γ, n, L0, L1), +ρ = ρ(p, γ, n, L0, L1) and C2 = C2(p, γ, n, L0, L1) such that if function u : B1 → R satisfies +(a) − (b) − (c) of Theorem 4.1 and furthermore +∥u − Hα,en∥L∞(B1) ≤ ǫ2, +with +α ≥ max(λ+, L0) + M∥u − Hα,en∥L∞(B1), +then there exist e ∈ Sn−1 and ˜α ≥ max(λ+, L0), for which (14) and (15) hold. +Proof of Theorem 4.1. The proof follows easily by combining the Lemmas 4.2 and +4.3. +□ +In order to prove Lemma 4.2 and Lemma 4.3, we will argue by contradiction. +Hence in the following, we consider a sequence uk of minimizers such that +(16) +ǫk := ∥uk − Hαk,en∥L∞(B1) → 0 +and +λ+ ≤ αk ≤ L. +We also set +(17) +ℓ := λp ++ lim +k→∞ +αp +k − λp ++ +pαp +kǫk += λp +− lim +k→∞ +βp +k − λp +− +pβp +kǫk +, +which we can assume to exist up to a subsequence. It might be useful to keep in +mind that ℓ = ∞ will correspond to Lemma 4.3 while 0 ≤ ℓ < ∞ (so αk → λ+ and +λ+ ≥ L0) to Lemma 4.2. +We first show that the sequence +(18) +vk(x) = + +v+,k(x) := uk(x) − αkx+ +n +ǫkαk +x ∈ Ω+ +uk ∩ B1 +v−,k(x) := uk(x) + βkx− +n +ǫkβk +x ∈ Ω− +uk ∩ B1 +is compact in some suitable sense. This will be mentioned in Lemma 4.4 below and +the proof will come in Subsection 4.1. Then, in Lemma 4.5, we obtain the limiting +problem which is solved by v, the limit of vk. Finally, in Subsection 4.3 we show +how to deduce Lemma 4.3 and Lemma 4.2 from Lemma 4.4 and Lemma 4.5. +In the following, we will denote with +B± +r := Br ∩ {x± +n > 0}, +for every r > 0. + +11 +Lemma 4.4 (Compactness of the linearizing sequence vk). Let uk be a sequence of +functions satisfying (a) − (b) − (c) of Theorem 4.1 uniformly in k and let ǫk and αk be as in +(16) and let vk be defined by (18). Then there are H¨older continuous functions +v+ : B+ +1 +2 → R +and +v− : B− +1 +2 → R, +with +v+ ≤ v− +on +B 1 +2 ∩ {xn = 0}, +v+(0) = v−(0) = 0, +and such that the sequence of closed graphs +Γ± +k := +� +(x, v±,k(x)) : x ∈ Ω± +uk ∩ B 1 +2 +� +, +converge, up to a subsequence, in the Hausdorff distance to the closed graphs +Γ± = +� +(x, v±(x)) : x ∈ B± +1 +2 +� +. +In particular, the following claims hold. +(i) For every δ > 0, v±,k converges uniformly to v± on B 1 +2 ∩ {±xn > δ}. +(ii) For every sequence xk ∈ Ω± +uk ∩ B1 converging to x ∈ B± +1 +2 , we have +v±(x) = lim +k→∞ v±,k(xk). +(iii) For every x ∈ {xn = 0} ∩ B 1 +2 , we have +v±(x) = − lim +k→∞ +xk · en +ǫk +for any sequence +∂Ω± +uk ∋ xk → x. +In particular, {xn = 0} ∩ B 1 +2 decomposes into an open jump set +J = {v+ < v−} ∩ {xn = 0} ∩ B 1 +2 , +and its complementary contact set +C = {v+ = v−} ∩ {xn = 0} ∩ B 1 +2 . +Furthermore, if x ∈ J, then +(19) +lim inf +k→∞ dist +� +x, ∂Ω+ +uk ∩ ∂Ω− +uk +� +> 0. +In particular for all x ∈ J, there exists two sequences x± +k ∈ Γ± +k,OP such that x± +k → x. +Now, in the next lemma, we determine the limiting problem for the function v +which is defined as +(20) +v(x) = + +v+(x) +for x ∈ B+ +1 +2 , +v−(x) +for x ∈ B− +1 +2 , +where v+ and v− are the functions defined in Lemma 4.4. +In what follows, we will denote with +Lp(u) := ∆u + (p − 2)∂nnu, +the frequently used operator which appears in the linearized problem. + +12 +M. BAYRAMI AND M. FOTOUHI +Lemma 4.5 (The ”linearized” problem). Let uk, ǫk and αk be as in (16), vk be defined +by (18) and ℓ as in (17). Let also v± be as in Lemma 4.4: +If ℓ = ∞, then J = ∅ and v± are viscosity solutions of the following transmission +problem: +(21) + +Lp(v±) = ∆v± + (p − 2)∂nnv± = 0, +in +B± +1 +2 , +αp +∞∂nv+ = βp +∞∂nv−, +on +B± +1 +2 ∩ {xn = 0}, +where α∞ = limk→∞ αk and β∞ = limk→∞ βk, which we can assume to exist up to extracting +a further subsequence. +If 0 ≤ ℓ < ∞, then v± are viscosity solutions of the following two membranes problem: +(22) + +Lp(v±) = ∆v± + (p − 2)∂nnv± = 0, +in +B± +1 +2 , +λp +±∂nv± + ℓ ≥ 0, +in +B 1 +2 ∩ {xn = 0}, +λp +±∂nv± + ℓ = 0, +in +J, +λp ++∂nv+ = λp +−∂nv−, +in +C, +v+ ≤ v−, +in +B 1 +2 ∩ {xn = 0}. +Remark 4.6. Here by viscosity solution of (21) and (22), we mean a function v as in +(20) such that v± are continuous in B± +1 +2 , Lp(v±) = 0 in B± +1 +2 (in viscosity or equivalently the +classical sense) and such that the following holds. +• If we are in case (21), let s, t ∈ R and let ˜P be a quadratic polynomial such that +∂n ˜P = 0. Suppose that Lp( ˜P) ≥ 0 (Lp( ˜P) ≤ 0) and that the function +P := sx+ +n − tx− +n + ˜P, +touches v strictly from below (above) at a point x0 ∈ B 1 +2 ∩ {xn = 0}, then +αp +∞s ≤ βp +∞t, +� +αp +∞s ≥ βp +∞t +� +. +• If we are in case (22) then +(1) if P± is a quadratic polynomial with Lp(P±) ≤ 0 in B± +1 +2 touching v± strictly +from above at x0 ∈ B 1 +2 ∩ {xn = 0}, then λp +±∂nP± ≥ 0; +(2) if P± is a quadratic polynomial with Lp(P±) ≥ 0 in B± +1 +2 touching v± strictly +from below at x0 ∈ J, then λp +±∂nP± ≤ 0; +(3) if s, t ∈ R and ˜P is a quadratic polynomial with Lp(P±) ≥ 0 (Lp(P±) ≤ 0) +such that ∂n ˜P = 0 and the function +P := sx+ +n − tx− +n + ˜P, +touches v strictly from below (above) at a point x0 ∈ B 1 +2 ∩ {xn = 0}, then +λp ++s ≤ λp +−t, +� +λp ++s ≥ λp +−t +� +. +4.1. Compactness of the linearizing sequence. As explained in [12, Subsection +3.1] for the case of classical two-phase Bernoulli problem, the authors declare that +the key point in establishing suitable compactness for vk is a ”partial Harnack” +inequality. We will follow the same approach and start with the following useful +lemma. + +13 +Lemma 4.7. There is a constant τ = τ(n, p) > 0 such that the following holds. Assume +that v : B1 → R is a continuous function with ∆pv = 0 in {v > 0} and +λ (xn + b)+ ≤ v(x) ≤ λ (xn + a)+ , +x ∈ B1, +for some λ > 0 and a, b ∈ (− 1 +100, +1 +100). Let P = (0, · · · , 0, 1 +2), then for all ǫ ∈ (0, 1 +2) +v(P) ≤ λ(1 − ǫ) +�1 +2 + a +�+ +=⇒ +v(x) ≤ λ(1 − τǫ) (xn + a)+ +in +B 1 +4 (0), +and +v(P) ≥ λ(1 + ǫ) +�1 +2 + b +�+ +=⇒ +v(x) ≥ λ(1 + τǫ) (xn + b)+ +in +B 1 +4 (0). +Proof. We prove only the first implication since the second statement can be ob- +tained by the same arguments. First, we notice that, since |b| < +1 +100, both v and +λ(xn + a)+ are positive and p-harmonic in B 1 +4 (P). Thus, +λ(xn + a)+ − v(x) ≥ 0, +x ∈ B 1 +4 (P), +and +λ +�1 +2 + a +�+ +− v(P) ≥ λǫ +�1 +2 + a +�+ +≥ 49 +100λǫ. +Now, we distinguish two cases: +Case (i). Suppose |∇v(P)| < λ +4 . Therefore, there exists r1 = r1(n, p) > 0 such that +|∇v(x)| ≤ λ +2 in B4r1(P) (note that v +λ is universally bounded and p-harmonic in B 1 +4 (P)). +It is easy to find that for ˜v := (xn + a)+ − 1 +λv, we have +div +� +|∇˜v − en|p−2(∇˜v − en) +� += 0, +in +B 1 +20 (P). +We now apply Harnack’s inequality for the above operator (see e.g. [17, Lemma +4.1]) in B4r1(P), to deduce that +(xn + a)+ − 1 +λv(x) ≥ C−1 +��1 +2 + a +�+ +− 1 +λv(P) +� +− r1, +in +Br1(P), +for an appropriate universal constant C = C(n, p) > 0. On the other hand, for all +x ∈ Br1(P), we obtain +C−1 49 +100ǫ − r1 ≤ (xn + a)+ − 1 +λv(x) +≤ (xn + 2r1 + a)+ − 2r1 − 1 +λv(x + 2r1en) + 2r1 +λ ∥∇v∥L∞(B4r1(P)) +≤ (xn + 2r1 + a)+ − 2r1 − 1 +λv(x + 2r1en) + r1 +≤ (xn + 2r1 + a)+ − 1 +λv(x + 2r1en) − r1. +Thus, with ˜P = P + 2r1en, we get +(23) +C−1 49 +100ǫ ≤ (xn + a)+ − 1 +λv (x) , +for all +x ∈ Br1( ˜P). +Hence, by considering the inequality (23) and also using the bound |a| ≤ +1 +100, there +is a constant c = c(n, p) such that +v(x) ≤ λ(1 − cǫ)(xn + a)+, +for all +x ∈ Br1( ˜P). + +14 +M. BAYRAMI AND M. FOTOUHI +We now let w be the solution to the following problem + +∆pw = 0 +in +� +B1(0) \ Br1( ˜P) +� +∩ {xn > −a} +w = 0 +on +B1 ∩ {xn = −a} +w = (xn + a)+ +on +∂B1(0) ∩ {xn > −a} +w = (1 − cǫ)(xn + a)+ +on +∂Br1( ˜P) ∩ {xn > −a}. +By the Hopf boundary lemma ([24, Proposition 3.2.1]), +w(x) ≤ (1 − τǫ)(xn + a)+, +for every x ∈ B 1 +4 ∩ {xn > −a}, +for a suitable constant τ = τ(n, p). On the other hand, by the comparison principle, +we have v ≤ λw in {v > 0} ∩ B1 \ Br1( ˜P), which concludes the proof in Case (i). +Case (ii). Suppose |∇v(P)| ≥ λ +4 . By the interior gradient estimate, we know that +∇v is bounded in B 1 +40 (P), and there exist a constant 0 < r0 = r0(n, p), with 8r0 ≤ +1 +40 +such that +λ +8 ≤ |∇v(x)| ≤ Cλ, +for all +x ∈ B8r0(P), +for an appropriate universal constant C = C(n, p) > 0. Now, v will be the weak +solution to the following uniformly elliptic equation +n +� +i,j=1 +θij∂xixjv = 0 +in +B4r0(P), +with θij = δij + (p − 2)|∇v|−2∂xiv∂xjv. Then, applying Harnack’s inequality (see e.g. +[20, Chapter 9]), we get +(24) +C−1 49 +100ǫ ≤ (xn + a)+ − 1 +λv (x) , +for all +x ∈ Br0(P). +Now, we can repeat the same argument of Case (i), by considering the inequality +(24) in the ball Br0(P) instead of inequality (23). This completes the proof of the +lemma. +□ +We next prove the two partial Harnack inequalities. The proof of these inequal- +ities is based on a comparison with suitable test functions. In order to build these +”barriers”, we will often use the following function ϕ. Let Q = (0, · · · , 0, 1 +5) and +define ϕ : B1 → R by +(25) +ϕ(x) = + +1, +if x ∈ B 1 +100 (Q), +κn +� +|x − Q|−n − ( 3 +4)−n� +, +if x ∈ B 3 +4 (Q) \ B 1 +100 (Q), +0, +otherwise, +where the dimensional constant κn is chosen in such a way that ϕ is continuous. +One can check that ϕ has the following properties: +(ϕ.1) 0 ≤ ϕ ≤ 1 in Rn, and ϕ = 0 on ∂B1; +(ϕ.2) For s > 0 small, +−div +����en − s∇ϕ +��� +p−2 �en − s∇ϕ�� +≥ c(n, p, s) > 0, +in +{ϕ > 0} \ B 1 +100 (Q), +(with fairly simple computations same as the ones which have been done +in [17, Lemma 4.2]); +(ϕ.3) ∂nϕ > 0 in {ϕ > 0} ∩ {|xn| ≤ +1 +100}; + +15 +(ϕ.4) ϕ ≥ cn > 0 in B 1 +6 ; +where c(n, p) and cn are constants. +Lemma 4.8 (Partial Boundary Harnack I). Given 1 < p < ∞ and λ+ ≥ λ− > 0, there +exist constants ǫ = ǫ(n, λ±, p) > 0 and c = c(n, λ±, p) ∈ (0, 1) such that, for every function +u : B4 → R satisfying (b) − (c) in Theorem 4.1, the following properties hold true. +Let a±, b± ∈ (− 1 +100, +1 +100) be such that +b+ ≤ b− ≤ a− ≤ a+, +and +(a− − b−) + (a+ − b+) ≤ ǫ. +Assume that for x ∈ B4 +λ+(xn + b+)+ ≤ u+(x) ≤ λ+(xn + a+)+, +and +−λ−(xn + b−)− ≤ −u−(x) ≤ −λ−(xn + a−)−. +Then, one can find new constants a±, b± ∈ (− 1 +100, +1 +100), with +b+ ≤ b− ≤ a− ≤ a+, +and +a− − b− ≤ c(a− − b−), +a+ − b+ ≤ c(a+ − b+) +such that for x ∈ B 1 +6 +λ+(xn + b+)+ ≤ u+(x) ≤ λ+(xn + a+)+, +and +−λ−(xn + b−)− ≤ −u−(x) ≤ −λ−(xn + a−)−. +Remark 4.9. We need to remark that the assumption λ+ ≥ λ− is not restrictive as one can +always replace u by −u in JTP. Also, when λ+ ≤ λ− the similar result holds if we replace +the order of a±, b± with a+ ≤ a− ≤ b− ≤ b+. +Proof of Lemma 4.8. Let us show how to improve the positive part. More precisely, +given a+, a−, b+, b− we will show how we can find a+ and b+. The proof for b− and +a− follows in the same way. We let +P = (0, · · · , 0, 2), +and distinguish two cases: +Case 1. Improvement from above. Assume that, at the point P, u+ is closer to +λ+(2 + b+)+ than to the upper barrier λ+(2 + a+)+. Precisely that +u+(P) ≤ λ+(2 + a+)+ − λ+(a+ − b+) +2 +. +In this case, we will show that u(x) is less than λ+(xn+a+)+ in a smaller ball centered +at the origin for a+ strictly smaller than a+. +We start by setting +ǫ := a+ − b+ ≤ ǫ. +Then +u+(P) ≤ λ+(2 + a+)+ − λ+ǫ +2 +≤ λ+(1 − cǫ)(2 + a+)+, + +16 +M. BAYRAMI AND M. FOTOUHI +for a suitable (universal) constant c. We can thus apply (the scaled version of) +Lemma 4.7 to u+, to infer the existence of a constant τ = τ(n, p) such that +(26) +u+(x) ≤ λ+(1 − τǫ)(xn + a+)+, +in +B1. +For ϕ as in (25) and t ∈ [0, 1], we set +ft = λ+ +� +1 − τǫ +2 +� +(xn + a+ − tcǫϕ)+, +where c = c(n, p) is a small constant chosen such that for all x ∈ B 1 +100 (Q) and t ∈ [0, 1), +(27) +u(x) ≤ λ+(1 − τǫ)(xn + a+)+ +≤ λ+ +� +1 − τǫ +2 +� +(xn + a+ − cǫ)+ < ft(x), +where we have used that (xn + a+) is within two universal constant for x ∈ B 1 +100 (Q). +We now let t ∈ (0, 1] the largest t such that ft ≥ u in B1 and we claim that t = 1. +Indeed assume that t < 1, then there exists x ∈ B1 such that +(28) +u(x) − ft(x) ≤ u(x) − ft(x) = 0, +for all +x ∈ B1. +Note that by (27), x � B 1 +100 (Q), while, by (26), x ∈ {ϕ > 0}. Moreover, if u(x) = ft(x) > +0, by (ϕ.2) we will have +∆p ft(x) = +� +λ+ +� +1 − τǫ +2 +��p−1 +div +����en − tcǫ∇ϕ(x) +��� +p−2 � +en − tcǫ∇ϕ(x) +�� +< 0, +but, since ∆pu(x) = 0, we reach a contradiction with (28) and the definition of +viscosity solution for the p-harmonic function u. Hence, u(x) = ft(x) = 0. Now +recall the free boundary condition (1) and apply (ϕ.3) to get +λp ++ ≤ |∇ft(x)|p = λp ++ +� +1 − τǫ +2 +�p � +1 − pctǫ∂nϕ(x) + O(ǫ2) +� +< λp ++, +provided ǫ ≤ ǫ(n, λ+, p) ≪ 1 (note that necessarily u(x) = 0 and thus x ∈ {|xn| ≤ +1 +100}). +This contradiction implies that t = 1. Hence, by (ϕ.4), we get for all x ∈ B 1 +6 +u(x) ≤ λ+ +� +1 − τǫ +2 +� +(xn + a+ − cǫϕ)+ ≤ λ+(xn + a+ − cǫ)+, +for a suitable constant c = c(n, p). Setting +a+ = a+ − cǫ, +b+ = b+, +and recalling that ǫ = a+ − b+ we finish the proof in this case. +Case 2. Improvement from below. We now assume that, at point P, u+ is closer to +λ+(2 + a+)+ than to λ+(2 + b+)+. Hence, we have +u+(P) ≥ λ+(2 + b+)+ + λ+(a+ − b+) +2 +, +and we set again +ǫ := a+ − b+ ≤ ǫ. +Arguing as in Case 1, by Lemma 4.7, there exists a constant τ = τ(n, p) such that +(29) +u+(x) ≥ λ+(1 + τǫ)(xn + b+)+, +in +B1. + +17 +We need now to distinguish two further sub-cases: +Case 2.1: Suppose that +ηǫ ≤ b− − b+, +where η ≪ τ is a small universal constant which we will choose at the end of the +proof. In this case, for x ∈ B1, +(30) +u(x) ≥ λ+(1 + τǫ)(xn + b+)+ − λ−(xn + b−)− +≥ λ+(1 + τǫ)(xn + b+)+ − λ−(1 − c1ηǫ)(xn + b+)−, +for a suitable universal constant c1. We now take ϕ as in (25) and set, for t ∈ [0, 1], +ft(x) = λ+ +� +1 + τǫ +2 +� +(xn + b+ + c2tǫϕ)+ − λ−(1 − c1ηǫ)(xn + b+ + c2tǫϕ)−, +for a suitably small universal constant 0 < c2 ≪ τ, chosen so that for all x ∈ B 1 +100 (Q) +(1 + τǫ)(xn + b+)+ ≥ +� +1 + τǫ +2 +� +(xn + b+ + c2ǫ)+. +This together with (29) implies that +(31) +u(x) ≥ λ+(1 + τǫ)(xn + b+)+ ≥ λ+ +� +1 + τǫ +2 +� +(xn + b+ + c2ǫ)+ +≥ f1(x) ≥ ft(x), +for all +x ∈ B 1 +100 (Q), t ∈ [0, 1]. +Furthermore u ≥ f0 in B1 thanks to (30). Similar to Case 1, let t be the biggest t such +that ft ≤ u in B1 and x be the first contact point, so that +u(x) − ft(x) ≥ u(x) − ft(x) = 0, +for all x ∈ B1. +Since, by using (ϕ.2), it can be checked that +∆p ft > 0, +on +{ft � 0} \ B 1 +100 (Q), +therefore, as in Case 1, x is a free boundary point. Moreover, since ft changes sign +in a neighborhood of x: +either +x ∈ Γ+ +OP = ∂Ω+ +u \ ∂Ω− +u, +or +x ∈ ΓTP = ∂Ω+ +u ∩ ∂Ω− +u. +In the first case, by definition of viscosity solution and (ϕ.3), +λp ++ ≥ |∇f + +t (x)|p = λp ++ +� +1 + τǫ +2 +�p � +1 + pc2tǫ∂nϕ(x) + O(ǫ2) +� +> λp ++, +a contradiction for ǫ ≪ 1. In the second case, we have a contradiction as well, since +(recall also the assumption λ+ − λ− ≥ 0) +λp ++ − λp +− ≥ |∇f + +t |p − |∇f − +t |p += +� +λp ++ +� +1 + τǫ +2 +�p +− λp +−(1 − c1ηǫ)p +� � +1 + pc2tǫ∂nϕ(x) + O(ǫ2) +� +> λp ++ − λp +−, +provided ǫ ≪ 1 (only depending on n, λ+ and p). Hence, t = 1, u ≥ f1 (so u+ ≥ f + +1 ) +which implies the desired conclusion by setting +a+ = a+, +b+ = b+ + c2ǫ, +for a suitable constant c2 = c2(n, p) and by recalling that ǫ = a+ − b+. + +18 +M. BAYRAMI AND M. FOTOUHI +Case 2.2: Assume instead that: +0 ≤ b− − b+ ≤ ηǫ, +where η = η(n, p) will be determined later. In this case we consider the family of +functions +ft(x) = λ+ +� +1 + τǫ +2 +� +(xn + b+ + ηtǫϕ)+ − λ−(xn + b−)−. +Since ϕ ≤ 1, this function is well defined due to b− ≤ b+ + ηǫ. Moreover, u ≥ f0 and, +thanks to (29) and by assuming η is sufficiently small (this can also be determined +universally depending only on the dimension and p) we will have, +u(x) ≥ f1(x) ≥ ft(x), +for all +x ∈ B 1 +100 (Q), t ∈ [0, 1]. +We consider again the first touching time t and the first touching point x. +By +arguing as in the previous cases, we get x ∈ {u = 0} ∩ {|xn| ≤ +1 +100}. +Also, the +definition of ft yields that x ∈ ∂{ft > 0}. This infer that x ∈ ∂Ω+ +u \ ∂Ω− +u (note that +ϕ(x) < 1). However, again by arguing as in Case 2.1, this is in contradiction with u +being a viscosity solution. We now conclude as in the previous cases. +□ +The following lemma addresses the situation in which the origin is not a branch- +ing point. +Lemma 4.10 (Partial Boundary Harnack II). Given 1 < p < ∞ and 0 < L0, L1 +and assume that 0 < λ− ≤ λ+ ≤ L1, then there exist constants ǫ = ǫ(n, L0, L1, p) > 0, +M = M(n, L0, L1, p) and c = c(n, L0, L1, p) ∈ (0, 1) such that for every function u : B4 → R +satisfying (b) − (c) in Theorem 4.1 the following property holds true. If there are constants +a, b ∈ (− 1 +100, +1 +100) with +0 ≤ a − b ≤ ǫ, +such that for x ∈ B4 +Hα,en(x + ben) ≤ u(x) ≤ Hα,en(x + aen), +and +max(λ+, L0) + Mǫ ≤ α ≤ L1, +then there are constants a, b ∈ (− 1 +100, +1 +100) with +0 ≤ a − b ≤ c(a − b), +such that for x ∈ B 1 +6 +Hα,en(x + ben) ≤ u(x) ≤ Hα,en(x + aen). +Proof. We consider the point P = (0, · · · , 0, 2) and distinguish two cases (note that +one of these inequalities is always satisfied): +either +Hα,en (P + ben) + α(a − b) +2 +≤ u(P), +or +Hα,en (P + aen) − α(a − b) +2 +≥ u(P). +Since the argument in both cases is completely symmetric we only consider the +second case. If we set +ǫ = a − b, + +19 +by Lemma 4.7 and by arguing as in Lemma 4.8 we deduce the existence of a +constant τ = τ(n, p) such that +u(x) ≤ α(1 − τǫ)(xn + a)+ − β(xn + a)−, +in B1. We let ϕ as in (25) and set +ft(x) = α +� +1 − τǫ +2 +� +(xn + a − ctǫϕ)+ − β(xn + a − ctǫϕ)−, +where c = c(n, p) is a constant chosen such that +u(x) ≤ f1(x) ≤ ft(x), +for all +x ∈ B 1 +100 (Q), t ∈ [0, 1], +where, Q = (0, · · · , 0, 1 +5). As in Lemma 4.8, we let t and x be the first contact time +and the first contact point and we aim to show that t = 1. For this purpose, we note +that, by the same arguments as in Lemma 4.8, necessarily x ∈ {u = 0}. We claim +that +x ∈ ΓTP = ∂Ω+ +u ∩ ∂Ω− +u. +Indeed, otherwise x ∈ ∂Ω− +u \ ∂Ω+ +u (the case x ∈ ∂Ω+ +u \ ∂Ω− +u will be impossible since +ft is negative in a neighborhood of x). And by definition of viscosity solution, this +along with (2) would imply +λp +− ≥ |∇f − +t (x)|p = βp(1 − pctǫ∂nϕ(x) + O(ǫ2)) +≥ (λp +− − λp ++ + αp)(1 − pctǫ∂nϕ(x) + O(ǫ2)) +≥ (λp +− − λp ++ + (max(λ+, L0) + Mǫ)p)(1 − pctǫ∂nϕ(x) + O(ǫ2)) += λp +− + p(Lp−1 +0 +M − ct∂nϕ(x))ǫ + O(ǫ2), +where the implicit constants in O(ǫ2) can control by L1, p and n. This inequality is +impossible if M is chosen sufficiently large. +Hence x ∈ ∂Ω+ +u ∩ ∂Ω− +u. This however implies: +λp ++ − λp +− ≤ |∇f + +t (x)|p − |∇f − +t (x)|p += +� +αp +� +1 − τǫ +2 +�p +− βp +� � +1 − pctǫ∂nϕ(x) + O(ǫ2) +� +< αp − βp = λp ++ − λp +−, +provided ǫ and as a consequence of ǫ = a − b ≤ ǫ, ǫ is chosen small enough, where +we have used (ϕ.3) and the equality +0 ≤ λp ++ − λp +− = αp − βp. +This contradiction shows that t = 1 and as in Lemma 4.8, this completes the +proof. +□ +With Lemmas 4.7 and 4.8 at hand the proof of Lemma 4.4 is as follows. +Proof of Lemma 4.4. We distinguish two cases: +Case 0 ≤ ℓ < +∞: By triangular inequality we have +∥uk − Hλ+,en∥L∞(B1) ≤ ǫk +� +1 + 2ℓ max(λ1−p ++ , λ1−p +− ) +� +, + +20 +M. BAYRAMI AND M. FOTOUHI +for k sufficiently large. Define the bounded sequence wk by +wk(x) = + +w+,k(x) := uk(x) − λ+x+ +n +αkǫk +x ∈ Ω+ +uk ∩ B1, +w−,k(x) := uk(x) + λ−x− +n +βkǫk +x ∈ Ω− +uk ∩ B1. +Now we can repeatedly apply Lemma 4.8 to deduce that wk satisfies +(32) +|wk(x) − wk(y)| ≤ C|x − y|γ, +when x, y ∈ B 1 +2 , and |x − y| ≥ ǫk +ǫ , +for some universal exponent 0 < γ < 1 and constant C; see [13, Corollary 4.2]. This +gives that the graphs of +˜Γ± +k := {(x, w±,k(x)) : x ∈ Ω± +uk ∩ B 1 +2 }, +converge, up to a subsequence, in the Hausdorff distance to the closed graphs +˜Γ± := {(x, w±(x)) : x ∈ B± +1 +2 }, +where w ∈ C0,α for some α > 0. Since +hk(x) := Hαk,en − Hλ+,en +ǫk +→ + +λ1−p ++ ℓxn +xn > 0, +λ1−p +− ℓxn +xn < 0, +the original sequence vk satisfies that their graphs, converges to the graph of a +limiting function v as we wanted, this in particular proves (i), (ii), and (iii). +Since 0 ∈ ∂Ω+ +uk ∩ ∂Ω− +uk then 0 is in the domain of v±,k and +v±,k(0) = 0, +which implies that v±(0) = 0. To show that v+(x) ≤ v−(x) for x = (x′, 0) ∈ B 1 +2 , we +simply exploit (iii) at the points x± +k = (x′, t± +k ) where +t+ +k = sup{t : (x′, t) ∈ ∂Ω+ +uk} +and +t− +k = inf{t : (x′, t) ∈ ∂Ω− +uk}, +and by noticing that t− +k ≤ t+ +k . +Finally, to see the last claim, (19), it is enough to note that if xk ∈ ∂Ω+ +uk ∩ ∂Ω− +uk is +converging to x then v+,k(xk) = v−,k(xk) and thus v+(x) = v−(x), yielding x ∈ C. +Case ℓ = ∞: In this case, the conclusion follows exactly with a similar argument +by using repeatedly Lemma 4.10 for function vk to obtain a relation similar to (32) +for functions vk. +□ +4.2. The linearized problem: proof of Lemma 4.5. Lemma 4.5 proves through the +following technical lemma, whose proof is easily obtained by adapting the one in +[12, Lemma 3.10] exactly. Then we present the statement without proof. +Lemma 4.11. Let uk, ǫk and αk be as in the statement of Lemma 4.4, vk be defined by (18) +and v± be as in Lemma 4.4. Then: +(1) Let P+ be a quadratic polynomial with Lp(P+) > 0 (or Lp(P+) < 0) on B+ +1 +2 touching +v+ strictly from below (above) at a point x0 ∈ {xn = 0} ∩ B 1 +2 . Then, there exists + +21 +a sequence of points ∂Ω+ +uk ∋ xk → x0 and a sequence of comparison functions Qk +such that Qk touches from below (above) u+ +k at xk, and such that +(33) +∇Q+ +k (xk) = αken + ǫkαk∇P+(x0) + o(ǫk). +(2) Let P− be a quadratic polynomial with Lp(P−) > 0 (Lp(P−) < 0) on B− +1 +2 touching +v− strictly from below (above) at a point x0 ∈ {xn = 0} ∩ B 1 +2 . Then, there exists +a sequence of points ∂Ω− +uk ∋ xk → x0 and a sequence of comparison functions Qk +such that Qk touches from below (above) −u− +k at xk, and such that +(34) +∇Q− +k (xk) = −βken + ǫkβk∇P−(x0) + o(ǫk). +(3) Let s, t ∈ R and ˜P be a quadratic polynomial on B 1 +2 such that ∂n ˜P = 0. Suppose +that Lp( ˜P) ≥ 0 (Lp( ˜P) ≤ 0) and that the function +P := sx+ +n − tx− +n + ˜P, +touches v strictly from below (above) at a point x0 ∈ C. Then, there exists a +sequence of points xk → x0 and a sequence of comparison functions Qk such that +Qk touches from below (above) the function uk at xk ∈ ∂Ωuk, and such that +(35) +∇Q+ +k (xk) = αk(1 + ǫks)en + o(ǫk), +∇Q− +k (xk) = −βk(1 + ǫkt)en + o(ǫk). +In particular, if s > 0 and Qk touches uk from below then xk � ∂Ω− +uk \ ∂Ω+ +uk, while +if t < 0 and Qk touches uk from above then xk � ∂Ω+ +uk \ ∂Ω− +uk. +Proof of Lemma 4.5. Step 1: In this step, we prove Lp(v±) = 0 in B± +1 +2 . +Let P(x) be a quadratic polynomial touching v = v+ at x ∈ B+ +1 +2 strictly from below. +We need to show that at this point +Lp(P) = ∆P + (p − 2)∂nnP ≤ 0. +Since v+,k → v+, there exist points xk ∈ Ω+ +uk ∩ B 1 +2 , xk → x and constants ck → 0 such +that +(36) +v+,k(xk) = P(xk) + ck, +and +(37) +v+,k ≥ P + ck, +in a neighborhood of xk. +From the definition of v+,k, (36) and (37) read +uk(xk) = Qk(xk), +and +uk(x) ≥ Qk(x), +in a neighborhood of xk, +where +Qk(x) = ǫkαk(P(x) + ck) + αkx+ +n. +Note that +(38) +∇Qk = ǫkαk∇P + αken, +thus, +(39) +∇Qk(xk) � 0, +for k large. + +22 +M. BAYRAMI AND M. FOTOUHI +Since uk is p-harmonic and Qk touches uk from below at xk, and ∇Qk(xk) � 0, by +the equivalence of weak and viscosity solutions of p-harmonic functions, we get +0 ≥ ∆pQk(xk) += div +� +|∇Qk(xk)|p−2∇Qk(xk) +� += |∇Qk(xk)|p−2 ∆Qk(xk) + (p − 2) |∇Qk(xk)|p−4 +n +� +i,j=1 +Qkxi(xk)Qkxj(xk)Qkxixj(xk) += ǫk |∇Qk(xk)|p−2 ∆P(xk) + ǫk(p − 2) |∇Qk(xk)|p−4 +n +� +i,j=1 +Qkxi(xk)Qkxj(xk)Pxixj(xk). +Now, dividing both sides by ǫk, and passing to the limit k → ∞, and recalling that +∇Qk(xk) → αken, +we conclude that +∆P(x) + (p − 2)∂nnP(x) ≤ 0. +Touching from above and reaching the opposite inequality is similar. Also, the +reasoning of the case v = v− in the negative half ball B− +1 +2 can be done similarly. +Step 2: In this step, we show that J = ∅, when ℓ = ∞. +Assume the contrary, since the set {v− > v+} is open in {xn = 0}, it contains a +(n − 1)-dimensional ball +B′ +ǫ(y′) := Bǫ((y′, 0)) ∩ {xn = 0} ⊂ J. +Next, let P be the polynomial +P(x) = A +� +n − 1 +2 +� +x2 +n − |x′ − y′|2 − Bxn, +where +x = (x′, xn), +for some constants A, B. We first choose suitable A = A(p) so that Lp(P) > 0. Notice +that +P < v+ +on +{|x′ − y′| = ǫ} ∩ {xn = 0}. +Moreover, we choose B ≫ A so that +P < v+ +on +Bǫ((y′, 0)). +Now we can translate P first down and then up to find that there exists C such +that P + C is touching v+ from below at a point x0 ∈ Bǫ((y′, 0)) ∩ {xn ≥ 0}. Since +Lp(P) > 0, the touching point can not be in the interior of the (half) ball, and thus +x0 ∈ B′ +ǫ(y′) ⊂ J. +By using Lemma 4.11, there exists a sequence of points ∂Ω+ +uk ∋ xk → x0 and of +functions Qk touching u+ +k from below at xk such that +∇Q+ +k (xk) = αken + ǫkαk∇P(x0) + o(ǫk). +Since x0 ∈ J, by (19) in Lemma 4.4, xk ∈ ∂Ω+ +uk \ ∂Ω− +uk. Hence, by (ii) in Lemma 3.2 +λp ++ ≥ |∇Q+ +k (xk)|p ≥ αp +k + pαp +kǫk∂nP(x0) + o(ǫk). +Now recalling (17), the definition of ℓ, +−B = ∂nP(x0) ≤ +λp ++ − αp +k +pαp +kǫk ++ o(1) → −∞. +This contradiction proves that J = ∅. + +23 +Step 3: In this step, we check the transmission condition in (21) when ℓ = ∞. +Let us show that +αp +∞∂nv+ − βp +∞∂nv− ≤ 0, +the opposite inequality can then be proved in a similar way. Suppose that there +exist s and t with αp +∞s > βp +∞t and a polynomial ˜P with Lp( ˜P) > 0 and ∂n ˜P = 0 such +that +P = sx+ +n − tx− +n + ˜P, +touches v strictly from below at a point x0 ∈ {xn = 0}∩B 1 +2 (note that {xn = 0}∩B 1 +2 = C +due to the previous step and Lemma 4.4). By Lemma 4.11 there exists a sequence of +points ∂Ω+ +uk ∪ ∂Ω− +uk ∋ xk → x0 and a sequence of comparison functions Qk touching +uk from below at xk and satisfying (35). In particular, xk � ∂Ω− +uk \ ∂Ω+ +uk. We claim +that xk ∈ ∂Ω+ +uk ∩ ∂Ω− +uk. Indeed, otherwise by (A.1) in Lemma 3.1, +λp ++ ≥ |∇Q+ +k (xk)|p, +and, by arguing as Step 2, this contradicts ℓ = +∞. Hence, by (A.3) in Lemma 3.1 +λp ++ − λp +− ≥ |∇Q+ +k (xk)|p − |∇Q− +k (xk)|p += αp +k − βp +k + pǫk(αp +ks − βp +kt) + o(ǫk) += λp ++ − λp +− + pǫk(αp +ks − βp +kt) + o(ǫk). +Dividing by ǫk and letting k → ∞, we obtain the desired contradiction. +Step 4: Here, we show that λp +±∂nv± ≥ −ℓ on B 1 +2 ∩ {xn = 0}, when 0 ≤ ℓ < ∞. +We focus on v− since the argument is symmetric. Let us assume that there exists +t ∈ R with λp +−t < −ℓ and a polynomial ˜P with Lp( ˜P) > 0 and ∂n ˜P = 0 such that +function +P = txn + ˜P = tx+ +n − tx− +n + ˜P, +touches v− strictly from below at a point x0 ∈ {xn = 0} ∩ B 1 +2 . Let now xk and Qk be +as in Lemma 4.11-(2). By optimality conditions +λp +− ≤ |∇Q− +k (xk)|p = βp +k + pǫkβp +kt + o(ǫk). +Since ℓ < ∞, we have βk = λ− + O(ǫk) and so the above inequality leads to +− ℓ +λp +− += lim +k→∞ +λp +− − βp +k +pǫkβp +k +≤ t < − ℓ +λp +− +, +which is a contradiction. +Step 5: We now show that λp +±∂nv± = −ℓ on J, when 0 ≤ ℓ < ∞. +By the previous step, it is enough to show that if there exists a polynomial ˜P with +Lp( ˜P) < 0 and ∂n ˜P = 0 such that +P = txn + ˜P = tx+ +n − tx− +n + ˜P, +touches v− strictly from above at a point x0 ∈ J, then λp +−t ≤ −ℓ. Again, by Lemma +4.11, we find points xk → x0 and functions Qk satisfying (34) and touching −u− +k +from below at xk. Since x0 ∈ J, by (19) in Lemma 4.4, xk ∈ ∂Ω− +uk \ ∂Ω+ +uk. Hence, by +Lemma 3.1, +λp +− ≥ |∇Q− +k (xk)|p = βp +k + pβp +kǫkt + o(ǫk), +which by arguing as above implies that λp +−t ≤ −ℓ. + +24 +M. BAYRAMI AND M. FOTOUHI +Step 6: In the last step, we show the transmission condition in (22) at points in +C. +Again by the symmetry of the arguments, we will only show that +λp ++∂nv+ − λp +−∂nv− ≤ 0 +on +C. +Let us hence assume that there exist s and t with λp ++s > λp +−t and a polynomial ˜P +with Lp( ˜P) > 0 and ∂n ˜P = 0 such that +P = sx+ +n − tx− +n + ˜P, +touches v+ and v− strictly from below at x0 ∈ C. By Lemma 4.11, we find points +xk → x0 and functions Qk satisfying (35). In particular xk � ∂Ω− +uk \ ∂Ω+ +uk. By the +previous step we know that λp +−t ≥ −ℓ and thus λp ++s > −ℓ, since we are assuming +λp ++s > λp +−t ≥ 0. We now distinguish two cases: +1) xk is one-phase point, namely xk ∈ ∂Ω+ +uk \ ∂Ω− +uk. In this case +λp ++ ≥ |∇Q+ +k (xk)|p = αp +k + pαp +kǫks + o(ǫk), +which implies that +λp ++s + ℓ = λp ++ lim +k→∞ + +s + +αp +k − λp ++ +pαp +kǫk + + ≤ 0, +in contradiction with λp ++s > −ℓ. +2) xk is two-phase point, namely xk ∈ ∂Ω+ +uk ∩ ∂Ω− +uk. Arguing as in Case 1), we +have that, by Lemma 3.1, +λp ++ − λp +− ≥ |∇Q+ +k (xk)|p − |∇Q− +k (xk)|p += αp +k − βp +k + pǫk(αp +ks − βp +kt) + o(ǫk) += λp ++ − λp +− + pǫk(λp ++s − λp +−t) + o(ǫk), +which gives a contradiction with λp ++s > λp +−t, as ǫk → 0. +□ +4.3. Proof of Lemmas 4.2 and 4.3. We recall the following regularity results for +the limiting problems. +Lemma 4.12 (Regularity for the transmission problem). There exists a universal +constant C = C(α∞, β∞, n, p) > 0 such that if v ∈ C0(B 1 +2 ) is a viscosity solution of (21) +with ∥v∥L∞(B 1 +2 ) ≤ 1 then there exist v ∈ Rn−1, s, t ∈ R with αp +∞s = βp +∞t such that +sup +x∈Br +���v(x) − v(0) − (v · x′ + sx+ +n − tx− +n) +��� ≤ Cr2, +for every r ≤ 1 +4. +Proof. For the proof when p = 2, we refer to [13, Theorem 3.2]. This result can be +extended easily to the general case (for any p) by changing the coordinate such that +the operator Lp = ∆ + (p − 2)∂nn transfer to the Laplacian. +□ +Lemma 4.13 (Regularity for the two-membrane problem). There exists a universal +constant C = C(λ±, n, p) > 0 such that if v is a viscosity solution of (22) with ∥v∥L∞(B 1 +2 ) ≤ 1 +then there exist v ∈ Rn−1, s, t ∈ R with λp ++s = λp +−t ≥ −ℓ such that +sup +x∈B± +r +���v(x) − v(0) − (v · x′ + sx+ +n − tx− +n) +��� ≤ C(1 + ℓ)r +3 +2 , +for every r ≤ rp, + +25 +where rp = 1 +4 for 1 < p ≤ 2 and rp = +1 +4√ +p−1 for 2 < p. +The proof of this lemma can be found in [12, Lemma 3.12] with a minor changes. +To keep the paper self-contained we will provide a complete proof for our case in +Appendix A. +Now, the proof of Lemmas 4.2 and 4.3 by the regularity theory for the limiting +problems and a classical compactness argument is available: +Proof of Lemma 4.2. Toward a contradiction assume that for fixed γ ∈ (0, 1 +2) and M, +we have a sequences of functions uk and numbers αk such that +ǫk = ∥uk − Hαk,en∥L∞(B1) → 0, +and +λ+ ≤ αk ≤ λ+ + Mǫk, +and fail (14) and (15) for some ρ and C which will be determined later. Note that +by the second assumption above +ℓ < Mλp−1 ++ +< ∞. +We let (vk)k be the sequence of functions defined in (18) and assume that they +converge to a function v as in Lemma 4.4, note that ∥v∥L∞(B 1 +2 ) ≤ max( 1 +λ+ , 1 +λ− ). By +Lemma 4.5, v solves (22) and thus by Lemma 4.13 there exist v ∈ Rn−1, s, t ∈ R +satisfying λp ++s = λp +−t ≥ −ℓ such that for all r ∈ (0, rp) +sup +x∈Br +���v(x) − (v · x′ + sx+ +n − tx− +n) +��� ≤ C(1 + M)r +3 +2 . +Hence, we can fix ρ = ρ(λ±, γ, L, M, p, n) < rp such that C(1 + M)ρ +1 +2 −γ ≤ 1 +2, so +(40) +sup +x∈Bρ +���v(x) − (v · x′ + sx+ +n − tx− +n) +��� ≤ ρ1+γ +2L . +We now set +˜αk := αk(1 + ǫks) + δkǫk +and +ek := +en + ǫkv +� +1 + ǫ2 +k|v|2 +, +where δk → 0 is chosen so that ˜αk ≥ λ+; note that the existence of such sequence is +due to the condition λp ++s ≥ −ℓ since +αk(1 + ǫks) = + +λ+ + +ℓ +λp−1 ++ +ǫk + o(ǫk) + + (1 + ǫks) ≥ λ+ + o(ǫk). +We let Hk := H ˜αk,ek and note that +| ˜αk − αk| + |ek − en| ≤ Cǫk, +for a universal constant C > 0; we also have used (40) to find out that s is universally +bounded. By the contradiction assumption we have +ρ1+γ < 1 +ǫk +sup +Bρ +|uk(x) − Hk(x)| +≤ max +� +αk∥v+ +k − Hk − Hαk,en +ǫkαk +∥L∞(Ω+uk ∩Bρ), βk∥v− +k − Hk − Hαk,en +ǫkβk +∥L∞(Ω− +uk ∩Bρ) +� +. + +26 +M. BAYRAMI AND M. FOTOUHI +To close the argument, we need to recall (40), the convergence of vk to v in the sense +of Lemma 4.4 and the convergence of (again in the sense of Lemma 4.4) + +Hk(x) − Hαk,en(x) +αkǫk +xn > 0, +Hk(x) − Hαk,en(x) +βkǫk +xn < 0, +to the function +(v · x′) + sx+ +n − tx− +n. +□ +Proof of Lemma 4.3. Arguing by contradiction one assume for fixed γ ∈ (0, 1) the +existence of a sequence of functions uk and numbers αk, Mk → ∞ such that +ǫk = ∥uk − Hαk,en∥L∞(B1) → 0, +and +αk − λ+ +ǫk +≥ Mk → ∞, +and fail (14) and (15) for some ρ and C which will be determined later. This implies +that ℓ = ∞ and that the limiting function v obtained in Lemma 4.4 is a solution +of (21). One then concludes the proof similar to the proof of Lemma 4.2 by using +Lemma 4.12. +□ +5. Regularity of the free boundary +The last step in achieving the desired regularity result is to demonstrate that +|∇u±| are C0,η for a suitable η > 0 up to the boundary, in the viscosity sense. +Indeed, this shows that u± are solutions to the classical one-phase free boundary +problem in its viscosity formulation and that the regularity will follow form [17]. +The arguments are similar to the ones in [25, Section 8] (see also [12, Section 4]). +Therefore we only sketch the main steps and refer the reader to that paper for more +details. +Before stating the main results, we introduce some notation. For every x0 ∈ F(u) +and every 0 < r < dist(x0, ∂D), we consider the function +ux0,r(x) := u(x0 + rx) +r +, +which is well-defined for |x| < +1 +r dist(x0, ∂D) and vanishes at the origin. When +x0 = 0, we denote u0,r by ur. Given a sequence rk > 0 such that rk → 0, we say that +the sequence of functions ux0,rk is a blow-up sequence of u at x0. If a subsequence of +ux0,rk convergs to v on every ball BR ⊂ Rn, we say that v is a blow-up limit of u at x0. +Lemma 5.1. There exists ¯ǫ > 0 such that if the minimizer u satisfies (3), then at every +point x0 ∈ ΓTP ∩ Br0 for a universal radius r0 > 0, there is a unique blow-up. Moreover, u +is Lipschitz in Br0/2 and there exists η > 0 and a constant C0(n, p, Λ0, Λ1) > 0 such that +for every x0, y0 ∈ ΓTP ∩ Br0/2, we have +(41) +|α(x0) − α(y0)| ≤ C0|x0 − y0|η +and +|e(x0) − e(y0)| ≤ C0|x0 − y0|η, +for any η ∈ (0, 1 +3), where Hα(x0),e(x0) and Hα(y0),e(y0) are the blow-ups at x0 and y0, respec- +tively. + +27 +Proof. Let L0 = Λ0 and L1 = 2Λ1 in Theorem 4.1 and find the universal constants +ǫ0 > 0, ρ0 > 0 and C > 0. Choose ¯ǫ < min{(1 − ργ) Λ1 +2C, ǫ +2} and r0 < +¯ǫ +Λ1 , then if the +minimizer u satisfies (3) for some e ∈ Sn−1 and λ+ ≤ α ≤ Λ1, then +∥ux0, 1 +2 − Hα,e∥L∞(B1) ≤ ¯ǫ + |Hα,e(x0)| ≤ 2¯ǫ, +for any x0 ∈ ΓTP ∩ Br0. Now we can thus repeatedly apply Theorem 4.1 to obtain +the sequences ux0, ρk +2 (x) = 2 +ρk u(x0 + ρk +2 x), max(L0, λ+) ≤ αk ≤ L1 and ek ∈ Sn−1 that +∥ux0, ρk +2 − Hαk,ek∥L∞(B1) ≤ 2¯ǫρkγ, +|ek+1 − ek| + |αk+1 − αk| ≤ 2C¯ǫρkγ. +This implies that αk and ek converge to some α = α(x0) and e = e(x0), respectively. +Now let r ≤ 1 +2 be arbitrary and choose k ∈ N such that ρk+1 ≤ 2r ≤ ρk, then +∥ux0,r − Hα(x0),e(x0)∥L∞(B1) ≤ 1 +ρ∥ux0, ρk +2 − Hα(x0),e(x0)∥L∞(B1) +≤ 1 +ρ +� +∥ux0, ρk +2 − Hαk,ek∥L∞(B1) + ∥Hαk,ek − Hα(x0),e(x0)∥L∞(B1) +� +≤ C¯ǫρkγ. +Therefore, there is ˜C = ˜C(n, p, Λ0, Λ1) such that for every r ≤ 1 +2 and x0 ∈ Br0, +(42) +∥ux0,r − Hα(x0),e(x0)∥L∞(B1) ≤ ˜Crγ, +where γ ∈ (0, 1 +2). +According to (42), +∥u∥L∞(Br(x0)) ≤ (L1 + ˜C)r, +r ≤ 1 +2, +for every x0 ∈ ΓTP ∩ Br0. From this and the Lipschitz regularity around the one- +phase points, Proposition 2.3, we conclude that u is Lipschitz in B r0 +2 ; see [3, Theorem +2.3] or [11, Theorem 2.1]. +Next, for x0, y0 ∈ ΓTP ∩ B r0 +2 set r := |x0 − y0|1−η and η := +γ +1+γ, and recall that u is +Lipschitz (with a constant ˜L) to get +∥Hα(x0),e(x0) − Hα(y0),e(y0)∥L∞(B1) +≤ ∥ux0,r − Hα(x0),e(x0)∥L∞(B1) + ∥ux0,r − uy0,r∥L∞(B1) + ∥uy0,r − Hα(y0),e(y0)∥L∞(B1) +≤ +� +C0rγ + +˜L +r |x0 − y0| + C0rγ +� += (˜L + 2C0)|x0 − y0|η. +The conclusion now follows easily from this inequality; see e.g. [25, Lemma 8.8] +for the details. +□ +Lemma 5.2. Under the same assumptions of Lemma 5.1, there are C0,η continuous func- +tions α : ∂Ω+ +u → R and β : ∂Ω− +u → R such that α ≥ λ+, β ≥ λ− and u± are viscosity +solutions of the one-phase problems +∆pu+ = 0 +in +Ω+ +u, +|∇u+| = α +on +∂Ω+ +u, +and +∆pu− = 0 +in +Ω− +u, +|∇u−| = β +on +∂Ω− +u. + +28 +M. BAYRAMI AND M. FOTOUHI +Proof. We will sketch the argument for u+. The proof of the case u− is similar. +Clearly ∆pu+ = 0 in Ω+ +u. By (42) we have that, if x0 ∈ ΓTP ∩ D′, then +(43) +���u+(x) − α(x0) ((x − x0) · e(x0))+��� ≤ C0|x − x0|1+γ, +for every x ∈ Br0(x0) ∩ Ω+ +u where r0 and C0 depends only on D′. In particular, u+ is +differentiable on Ω+ +u up to x0 (in the classical sense) and |∇u+(x0)| = α(x0). On the +other hand if x0 ∈ Γ+ +OP, then |∇u+(x0)| = λ+ is constant, in the viscosity sense. +To close the argument, we only need to prove that α ∈ C0,η(∂Ω+ +u). Since α is +η-H¨older continuous on ΓTP by Lemma 5.1, and constant on Γ+ +OP (in the viscosity +sense), we just need to show that if x0 ∈ ΓTP is such that there is a sequence xk ∈ Γ+ +OP +converging to x0, then α(x0) = λ+. To this end, let yk ∈ ΓTP be such that +dist(xk, ΓTP) = |xk − yk|, +and denote +rk = |xk − yk| +and +uk(x) = 1 +rk +u+(xk + rkx), +and note that uk is a viscosity solution of the free boundary problem +∆puk = 0 +in +Ω+ +uk ∩ B1, +|∇uk| = λ+ +on +∂{uk > 0} ∩ B1. +Since uk are uniformly Lipschitz in B 1 +2 (Proposition 2.3) they converge to a function +u∞ which is also a viscosity solution of the same problem (see e.g. [17]). On the +other hand, by (43) for two-phase point yk ∈ ΓTP and letting zk := xk−yk +rk , we have +that +���uk(x) − α(yk) �(x − zk) · e(yk)�+��� ≤ C0rγ +k|x − zk|1+γ. +Suppose zk → z0 and passing to the limit +u∞(x) = α(x0) ((x − z0) · e(x0))+ , +in B 1 +2 , +which gives that α(x0) = |∇u∞(0)| = λ+. +□ +Proof of Theorem 1.1. Let x0 ∈ ΓTP = ∂Ω+ +u ∩ ∂Ω− +u and let ǫ be the constant satisfies +in [17, Theorem 1.1] and Lemma 5.1. By virtue of Lemma 5.2, we can apply [17, +Theorem 1.1] to conclude that locally at x0 ∈ ΓTP the free boundaries ∂Ω± +u are C1,η +graphs. Since x0 is arbitrary, we conclude the proof. +□ +6. Lipschitz regularity of solutions +In this section, we are going to prove Theorem 1.2. We will follow the idea in +[14]. +Proposition 6.1. Let u : D → R be a minimizer of JTP that 0 ∈ F(u) ∩ B1 ⊂ D. Then +there exists constants L and δ such that one of the following alternative holds: +(1) u is Lipschitz in Bδ and +|∇u| ≤ C max(∥u∥L∞(B1), L), +in Bδ, +for some universal constant C. +(2) +1 +δ∥u∥L∞(Bδ) ≤ 1 +2 max(∥u∥L∞(B1), L). + +29 +Proof. Let δ be fixed, to be specified later. Assume by contradiction that there exist +a sequence of Lj → ∞ and a sequence of solutions uj such that does not satisfy +either (1) nor (2). Let Cj := max(∥uj∥L∞(B1), Lj) and define +˜uj := +uj +Cj +, +which satisfy +∥ ˜uj∥L∞(B1) ≤ 1, +and +∥ ˜uj∥L∞(Bδ) ≥ δ +2. +By using Proposition 2.5, we get that ˜uj is a minimizer of the scaled functional (4) +for σj = +1 +Cj → 0. Thus up to a subsequence, ˜uj converges uniformly to a p-harmonic +function u0. Hence by C1,α regularity for p-harmonic functions we get that +(44) +sup +Br +|u0(x) − ∇u0(0) · x| ≤ ˜Cr1+α, +for all r ≤ 1, +where the constant C is universal and also |∇u0(0)| ≤ ˜C. Now we distinguish two +cases: +Case I: |∇u0(0)| ≤ 1 +4. +In this case, from (44) we deduce that +1 +δ∥u0∥L∞(Bδ) ≤ 1 +4 + ˜Cδα ≤ 1 +3, +if we choose δ small enough. Thus all uj for sufficiently large j will satisfy (2), +which is a contradiction. +Case II: |∇u0(0)| ≥ 1 +4. +In this case we will use our flatness result in Theorem 1.1. Put ˜r = 2δ +r0 in (44) where +r0 is the radius obtained in Theorem 1.1 (we have also assumed 2δ ≤ r0) +sup +B1 +��� ˜uj,˜r(x) − ∇u0(0) · x +��� ≤ +˜C +rα +0 +δα. +Now let e = ∇u0(0) +|∇u0(0)|, α = |∇u(0)| and βj = +1 +Cj (λp +− − λp ++ + αpCp +j) +1 +p , then +∥uj,˜r − Hα,e∥L∞(B1) ≤ ∥uj,˜r − ∇u0(0) · x∥L∞(B1) + |α − βj| ≤ 2 ˜C +rα +0 +δα, +for sufficiently large j. +Applying Theorem 1.1 for some Λ0 ≤ +1 +4 and Λ1 ≥ ˜C +and notice that uj,˜r is a minimizer of JTP for coefficients 1 +Cj λ±. Note that the critical +flatness in Theorem 1.1 or Lemma 5.1 depends on Λ0 and Λ1 rather than coefficients +λ±. Then we can find δ universally small such that uj,˜r satisfy in Lemma 5.1. In +particular, uj,˜r is Lipschitz in B r0 +2 with a universal constant. It proves that uj is +Lipschitz in Bδ. +□ +Proof of Theorem 1.2. Let δ, C and L be the universal constants in Proposition 6.1. +Assume 0 ∈ F(u) and let ˜L := max(∥u∥L∞(B1), L). We first show +(45) +∥u∥L∞(Bδk) ≤ C˜Lδk, +∀k ≥ 0. +By Proposition 6.1 either (1) or (2) holds. In the first case, u is Lipschitz in Bδ and +|∇u| ≤ C˜L +in Bδ. +Thus (45) holds for all k ≥ 1. + +30 +M. BAYRAMI AND M. FOTOUHI +If (2) holds, then +∥u∥L∞(Bδ) ≤ +˜L +2 δ. +We now rescale and iterate. Define +uk(x) := u(δkx) +δk +, +which is also a minimizer of JTP and we can apply Proposition 6.1. If k0 is the +smallest k for which uk satisfies (1), then for 0 ≤ k < k0 the item (2) holds and so +∥u∥L∞(Bδk) ≤ ˜Lδk, +for 0 ≤ k ≤ k0. +Moreover, uk0 is Lipschitz in Bδ, with +|∇uk0| ≤ C max(∥uk0∥L∞(B1), L) ≤ C max(˜L, L) = C˜L, +in Bδ. +Hence, (45) holds for all k ≥ k0. If uk satisfy the alternative (2) for all k, the estimate +(45) will be obtained easily. +Now for an arbitrary r choose k such that δk+1 ≤ r ≤ δk, then by (45) we get +∥u∥L∞(Br) ≤ ∥u∥L∞(Bδk) ≤ C˜Lδk ≤ C˜L +δ r. +This is enough to obtain the Lipschitz continuity locally in D. +□ +Appendix A. Proof of Proposition 2.5 +Proof of Proposition 2.5. By definition of vj and an easy computation, we get +∇vj(x) = +rj +Sj +∇uj(xj + rjx) = σj∇uj(xj + rjx). +In order to show that vj is a minimizer of ˆJTP in BR, consider w that +ˆJTP(w; BR) < ˆJTP(vj; BR), +and +w = v +on ∂BR. +Then ˆw(x) = w( +x−xj +rj ) will satisfy ˆw = uj on ∂Brj(xj) and by a simple calculation, we +get that +JTP( ˆw; Brj) = +rn +j +σp +j +ˆJTP(w; BR) < +rn +j +σp +j +ˆJTP(vj; BR) = ˆJTP(uj; Brj). +This is a contradiction with the minimality of uj. +Moreover, using |vj| ≤ M in B 4R +3 and Caccioppoli’s inequality, we conclude that +� +BR +|∇v± +j |p dx ≤ 4pC(n) +� +B 4R +3 +(v± +j )p dx ≤ (4M)pC(n), +for some C(n) > 0, indicating that ∥vj∥W1,p(BR) are uniformly bounded. Hence, from +Proposition 2.4, i.e. the BMO estimate for the gradient, we obtain that for any q > 1 +and 0 < R < 1 +rj there exists a constant C = C(R, q) > 0 independent of j such that +max +� +∥vj∥Cα(BR), ∥∇vj∥Lq(BR) +� +≤ C, +for some α ∈ (0, 1) (if q > n, one can take α = 1 − n +q by the Morrey’s inequality). +Therefore, by a standard compactness argument, we have that, up to a subse- +quence, vj converges to some function v0 as j → +∞ in Cα(BR) and weakly in +W1,q(BR) for any q > 1, and for any fixed R. This completes the proof of (i). + +31 +For obtaining (ii), firstly, we prove that ∆pv0 = 0 in the positivity set of v0. Let +E ⋐ {v0 > 0}. Then, there exists c > 0 such that v0 ≥ 2c in E. By the uniform +convergence of vj to v0, we will have vj > c in E for large j. This implies that v0 is +p-harmonic in E. Since E was arbitrary, we are done. Now, take 0 ≤ ϕ ∈ C1 +c(Rn) +and s > 0. By using (v0 −s)+ϕ as a test function in the weak formulation of ∆pv0 = 0 +in the set {v0 > 0}, we have +� +{v0>s} +|∇v0|pϕ dx = − +� +{v0>s} +|∇v0|p−2 �∇v0 · ∇ϕ� v0 dx + s +� +{v0>s} +|∇v0|p−2∇v0 · ∇ϕ dx. +Letting s → 0 gives that +� +{v0>0} +|∇v0|pϕ dx = − +� +{v0>0} +|∇v0|p−2 �∇v0 · ∇ϕ� v0 dx. +Similar argument holds for using test function (v0 + s)−ϕ and finally we get +(46) +� +Rn |∇v0|pϕ dx = − +� +Rn |∇v0|p−2 �∇v0 · ∇ϕ� v0 dx. +On the other hand since vjϕ∆pvj ≥ 0 (Theorem 2.1), we have +(47) +� +Rn |∇vj|pϕ dx ≤ − +� +Rn |∇vj|p−2 � +∇vj · ∇ϕ +� +vj dx. +Usingtheuniformconvergenceofvj tov0 andtheweakconvergenceof|∇vj|p−2∇vj ⇀ +|∇v0|p−2∇v0 in L +p +p−1 +loc (Rn) (see [15]), we infer from (46) and (47) that +(48) +lim sup +j→+∞ +� +Rn |∇vj|pϕ dx ≤ +� +Rn |∇v0|pϕ dx. +Since also ∇vj ⇀ ∇v0 weakly in Lp +loc(Rn), we have +(49) +� +Rn |∇v0|pϕ dx ≤ lim inf +j→+∞ +� +Rn |∇vj|pϕ dx. +It follows from (48), (49), and a simple compactness argument that +(50) +|∇vj|p → |∇v0|p, +strongly in +L1 +loc(Rn), +and we get (ii). +Finally, we prove the claim (iii). For this, notice that for any ψ ∈ C∞ +c (BR) +(51) +� +BR +|∇vj|p + σp +j(p − 1)λp ++χ{vj>0} + σp +j(p − 1)λp +−χ{vj<0} dx +≤ +� +BR +|∇(vj + ψ)|p + σp +j(p − 1)λp ++χ{vj+ψ>0} + σp +j(p − 1)λp +−χ{vj+ψ<0} dx, +because vj is a minimizer for ˆJTP defined in (4). Recall the strong convergence (50) +along with the following standard inequality +|∇(vj + ψ)|p ≤ 2p−1 � +|∇vj|p + |∇ψ|p� +, +we get +� +BR +|∇(vj + ψ)|p dx → +� +BR +|∇(v0 + ψ)|p dx, + +32 +M. BAYRAMI AND M. FOTOUHI +as j → +∞. Thus passing (51) to limit, we have +� +BR +|∇v0|p + σp(p − 1)λp ++χ{v0>0} + σp(p − 1)λp +−χ{v0<0} dx +≤ +� +BR +|∇(v0 + ψ)|p + σp(p − 1)λp ++χ{v0+ψ>0} + σp(p − 1)λp +−χ{v0+ψ<0} dx, +for any ψ ∈ C∞ +c (BR). This implies (iii) and the proof of proposition finishes. +□ +Appendix B. Proof of the regularity for two-membrane problem +Proof of Lemma 4.13. For the given solution v, define w +w±(x) = v±(x′, (p − 1) +1 +2 xn) + ℓ(p − 1) +1 +2 +λp +± +xn, +x ∈ B± +2rp. +It is straightforward to check that w± is a viscosity solution of + +Lp(w±) = 0, +in +B± +2rp, +∂nw± ≥ 0, +in +B2rp ∩ {xn = 0}, +∂nw± = 0, +in +J = {w+ < w−} ∩ {xn = 0}, +λp ++∂nw+ = λp +−∂nw−, +in +C = {w+ = w−} ∩ {xn = 0}, +w+ ≤ w−, +in +B2rp ∩ {xn = 0}. +Furthermore one can easily check that +(52) +w±(x′, xn) = ˜w(x′, ∓xn) ∓ 1 +λp +± +wS(x′, ∓xn), +where ˜w solves the following Neumann problem + +∆ ˜w = 0, +on +B− +2rp, +∂n ˜w = 0, +on +B− +2rp ∩ {xn = 0}, +and wS is a solution to the thin obstacle (the Signorini) problem + +∆wS = 0, +on +B− +2rp, +wS ≥ 0, +on +B− +2rp ∩ {xn = 0}, +∂nwS ≥ 0, +on +B− +2rp ∩ {xn = 0}, +wS∂nwS = 0, +on +B− +2rp ∩ {xn = 0}. +The boundary data of ˜w and wS on ∂B2rp ∩ {xn < 0} will be obtained uniquely from +(52). Clearly ˜w ∈ C∞(B− +rp) with +∥ ˜w∥Ck(B− +rp) ≤ Ck∥ ˜w∥L∞(B− +2rp). +On the other hand, by [8], wS ∈ C1, 1 +2 (B− +rp) with +∥wS∥C1, 1 +2 (B− +rp) ≤ C∥wS∥L∞(B− +2rp). + +33 +From the last two estimates and the definition of w, it is easy to deduce the con- +clusion of the lemma for (note that the positivity of wS along with its regularity +necessitates that ∇′wS(0) = 0) +v := ∇′ ˜w(0) +and +s± := (p − 1)− 1 +2 +λp +± +∂nwS(0) − ℓ +λp +± +. +□ +Appendix C. Proof of non-degeneracy +Proof of Proposition 2.2. We will prove that for any k ∈ (0, 1), there exists a constant +ck > 0 such that for any local minimizer of JTP and for any small ball Br(x0) ⊂ D +if +1 +r +� +⧸ +� +Br(x0) +(u±)p dx +� 1 +p +< ck +then u± ≡ 0 in Bkr(x0). +By symmetry of the problem, we prove only the case u+. +Also, by the scale +invariance, we can take r = 1 and x0 = 0 for simplicity. Now, let define +ε := +1√ +k +sup +B √ +k +u+. +Since u+ is p-subharmonic, then by [22, Theorem 3.9] +ε ≤ +1√ +k +C(n, p) +(1 − +√ +k) +n +p +� +⧸ +� +B1 +(u+)p dx +� 1 +p +. +Also, let +v(x) := + +C1ε +� +e−µ|x|2 − e−µk2� +in +B √ +k \ Bk, +0 +in +Bk, +where µ > 0 and C1 are such that +(53) +v��� +∂B √ +k +:= +√ +kε = sup +B √ +k +u+ ≥ u��� +∂B √ +k +. +By direct computation, it is straightforward to check that +∇v(x) = −2C1εµxe−µ|x|2 +in +B √ +k \ Bk, +and +∆pv(x) = C1ε(p − 1)(2µ)2|∇v|p−2e−µ|x|2 +� +|x|2 − n + p − 2 +2µ(p − 1) +� +. +Thus v is nonnegative p-superharmonic in B √ +k \ Bk, if µ is sufficiently small, say, +(54) +µ < n + p − 2 +2k(p − 1). +On the other hand, since +w := min(u, v) = u on ∂B √ +k, +thanks to (53), by invoking the minimality of u we get +(55) +JTP(u, B √ +k) ≤ JTP(w, B √ +k). + +34 +M. BAYRAMI AND M. FOTOUHI +Now, since we have +JTP(w, B √ +k) = +� +Bk +|∇w|p + (p − 1)λp ++χ{w>0} + (p − 1)λp +−χ{w<0} dx ++ +� +B √ +k\Bk +|∇w|p + (p − 1)λp ++χ{w>0} + (p − 1)λp +−χ{w<0} dx += +� +Bk∩{u≤0} +|∇u|p + (p − 1)λp ++χ{u>0} + (p − 1)λp +−χ{u<0} dx ++ +� +B √ +k\Bk +|∇w|p + (p − 1)λp ++χ{w>0} + (p − 1)λp +−χ{w<0} dx. +Therefore, from (55), {w > 0} ⊆ {u > 0} and {w < 0} = {u < 0}, we have that +� +Bk∩{u>0} +|∇u|p + (p − 1)λp ++χ{u>0} + (p − 1)λp +−χ{u<0} dx +≤ +� +B √ +k\Bk +|∇w|p + (p − 1)λp ++χ{w>0} + (p − 1)λp +−χ{w<0} dx +− +� +B √ +k\Bk +|∇u|p + (p − 1)λp ++χ{u>0} + (p − 1)λp +−χ{u<0} dx +≤ +� +B √ +k\Bk +|∇w|p − |∇u|p dx = +� +(B √ +k\Bk)∩{u>v} +|∇v|p − |∇u|p dx +≤ − p +� +B √ +k\Bk +|∇v|p−2∇v · ∇ max(u − v, 0) dx += − p +� +B √ +k\Bk +−∆pv max(u − v, 0) + div +� +|∇v|p−2∇v max(u − v, 0) +� +dx +≤ − p +� +B √ +k\Bk +div +� +|∇v|p−2∇v max(u − v, 0) +� +dx +=p +� +∂Bk +|∇v|p−2(∇v · ν)u+, +where to get the last inequality we have used the fact that v is a p-superharmonic +in B √ +k \ Bk. Moreover, by (54), we have that |∇v| = 2C1εµke−µk2 ≤ Cε on ∂Bk, for +some C > 0. Thus +(56) +� +Bk∩{u>0} +|∇u|p + (p − 1)λp ++χ{u>0} dx ≤ p(Cε)p−1 +� +∂Bk +u+. + +35 +On the other hand, from trace estimate, Young’s inequality, we get +(57) +� +∂Bk +u+ ≤ C(n, k) +�� +Bk +u+ dx + +� +Bk +|∇u+| dx +� +≤ C(n, k) + +sup +Bk +u+ +� +Bk +χ{u>0} dx + +� +Bk +1 +p|∇u+|p + 1 +p′ χ{u>0} dx + + +≤ C(n, k) +� +(ε +√ +k + 1 +p′ ) +� +Bk +χ{u>0} dx + 1 +p +� +Bk +|∇u+|p dx +� +≤ C0 +� +Bk∩{u>0} +|∇u|p + (p − 1)λp ++χ{u>0} dx, +where p′ is the conjugate of p and +C0 := C(n, k) +� +ε +√ +k + 1 +� +. +Finally, putting together (56) and (57), we reach to +� +Bk(x0)∩{u>0} +|∇u|p + (p − 1)λp ++χ{u>0} dx +≤ p(Cε)p−1C0 +� +Bk(x0)∩{u>0} +|∇u|p + (p − 1)λp ++χ{u>0} dx, +which implies that u ≡ 0 in Bk(x0) if ε is small enough. This completes the proof of +the non-degeneracy property. +□ +Declarations +Data availability statement: All data needed are contained in the manuscript. +Funding and/or Conflicts of interests/Competing interests: The authors declare +that there are no financial, competing or conflict of interests. +References +[1] Alt, H. W., and Caffarelli, L. A. Existence and regularity for a minimum problem with free +boundary. J. Reine Angew. Math. 325 (1981), 105–144. +[2] Alt, H. W., Caffarelli, L. A., and Friedman, A. Axially symmetric jet flows. Arch. Rational Mech. +Anal. 81, 2 (1983), 97–149. +[3] Alt, H. W., Caffarelli, L. A., and Friedman, A. A free boundary problem for quasilinear elliptic +equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11, 1 (1984), 1–44. +[4] Alt, H. W., Caffarelli, L. A., and Friedman, A. Jets with two fluids. I. One free boundary. Indiana +Univ. Math. J. 33, 2 (1984), 213–247. +[5] Alt, H. W., Caffarelli, L. A., and Friedman, A. Jets with two fluids. II. 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Stratification of free boundary points for a two-phase +variational problem. Adv. Math. 328 (2018), 40–81. +[17] Ferrari, F., and Lederman, C. Regularity of flat free boundaries for a p(x)-Laplacian problem with +right hand side. Nonlinear Anal. 212 (2021), Paper No. 112444, 25. +[18] Ferrari, F., and Lederman, C. Regularity of lipschitz free boundaries for a p(x)-laplacian problem +with right hand side. Journal de Math´ematiques Pures et Appliqu´ees (2022). +[19] Fotouhi, M., and Shahgholian, H. A minimization problem with free boundary for p-laplacian +weakly coupled system. (2023) arxiv:2301.02236 (preprint). +[20] Gilbarg, D., and Trudinger, N. S. Elliptic partial differential equations of second order. Classics +in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. +[21] Karakhanyan, A. Regularity for the two-phase singular perturbation problems. Proceedings of +the London Mathematical Society 123, 5 (2021), 433–459. +[22] Mal´y, J., and Ziemer, W. P. 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Lecture notes available at http://cvgmt. +sns. it/paper/4367 (2019). +Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran +Email address: masoud.bayrami1990@sharif.edu +Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran +Email address: fotouhi@sharif.edu + diff --git a/5NFKT4oBgHgl3EQfSS3g/content/tmp_files/load_file.txt b/5NFKT4oBgHgl3EQfSS3g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a1ec958663e5f946e6c66d0dbc1dc214f5b5a624 --- /dev/null +++ b/5NFKT4oBgHgl3EQfSS3g/content/tmp_files/load_file.txt @@ -0,0 +1,1100 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf,len=1099 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='11775v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='AP] 11 Jan 2023 REGULARITY IN THE TWO-PHASE BERNOULLI PROBLEM FOR THE p-LAPLACE OPERATOR MASOUD BAYRAMI AND MORTEZA FOTOUHI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We show that any minimizer of the well-known ACF functional (for the p-Laplacian) is a viscosity solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This allows us to establish a uniform flatness decay at the two-phase free boundary points to improve the flatness, that boils down to C1,η regularity of the flat part of the free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This result, in turn, is used to prove the Lipschitz regularity of minimizers by a dichotomy argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Introduction and main result We study the problem of minimizing the following two-phase functional JTP(v, D) := � D |∇v|p + (p − 1)λp +χ{v>0} + (p − 1)λp −χ{v<0} dx, v ∈ K, where D is a bounded and smooth domain in Rn, χA is the characteristic function of the set A, 1 < p < ∞, and λ± > 0 are given constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The class of admissible functions K, consists of all functions v ∈ g + W1,p 0 (D), where g ∈ W1,p(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Any minimizer u satisfies, in a certain weak sense, the following system of equations (1) \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 ∆pu := div(|∇u|p−2∇u) = 0, in Ω+ u ∪ Ω− u, |∇u+|p − |∇u−|p = λp + − λp −, on �∂Ω+ u ∩ ∂Ω− u � ∩ D, |∇u+| ≥ λ+, |∇u−| ≥ λ−, on �∂Ω+ u ∩ ∂Ω− u � ∩ D, |∇u+| = λ+, on �∂Ω+ u \\ ∂Ω− u � ∩ D, |∇u−| = λ−, on �∂Ω− u \\ ∂Ω+ u � ∩ D, where Ω± u = {x ∈ D : ±u(x) > 0}, u± := max{±u, 0}, and ∆pu = div(|∇u|p−2∇u) is the p-Laplace operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' These types of problems are known as Bernoulli-type free boundary problems which appear in various models of fluid mechanics or heat conduction (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [2, 4, 5, 7, 21, 18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' For the admissible functions in K+ := {v ∈ K : v ≥ 0}, the analogous one-phase functional and the corresponding overdetermined problem called the one-phase Bernoulli problem, was first studied in [1] for the case p = 2, and then in [6] for the two-phase problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Also, the case of uniformly elliptic quasilinear equations in the one-phase case has been treated in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The difficulty of the problem (1) is that the governing operator, ∆pu = div(|∇u|p−2∇u), is not Date: January 30, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 1991 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 35R35, 35J92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Free boundary regularity, Two-phase Bernoulli problem, p-Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Bayrami and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Fotouhi was supported by Iran National Science Foundation (INSF) under project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 4001885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 1 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BAYRAMI AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' FOTOUHI uniformly elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Obviously, close to regular free boundary points one expects that |∇u| > 0 implying uniform ellipticity of the p-Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' However, without such a regularity assumption, it is difficult to prove non-degeneracy up to the free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In [10], the authors circumvent this issue by simultaneously showing the non-degeneracy of the gradient and the regularity of the free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Here below we list terminologies and definitions that are frequently used in this paper: A function u : D → R is said to be a minimizer of JTP in D if and only if JTP(u, D) ≤ JTP(v, D), for all v ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' F(u) := �∂Ω+ u ∪ ∂Ω− u � ∩ D, denotes the free boundary of the minimizer u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The set ΓTP := ∂Ω+ u ∩ ∂Ω− u ∩ D is the two-phase points of the free boundary F(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The boundary of positive and negative phases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' ∂Ω± u ∩ D can be decom- posed as ∂Ω± u ∩ D = Γ± OP ∪ ΓTP, where Γ+ OP := �∂Ω+ u \\ ∂Ω− u �∩D and Γ− OP := �∂Ω− u \\ ∂Ω+ u �∩D are the one-phase parts of F(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We will say that x0 ∈ ΓTP is an interior two-phase point and will denote it by x0 ∈ Γint TP, if |Br(x0) ∩ {u = 0}| = 0, for some r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We will say that x0 ∈ ΓTP is a branching point and will denote it by x0 ∈ Γbr TP, if |Br(x0) ∩ {u = 0}| > 0, for every r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We denote by Hα,e the following one-dimensional function Hα,e(x) = α (x · e)+ − β (x · e)− , with a unit vector e ∈ Sn−1 and the constants α and β satisfying the condi- tions (2) α ≥ λ+, β ≥ λ−, αp − βp = λp + − λp −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Hα,e is the so-called two-plane solution to (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Our goal is to study the regularity of the free boundary F(u) = �∂Ω+ u ∪ ∂Ω− u �∩D, for minimizers of JTP in D, around the two-phase points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' More precisely, we prove that in a suitable neighborhood of the two-phase points, the sets Ω+ u and Ω− u are two C1,η-regular domains touching along the closed set of two-phase points ΓTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' For the special case p = 2, this result has been recently obtained in [12], by invoking the linearization technique and we will closely follow this technique in order to generalize this result to any 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' As is usual for problems of this type, prior to applying any method to determine the regularity of the free boundary, the Lipschitz continuity of the minimizers across the free boundary is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Our partial result for the regularity of the free boundary, however, gives us the Lipschitz regularity of the solution as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We first show C1,η-regularity of the free boundary with a flatness assumption in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 3 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1 (Flatness implies C1,η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let u : D → R be a minimizer of JTP in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' For any positive constants Λ0 and Λ1, there exists a constant ¯ǫ = ¯ǫ(n, p, Λ0, Λ1) such that if (3) ∥u − Hα,e∥L∞(B1) ≤ ¯ǫ, for some e ∈ Sn−1 and max(Λ0, λ+) ≤ α ≤ Λ1, then ∂Ω± u ∩ Br0 are C1,η graphs for some r0 > 0 and for any η ∈ (0, 1 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We need to remark that the critical flatness to obtain the regularity does not depend on λ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Indeed, as long as we are close enough to a two-plane solution with coefficient α ∈ [Λ0, Λ1], we obtain the regularity of the free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This result in turn is crucial to obtain the Lipschitz regularity of minimizers in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2 (Lipschitz regularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let u : D → R be a minimizer of JTP in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then u is locally Lipschitz continuous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' u ∈ C0,1 loc(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Basic properties of minimizers In this section, we gather some basic properties of minimizers of JTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1 (Existence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' If the admissible set K is nonempty, then there exists a mini- mizer u of JTP over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Moreover, every minimizer satisfies \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 ∆pu = 0, in Ω+ u ∪ Ω− u, ∆pu± ≥ 0, in D, ∥u∥L∞(D) ≤ ∥g∥L∞(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The existence of a bounded minimizer u of the functional JTP can be easily established using the semi-continuity of the p-Dirichlet energy and the weak con- vergence in W1,p, and can be obtained in the standard way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [6, 23] for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Also, notice that by comparison of u and u + tϕ, where ϕ is a suitable smooth that supp ϕ ⊂ Ω+ u ∪ Ω− u, it is easy to find that ∆pu = 0 in Ω+ u ∪ Ω− u in the sense of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' To prove that u+ are p-subharmonic, we first note that since ∆pu = 0 in Ω+ u, we may choose ǫk → 0 such that {u = ǫk} to be a C1 manifold by the Sard’s Theorem, resulting in −∇u |∇u| to be the outer normal vector on ∂{u > ǫk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now take 0 ≤ ϕ ∈ C∞ c (D), the integration by parts implies that � D |∇u+|p−2∇u+ · ∇ϕ dx = � {u>0} |∇u+|p−2∇u+ · ∇ϕ dx = lim ǫk→0 � {u>ǫk} |∇u+|p−2∇u+ · ∇ϕ dx = lim ǫk→0 � {u=ǫk} |∇u+|p−2 � ∇u+ · −∇u |∇u| � ϕ dx − � {u>ǫk} ∆pu ϕ dx = − lim ǫk→0 � {u=ǫk} |∇u+|p−1ϕ dx ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The proof of ∆pu− ≥ 0 is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Finally, the last estimate ∥u∥L∞(D) ≤ ∥g∥L∞(D) 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BAYRAMI AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' FOTOUHI is the consequence of the p-subharmonicity of u± in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ In the following proposition we show the non-degeneracy property for the minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' It reveals the fact that each of the two phases Ω+ u and Ω− u are optimal with respect to one-sided inwards perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The proof is the same as the proof of non-degeneracy for one-phase problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' see [10, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We postpone the proof to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2 (Non-degeneracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let D ⊂ Rn be an open set, and u be a minimizer of JTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then, u is non-degenerate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' there is a constant C = C(n, λ±, p) > 0 such that ⧸ � Br(x0) �u±�p dx ≥ Crp, for every x0 ∈ Ω± u ∩ D and every 0 < r < dist (x0, ∂D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The next proposition concerns the Lipschitz regularity of the minimizers around the one-phase points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3 (Lipschitz regularity at one-phase points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let u : D → R be a minimizer of JTP in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' There there is constant C = C(n, p, ±λ) such that if x0 ∈ Γ+ OP (or x0 ∈ Γ− OP) is one-phase point and Br(x0) ∩ Ω− u = ∅ (Br(x0) ∩ Ω+ u = ∅), then ∥∇u∥L∞(B r 2 (x0)) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We remark that the condition Br(x0) ∩ Ω− u = ∅ always holds for some r > 0 by the definition of one-phase points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We know that ux0,r(x) = u(x0+rx) r is a minimizer of the following one phase functional in B1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' minimizer of JOP(v, B1) := � B1 |∇v|p dx + (p − 1)λp +|{v > 0} ∩ B1|, over the class of nonnegative functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then the boundedness of the gradient ∥∇u∥L∞(B r 2 (x0)) = ∥∇ux0,r∥L∞(B 1 2 ) ≤ C(n, p, λ+), follows from [10, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We shall remark that the Lipschitz constant for one-phase problems does not depend on the boundary values of the minimizer as long as we stay uniformly far from the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ Next, we mention the following continuity result for minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4 (BMO estimates for the gradient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let u be a minimizer of JTP and D′ ⋐ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then, (i) for 1 < p < 2, we have that |∇u| p−2 2 ∇u ∈ BMO(D′), and consequently u ∈ Cσ(D′) for any σ ∈ (0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (ii) for 2 < p < ∞, we have that ∇u ∈ BMO(D′) and thus u is locally log-Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In particular, ∇u ∈ Lq(D′) for any 1 < q < ∞ and for any 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The proof is the same as the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1 in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ 5 The BMO estimate for the gradient of minimizers is sufficient to obtain the fol- lowing compactness result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Since we have not yet proved the Lipschitz continuity, this result will be extremely valuable for our argument in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We postpone the proof to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let uj be a bounded minimizer of JTP in B2 with the points xj ∈ B1 such that uj(xj) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Also, set vj(x) = uj(xj+rjx) Sj , for any x ∈ BR, with 0 < R < 1 rj , where rj → 0, as j → +∞ and Sj > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then, vj is the minimizer (according to its own boundary values) of the following scaled functional (4) ˆJTP(v) := � BR |∇v|p + (p − 1)σp jλp +χ{v>0} + (p − 1)σp jλp −χ{v<0} dx, where σj := rj Sj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Moreover, if |vj| ≤ M in BR, for any fixed 0 < R < 1 rj and for some M = M(R) > 0, then up to a subsequence, the followings hold: (i) For any q > 1, and some α ∈ (0, 1) (if q > n, one can take α = 1 − n q), vj converges to some function v0 as j → +∞ in Cα(BR) and weakly in W1,q(BR);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (ii) vj → v0 strongly in W1,p(BR);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (iii) If moreover, σj := rj Sj → σ, as j → +∞, then v0 is a minimizer of ˆJTP(v) := � BR |∇v|p + (p − 1)σpλp +χ{v>0} + (p − 1)σpλp −χ{v<0} dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In particular, if σ = 0, then v0 is p-harmonic in BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The following lemma states that u+ and u− have coherent growth at two-phase points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This is essential to show that the minimizers are the viscosity solution of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let u be a bounded minimizer of JTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let x0 ∈ ΓTP and r0 > 0 be small such that Br0(x0) ⊂ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Assume that supBr(x0) u− ≤ C0r (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' supBr(x0) u+ ≤ C0r) for all r ∈ (0, r0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then there exist constant C1 > 0 such that supBr(x0) u+ ≤ C1r (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' supBr(x0)u− ≤ C1r) for all r ∈ (0, r0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We will just demonstrate one of the claims;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' the other can be demonstrated similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By the assumption of the lemma (5) sup Br(x0) u− ≤ C0r, ∀r ∈ (0, r0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We claim that there is ˜C1 > 0 such that (6) S(k + 1) ≤ max � ˜C1 2k+1 , 1 2S(k) � , where S(k) := ∥u∥L∞(B2−k(x0)), for any k ∈ N that 2−k ≤ r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' To prove this, we argue by contradiction and suppose that (6) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then there is a sequence of integers kj, with j = 1, 2, · · · such that (7) S(kj + 1) > max � j 2kj+1 , 1 2S(kj) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BAYRAMI AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' FOTOUHI Observe that since u is a bounded minimizer, then (7) implies that kj → +∞ as j → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Also, notice that (7) implies that (8) σj := 2−kj S(kj + 1) ≤ 2 j → 0 as j → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now, we introduce the scaled functions vj(x) := u(x0+2−kjx) S(kj+1) , for any x ∈ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then, from (5) and (8), it follows that vj(0) = 0 and (9) v− j (x) = u−(x0 + 2−kjx) S(kj + 1) ≤ 2−kjC0 S(kj + 1) ≤ 2C0 j → 0, as j → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Furthermore, it is simple to show that (7) implies that (10) sup B1 |vj| ≤ 2, and sup B 1 2 |vj| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Also, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='5 entails that vj is a minimizer of the scaled functional (4) for R = 3 4 and we can extract a converging subsequence such that vj → v0 uniformly in B 3 4 that v0 is p-harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The uniform convergence of vj to v0 along with (9), (10), give that ∆pv0(x) = 0, v0(x) ≥ 0 if x ∈ B 3 4 , v(0) = 0, sup B 1 2 v0 = 1, which is in contradiction with the strong minimum principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Thus (6) obtains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now we show how (6) implies the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Assume that k0 is the smallest integer k that 2−k ≤ r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let ¯C1 = max( ˜C1, 2k0S(k0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' It is not difficult to observe from (6) that S(k) ≤ ¯C12−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' For an arbitrary r ∈ (0, r0] choose k ≥ k0 such that 2−(k+1) < r ≤ 2−k, then ∥u∥L∞(Br(x0)) ≤ ∥u∥L∞(B2−k(x0)) = S(k) ≤ ¯C12−k ≤ 2 ¯C1r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Thus the statement in the lemma holds for C1 = 2 ¯C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ The following theorem roughly says that, in a very weak sense,the free boundary conditions (1) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Suppose that u is a minimizer of JTP in D and D′ ⊂ D be such that |D′ ∩ {u = 0}| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then, we have the following free boundary condition in the very weak sense lim ǫ→0+ � ∂{u>ǫ}∩D′ � |∇u+|p − λp + � η · ν + lim δ→0+ � ∂{u<−δ}∩D′ � |∇u−|p − λp − � η · ν = 0, for any η ∈ W1,p 0 (D′, Rn), and where ν is the outward normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The proof can be established precisely as in [6, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Suppose u(x) = α (x · e)+ − β (x · e)− is a global minimizer of JTP for some unite vector e ∈ Sn−1 and the positive constants α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then α and β satisfy conditions (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The equality αp − βp = λp + − λp − is obvious by invoking Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Besides, conditions α ≥ λ+, and β ≥ λ−, 7 can be obtained by a smooth variation of the free boundary {u = 0} = {x · e = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Indeed, by considering competitors of the form ut(x) = u+(x) − u−(x + tξ(x)) for vector fields ξ ∈ C∞ c (Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Rn) with ξ · e ≤ 0 so that it moves negative phase only inwards, that is, {ut < 0} ⊂ {u < 0}, and taking the derivative of JTP(ut, BR) at t > 0 and letting t → 0 (where R is sufficiently large such that supp ξ ⊂ BR), we get � {u=0}∩BR (ξ · e) � |∇u−|p − λp − � ≤ 0, which gives β ≥ λ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The estimate on α is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Free boundary conditions in the viscosity sense Let u : D → R be a local minimizer of JTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In this section, we will show that u satisfies the free boundary conditions (1) in a weak (viscosity) sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let D be an open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We say that a function Q : D → R touches a function w : D → R from below (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' from above) at a point x0 ∈ D if Q(x0) = w(x0) and Q(x) − w(x) ≤ 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Q(x) − w(x) ≥ 0), for every x in a neighborhood of x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We will say that Q touches w strictly from below (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' above) if the above inequalities are strict for x � x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' A function Q is an (admissible) comparison function in D if (a) Q ∈ C1({Q > 0} ∩ D) ∩ C1({Q < 0} ∩ D);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (b) Q ∈ C2({Q > 0} ∩ D) ∩ C2({Q < 0} ∩ D);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (c) ∂{Q > 0} and ∂{Q < 0} are smooth manifolds in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We should remark that if ∇Q � 0 on ∂{Q > 0} ∪ ∂{Q < 0}, the condition (c) above holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let u be a local minimizer of JTP in the open set D ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then the following optimality conditions on the free boundary F(u) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (A) Suppose that Q is a comparison function that touches u from below at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1) If x0 ∈ Γ+ OP, then |∇Q+(x0)| ≤ λ+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2) if x0 ∈ Γ− OP, then Q+ ≡ 0 in a neighborhood of x0 and |∇Q−(x0)| ≥ λ−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3) if x0 ∈ ΓTP, then |∇Q−(x0)| ≥ λ− and |∇Q+(x0)|p − |∇Q−(x0)|p ≤ λp + − λp −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (B) Suppose that Q is a comparison function that touches u from above at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1) If x0 ∈ Γ+ OP, then Q− ≡ 0 in a neighborhood of x0 and |∇Q+(x0)| ≥ λ+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2) if x0 ∈ Γ− OP, then |∇Q−(x0)| ≤ λ−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3) if x0 ∈ ΓTP, then |∇Q+(x0)| ≥ λ+ and |∇Q+(x0)|p − |∇Q−(x0)|p ≥ λp + − λp −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' First, we will prove the gradient bounds in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The case x0 ∈ Γ− OP, and the proofs of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2) can be obtained similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let x0 ∈ Γ+ OP be a one-phase point and let Q touches u from below at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then, Q+ touches u from below at x0, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Consider ux0,rk(x) = u(x0+rkx) rk and Q+ x0,rk(x) = Q+(x0+rkx) rk as the blow-up sequences of u and Q+ at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By virtue of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3, the functions ux0,rk are uniformly Lipschitz for sufficiently rk small and up to extracting a subsequence, we can assume that ux0,rk converges uniformly to a blow-up limit v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The limit v is a minimizer of one-phase functional JOP and so ∆pv = 0 in {v > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BAYRAMI AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' FOTOUHI On the other hand, since Q+ is differentiable at x0 in Ω+ Q, we get that Q+ x0,rk converges to the function (11) HQ+(x) = (x · e′)+ with e′ = ∇Q+(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' If ∇Q+(x0) = 0, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1) is trivially valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We assume that e′ � 0, and since HQ+ touches v from below at x = 0, we get v(x) = α(x · e′)+ + o(|x|), α ≥ 1, for some α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' see [19, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We get that any blow-ups of v will be v0(x) = α(x ·e′)+ which is also a minimizer of JOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Thus α|e′| = λ+ due to the free boundary condition for one-phase minimizers (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='7 or see [10, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1]) and so |∇Q+(x0)| ≤ λ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Similarly, when Q touches u from above at x0, then also Q+ touches u from above at x0, and the claim Q− ≡ 0 in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1) is trivially true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Again, consider ux0,rk, which up to extracting a subsequence, converges uniformly to a blow-up limit v and Q+ x0,rk, as the blow-up sequences of Q+ at x0, which converges to the function (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now, we argue similar to the proof of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1) to get |∇Q+(x0)| ≥ λ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now, we prove (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Suppose x0 ∈ ΓTP and assume that Q touches u from below at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then u− ≤ Q− and u−(x) ≤ C0|x − x0| for C0 = 2|∇Q−(x0)| if |x − x0| is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now we employ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='6 to deduce that |u(x)| ≤ C1|x − x0| in a neighborhood of x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let ux0,rk and Qx0,rk be the blow-up sequences of u and Q at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then, by using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='5, up to extracting a subsequence, we can assume that ux0,rk converges uniformly to some function v which is also a minimizer of JTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Moreover, it satisfies (12) |v(x)| ≤ C1|x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' On the other hand, since Q+ and Q− are differentiable at x0 (respectively in Ω+ Q and Ω− Q), we get that Qx0,rk converges to the function HQ(x) = (x · ˜e+)+ − �x · ˜e−�− , where ˜e± = ∇Q±(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Since HQ touches v from below at x = 0, we have ([19, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1]) v+(x) = α(x · ˜e−)+ + o(|x|), |˜e+| ≤ α|˜e−|, v−(x) = β(x · ˜e−)− + o(|x|), β ≤ 1, for some α, β ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Note that by virtue of the non-degeneracy, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2, v− � 0 and so ˜e− � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' If v0 is a blowup of v (recall (12) and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='5), it will be v0(x) = α(x · ˜e−)+ − β(x · ˜e−)− which is also a minimizer of JTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now apply Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='8, we get (αp − βp)|˜e−|p = λp + − λp −, β|˜e−| ≥ λ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Hence, |∇Q+(x0)|p − |∇Q−(x0)|p ≤ αp − βp = λp + − λp −, as well as |∇Q−(x0)| ≥ λ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The proof of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3) is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ 9 If u : D → R is a continuous function such that the claims (A) and (B) hold for every comparison function Q, then we say that u satisfies the boundary condition (1) on the free boundary in viscosity sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We need the following straightforward consequence of the definition of viscosity solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' It emphasizes what happens when a function is touching only one of the two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let u : D → R be a continuous function that satisfies (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (i) Assume that Q is a comparison function touching u+ from above at x0 ∈ ∂Ω+ u (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' −u− from below at x0 ∈ ∂Ω− u), then |∇Q+(x0)| ≥ λ+ �resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' |∇Q−(x0)| ≥ λ− � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (ii) Assume that Q is a comparison function touching u+ from below at x0 ∈ Γ+ OP (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' −u− from above at x0 ∈ Γ− OP), then |∇Q+(x0)| ≤ λ+ �resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' |∇Q−(x0)| ≤ λ− � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Statement (i) will be obtained directly from (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The proof of (ii) follows the same lines of arguments as the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Flatness decay at two-phase points In this section, we will follow the method of improvement of flatness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In fact, we will prove that at two-phase points x0 ∈ ΓTP, there is a constant ǫ0 > 0 such that if u is ǫ0-flat in Br(x0) with respect to H = Hα,e, then it has excess flatness in smaller scales with respect to another ˜H = H ˜α,˜e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' For every 1 < p < ∞, 0 < L0, L1 and γ ∈ (0, 1 2), there exist ǫ0 > 0, C > 0 and ρ > 0 such that if the function u : B1 → R satisfies: (a) the origin is on the two-phase free boundary, 0 ∈ ΓTP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (b) u is p-harmonic in Ω+ u ∪ Ω− u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (c) u satisfies the free boundary condition (1) in viscosity sense;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (d) u is ǫ0-flat in B1, that is, (13) ∥u − Hα,en∥L∞(B1) ≤ ǫ0, for some max(λ+, L0) ≤ α ≤ L1, then, there are e ∈ Sn−1 and ˜α ≥ max(λ+, L0), such that (14) |e − en| + | ˜α − α| ≤ C∥u − Hα,en∥L∞(B1), and (15) ∥uρ − H ˜α,e∥L∞(B1) ≤ ργ∥u − Hα,en∥L∞(B1), where uρ(x) denotes u0,ρ(x) = u(ρx) ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1 is an easy consequence of the two upcoming lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In the first one, we deal with the situation where the two-plane is, roughly, Hλ+,e for some e ∈ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Note that this is the case where one might expect the presence of branching points and it is indeed in this setting that we will obtain the two membrane problems as ”linearization” (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [12, Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3] for a presentation of the linearization method in studying the regularity of free boundaries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In the second lemma, we deal with the case when the closest half-plane solution has a gradient much larger than λ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In this case, the origin will be an interior two-phase point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In fact, in one-phase problems, it is possible to obtain universal interior bounds, in 10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BAYRAMI AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' FOTOUHI the sense that, if u is a solution in a ball B1 and 0 ∈ F(u), then |∇u| is bounded in B 1 2 by a universal constant, no matter what the boundary data are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' However, in two- phase problems, this is generally not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' For instance, in the one-dimensional minimization scenario, increasing the boundary data leads to the appearance of a solution with a large gradient near the origin, see [9, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2 (Improvement of flatness: branching points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' For every 1 < p < ∞, 0 < L0, L1, γ ∈ (0, 1 2), and M > 0, there exist ǫ1 = ǫ1(p, γ, n, L0, L1, M), C1 = C1(p, γ, n, L0, L1, M) and ρ = ρ(p, γ, n, L0, L1, M) such that if function u : B1 → R satisfies (a) − (b) − (c) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1 and furthermore ∥u − Hα,en∥L∞(B1) ≤ ǫ1, with L0 ≤ λ+ ≤ α ≤ λ+ + M∥u − Hα,en∥L∞(B1), then there exist e ∈ Sn−1 and ˜α ≥ λ+, for which (14) and (15) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3 (Improvement of flatness: non-branching points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' For every 1 < p < ∞, 0 < L0, L1 and γ ∈ (0, 1), there exist ǫ2 = ǫ2(p, γ, n, L0, L1), M = M(p, γ, n, L0, L1), ρ = ρ(p, γ, n, L0, L1) and C2 = C2(p, γ, n, L0, L1) such that if function u : B1 → R satisfies (a) − (b) − (c) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1 and furthermore ∥u − Hα,en∥L∞(B1) ≤ ǫ2, with α ≥ max(λ+, L0) + M∥u − Hα,en∥L∞(B1), then there exist e ∈ Sn−1 and ˜α ≥ max(λ+, L0), for which (14) and (15) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The proof follows easily by combining the Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ In order to prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3, we will argue by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Hence in the following, we consider a sequence uk of minimizers such that (16) ǫk := ∥uk − Hαk,en∥L∞(B1) → 0 and λ+ ≤ αk ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We also set (17) ℓ := λp + lim k→∞ αp k − λp + pαp kǫk = λp − lim k→∞ βp k − λp − pβp kǫk , which we can assume to exist up to a subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' It might be useful to keep in mind that ℓ = ∞ will correspond to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3 while 0 ≤ ℓ < ∞ (so αk → λ+ and λ+ ≥ L0) to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We first show that the sequence (18) vk(x) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 v+,k(x) := uk(x) − αkx+ n ǫkαk x ∈ Ω+ uk ∩ B1 v−,k(x) := uk(x) + βkx− n ǫkβk x ∈ Ω− uk ∩ B1 is compact in some suitable sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This will be mentioned in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4 below and the proof will come in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then, in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='5, we obtain the limiting problem which is solved by v, the limit of vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Finally, in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3 we show how to deduce Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2 from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In the following, we will denote with B± r := Br ∩ {x± n > 0}, for every r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 11 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4 (Compactness of the linearizing sequence vk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let uk be a sequence of functions satisfying (a) − (b) − (c) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1 uniformly in k and let ǫk and αk be as in (16) and let vk be defined by (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then there are H¨older continuous functions v+ : B+ 1 2 → R and v− : B− 1 2 → R, with v+ ≤ v− on B 1 2 ∩ {xn = 0}, v+(0) = v−(0) = 0, and such that the sequence of closed graphs Γ± k := � (x, v±,k(x)) : x ∈ Ω± uk ∩ B 1 2 � , converge, up to a subsequence, in the Hausdorff distance to the closed graphs Γ± = � (x, v±(x)) : x ∈ B± 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In particular, the following claims hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (i) For every δ > 0, v±,k converges uniformly to v± on B 1 2 ∩ {±xn > δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (ii) For every sequence xk ∈ Ω± uk ∩ B1 converging to x ∈ B± 1 2 , we have v±(x) = lim k→∞ v±,k(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (iii) For every x ∈ {xn = 0} ∩ B 1 2 , we have v±(x) = − lim k→∞ xk · en ǫk for any sequence ∂Ω± uk ∋ xk → x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In particular, {xn = 0} ∩ B 1 2 decomposes into an open jump set J = {v+ < v−} ∩ {xn = 0} ∩ B 1 2 , and its complementary contact set C = {v+ = v−} ∩ {xn = 0} ∩ B 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Furthermore, if x ∈ J, then (19) lim inf k→∞ dist � x, ∂Ω+ uk ∩ ∂Ω− uk � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In particular for all x ∈ J, there exists two sequences x± k ∈ Γ± k,OP such that x± k → x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now, in the next lemma, we determine the limiting problem for the function v which is defined as (20) v(x) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3 v+(x) for x ∈ B+ 1 2 , v−(x) for x ∈ B− 1 2 , where v+ and v− are the functions defined in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In what follows, we will denote with Lp(u) := ∆u + (p − 2)∂nnu, the frequently used operator which appears in the linearized problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BAYRAMI AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' FOTOUHI Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='5 (The ”linearized” problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let uk, ǫk and αk be as in (16), vk be defined by (18) and ℓ as in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let also v± be as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4: If ℓ = ∞, then J = ∅ and v± are viscosity solutions of the following transmission problem: (21) \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3 Lp(v±) = ∆v± + (p − 2)∂nnv± = 0, in B± 1 2 , αp ∞∂nv+ = βp ∞∂nv−, on B± 1 2 ∩ {xn = 0}, where α∞ = limk→∞ αk and β∞ = limk→∞ βk, which we can assume to exist up to extracting a further subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' If 0 ≤ ℓ < ∞, then v± are viscosity solutions of the following two membranes problem: (22) \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 Lp(v±) = ∆v± + (p − 2)∂nnv± = 0, in B± 1 2 , λp ±∂nv± + ℓ ≥ 0, in B 1 2 ∩ {xn = 0}, λp ±∂nv± + ℓ = 0, in J, λp +∂nv+ = λp −∂nv−, in C, v+ ≤ v−, in B 1 2 ∩ {xn = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Here by viscosity solution of (21) and (22), we mean a function v as in (20) such that v± are continuous in B± 1 2 , Lp(v±) = 0 in B± 1 2 (in viscosity or equivalently the classical sense) and such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' If we are in case (21), let s, t ∈ R and let ˜P be a quadratic polynomial such that ∂n ˜P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Suppose that Lp( ˜P) ≥ 0 (Lp( ˜P) ≤ 0) and that the function P := sx+ n − tx− n + ˜P, touches v strictly from below (above) at a point x0 ∈ B 1 2 ∩ {xn = 0}, then αp ∞s ≤ βp ∞t, � αp ∞s ≥ βp ∞t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' If we are in case (22) then (1) if P± is a quadratic polynomial with Lp(P±) ≤ 0 in B± 1 2 touching v± strictly from above at x0 ∈ B 1 2 ∩ {xn = 0}, then λp ±∂nP± ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (2) if P± is a quadratic polynomial with Lp(P±) ≥ 0 in B± 1 2 touching v± strictly from below at x0 ∈ J, then λp ±∂nP± ≤ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (3) if s, t ∈ R and ˜P is a quadratic polynomial with Lp(P±) ≥ 0 (Lp(P±) ≤ 0) such that ∂n ˜P = 0 and the function P := sx+ n − tx− n + ˜P, touches v strictly from below (above) at a point x0 ∈ B 1 2 ∩ {xn = 0}, then λp +s ≤ λp −t, � λp +s ≥ λp −t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Compactness of the linearizing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' As explained in [12, Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1] for the case of classical two-phase Bernoulli problem, the authors declare that the key point in establishing suitable compactness for vk is a ”partial Harnack” inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We will follow the same approach and start with the following useful lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 13 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' There is a constant τ = τ(n, p) > 0 such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Assume that v : B1 → R is a continuous function with ∆pv = 0 in {v > 0} and λ (xn + b)+ ≤ v(x) ≤ λ (xn + a)+ , x ∈ B1, for some λ > 0 and a, b ∈ (− 1 100, 1 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let P = (0, · · · , 0, 1 2), then for all ǫ ∈ (0, 1 2) v(P) ≤ λ(1 − ǫ) �1 2 + a �+ =⇒ v(x) ≤ λ(1 − τǫ) (xn + a)+ in B 1 4 (0), and v(P) ≥ λ(1 + ǫ) �1 2 + b �+ =⇒ v(x) ≥ λ(1 + τǫ) (xn + b)+ in B 1 4 (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We prove only the first implication since the second statement can be ob- tained by the same arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' First, we notice that, since |b| < 1 100, both v and λ(xn + a)+ are positive and p-harmonic in B 1 4 (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Thus, λ(xn + a)+ − v(x) ≥ 0, x ∈ B 1 4 (P), and λ �1 2 + a �+ − v(P) ≥ λǫ �1 2 + a �+ ≥ 49 100λǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now, we distinguish two cases: Case (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Suppose |∇v(P)| < λ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Therefore, there exists r1 = r1(n, p) > 0 such that |∇v(x)| ≤ λ 2 in B4r1(P) (note that v λ is universally bounded and p-harmonic in B 1 4 (P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' It is easy to find that for ˜v := (xn + a)+ − 1 λv, we have div � |∇˜v − en|p−2(∇˜v − en) � = 0, in B 1 20 (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We now apply Harnack’s inequality for the above operator (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [17, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1]) in B4r1(P), to deduce that (xn + a)+ − 1 λv(x) ≥ C−1 ��1 2 + a �+ − 1 λv(P) � − r1, in Br1(P), for an appropriate universal constant C = C(n, p) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' On the other hand, for all x ∈ Br1(P), we obtain C−1 49 100ǫ − r1 ≤ (xn + a)+ − 1 λv(x) ≤ (xn + 2r1 + a)+ − 2r1 − 1 λv(x + 2r1en) + 2r1 λ ∥∇v∥L∞(B4r1(P)) ≤ (xn + 2r1 + a)+ − 2r1 − 1 λv(x + 2r1en) + r1 ≤ (xn + 2r1 + a)+ − 1 λv(x + 2r1en) − r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Thus, with ˜P = P + 2r1en, we get (23) C−1 49 100ǫ ≤ (xn + a)+ − 1 λv (x) , for all x ∈ Br1( ˜P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Hence, by considering the inequality (23) and also using the bound |a| ≤ 1 100, there is a constant c = c(n, p) such that v(x) ≤ λ(1 − cǫ)(xn + a)+, for all x ∈ Br1( ˜P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 14 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BAYRAMI AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' FOTOUHI We now let w be the solution to the following problem \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 ∆pw = 0 in � B1(0) \\ Br1( ˜P) � ∩ {xn > −a} w = 0 on B1 ∩ {xn = −a} w = (xn + a)+ on ∂B1(0) ∩ {xn > −a} w = (1 − cǫ)(xn + a)+ on ∂Br1( ˜P) ∩ {xn > −a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By the Hopf boundary lemma ([24, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1]), w(x) ≤ (1 − τǫ)(xn + a)+, for every x ∈ B 1 4 ∩ {xn > −a}, for a suitable constant τ = τ(n, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' On the other hand, by the comparison principle, we have v ≤ λw in {v > 0} ∩ B1 \\ Br1( ˜P), which concludes the proof in Case (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Case (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Suppose |∇v(P)| ≥ λ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By the interior gradient estimate, we know that ∇v is bounded in B 1 40 (P), and there exist a constant 0 < r0 = r0(n, p), with 8r0 ≤ 1 40 such that λ 8 ≤ |∇v(x)| ≤ Cλ, for all x ∈ B8r0(P), for an appropriate universal constant C = C(n, p) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now, v will be the weak solution to the following uniformly elliptic equation n � i,j=1 θij∂xixjv = 0 in B4r0(P), with θij = δij + (p − 2)|∇v|−2∂xiv∂xjv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then, applying Harnack’s inequality (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [20, Chapter 9]), we get (24) C−1 49 100ǫ ≤ (xn + a)+ − 1 λv (x) , for all x ∈ Br0(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now, we can repeat the same argument of Case (i), by considering the inequality (24) in the ball Br0(P) instead of inequality (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ We next prove the two partial Harnack inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The proof of these inequal- ities is based on a comparison with suitable test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In order to build these ”barriers”, we will often use the following function ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let Q = (0, · · · , 0, 1 5) and define ϕ : B1 → R by (25) ϕ(x) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 1, if x ∈ B 1 100 (Q), κn � |x − Q|−n − ( 3 4)−n� , if x ∈ B 3 4 (Q) \\ B 1 100 (Q), 0, otherwise, where the dimensional constant κn is chosen in such a way that ϕ is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' One can check that ϕ has the following properties: (ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1) 0 ≤ ϕ ≤ 1 in Rn, and ϕ = 0 on ∂B1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2) For s > 0 small, −div ����en − s∇ϕ ��� p−2 �en − s∇ϕ�� ≥ c(n, p, s) > 0, in {ϕ > 0} \\ B 1 100 (Q), (with fairly simple computations same as the ones which have been done in [17, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3) ∂nϕ > 0 in {ϕ > 0} ∩ {|xn| ≤ 1 100};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 15 (ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4) ϕ ≥ cn > 0 in B 1 6 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' where c(n, p) and cn are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='8 (Partial Boundary Harnack I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Given 1 < p < ∞ and λ+ ≥ λ− > 0, there exist constants ǫ = ǫ(n, λ±, p) > 0 and c = c(n, λ±, p) ∈ (0, 1) such that, for every function u : B4 → R satisfying (b) − (c) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1, the following properties hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let a±, b± ∈ (− 1 100, 1 100) be such that b+ ≤ b− ≤ a− ≤ a+, and (a− − b−) + (a+ − b+) ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Assume that for x ∈ B4 λ+(xn + b+)+ ≤ u+(x) ≤ λ+(xn + a+)+, and −λ−(xn + b−)− ≤ −u−(x) ≤ −λ−(xn + a−)−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then, one can find new constants a±, b± ∈ (− 1 100, 1 100), with b+ ≤ b− ≤ a− ≤ a+, and a− − b− ≤ c(a− − b−), a+ − b+ ≤ c(a+ − b+) such that for x ∈ B 1 6 λ+(xn + b+)+ ≤ u+(x) ≤ λ+(xn + a+)+, and −λ−(xn + b−)− ≤ −u−(x) ≤ −λ−(xn + a−)−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We need to remark that the assumption λ+ ≥ λ− is not restrictive as one can always replace u by −u in JTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Also, when λ+ ≤ λ− the similar result holds if we replace the order of a±, b± with a+ ≤ a− ≤ b− ≤ b+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let us show how to improve the positive part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' More precisely, given a+, a−, b+, b− we will show how we can find a+ and b+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The proof for b− and a− follows in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We let P = (0, · · · , 0, 2), and distinguish two cases: Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Improvement from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Assume that, at the point P, u+ is closer to λ+(2 + b+)+ than to the upper barrier λ+(2 + a+)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Precisely that u+(P) ≤ λ+(2 + a+)+ − λ+(a+ − b+) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In this case, we will show that u(x) is less than λ+(xn+a+)+ in a smaller ball centered at the origin for a+ strictly smaller than a+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We start by setting ǫ := a+ − b+ ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then u+(P) ≤ λ+(2 + a+)+ − λ+ǫ 2 ≤ λ+(1 − cǫ)(2 + a+)+, 16 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BAYRAMI AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' FOTOUHI for a suitable (universal) constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We can thus apply (the scaled version of) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='7 to u+, to infer the existence of a constant τ = τ(n, p) such that (26) u+(x) ≤ λ+(1 − τǫ)(xn + a+)+, in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' For ϕ as in (25) and t ∈ [0, 1], we set ft = λ+ � 1 − τǫ 2 � (xn + a+ − tcǫϕ)+, where c = c(n, p) is a small constant chosen such that for all x ∈ B 1 100 (Q) and t ∈ [0, 1), (27) u(x) ≤ λ+(1 − τǫ)(xn + a+)+ ≤ λ+ � 1 − τǫ 2 � (xn + a+ − cǫ)+ < ft(x), where we have used that (xn + a+) is within two universal constant for x ∈ B 1 100 (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We now let t ∈ (0, 1] the largest t such that ft ≥ u in B1 and we claim that t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Indeed assume that t < 1, then there exists x ∈ B1 such that (28) u(x) − ft(x) ≤ u(x) − ft(x) = 0, for all x ∈ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Note that by (27), x � B 1 100 (Q), while, by (26), x ∈ {ϕ > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Moreover, if u(x) = ft(x) > 0, by (ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2) we will have ∆p ft(x) = � λ+ � 1 − τǫ 2 ��p−1 div ����en − tcǫ∇ϕ(x) ��� p−2 � en − tcǫ∇ϕ(x) �� < 0, but, since ∆pu(x) = 0, we reach a contradiction with (28) and the definition of viscosity solution for the p-harmonic function u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Hence, u(x) = ft(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now recall the free boundary condition (1) and apply (ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3) to get λp + ≤ |∇ft(x)|p = λp + � 1 − τǫ 2 �p � 1 − pctǫ∂nϕ(x) + O(ǫ2) � < λp +, provided ǫ ≤ ǫ(n, λ+, p) ≪ 1 (note that necessarily u(x) = 0 and thus x ∈ {|xn| ≤ 1 100}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This contradiction implies that t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Hence, by (ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4), we get for all x ∈ B 1 6 u(x) ≤ λ+ � 1 − τǫ 2 � (xn + a+ − cǫϕ)+ ≤ λ+(xn + a+ − cǫ)+, for a suitable constant c = c(n, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Setting a+ = a+ − cǫ, b+ = b+, and recalling that ǫ = a+ − b+ we finish the proof in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Improvement from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We now assume that, at point P, u+ is closer to λ+(2 + a+)+ than to λ+(2 + b+)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Hence, we have u+(P) ≥ λ+(2 + b+)+ + λ+(a+ − b+) 2 , and we set again ǫ := a+ − b+ ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Arguing as in Case 1, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='7, there exists a constant τ = τ(n, p) such that (29) u+(x) ≥ λ+(1 + τǫ)(xn + b+)+, in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 17 We need now to distinguish two further sub-cases: Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1: Suppose that ηǫ ≤ b− − b+, where η ≪ τ is a small universal constant which we will choose at the end of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In this case, for x ∈ B1, (30) u(x) ≥ λ+(1 + τǫ)(xn + b+)+ − λ−(xn + b−)− ≥ λ+(1 + τǫ)(xn + b+)+ − λ−(1 − c1ηǫ)(xn + b+)−, for a suitable universal constant c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We now take ϕ as in (25) and set, for t ∈ [0, 1], ft(x) = λ+ � 1 + τǫ 2 � (xn + b+ + c2tǫϕ)+ − λ−(1 − c1ηǫ)(xn + b+ + c2tǫϕ)−, for a suitably small universal constant 0 < c2 ≪ τ, chosen so that for all x ∈ B 1 100 (Q) (1 + τǫ)(xn + b+)+ ≥ � 1 + τǫ 2 � (xn + b+ + c2ǫ)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This together with (29) implies that (31) u(x) ≥ λ+(1 + τǫ)(xn + b+)+ ≥ λ+ � 1 + τǫ 2 � (xn + b+ + c2ǫ)+ ≥ f1(x) ≥ ft(x), for all x ∈ B 1 100 (Q), t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Furthermore u ≥ f0 in B1 thanks to (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Similar to Case 1, let t be the biggest t such that ft ≤ u in B1 and x be the first contact point, so that u(x) − ft(x) ≥ u(x) − ft(x) = 0, for all x ∈ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Since, by using (ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2), it can be checked that ∆p ft > 0, on {ft � 0} \\ B 1 100 (Q), therefore, as in Case 1, x is a free boundary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Moreover, since ft changes sign in a neighborhood of x: either x ∈ Γ+ OP = ∂Ω+ u \\ ∂Ω− u, or x ∈ ΓTP = ∂Ω+ u ∩ ∂Ω− u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In the first case, by definition of viscosity solution and (ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3), λp + ≥ |∇f + t (x)|p = λp + � 1 + τǫ 2 �p � 1 + pc2tǫ∂nϕ(x) + O(ǫ2) � > λp +, a contradiction for ǫ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In the second case, we have a contradiction as well, since (recall also the assumption λ+ − λ− ≥ 0) λp + − λp − ≥ |∇f + t |p − |∇f − t |p = � λp + � 1 + τǫ 2 �p − λp −(1 − c1ηǫ)p � � 1 + pc2tǫ∂nϕ(x) + O(ǫ2) � > λp + − λp −, provided ǫ ≪ 1 (only depending on n, λ+ and p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Hence, t = 1, u ≥ f1 (so u+ ≥ f + 1 ) which implies the desired conclusion by setting a+ = a+, b+ = b+ + c2ǫ, for a suitable constant c2 = c2(n, p) and by recalling that ǫ = a+ − b+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 18 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BAYRAMI AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' FOTOUHI Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2: Assume instead that: 0 ≤ b− − b+ ≤ ηǫ, where η = η(n, p) will be determined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In this case we consider the family of functions ft(x) = λ+ � 1 + τǫ 2 � (xn + b+ + ηtǫϕ)+ − λ−(xn + b−)−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Since ϕ ≤ 1, this function is well defined due to b− ≤ b+ + ηǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Moreover, u ≥ f0 and, thanks to (29) and by assuming η is sufficiently small (this can also be determined universally depending only on the dimension and p) we will have, u(x) ≥ f1(x) ≥ ft(x), for all x ∈ B 1 100 (Q), t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We consider again the first touching time t and the first touching point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By arguing as in the previous cases, we get x ∈ {u = 0} ∩ {|xn| ≤ 1 100}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Also, the definition of ft yields that x ∈ ∂{ft > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This infer that x ∈ ∂Ω+ u \\ ∂Ω− u (note that ϕ(x) < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' However, again by arguing as in Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1, this is in contradiction with u being a viscosity solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We now conclude as in the previous cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ The following lemma addresses the situation in which the origin is not a branch- ing point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='10 (Partial Boundary Harnack II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Given 1 < p < ∞ and 0 < L0, L1 and assume that 0 < λ− ≤ λ+ ≤ L1, then there exist constants ǫ = ǫ(n, L0, L1, p) > 0, M = M(n, L0, L1, p) and c = c(n, L0, L1, p) ∈ (0, 1) such that for every function u : B4 → R satisfying (b) − (c) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1 the following property holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' If there are constants a, b ∈ (− 1 100, 1 100) with 0 ≤ a − b ≤ ǫ, such that for x ∈ B4 Hα,en(x + ben) ≤ u(x) ≤ Hα,en(x + aen), and max(λ+, L0) + Mǫ ≤ α ≤ L1, then there are constants a, b ∈ (− 1 100, 1 100) with 0 ≤ a − b ≤ c(a − b), such that for x ∈ B 1 6 Hα,en(x + ben) ≤ u(x) ≤ Hα,en(x + aen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We consider the point P = (0, · · · , 0, 2) and distinguish two cases (note that one of these inequalities is always satisfied): either Hα,en (P + ben) + α(a − b) 2 ≤ u(P), or Hα,en (P + aen) − α(a − b) 2 ≥ u(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Since the argument in both cases is completely symmetric we only consider the second case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' If we set ǫ = a − b, 19 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='7 and by arguing as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='8 we deduce the existence of a constant τ = τ(n, p) such that u(x) ≤ α(1 − τǫ)(xn + a)+ − β(xn + a)−, in B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We let ϕ as in (25) and set ft(x) = α � 1 − τǫ 2 � (xn + a − ctǫϕ)+ − β(xn + a − ctǫϕ)−, where c = c(n, p) is a constant chosen such that u(x) ≤ f1(x) ≤ ft(x), for all x ∈ B 1 100 (Q), t ∈ [0, 1], where, Q = (0, · · · , 0, 1 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' As in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='8, we let t and x be the first contact time and the first contact point and we aim to show that t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' For this purpose, we note that, by the same arguments as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='8, necessarily x ∈ {u = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We claim that x ∈ ΓTP = ∂Ω+ u ∩ ∂Ω− u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Indeed, otherwise x ∈ ∂Ω− u \\ ∂Ω+ u (the case x ∈ ∂Ω+ u \\ ∂Ω− u will be impossible since ft is negative in a neighborhood of x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' And by definition of viscosity solution, this along with (2) would imply λp − ≥ |∇f − t (x)|p = βp(1 − pctǫ∂nϕ(x) + O(ǫ2)) ≥ (λp − − λp + + αp)(1 − pctǫ∂nϕ(x) + O(ǫ2)) ≥ (λp − − λp + + (max(λ+, L0) + Mǫ)p)(1 − pctǫ∂nϕ(x) + O(ǫ2)) = λp − + p(Lp−1 0 M − ct∂nϕ(x))ǫ + O(ǫ2), where the implicit constants in O(ǫ2) can control by L1, p and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This inequality is impossible if M is chosen sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Hence x ∈ ∂Ω+ u ∩ ∂Ω− u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This however implies: λp + − λp − ≤ |∇f + t (x)|p − |∇f − t (x)|p = � αp � 1 − τǫ 2 �p − βp � � 1 − pctǫ∂nϕ(x) + O(ǫ2) � < αp − βp = λp + − λp −, provided ǫ and as a consequence of ǫ = a − b ≤ ǫ, ǫ is chosen small enough, where we have used (ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3) and the equality 0 ≤ λp + − λp − = αp − βp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This contradiction shows that t = 1 and as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='8, this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ With Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='7 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='8 at hand the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4 is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We distinguish two cases: Case 0 ≤ ℓ < +∞: By triangular inequality we have ∥uk − Hλ+,en∥L∞(B1) ≤ ǫk � 1 + 2ℓ max(λ1−p + , λ1−p − ) � , 20 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BAYRAMI AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' FOTOUHI for k sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Define the bounded sequence wk by wk(x) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 w+,k(x) := uk(x) − λ+x+ n αkǫk x ∈ Ω+ uk ∩ B1, w−,k(x) := uk(x) + λ−x− n βkǫk x ∈ Ω− uk ∩ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now we can repeatedly apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='8 to deduce that wk satisfies (32) |wk(x) − wk(y)| ≤ C|x − y|γ, when x, y ∈ B 1 2 , and |x − y| ≥ ǫk ǫ , for some universal exponent 0 < γ < 1 and constant C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' see [13, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This gives that the graphs of ˜Γ± k := {(x, w±,k(x)) : x ∈ Ω± uk ∩ B 1 2 }, converge, up to a subsequence, in the Hausdorff distance to the closed graphs ˜Γ± := {(x, w±(x)) : x ∈ B± 1 2 }, where w ∈ C0,α for some α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Since hk(x) := Hαk,en − Hλ+,en ǫk → \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 λ1−p + ℓxn xn > 0, λ1−p − ℓxn xn < 0, the original sequence vk satisfies that their graphs, converges to the graph of a limiting function v as we wanted, this in particular proves (i), (ii), and (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Since 0 ∈ ∂Ω+ uk ∩ ∂Ω− uk then 0 is in the domain of v±,k and v±,k(0) = 0, which implies that v±(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' To show that v+(x) ≤ v−(x) for x = (x′, 0) ∈ B 1 2 , we simply exploit (iii) at the points x± k = (x′, t± k ) where t+ k = sup{t : (x′, t) ∈ ∂Ω+ uk} and t− k = inf{t : (x′, t) ∈ ∂Ω− uk}, and by noticing that t− k ≤ t+ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Finally, to see the last claim, (19), it is enough to note that if xk ∈ ∂Ω+ uk ∩ ∂Ω− uk is converging to x then v+,k(xk) = v−,k(xk) and thus v+(x) = v−(x), yielding x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Case ℓ = ∞: In this case, the conclusion follows exactly with a similar argument by using repeatedly Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='10 for function vk to obtain a relation similar to (32) for functions vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The linearized problem: proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='5 proves through the following technical lemma, whose proof is easily obtained by adapting the one in [12, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='10] exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then we present the statement without proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let uk, ǫk and αk be as in the statement of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4, vk be defined by (18) and v± be as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then: (1) Let P+ be a quadratic polynomial with Lp(P+) > 0 (or Lp(P+) < 0) on B+ 1 2 touching v+ strictly from below (above) at a point x0 ∈ {xn = 0} ∩ B 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then, there exists 21 a sequence of points ∂Ω+ uk ∋ xk → x0 and a sequence of comparison functions Qk such that Qk touches from below (above) u+ k at xk, and such that (33) ∇Q+ k (xk) = αken + ǫkαk∇P+(x0) + o(ǫk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (2) Let P− be a quadratic polynomial with Lp(P−) > 0 (Lp(P−) < 0) on B− 1 2 touching v− strictly from below (above) at a point x0 ∈ {xn = 0} ∩ B 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then, there exists a sequence of points ∂Ω− uk ∋ xk → x0 and a sequence of comparison functions Qk such that Qk touches from below (above) −u− k at xk, and such that (34) ∇Q− k (xk) = −βken + ǫkβk∇P−(x0) + o(ǫk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (3) Let s, t ∈ R and ˜P be a quadratic polynomial on B 1 2 such that ∂n ˜P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Suppose that Lp( ˜P) ≥ 0 (Lp( ˜P) ≤ 0) and that the function P := sx+ n − tx− n + ˜P, touches v strictly from below (above) at a point x0 ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then, there exists a sequence of points xk → x0 and a sequence of comparison functions Qk such that Qk touches from below (above) the function uk at xk ∈ ∂Ωuk, and such that (35) ∇Q+ k (xk) = αk(1 + ǫks)en + o(ǫk), ∇Q− k (xk) = −βk(1 + ǫkt)en + o(ǫk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In particular, if s > 0 and Qk touches uk from below then xk � ∂Ω− uk \\ ∂Ω+ uk, while if t < 0 and Qk touches uk from above then xk � ∂Ω+ uk \\ ∂Ω− uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Step 1: In this step, we prove Lp(v±) = 0 in B± 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let P(x) be a quadratic polynomial touching v = v+ at x ∈ B+ 1 2 strictly from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We need to show that at this point Lp(P) = ∆P + (p − 2)∂nnP ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Since v+,k → v+, there exist points xk ∈ Ω+ uk ∩ B 1 2 , xk → x and constants ck → 0 such that (36) v+,k(xk) = P(xk) + ck, and (37) v+,k ≥ P + ck, in a neighborhood of xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' From the definition of v+,k, (36) and (37) read uk(xk) = Qk(xk), and uk(x) ≥ Qk(x), in a neighborhood of xk, where Qk(x) = ǫkαk(P(x) + ck) + αkx+ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Note that (38) ∇Qk = ǫkαk∇P + αken, thus, (39) ∇Qk(xk) � 0, for k large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 22 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BAYRAMI AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' FOTOUHI Since uk is p-harmonic and Qk touches uk from below at xk, and ∇Qk(xk) � 0, by the equivalence of weak and viscosity solutions of p-harmonic functions, we get 0 ≥ ∆pQk(xk) = div � |∇Qk(xk)|p−2∇Qk(xk) � = |∇Qk(xk)|p−2 ∆Qk(xk) + (p − 2) |∇Qk(xk)|p−4 n � i,j=1 Qkxi(xk)Qkxj(xk)Qkxixj(xk) = ǫk |∇Qk(xk)|p−2 ∆P(xk) + ǫk(p − 2) |∇Qk(xk)|p−4 n � i,j=1 Qkxi(xk)Qkxj(xk)Pxixj(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now, dividing both sides by ǫk, and passing to the limit k → ∞, and recalling that ∇Qk(xk) → αken, we conclude that ∆P(x) + (p − 2)∂nnP(x) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Touching from above and reaching the opposite inequality is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Also, the reasoning of the case v = v− in the negative half ball B− 1 2 can be done similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Step 2: In this step, we show that J = ∅, when ℓ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Assume the contrary, since the set {v− > v+} is open in {xn = 0}, it contains a (n − 1)-dimensional ball B′ ǫ(y′) := Bǫ((y′, 0)) ∩ {xn = 0} ⊂ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Next, let P be the polynomial P(x) = A � n − 1 2 � x2 n − |x′ − y′|2 − Bxn, where x = (x′, xn), for some constants A, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We first choose suitable A = A(p) so that Lp(P) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Notice that P < v+ on {|x′ − y′| = ǫ} ∩ {xn = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Moreover, we choose B ≫ A so that P < v+ on Bǫ((y′, 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now we can translate P first down and then up to find that there exists C such that P + C is touching v+ from below at a point x0 ∈ Bǫ((y′, 0)) ∩ {xn ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Since Lp(P) > 0, the touching point can not be in the interior of the (half) ball, and thus x0 ∈ B′ ǫ(y′) ⊂ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='11, there exists a sequence of points ∂Ω+ uk ∋ xk → x0 and of functions Qk touching u+ k from below at xk such that ∇Q+ k (xk) = αken + ǫkαk∇P(x0) + o(ǫk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Since x0 ∈ J, by (19) in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4, xk ∈ ∂Ω+ uk \\ ∂Ω− uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Hence, by (ii) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2 λp + ≥ |∇Q+ k (xk)|p ≥ αp k + pαp kǫk∂nP(x0) + o(ǫk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now recalling (17), the definition of ℓ, −B = ∂nP(x0) ≤ λp + − αp k pαp kǫk + o(1) → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This contradiction proves that J = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 23 Step 3: In this step, we check the transmission condition in (21) when ℓ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let us show that αp ∞∂nv+ − βp ∞∂nv− ≤ 0, the opposite inequality can then be proved in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Suppose that there exist s and t with αp ∞s > βp ∞t and a polynomial ˜P with Lp( ˜P) > 0 and ∂n ˜P = 0 such that P = sx+ n − tx− n + ˜P, touches v strictly from below at a point x0 ∈ {xn = 0}∩B 1 2 (note that {xn = 0}∩B 1 2 = C due to the previous step and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='11 there exists a sequence of points ∂Ω+ uk ∪ ∂Ω− uk ∋ xk → x0 and a sequence of comparison functions Qk touching uk from below at xk and satisfying (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In particular, xk � ∂Ω− uk \\ ∂Ω+ uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We claim that xk ∈ ∂Ω+ uk ∩ ∂Ω− uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Indeed, otherwise by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1, λp + ≥ |∇Q+ k (xk)|p, and, by arguing as Step 2, this contradicts ℓ = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Hence, by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1 λp + − λp − ≥ |∇Q+ k (xk)|p − |∇Q− k (xk)|p = αp k − βp k + pǫk(αp ks − βp kt) + o(ǫk) = λp + − λp − + pǫk(αp ks − βp kt) + o(ǫk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Dividing by ǫk and letting k → ∞, we obtain the desired contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Step 4: Here, we show that λp ±∂nv± ≥ −ℓ on B 1 2 ∩ {xn = 0}, when 0 ≤ ℓ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We focus on v− since the argument is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let us assume that there exists t ∈ R with λp −t < −ℓ and a polynomial ˜P with Lp( ˜P) > 0 and ∂n ˜P = 0 such that function P = txn + ˜P = tx+ n − tx− n + ˜P, touches v− strictly from below at a point x0 ∈ {xn = 0} ∩ B 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let now xk and Qk be as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='11-(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By optimality conditions λp − ≤ |∇Q− k (xk)|p = βp k + pǫkβp kt + o(ǫk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Since ℓ < ∞, we have βk = λ− + O(ǫk) and so the above inequality leads to − ℓ λp − = lim k→∞ λp − − βp k pǫkβp k ≤ t < − ℓ λp − , which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Step 5: We now show that λp ±∂nv± = −ℓ on J, when 0 ≤ ℓ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By the previous step, it is enough to show that if there exists a polynomial ˜P with Lp( ˜P) < 0 and ∂n ˜P = 0 such that P = txn + ˜P = tx+ n − tx− n + ˜P, touches v− strictly from above at a point x0 ∈ J, then λp −t ≤ −ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Again, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='11, we find points xk → x0 and functions Qk satisfying (34) and touching −u− k from below at xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Since x0 ∈ J, by (19) in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4, xk ∈ ∂Ω− uk \\ ∂Ω+ uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Hence, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1, λp − ≥ |∇Q− k (xk)|p = βp k + pβp kǫkt + o(ǫk), which by arguing as above implies that λp −t ≤ −ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 24 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BAYRAMI AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' FOTOUHI Step 6: In the last step, we show the transmission condition in (22) at points in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Again by the symmetry of the arguments, we will only show that λp +∂nv+ − λp −∂nv− ≤ 0 on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let us hence assume that there exist s and t with λp +s > λp −t and a polynomial ˜P with Lp( ˜P) > 0 and ∂n ˜P = 0 such that P = sx+ n − tx− n + ˜P, touches v+ and v− strictly from below at x0 ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='11, we find points xk → x0 and functions Qk satisfying (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In particular xk � ∂Ω− uk \\ ∂Ω+ uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By the previous step we know that λp −t ≥ −ℓ and thus λp +s > −ℓ, since we are assuming λp +s > λp −t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We now distinguish two cases: 1) xk is one-phase point, namely xk ∈ ∂Ω+ uk \\ ∂Ω− uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In this case λp + ≥ |∇Q+ k (xk)|p = αp k + pαp kǫks + o(ǫk), which implies that λp +s + ℓ = λp + lim k→∞ \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8eds + αp k − λp + pαp kǫk \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ≤ 0, in contradiction with λp +s > −ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 2) xk is two-phase point, namely xk ∈ ∂Ω+ uk ∩ ∂Ω− uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Arguing as in Case 1), we have that, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1, λp + − λp − ≥ |∇Q+ k (xk)|p − |∇Q− k (xk)|p = αp k − βp k + pǫk(αp ks − βp kt) + o(ǫk) = λp + − λp − + pǫk(λp +s − λp −t) + o(ǫk), which gives a contradiction with λp +s > λp −t, as ǫk → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proof of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We recall the following regularity results for the limiting problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='12 (Regularity for the transmission problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' There exists a universal constant C = C(α∞, β∞, n, p) > 0 such that if v ∈ C0(B 1 2 ) is a viscosity solution of (21) with ∥v∥L∞(B 1 2 ) ≤ 1 then there exist v ∈ Rn−1, s, t ∈ R with αp ∞s = βp ∞t such that sup x∈Br ���v(x) − v(0) − (v · x′ + sx+ n − tx− n) ��� ≤ Cr2, for every r ≤ 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' For the proof when p = 2, we refer to [13, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This result can be extended easily to the general case (for any p) by changing the coordinate such that the operator Lp = ∆ + (p − 2)∂nn transfer to the Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='13 (Regularity for the two-membrane problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' There exists a universal constant C = C(λ±, n, p) > 0 such that if v is a viscosity solution of (22) with ∥v∥L∞(B 1 2 ) ≤ 1 then there exist v ∈ Rn−1, s, t ∈ R with λp +s = λp −t ≥ −ℓ such that sup x∈B± r ���v(x) − v(0) − (v · x′ + sx+ n − tx− n) ��� ≤ C(1 + ℓ)r 3 2 , for every r ≤ rp, 25 where rp = 1 4 for 1 < p ≤ 2 and rp = 1 4√ p−1 for 2 < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The proof of this lemma can be found in [12, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='12] with a minor changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' To keep the paper self-contained we will provide a complete proof for our case in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now, the proof of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3 by the regularity theory for the limiting problems and a classical compactness argument is available: Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Toward a contradiction assume that for fixed γ ∈ (0, 1 2) and M, we have a sequences of functions uk and numbers αk such that ǫk = ∥uk − Hαk,en∥L∞(B1) → 0, and λ+ ≤ αk ≤ λ+ + Mǫk, and fail (14) and (15) for some ρ and C which will be determined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Note that by the second assumption above ℓ < Mλp−1 + < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We let (vk)k be the sequence of functions defined in (18) and assume that they converge to a function v as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4, note that ∥v∥L∞(B 1 2 ) ≤ max( 1 λ+ , 1 λ− ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='5, v solves (22) and thus by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='13 there exist v ∈ Rn−1, s, t ∈ R satisfying λp +s = λp −t ≥ −ℓ such that for all r ∈ (0, rp) sup x∈Br ���v(x) − (v · x′ + sx+ n − tx− n) ��� ≤ C(1 + M)r 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Hence, we can fix ρ = ρ(λ±, γ, L, M, p, n) < rp such that C(1 + M)ρ 1 2 −γ ≤ 1 2, so (40) sup x∈Bρ ���v(x) − (v · x′ + sx+ n − tx− n) ��� ≤ ρ1+γ 2L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We now set ˜αk := αk(1 + ǫks) + δkǫk and ek := en + ǫkv � 1 + ǫ2 k|v|2 , where δk → 0 is chosen so that ˜αk ≥ λ+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' note that the existence of such sequence is due to the condition λp +s ≥ −ℓ since αk(1 + ǫks) = \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8edλ+ + ℓ λp−1 + ǫk + o(ǫk) \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 (1 + ǫks) ≥ λ+ + o(ǫk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We let Hk := H ˜αk,ek and note that | ˜αk − αk| + |ek − en| ≤ Cǫk, for a universal constant C > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' we also have used (40) to find out that s is universally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By the contradiction assumption we have ρ1+γ < 1 ǫk sup Bρ |uk(x) − Hk(x)| ≤ max � αk∥v+ k − Hk − Hαk,en ǫkαk ∥L∞(Ω+uk ∩Bρ), βk∥v− k − Hk − Hαk,en ǫkβk ∥L∞(Ω− uk ∩Bρ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 26 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BAYRAMI AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' FOTOUHI To close the argument, we need to recall (40), the convergence of vk to v in the sense of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4 and the convergence of (again in the sense of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4) \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 Hk(x) − Hαk,en(x) αkǫk xn > 0, Hk(x) − Hαk,en(x) βkǫk xn < 0, to the function (v · x′) + sx+ n − tx− n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Arguing by contradiction one assume for fixed γ ∈ (0, 1) the existence of a sequence of functions uk and numbers αk, Mk → ∞ such that ǫk = ∥uk − Hαk,en∥L∞(B1) → 0, and αk − λ+ ǫk ≥ Mk → ∞, and fail (14) and (15) for some ρ and C which will be determined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This implies that ℓ = ∞ and that the limiting function v obtained in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4 is a solution of (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' One then concludes the proof similar to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2 by using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Regularity of the free boundary The last step in achieving the desired regularity result is to demonstrate that |∇u±| are C0,η for a suitable η > 0 up to the boundary, in the viscosity sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Indeed, this shows that u± are solutions to the classical one-phase free boundary problem in its viscosity formulation and that the regularity will follow form [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The arguments are similar to the ones in [25, Section 8] (see also [12, Section 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Therefore we only sketch the main steps and refer the reader to that paper for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Before stating the main results, we introduce some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' For every x0 ∈ F(u) and every 0 < r < dist(x0, ∂D), we consider the function ux0,r(x) := u(x0 + rx) r , which is well-defined for |x| < 1 r dist(x0, ∂D) and vanishes at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' When x0 = 0, we denote u0,r by ur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Given a sequence rk > 0 such that rk → 0, we say that the sequence of functions ux0,rk is a blow-up sequence of u at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' If a subsequence of ux0,rk convergs to v on every ball BR ⊂ Rn, we say that v is a blow-up limit of u at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' There exists ¯ǫ > 0 such that if the minimizer u satisfies (3), then at every point x0 ∈ ΓTP ∩ Br0 for a universal radius r0 > 0, there is a unique blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Moreover, u is Lipschitz in Br0/2 and there exists η > 0 and a constant C0(n, p, Λ0, Λ1) > 0 such that for every x0, y0 ∈ ΓTP ∩ Br0/2, we have (41) |α(x0) − α(y0)| ≤ C0|x0 − y0|η and |e(x0) − e(y0)| ≤ C0|x0 − y0|η, for any η ∈ (0, 1 3), where Hα(x0),e(x0) and Hα(y0),e(y0) are the blow-ups at x0 and y0, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 27 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let L0 = Λ0 and L1 = 2Λ1 in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1 and find the universal constants ǫ0 > 0, ρ0 > 0 and C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Choose ¯ǫ < min{(1 − ργ) Λ1 2C, ǫ 2} and r0 < ¯ǫ Λ1 , then if the minimizer u satisfies (3) for some e ∈ Sn−1 and λ+ ≤ α ≤ Λ1, then ∥ux0, 1 2 − Hα,e∥L∞(B1) ≤ ¯ǫ + |Hα,e(x0)| ≤ 2¯ǫ, for any x0 ∈ ΓTP ∩ Br0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now we can thus repeatedly apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1 to obtain the sequences ux0, ρk 2 (x) = 2 ρk u(x0 + ρk 2 x), max(L0, λ+) ≤ αk ≤ L1 and ek ∈ Sn−1 that ∥ux0, ρk 2 − Hαk,ek∥L∞(B1) ≤ 2¯ǫρkγ, |ek+1 − ek| + |αk+1 − αk| ≤ 2C¯ǫρkγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This implies that αk and ek converge to some α = α(x0) and e = e(x0), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now let r ≤ 1 2 be arbitrary and choose k ∈ N such that ρk+1 ≤ 2r ≤ ρk, then ∥ux0,r − Hα(x0),e(x0)∥L∞(B1) ≤ 1 ρ∥ux0, ρk 2 − Hα(x0),e(x0)∥L∞(B1) ≤ 1 ρ � ∥ux0, ρk 2 − Hαk,ek∥L∞(B1) + ∥Hαk,ek − Hα(x0),e(x0)∥L∞(B1) � ≤ C¯ǫρkγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Therefore, there is ˜C = ˜C(n, p, Λ0, Λ1) such that for every r ≤ 1 2 and x0 ∈ Br0, (42) ∥ux0,r − Hα(x0),e(x0)∥L∞(B1) ≤ ˜Crγ, where γ ∈ (0, 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' According to (42), ∥u∥L∞(Br(x0)) ≤ (L1 + ˜C)r, r ≤ 1 2, for every x0 ∈ ΓTP ∩ Br0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' From this and the Lipschitz regularity around the one- phase points, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3, we conclude that u is Lipschitz in B r0 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' see [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3] or [11, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Next, for x0, y0 ∈ ΓTP ∩ B r0 2 set r := |x0 − y0|1−η and η := γ 1+γ, and recall that u is Lipschitz (with a constant ˜L) to get ∥Hα(x0),e(x0) − Hα(y0),e(y0)∥L∞(B1) ≤ ∥ux0,r − Hα(x0),e(x0)∥L∞(B1) + ∥ux0,r − uy0,r∥L∞(B1) + ∥uy0,r − Hα(y0),e(y0)∥L∞(B1) ≤ � C0rγ + ˜L r |x0 − y0| + C0rγ � = (˜L + 2C0)|x0 − y0|η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The conclusion now follows easily from this inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [25, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='8] for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Under the same assumptions of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1, there are C0,η continuous func- tions α : ∂Ω+ u → R and β : ∂Ω− u → R such that α ≥ λ+, β ≥ λ− and u± are viscosity solutions of the one-phase problems ∆pu+ = 0 in Ω+ u, |∇u+| = α on ∂Ω+ u, and ∆pu− = 0 in Ω− u, |∇u−| = β on ∂Ω− u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 28 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BAYRAMI AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' FOTOUHI Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We will sketch the argument for u+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The proof of the case u− is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Clearly ∆pu+ = 0 in Ω+ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By (42) we have that, if x0 ∈ ΓTP ∩ D′, then (43) ���u+(x) − α(x0) ((x − x0) · e(x0))+��� ≤ C0|x − x0|1+γ, for every x ∈ Br0(x0) ∩ Ω+ u where r0 and C0 depends only on D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In particular, u+ is differentiable on Ω+ u up to x0 (in the classical sense) and |∇u+(x0)| = α(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' On the other hand if x0 ∈ Γ+ OP, then |∇u+(x0)| = λ+ is constant, in the viscosity sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' To close the argument, we only need to prove that α ∈ C0,η(∂Ω+ u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Since α is η-H¨older continuous on ΓTP by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1, and constant on Γ+ OP (in the viscosity sense), we just need to show that if x0 ∈ ΓTP is such that there is a sequence xk ∈ Γ+ OP converging to x0, then α(x0) = λ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' To this end, let yk ∈ ΓTP be such that dist(xk, ΓTP) = |xk − yk|, and denote rk = |xk − yk| and uk(x) = 1 rk u+(xk + rkx), and note that uk is a viscosity solution of the free boundary problem ∆puk = 0 in Ω+ uk ∩ B1, |∇uk| = λ+ on ∂{uk > 0} ∩ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Since uk are uniformly Lipschitz in B 1 2 (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='3) they converge to a function u∞ which is also a viscosity solution of the same problem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' On the other hand, by (43) for two-phase point yk ∈ ΓTP and letting zk := xk−yk rk , we have that ���uk(x) − α(yk) �(x − zk) · e(yk)�+��� ≤ C0rγ k|x − zk|1+γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Suppose zk → z0 and passing to the limit u∞(x) = α(x0) ((x − z0) · e(x0))+ , in B 1 2 , which gives that α(x0) = |∇u∞(0)| = λ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let x0 ∈ ΓTP = ∂Ω+ u ∩ ∂Ω− u and let ǫ be the constant satisfies in [17, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1] and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By virtue of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2, we can apply [17, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1] to conclude that locally at x0 ∈ ΓTP the free boundaries ∂Ω± u are C1,η graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Since x0 is arbitrary, we conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Lipschitz regularity of solutions In this section, we are going to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We will follow the idea in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let u : D → R be a minimizer of JTP that 0 ∈ F(u) ∩ B1 ⊂ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then there exists constants L and δ such that one of the following alternative holds: (1) u is Lipschitz in Bδ and |∇u| ≤ C max(∥u∥L∞(B1), L), in Bδ, for some universal constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (2) 1 δ∥u∥L∞(Bδ) ≤ 1 2 max(∥u∥L∞(B1), L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 29 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let δ be fixed, to be specified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Assume by contradiction that there exist a sequence of Lj → ∞ and a sequence of solutions uj such that does not satisfy either (1) nor (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let Cj := max(∥uj∥L∞(B1), Lj) and define ˜uj := uj Cj , which satisfy ∥ ˜uj∥L∞(B1) ≤ 1, and ∥ ˜uj∥L∞(Bδ) ≥ δ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='5, we get that ˜uj is a minimizer of the scaled functional (4) for σj = 1 Cj → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Thus up to a subsequence, ˜uj converges uniformly to a p-harmonic function u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Hence by C1,α regularity for p-harmonic functions we get that (44) sup Br |u0(x) − ∇u0(0) · x| ≤ ˜Cr1+α, for all r ≤ 1, where the constant C is universal and also |∇u0(0)| ≤ ˜C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now we distinguish two cases: Case I: |∇u0(0)| ≤ 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In this case, from (44) we deduce that 1 δ∥u0∥L∞(Bδ) ≤ 1 4 + ˜Cδα ≤ 1 3, if we choose δ small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Thus all uj for sufficiently large j will satisfy (2), which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Case II: |∇u0(0)| ≥ 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In this case we will use our flatness result in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Put ˜r = 2δ r0 in (44) where r0 is the radius obtained in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1 (we have also assumed 2δ ≤ r0) sup B1 ��� ˜uj,˜r(x) − ∇u0(0) · x ��� ≤ ˜C rα 0 δα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now let e = ∇u0(0) |∇u0(0)|, α = |∇u(0)| and βj = 1 Cj (λp − − λp + + αpCp j) 1 p , then ∥uj,˜r − Hα,e∥L∞(B1) ≤ ∥uj,˜r − ∇u0(0) · x∥L∞(B1) + |α − βj| ≤ 2 ˜C rα 0 δα, for sufficiently large j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Applying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1 for some Λ0 ≤ 1 4 and Λ1 ≥ ˜C and notice that uj,˜r is a minimizer of JTP for coefficients 1 Cj λ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Note that the critical flatness in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1 or Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1 depends on Λ0 and Λ1 rather than coefficients λ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then we can find δ universally small such that uj,˜r satisfy in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In particular, uj,˜r is Lipschitz in B r0 2 with a universal constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' It proves that uj is Lipschitz in Bδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let δ, C and L be the universal constants in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Assume 0 ∈ F(u) and let ˜L := max(∥u∥L∞(B1), L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We first show (45) ∥u∥L∞(Bδk) ≤ C˜Lδk, ∀k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1 either (1) or (2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In the first case, u is Lipschitz in Bδ and |∇u| ≤ C˜L in Bδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Thus (45) holds for all k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 30 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BAYRAMI AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' FOTOUHI If (2) holds, then ∥u∥L∞(Bδ) ≤ ˜L 2 δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We now rescale and iterate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Define uk(x) := u(δkx) δk , which is also a minimizer of JTP and we can apply Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' If k0 is the smallest k for which uk satisfies (1), then for 0 ≤ k < k0 the item (2) holds and so ∥u∥L∞(Bδk) ≤ ˜Lδk, for 0 ≤ k ≤ k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Moreover, uk0 is Lipschitz in Bδ, with |∇uk0| ≤ C max(∥uk0∥L∞(B1), L) ≤ C max(˜L, L) = C˜L, in Bδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Hence, (45) holds for all k ≥ k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' If uk satisfy the alternative (2) for all k, the estimate (45) will be obtained easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now for an arbitrary r choose k such that δk+1 ≤ r ≤ δk, then by (45) we get ∥u∥L∞(Br) ≤ ∥u∥L∞(Bδk) ≤ C˜Lδk ≤ C˜L δ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This is enough to obtain the Lipschitz continuity locally in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='5 Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By definition of vj and an easy computation, we get ∇vj(x) = rj Sj ∇uj(xj + rjx) = σj∇uj(xj + rjx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' In order to show that vj is a minimizer of ˆJTP in BR, consider w that ˆJTP(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BR) < ˆJTP(vj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BR), and w = v on ∂BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then ˆw(x) = w( x−xj rj ) will satisfy ˆw = uj on ∂Brj(xj) and by a simple calculation, we get that JTP( ˆw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Brj) = rn j σp j ˆJTP(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BR) < rn j σp j ˆJTP(vj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BR) = ˆJTP(uj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Brj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This is a contradiction with the minimality of uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Moreover, using |vj| ≤ M in B 4R 3 and Caccioppoli’s inequality, we conclude that � BR |∇v± j |p dx ≤ 4pC(n) � B 4R 3 (v± j )p dx ≤ (4M)pC(n), for some C(n) > 0, indicating that ∥vj∥W1,p(BR) are uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Hence, from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' the BMO estimate for the gradient, we obtain that for any q > 1 and 0 < R < 1 rj there exists a constant C = C(R, q) > 0 independent of j such that max � ∥vj∥Cα(BR), ∥∇vj∥Lq(BR) � ≤ C, for some α ∈ (0, 1) (if q > n, one can take α = 1 − n q by the Morrey’s inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Therefore, by a standard compactness argument, we have that, up to a subse- quence, vj converges to some function v0 as j → +∞ in Cα(BR) and weakly in W1,q(BR) for any q > 1, and for any fixed R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This completes the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 31 For obtaining (ii), firstly, we prove that ∆pv0 = 0 in the positivity set of v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Let E ⋐ {v0 > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Then, there exists c > 0 such that v0 ≥ 2c in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By the uniform convergence of vj to v0, we will have vj > c in E for large j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This implies that v0 is p-harmonic in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Since E was arbitrary, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now, take 0 ≤ ϕ ∈ C1 c(Rn) and s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By using (v0 −s)+ϕ as a test function in the weak formulation of ∆pv0 = 0 in the set {v0 > 0}, we have � {v0>s} |∇v0|pϕ dx = − � {v0>s} |∇v0|p−2 �∇v0 · ∇ϕ� v0 dx + s � {v0>s} |∇v0|p−2∇v0 · ∇ϕ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Letting s → 0 gives that � {v0>0} |∇v0|pϕ dx = − � {v0>0} |∇v0|p−2 �∇v0 · ∇ϕ� v0 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Similar argument holds for using test function (v0 + s)−ϕ and finally we get (46) � Rn |∇v0|pϕ dx = − � Rn |∇v0|p−2 �∇v0 · ∇ϕ� v0 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' On the other hand since vjϕ∆pvj ≥ 0 (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='1), we have (47) � Rn |∇vj|pϕ dx ≤ − � Rn |∇vj|p−2 � ∇vj · ∇ϕ � vj dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Usingtheuniformconvergenceofvj tov0 andtheweakconvergenceof|∇vj|p−2∇vj ⇀ |∇v0|p−2∇v0 in L p p−1 loc (Rn) (see [15]), we infer from (46) and (47) that (48) lim sup j→+∞ � Rn |∇vj|pϕ dx ≤ � Rn |∇v0|pϕ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Since also ∇vj ⇀ ∇v0 weakly in Lp loc(Rn), we have (49) � Rn |∇v0|pϕ dx ≤ lim inf j→+∞ � Rn |∇vj|pϕ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' It follows from (48), (49), and a simple compactness argument that (50) |∇vj|p → |∇v0|p, strongly in L1 loc(Rn), and we get (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Finally, we prove the claim (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' For this, notice that for any ψ ∈ C∞ c (BR) (51) � BR |∇vj|p + σp j(p − 1)λp +χ{vj>0} + σp j(p − 1)λp −χ{vj<0} dx ≤ � BR |∇(vj + ψ)|p + σp j(p − 1)λp +χ{vj+ψ>0} + σp j(p − 1)λp −χ{vj+ψ<0} dx, because vj is a minimizer for ˆJTP defined in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Recall the strong convergence (50) along with the following standard inequality |∇(vj + ψ)|p ≤ 2p−1 � |∇vj|p + |∇ψ|p� , we get � BR |∇(vj + ψ)|p dx → � BR |∇(v0 + ψ)|p dx, 32 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BAYRAMI AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' FOTOUHI as j → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Thus passing (51) to limit, we have � BR |∇v0|p + σp(p − 1)λp +χ{v0>0} + σp(p − 1)λp −χ{v0<0} dx ≤ � BR |∇(v0 + ψ)|p + σp(p − 1)λp +χ{v0+ψ>0} + σp(p − 1)λp −χ{v0+ψ<0} dx, for any ψ ∈ C∞ c (BR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This implies (iii) and the proof of proposition finishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proof of the regularity for two-membrane problem Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' For the given solution v, define w w±(x) = v±(x′, (p − 1) 1 2 xn) + ℓ(p − 1) 1 2 λp ± xn, x ∈ B± 2rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' It is straightforward to check that w± is a viscosity solution of \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 Lp(w±) = 0, in B± 2rp, ∂nw± ≥ 0, in B2rp ∩ {xn = 0}, ∂nw± = 0, in J = {w+ < w−} ∩ {xn = 0}, λp +∂nw+ = λp −∂nw−, in C = {w+ = w−} ∩ {xn = 0}, w+ ≤ w−, in B2rp ∩ {xn = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Furthermore one can easily check that (52) w±(x′, xn) = ˜w(x′, ∓xn) ∓ 1 λp ± wS(x′, ∓xn), where ˜w solves the following Neumann problem \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3 ∆ ˜w = 0, on B− 2rp, ∂n ˜w = 0, on B− 2rp ∩ {xn = 0}, and wS is a solution to the thin obstacle (the Signorini) problem \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 ∆wS = 0, on B− 2rp, wS ≥ 0, on B− 2rp ∩ {xn = 0}, ∂nwS ≥ 0, on B− 2rp ∩ {xn = 0}, wS∂nwS = 0, on B− 2rp ∩ {xn = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' The boundary data of ˜w and wS on ∂B2rp ∩ {xn < 0} will be obtained uniquely from (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Clearly ˜w ∈ C∞(B− rp) with ∥ ˜w∥Ck(B− rp) ≤ Ck∥ ˜w∥L∞(B− 2rp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' On the other hand, by [8], wS ∈ C1, 1 2 (B− rp) with ∥wS∥C1, 1 2 (B− rp) ≤ C∥wS∥L∞(B− 2rp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 33 From the last two estimates and the definition of w, it is easy to deduce the con- clusion of the lemma for (note that the positivity of wS along with its regularity necessitates that ∇′wS(0) = 0) v := ∇′ ˜w(0) and s± := (p − 1)− 1 2 λp ± ∂nwS(0) − ℓ λp ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proof of non-degeneracy Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' We will prove that for any k ∈ (0, 1), there exists a constant ck > 0 such that for any local minimizer of JTP and for any small ball Br(x0) ⊂ D if 1 r � ⧸ � Br(x0) (u±)p dx � 1 p < ck then u± ≡ 0 in Bkr(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By symmetry of the problem, we prove only the case u+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Also, by the scale invariance, we can take r = 1 and x0 = 0 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Now, let define ε := 1√ k sup B √ k u+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Since u+ is p-subharmonic, then by [22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='9] ε ≤ 1√ k C(n, p) (1 − √ k) n p � ⧸ � B1 (u+)p dx � 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Also, let v(x) := \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3 C1ε � e−µ|x|2 − e−µk2� in B √ k \\ Bk, 0 in Bk, where µ > 0 and C1 are such that (53) v��� ∂B √ k := √ kε = sup B √ k u+ ≥ u��� ∂B √ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' By direct computation, it is straightforward to check that ∇v(x) = −2C1εµxe−µ|x|2 in B √ k \\ Bk, and ∆pv(x) = C1ε(p − 1)(2µ)2|∇v|p−2e−µ|x|2 � |x|2 − n + p − 2 2µ(p − 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Thus v is nonnegative p-superharmonic in B √ k \\ Bk, if µ is sufficiently small, say, (54) µ < n + p − 2 2k(p − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' On the other hand, since w := min(u, v) = u on ∂B √ k, thanks to (53), by invoking the minimality of u we get (55) JTP(u, B √ k) ≤ JTP(w, B √ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 34 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BAYRAMI AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' FOTOUHI Now, since we have JTP(w, B √ k) = � Bk |∇w|p + (p − 1)λp +χ{w>0} + (p − 1)λp −χ{w<0} dx + � B √ k\\Bk |∇w|p + (p − 1)λp +χ{w>0} + (p − 1)λp −χ{w<0} dx = � Bk∩{u≤0} |∇u|p + (p − 1)λp +χ{u>0} + (p − 1)λp −χ{u<0} dx + � B √ k\\Bk |∇w|p + (p − 1)λp +χ{w>0} + (p − 1)λp −χ{w<0} dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' from (55),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' {w > 0} ⊆ {u > 0} and {w < 0} = {u < 0},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' we have that � Bk∩{u>0} |∇u|p + (p − 1)λp +χ{u>0} + (p − 1)λp −χ{u<0} dx ≤ � B √ k\\Bk |∇w|p + (p − 1)λp +χ{w>0} + (p − 1)λp −χ{w<0} dx − � B √ k\\Bk |∇u|p + (p − 1)λp +χ{u>0} + (p − 1)λp −χ{u<0} dx ≤ � B √ k\\Bk |∇w|p − |∇u|p dx = � (B √ k\\Bk)∩{u>v} |∇v|p − |∇u|p dx ≤ − p � B √ k\\Bk |∇v|p−2∇v · ∇ max(u − v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 0) dx = − p � B √ k\\Bk −∆pv max(u − v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 0) + div � |∇v|p−2∇v max(u − v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 0) � dx ≤ − p � B √ k\\Bk div � |∇v|p−2∇v max(u − v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 0) � dx =p � ∂Bk |∇v|p−2(∇v · ν)u+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' where to get the last inequality we have used the fact that v is a p-superharmonic in B √ k \\ Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Moreover, by (54), we have that |∇v| = 2C1εµke−µk2 ≤ Cε on ∂Bk, for some C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Thus (56) � Bk∩{u>0} |∇u|p + (p − 1)λp +χ{u>0} dx ≤ p(Cε)p−1 � ∂Bk u+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 35 On the other hand, from trace estimate, Young’s inequality, we get (57) � ∂Bk u+ ≤ C(n, k) �� Bk u+ dx + � Bk |∇u+| dx � ≤ C(n, k) \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8edsup Bk u+ � Bk χ{u>0} dx + � Bk 1 p|∇u+|p + 1 p′ χ{u>0} dx \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ≤ C(n, k) � (ε √ k + 1 p′ ) � Bk χ{u>0} dx + 1 p � Bk |∇u+|p dx � ≤ C0 � Bk∩{u>0} |∇u|p + (p − 1)λp +χ{u>0} dx, where p′ is the conjugate of p and C0 := C(n, k) � ε √ k + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Finally, putting together (56) and (57), we reach to � Bk(x0)∩{u>0} |∇u|p + (p − 1)λp +χ{u>0} dx ≤ p(Cε)p−1C0 � Bk(x0)∩{u>0} |∇u|p + (p − 1)λp +χ{u>0} dx, which implies that u ≡ 0 in Bk(x0) if ε is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' This completes the proof of the non-degeneracy property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' □ Declarations Data availability statement: All data needed are contained in the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Funding and/or Conflicts of interests/Competing interests: The authors declare that there are no financial, competing or conflict of interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' References [1] Alt, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', and Caffarelli, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Existence and regularity for a minimum problem with free boundary.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', and Friedman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Axially symmetric jet flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Rational Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Scuola Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Sup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Pisa Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (4) 11, 1 (1984), 1–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [4] Alt, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', Caffarelli, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', and Friedman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Jets with two fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} 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Indiana Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 33, 3 (1984), 367–391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [6] Alt, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', Caffarelli, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} 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Compressible flows of jets and cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Differential Equations 56, 1 (1985), 82–141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [8] Athanasopoulos, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', and Caffarelli, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Optimal regularity of lower dimensional obstacle problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Zap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Nauchn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Sem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='-Peterburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Otdel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Steklov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (POMI) 310, Kraev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Zadachi Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' i Smezh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Vopr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Funkts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 35 [34] (2004), 49–66, 226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [9] Caffarelli, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', and Salsa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' A geometric approach tofree boundary problems,vol.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Partial Differential Equations 23, 1 (2005), 97–124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 36 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' BAYRAMI AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' FOTOUHI [11] Danielli, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', Petrosyan, A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [12] De Philippis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', Spolaor, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', and Velichkov, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Regularity of the free boundary for the two-phase Bernoulli problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 225, 2 (2021), 347–394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [13] De Silva, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', Ferrari, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', and Salsa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Two-phase problems with distributed sources: regularity of the free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' PDE 7, 2 (2014), 267–310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [14] De Silva, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', and Savin, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Lipschitz regularity of solutions to two-phase free boundary problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' IMRN, 7 (2019), 2204–2222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [15] DiBenedetto, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', and Manfredi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 115, 5 (1993), 1107–1134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [16] Dipierro, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', and Karakhanyan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Stratification of free boundary points for a two-phase variational problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 328 (2018), 40–81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [17] Ferrari, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', and Lederman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Regularity of flat free boundaries for a p(x)-Laplacian problem with right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Nonlinear Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 212 (2021), Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 112444, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [18] Ferrari, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', and Lederman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Regularity of lipschitz free boundaries for a p(x)-laplacian problem with right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Journal de Math´ematiques Pures et Appliqu´ees (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [19] Fotouhi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', and Shahgholian, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' A minimization problem with free boundary for p-laplacian weakly coupled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' (2023) arxiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='02236 (preprint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [20] Gilbarg, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', and Trudinger, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Elliptic partial differential equations of second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Classics in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Springer-Verlag, Berlin, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Reprint of the 1998 edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [21] Karakhanyan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Regularity for the two-phase singular perturbation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Proceedings of the London Mathematical Society 123, 5 (2021), 433–459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [22] Mal´y, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', and Ziemer, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Fine regularity of solutions of elliptic partial differential equations, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 51 of Mathematical Surveys and Monographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' American Mathematical Society, Providence, RI, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [23] Petrosyan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=', and Valdinoci, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Geometric properties of Bernoulli-type minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Interfaces Free Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' 7, 1 (2005), 55–77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [24] Tolksdorf, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' On the Dirichlet problem for quasilinear equations in domains with conical boundary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Partial Differential Equations 8, 7 (1983), 773–817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' [25] Velichkov, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Regularity of the one-phase free boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Lecture notes available at http://cvgmt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' sns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' it/paper/4367 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content=' Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran Email address: masoud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='bayrami1990@sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='edu Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran Email address: fotouhi@sharif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFKT4oBgHgl3EQfSS3g/content/2301.11775v1.pdf'} diff --git a/69FKT4oBgHgl3EQf_y40/content/tmp_files/load_file.txt b/69FKT4oBgHgl3EQf_y40/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eabaf998790ea3cf00d417aa0d006922031949b5 --- /dev/null +++ b/69FKT4oBgHgl3EQf_y40/content/tmp_files/load_file.txt @@ -0,0 +1,508 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf,len=507 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='11963v1 [hep-th] 27 Jan 2023 On 10 dimensional Exceptional Drinfel’d Algebras Sameer Kumar1, Edvard T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Musaev2 Moscow Institute of Physics and Technology, Institutskii pereulok 9, Dolgoprudny, 141700, Russia Abstract Based on the Mubarakzyanov’s classification of four-dimensional real Lie Algebras, we classify ten-dimensional Exceptional Drinfel’d Algebras (EDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The classifica- tion is restricted to EDAs whose maximal isotropic (geometric) subalgebras cannot be represented as a product of a 3D Lie algebra and a 1D abelian factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' We show that all obtained EDAs are inequivalent and conclude that there are no Nambu-Lie U-dualities between 11D supergravity backgrounds within 10D EDAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' 1kumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='samip@phystech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='edu 2musaev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='et@phystech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='edu 1 Introduction String theory is a background-dependent theory meaning that dynamics of the string is defined on a fixed background of space-time fields including the metric, the dilaton, Kalb- Ramond 2-form field, and Ramond-Ramond p-form fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The moduli space of these vacua appears to be highly degenerate due to duality symmetries of string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Some of them, such as (abelian) T-dualities are exact perturbative symmetries of the superstring partition function at all orders in α′ and gs [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' This implies that physics of the string does does not change if the underlying space-time background is transformed by T-duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Given a non-abelian algebra of isometries of a string background, abelian T-duality transformation rules can be generalized to what is called non-abelian T-duality (NATD) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' In contrast to the abelian case NATD is not an exact quantum symmetry of the conformal theory due to problems with definition of winding modes [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' However, the NATD transformation map can be corrected to be a valid symmetry at the leading order in α′ [6,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Using the notion of non-commutative currents, the non-abelian T-duality transformations can be extended to Poisson-Lie T-dualities that are symmetries of string theory in the same sense [8,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' While abelian T-duality starts from a background with certain abelian isometries and preserves them, non-abelian T-duality breaks the non-abelian algebra of initial isometries naively preventing from performing the inverse transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The algebraic structure behind non-abelian T-duality symmetries, that is classical Drinfeld algebras, reveals that the initial isometry becomes hidden inside the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' More specifically classical Drinfeld algebra D is defined in terms of Manin triple (D, g, ˜g), where D is a Lie algebra with non-degenerate quadratic form η, and g and ˜g are subalgebras maximally isotropic with respect to the form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The algebra g is commonly referred to as the geometric subalgebra, and is responsible for the background space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' a group manifold or a coset space, while ˜g is commonly referred to as the dual algebra and it is responsible for conservation laws of the sigma model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' To illustrate that, denote fabc and ˜fabc as structure constants of the algebras - g and ˜g, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Then the following holds, [va, vb] = fab cvc, dJa = ˜fa bcJb ∧ Jc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='1) Here, vectors va define action of G = exp g on itself or on a coset space as δxi = vaiǫa, where xi denote coordinates on the group (coset) manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Noether currents Ja = Ja idxi satisfy the non-commutative conservation law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' When ˜fabc = 0, the currents are conserved in the usual sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Non-abelian T-duality simply maps g ↔ ˜g, hence vanishing ˜fabc get replaced by 2 non-vanishing fabc and the conservation law becomes non-commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The initial isometry becomes hidden in g′ = ˜g and is no longer manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' In this language the condition for classical equations of motion for the string to satisfy is simply the Leibniz identity [X, [Y, Z]] = [[X, Y ], Z] + [Y, [X, Z]], X, Y ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='2) Here, the brackets are given by the following relations in terms of the generators (Ta, ˜T a) = bas D: [Ta, Tb] = fab cTc, [ ˜T a, ˜T a] = fc ab ˜T c, [ ˜T a, Tb] = ˜fc ab ˜T c + fab cTc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='3) In terms of structure constants, Leibniz identity is equivalent to Jacobi identities for fabc and ˜fabc along with the following mixed identity ˜fl jkfmi l + ˜fm klfli j + ˜fi jlflm k + ˜fm jlfil k + ˜fi klflm j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='4) For a review of the algebraic construction behind Poisson-Lie T-dualities see [10], for a review of applications of NATD see [11,12], for formulation of Poisson-Lie T-dualities in the supergravity language see [13,14], for geometric aspects see [15,16] In the most general case when both sets of structure constants are non-zero, one is able to define the so-called Poisson-Lie duality transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' When dim g = d, these are such O(d, d) maps CAB that preserve the structure of classical Drinfel’d double: TA → CA BTB, TA = (Ta, ˜T a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='5) There is a distinguished set of such transformations called Poisson-Lie (PL) T-dualities (plu- ralities) when the map CAB relates different realization of the same Drinfeld algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The simplest example is the swapping g ↔ ˜g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' For lower dimensional Lie algebras full classification of all possible Poisson-Lie T-dualities or likewise of all equivalent Manin triples is available [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' This is based on classification of all possible dual algebras ˜g for each g belonging to the Bianchi classification of three-dimensional real Lie algebras (for more on classification of Lie Algebras, see for example [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' More generally, one may have maps CAB that relate different Drinfeld algebras, for example, Yang-Baxter deformations that draw the interest since they preserve integrability of the underlying sigma-model [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' When extending abelian T-duality symmetries by S-dualities that are non-perturbative 3 transformations exchanging gs with g−1 s , one arrives at U-duality transformations that are symmetries of M-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Speaking more concretely, U-duality is a symmetry of classical field equations of 11D supergravity compactified on a d-torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' These are known as Cremmer-Julia symmetries and are given by the exceptional groups Ed(d) [20,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' In M-theory, whose low-energy approximation is given by 11D supergravity, U-duality can be thought of as symmetries of BPS states [22] or in terms of a Buscher-like procedure for M2-brane wrapping a 4-torus [23,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The algebraic structure behind Poisson-Lie T-dualities can be extended to the so-called Exceptional Drinfeld Algebras (EDA), that include the usual abelian U-dualities (Cremmer-Julia symme- tries) [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Keeping the more detailed description of EDAs to the next section, we mention that these are Leibniz algebras with generators TA on which exceptional group Ed(d) acts in the same sense as the orthogonal group O(d, d) acts on generators of the classical Drinfeld double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Nambu-Lie U-dualities are then transformations that preserve the structure of the EDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' What differs these from the PL T-duality case is that there is no naturally defined analogue of the swapping g ↔ ˜g, simply due to the following two facts: i) dimension of the geometric subalgebra g of an EDA is never half of dimension of the EDA itself, ii) orthogonal completion of g inside the EDA is not an algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' For this reason, searching for pairs of 11D geometries related by a Nambu-Lie U-duality is an extremely complicated task for a general EDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' At the moment few examples of such dualities between 11D backgrounds and solutions to Type IIB supergravity equations are known [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' In [30] a general procedure has been suggested similar to the natural swapping g ↔ ˜g based on external automorphisms of Ed(d) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Further it has been used to generate few examples of mutually dual backgrounds in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' In this work, we elaborate further on the results of [30,31] that in particular state that there are no non-abelian U-dualities in the defined sense between 11D background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The narrative we follow is along the same lines as [32] where a full classification of 6D Exceptional Drinfeld Doubles based on 3D geometric algebras has been presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Starting from the classification of four-dimensional real Lie algebras [33], we construct all possible EDAs for a representative of each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' For each pair of such obtained EDAs we search for an SL(5) transformation relating them, that would mean existence of a Nambu-Lie U-duality between backgrounds that geometrically realize the corresponding geometric algebras g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Restricting ourselves to only such 4D real Lie algebras that do not contain a 1d (abelian) factor we find no such transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The restriction is motivated by the interest only in dualities between 11D background as maps from 11D→IIB are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' In the beginning of Section 2 we briefly review the con- struction of Exceptional Drinfel’s Algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='1 we discuss the geometric realization of EDAs and Nambu-Lie U-dualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='2, we present classification of 10D EDAs, 4 given the conditions stated in the preceding section and state the main results of the paper 2 Exceptional Drinfel’d Algebras Before proceeding with the classification of 10d EDAs, let us briefly review the algebraic construction following [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' We will be focusing on the 10d case where generators of the exceptional Drinfeld algebra ED4 are collected into the 10-dimensional representation of the SL(5) group basED4 = {TAB}, where A, B = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' , 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Multiplication table is then given by TAB ◦ TCD = i 2FAB,CD GHTGH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='1) The structures constants FAB,CDGH are defined by the following relations FAB,CD GH = 4FAB,[C [GδH] D] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='2) FAB,C D = 1 2ǫABCGHZGHD + 1 2δD [ASB]C + 1 3δD [AτB]C + 1 6δD C τAB, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='3) where τ is antisymmetric and S is a symmetric tensor, while Z[ABC] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' For the algebra to be an EDA, components of the constants ZABC, SAB and τAB under decomposition SL(5) ←֓ GL(4) must be defined as Zabc = 1 6ǫabcdfde e + 1 4ǫabeffef c, S5a = fab b − 3Za, τ5a = 9 2Za − 1 2fab b Z5[a,b] = 1 6 ˜fc abc, Sab = 1 3 ˜f(a cdeǫb)cde, τab = −1 6 ˜f[a cdeǫb]cde Zab,5 = −Z5a,b + Z5b,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='4) The constants FAB,CD have the same structure as the embedding tensor of [34], and in this language the above construction implies that only the geometric flux (anholonomy coefficients) and Q-flux are turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The former is given by the structure constants fabc of the geometric subalgebra g and the latter is given by ˜fabcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The algebra is Leibniz with the fundamental identity given by the quadratic relations analogous to those of 7d maximal gauged SUGRA [34]: 2F G AB[CF I GD],H − F I ABGF G CDH + F G ABHF I CDG = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='5) 5 In terms of structure constants fabc and dual constants fabcd, the conditions become 6ff[a [c ˜fb] de]f + fab f ˜ff cde − 1 3 ˜f[a cdefb]f f = 0 ˜fc abcfbd d = 0, fde a ˜f bde c − 1 3 ˜fc abdfde e = 0 ˜fc abg ˜fg def − 3 ˜fc g[de ˜fg f]ab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='6) The last of the above equations is also referred to as the dual Jacobi condition, just as the dual conditions in the Manin triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' It describes the internal (isolated) relations between the structure constants of the dual algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' As in the case of Classical Drinfeld Algebra, in general, there might exist multiple equivalent choices of the geometric subalgebra g inside an EDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Proper generalization of the isometry condition to the case of exceptional structures has been given in [25,26] and can be written as follows ǫABCDETAB ⊗ TCD ���� g⊗g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='7) In other words, for a given EDA, its geometric subalgebra g is spanned by such a subset of the whole set of generators {TAB} that satisfy the above condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' For Classical Drinfel’d Double the condition is ηABTA ⊗ TB = 0, implying that one may, for example, take bas g = {Ta}, or bas g = { ˜T a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' For EDAs, one choice is self-evident - bas g = {T5a}, while presenting an alternative choice is usually a hard task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' This implies that there is no natural generalization of the Non-Abelian T-duality transformation swapping g ↔ ˜g in the case of EDAs, although certain progress in defining an analogue of these swappings has been done in [30,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='1 Geometric realization and dualities The algebraic structure of EDAs stands behind Nambu-Lie U-dualities of supergravity so- lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' These can map solutions to 11D supergravity equations into each other or into Type IIB supergravity equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Such duality transformations map the group manifolds correspond- ing to different choices of the geometric subalgebra g into each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' For more detailed and concrete algorithm of constructing mutually dual backgrounds see [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Below, we will briefly recall the overall construction and highlight relations to Exceptional Field Theory (ExFT) that provide convenient variables for writing such duality maps [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' These are Ed(d)-covariant field theories defined in 11-dimensional space-time with an explicit split - 11 = D + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The D-dimensional space-time is usually referred to as the external, the d-dimensional space is 6 usually referred to as internal, although no compactification is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' In the d = 4 case relevant to the present discussion, field content of the theory includes the external metric gµν, ten vector fields AµMN, five 2-form fields BµνM, and 14 scalar fields parametrized by a coset element MMN ∈ SL(5)/SO(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The indices µ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' , 6 parameterize directions of the external space-time whereas the indices M, N = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' , 5 belong to the 5 of SL(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' For more details of the construction see [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Here we are interested in the special case where all fields transform- ing in irreps of SL(5) can be decomposed in terms of matrices EABMN (generalized vielbeins) geometrically realizing an EDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' In compact notation one writes [EAB, ECD] = FAB,CD EFEEF, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='8) where the constants FAB,CDEF are precisely the structure constants of the EDA and the brackets denote the so-called generalized Lie derivative of ExFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Generalized vielbeins are parametrized by fields of 11D supergravity in the 11 = 7 + 4 split transforming as scalars under 7-dimensional diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Introducing a unity matrix MAB compose MMN,KL = 2EMN ABEKL CDMACMBD = MMKMNL − MMLMNK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='9) The symmetric matrix mMN = e− φ 2 � |g|− 1 2gij Vi Vj |g| 1 2(1 + V 2) � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='10) is then defined in terms of the 4d metric gmn on the group manifold, the vector V m = 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='ǫmnklCnkl and a scalar field eφ = |g7|1/7 which is the determinant |g7| of external 7 dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The metric gmn on the group manifold is defined as usual in terms of Maurer-Cartan forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Let g ∈ G = exp g be an element of the group G whose Lie algebra is g, then 1-forms on the group manifold g−1dg ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' In components we have g−1dg = rm aTadxm, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='11) where xm are some coordinates on the group manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Given an EDA and a choice of the isotropic subalgebra g one can explicitly construct the corresponding generalized vielbein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' A step-by-step algorithm of this procedure based on constructing adjoint action of eh ∈ G for some h ∈ g on an element of EDA can be found in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' An alternative choice of the isotropic subalgebra, if exists, is related to the given one by an SL(5) transformation T ′ AB = CA CCB DTCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='12) 7 If this transformation respects the structure of EDA, then the alternative isotropic subalgebra is spanned by T ′ 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Structure constants of the EDA then transform as F ′ A′B′,C′D′ = CA′ACB′BCC′CCD D′FAB,C D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='13) Note that not any such matrix corresponds to a Nambu-Lie U-duality transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Indeed, one can always perform a GL(4) transformation on generators of a given algebra g thus changing explicit realization of the corresponding EDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Two EDA’s related by such transformation then correspond to 11D backgrounds related by a coordinate transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Another trivial choice is CA B = � 14×4 λm 0 1 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='14) that corresponds to simply a gauge transformation of the 3-form Cmnk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' To avoid counting of EDA’s related by a rotation of the basis of their isotropic subalgebras we first classify Exceptional Drinfeld Algebras using classification of 4D real Lie Algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='2 Classification of 10 dimensional EDA’s The main goal of this work is to investigate relations between 10d EDAs that correspond to Nambu-Lie U-duality transformations of 11-dimensional supergravity backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' For this purpose, we start with a classification of 10D EDAs of certain class based on the classification of 4-dimensional real Lie Algebras by Mubarakzyanov [33] (for a review in English see [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Since explicit examples of Nambu-Lie U-dualities between 11D and Type IIB backgrounds are known in the literature, we are interested here only in EDAs constructed on 4d real Lie Algebras g4 that cannot be decomposed into a sum g4 = g4 ⊕ g1, where g3 is a 3d Lie algebra and g1 is 1-dimensional Abelan factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' We list all relevant 4d real Lie Algebras in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='1 [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = T1 [T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = T2 g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='5 [T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = AT1 [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = BT2 [T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = CT3 ABC̸= 0 g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='9 [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T3] = T1 [T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = 2AT1 [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = AT2 − T3 [T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = T2 + AT3 A > 0 g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='2 [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = βT1 [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = T2 [T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = T2 + T3 g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='6 [T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = AT1 [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = BT2 − T3 [T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = T2 + BT3 A > 0 g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='10 [T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T3] = T1 [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T3] = T2 [T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = −T2 [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = T1 8 g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='3 [T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = T1 [T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = T2 g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='7 [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T3] = T1 [T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = 2T1 [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = T2 [T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = T2 + T3 2g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='1 [T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T2] = T1 [T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = T3 g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='4 [T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = T1 [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = T1 + T2 [T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = T2 + T3 g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='8 [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T3] = T1 [T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = (1 + β)T1 [T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = T2 [T3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' T4] = βT3 β ∈ [−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' 1] Table 1: Classification of 4-dimensional indecomposable real Lie algebras g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='n with n = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' , 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The algebra 2g2,1 is decomposable, however does not have a u(1) fac- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' To arrive at the corresponding classification of 10d EDAs, we solve quadratic constraints for each class in the table above to find all possible sets of the dual structure coefficients ˜fdabc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' To solve the equations we use mathematical software Mathematica , that gives us all the 4 dimensional EDAs in the chosen class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The result is listed in Table 2, where only unique combinations of indices are explicitly given in the coefficients of the underlying algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The rest of the indices are obtained by the antisymmetric property of the structure coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' EDA Structure Constants ˜f abcd g4,1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' ˜f 1232 = ˜f 1344, ˜f 1242 = ˜f 1343 ˜f 1234 = ˜f 1233 ˜f 1344 − ˜f 1244 ˜f 1344 2 ˜f 1343 ˜f 1243 = ( ˜f 1244 − ˜f 1233) ˜f 1343 2 ˜f 1344 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' ˜f 1232 = − ˜f 1344, ˜f 1244 = ˜f 1233, ˜f 2344 = ˜f 1231 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' ˜f 1242 = ˜f 1343 ˜f 1244 = ˜f 1233 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' ˜f 1244 = ˜f 1233, ˜f 2344 = ˜f 1231 g4,2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' ˜f 1244 = −1 3 (1 + 2β) ˜f 1233 6 ˜f 2344 = 1 3β(β − 4) ˜f 1231 g4,3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' ˜f 2344 = 1 3 ˜f 1231 g4,4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' ˜f 1244 = − ˜f 123 3 g4,5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' ˜f 1244 = −2A−2B+C 3C ˜f 1233 9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' ˜f 1344 = 1 3B(2A − B + 2C) ˜f 1232 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' ˜f 2344 = 1 3A(A − 2B − 2C) ˜f 1231 g4,6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' ˜f 2344 = 1 3A(A − 2B − 2C) ˜f 1231 g4,7 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' ˜f 1244 = −5 3 ˜f 1233 g4,8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' ˜f 1244 = − 1 3β(4 − β) ˜f 1233 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' ˜f 1344 = 1 3(1 + 4B) ˜f 1232 g4,9 ˜f abcd = 0 or imaginary g4,10 ˜f abcd = 0 2g2,1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' ˜f 1234 = ˜f 1232, ˜f 1342 = ˜f 1344 Table 2: All possible structure constant of 10d EDAs for each g4,n with n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' , 10 and 2g2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The constants A, B, C, β are the same as in the previous table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Hence, given we are interested only in real non-trivial EDAs, we end up with 16 examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' A natural question would be: whether there exists a pair of EDAs in this set that are equivalent up to an SL(5) transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' This would mean that the same EDA can be generated by two 4d Lie Algebras that belong to different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' In the supergravity language this would mean existence of a Nambu-Lie U-duality between 11D backgrounds geometrically realizing this pair of 4d Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Result of our calculations is that there are no such pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' To arrive at this statement we used Mathematica software and explicitly solve equations on components of the matrix CAB for each pair of 16 algebras with no further restrictions on the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' This means, that although in Table 2 we list algebras as though all explicitly written dual structure constants are non-vanishing, our code does not assume that [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' 3 Discussion In this work we obtain a classification of 10-dimensional EDA based on the classification of 4-dimensional real Lie Algebras by Mubarakzyanov [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' We intentionally restrict only to such 4d algebras that cannot be decomposed into a 3d algebra and a 1d abelian factor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='e, we are interested in Nambu-Lie U-dualities between 11d backgrounds, rather than dualities between 11D and Type IIB solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' More specifically, we look only at EDAs whose isotropic (geomet- ric) subalgebra is given by g4,n with n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=', 10 and 2g2,1 in terms of the Mubarakzyanov’s classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Given these restrictions the classification of EDAs is summarized in Table 2, 10 where 16 non-trivial EDAs are listed in terms of dual structure constants ˜fabcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The important question we were interested in is whether there exists a Nambu-Lie U-duality between 11D solutions and supergravity equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Equivalently, in the algebraic language: whether any of the sixteen exceptional Drinfeld algebras are equivalent up to an SL(5) trans- formation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' For that we computed the explicit form of all possible transformations between all possible pairs of EDA listed in Table 2 of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' In our findings, we discovered that none of the EDA pairs except the (ED2, ED4) possess transformation matrices, taking the basis of one EDA to another, with a non-zero determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Moreover, the transformation relating the aforesaid algebras - ED2 and ED4 is simply a GL(4) transformation rotating the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Hence, these two solutions are equivalent and their geometric realizations can be mapped into each other by a 4D coordinate transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Hence, there are no Nambu-Lie U-dualities inside SL(5) exceptional Drinfeld algebras relating 11D backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Note however, this does not rule out transformations between 11D and Type IIB backgrounds, explicit examples of which are known [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Previously in [30] the same has been shown for transformations involving external automorphisms of the algebra sl(5), suggested as the natural analogue of Non-Abelian T-duality transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Here we complete the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' There are further directions to extend this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' The most obvious task is to complete the classification including all 4D real Lie algebras and list sets of EDA’s mutually Nambu- Lie U-dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Less straightforward is to increase the dimension of the geometric subalgebra g by one and consider 16D Exceptional Drinfeld Algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Unfortunately, there is no ready to use classification of 5D real Lie algebras, but certain restricted classifications are present in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Some useful examples can be found in [40–43], for a review see [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' Another interesting direction of further research is to list those EDAs from our classification that can be obtained as generalized Yang-Baxter deformations of the trivial EDA when all dual structure constants are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' In other words, to answer the question: for which algebras in Table 2 dual structre constants can be represented in the form ˜fa bcd = re[bcfae d], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content='1) where rabc is completely antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' In the case of classical Drinfeld algebras such trans- formations are known to preserve integrability of the 2d sigma-model on the corresponding background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' There is no analogous statement for 3d sigma-models describing membranes propagating on 11d supergravity backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' However, such defined generalized Yang-Baxter deformations are of certain interest (see [45] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FKT4oBgHgl3EQf_y40/content/2301.11963v1.pdf'} +page_content=' 11 Acknowledgments This work has been supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”, grant No 21-1-2-3-1 and by Russian Ministry of Education and Science.' metadata={'source': 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There had been published +many proofs and many papers claimed to provide a simplified proof, but most of them are long and +still complicated. We provide the proof that is really short, simple and discloses the nature of this +fundamental result. We follow the technique closed to the original Parikh’s paper and our proof is +similar to the proof by Ryoma Sin’ya 2019, but we provide more detailed exposition and pretend to +more simplicity as well. We achieve the simplicity via nonconstructivenes that allows us avoiding +many difficulties met by other proofs. +2012 ACM Subject Classification Theory of computation → Grammars and context-free languages +Keywords and phrases Formal Languages, Context-Free Languages, Parikh’s Theorem +Funding The paper was supported by RFBR grant 20-01-00645 +Acknowledgements I want to thank Dmitry Chistikov for the helpfull discussion of the paper +1 +Introduction +Parikh’s theorem [6] is a fundamental theorem of the formal language’s theory. There had +been published many proofs (see [1], [4] for the survey of different proofs and detailed +exposition on the topic). Many papers claimed to provide a simplified proof ([2] looks to +be the most known), but most of them are long and still complicated. Despite really short +and simple proofs are already known (e.g., [7]), papers with another proofs continues to be +published [5]. We provide the proof that is really short, simple and discloses the nature of +this fundamental result. Our proof is based on derivations trees so as the original Parikh’s +proof [6]. As Parikh, we decompose derivation trees into small ones; we consider similar +kinds of trees: ordinary derivation trees and auxiliary ones (with only nonterminal in the +crown that is the same as the root). We get rid of duplicates in auxiliary trees and pump +them. While our technique is similar to the original proof in general, it is simpler since we +do not have restrictions on grammar and other technical issues (like considering derivation +trees that contains nonterminals from a fixed subset). The simplicity of our construction is +based on nonconstructivenes that allows us avoiding many technical issues. Also trees (in +the decomposition) in our construction have linear height (in the number of nonterminals) +while in the Parikh’s construction trees have quadratic height. In our proof we generalize the +idea of derivation from words to trees and translate the idea of the pumping lemma to the +trees as well. It makes our construction clear enough to explain students during a lecture in +a basic course of formal languages and automata theory, and so we hope that the detailed +version will help to spread it in the community. +Our technique is similar to [8]. Unlike concise exposition in [8], we provide more detailed +exposition and explain the intuition of the proof. And we do not use auxiliary results for our +proof (thanks to nonconstructivenes). +arXiv:2301.00047v1 [cs.FL] 30 Dec 2022 + +2 +The Simplest Proof of Parikh’s Theorem via Derivation Trees +2 +Definitions +We denote by N non-negative integers. Let Σk = {a1, . . . , ak} be an alphabet, k ⩾ 1. We +denote by Ψ : Σ∗ +k → Nk the Parikh mapping that maps a word w to its Parikh’s image the +vector (|w|a1, |w|a2, . . . , |w|ak), where |w|aj is the number of letters aj in w. We denote the +Parikh’s image of a language L ⊆ Σ∗ +k in the natural way: +Ψ(L) = {Ψ(w) | w ∈ L}. +A set S ⊆ Nk is linear if there exist vectors v0, v1, . . . , vm ∈ Nk such that +S = {v0 + v1t1 + v2t2 + . . . vmtm | t1, . . . tm ∈ N}. +A set is semi-linear if it is a union of finitely many linear sets. +▶ Theorem 1 (Parikh). For each context-free language L the set Ψ(L) is semi-linear. +We use classical notation for context-free grammars [3]. We denote by N the set of +nonterminals, and denote nonterminals by capital letters. Small letters from the end of +the alphabet denote words and Greek letters denote words over the alphabet Σ ∪ N, called +sentential forms. +Our proof uses derivation trees and exploits the idea similar to the classical proof of the +pumping lemma. We use derivation trees not only for words, but also for sentential forms of +a special kind. We call a tree that corresponds to a derivation of the form A +∗=⇒ uAv a block +tree. We call derivation trees for words ground trees. We say that a tree T (a ground or a +block one) is minimal if it does not have a block tree as a subtree, i.e. T does not have the +form (S +∗=⇒ xAz +∗=⇒ xuAvz +∗=⇒ xuβvz). Hereinafter, we describe a tree by any derivation +corresponding to the tree. +S +A +A +x +u +y +v +z +S +A +y +x +z +A +A +u +v +Figure 1 Decomposition of tree T into the pair of trees (T1, T2) +A non-minimal ground tree has the form S +∗=⇒ xAz +∗=⇒ xuAvz +∗=⇒ xuyvz. It can be +decomposed into a pair of the ground tree S +∗=⇒ xAz +∗=⇒ xyz and the block tree A +∗=⇒ uAv, +see Fig. 1. Formally, we say that a tree T is decomposed to the pair of trees (T1, T2) if +for some B ∈ N : T = A +∗=⇒ xBz +∗=⇒ xuBvz +∗=⇒ xuβvz, T1 = A +∗=⇒ xBz +∗=⇒ xβz and +T2 = B +∗=⇒ uBv; here β is either A if T is a block tree or β ∈ Σ∗ if T is a ground tree (and +in this case A is the axiom). We say that T can be composed from T1 and T2. Note that T1 +and T2 can be composed in several ways if T1 has several B nodes. We denote +T1 ◦ T2 = {T | T can be decomposed into (T1, T2)} + +A. Rubtsov +3 +In addition to derivation of words we consider derivation of trees as follows. +▶ Definition 2. A derivation of a tree starts with a minimal ground tree and each derivation +step leads to a ground tree as well. At a derivation step we choose a node A and a minimal +block tree A +∗=⇒ xAy, replace the chosen node by the block tree; the subtree of the chosen +node is glued into A from the crown of the block tree. +So in the trees derivation a minimal ground tree has the role of the axiom (there can +be several ones), and a replacement A → Ti where Ti = (A +∗=⇒ xAz) has the role of the +production rule. +▶ Definition 3. Let S be a multiset of trees. We say that S′ is derived from S if S′ is +obtained from S by the replacement of two trees T1 and T2 by a tree T ∈ T1 ◦ T2. We denote +it as S ⊢ S′. We say that a multiset of trees S is well-formed such that S ⊢ +∗ {T} for some +ground tree T. +It is easy to see, that if S contains only minimal trees and S ⊢ +∗ {T}, then there exists a +derivation of T such that S = {T0, T1, . . . , Tn} where T0 is the ground tree and Ti, i ⩾ 1 is +the block tree that was chosen at the i-th derivation step. +We define the Parikh image for the trees in a natural way. Ψ(T) = Ψ(w) if T = (S +∗=⇒ w) +and Ψ(T) = Ψ(xz) if T = (A +∗=⇒ xAz). The following lemma directly follows from the +definitions. +▶ Lemma 4. +If T, T ′ ∈ T1 ◦ T2, then Ψ(T) = Ψ(T ′) = Ψ(T1) + Ψ(T2). +If S ⊢ +∗ {T} and S ⊢ +∗ {T ′} then Ψ(T) = Ψ(T ′). +Lemma 4 implies that we can extend the definition of the Parikh map to well-formed +multisets: Ψ(S) = � +T ∈S Ψ(T). +By the pigeonhole principle each minimal tree has depth at most |N| − 1 (otherwise on +the longest path from the root to a leaf there would be a repetition of nonterminals). Since +(for a fixed grammar) each node of a tree has a bounded degree, there is only finitely many +minimal trees. Let us enumerate them and denote the number of minimal trees by m. So, +each multiset of minimal trees S has a corresponding vector ⃗v ∈ Nm where ⃗vi is the number +of occurrences of Ti in S. +3 +Proof +We begin with the proof idea. Any word w derived from a grammar G has some derivation +tree Tw and Ψ(Tw) = Ψ(w). We decompose Tw into a multiset Sw of minimal trees (Sw ⊢ +∗ Tw). +Denote by Tg the ground tree from Sw. Obtain the set S′ +w from Sw by removing repetitions. +Note that S′ +w ⊆ MG where MG is the finite set of minimal trees of grammar G. So we have +defined a mapping w �→ S′ +w with a finite codomain 2MG. For any multiset S obtained from +S′ +w by repetition of non-ground trees, there exists a derivation tree T (of G) such that S ⊢ +∗ T +(S is well-formed). Therefore Ψ(T) ∈ Ψ(L(G)). In other words, +� +�Ψ(S′ +w) + +� +T ′∈S′w\{Tg} +tT ′ · Ψ(T ′) +� +� ∈ Ψ(L(G)) + +4 +The Simplest Proof of Parikh’s Theorem via Derivation Trees +no matter how we choose tT ′ ∈ N for each T ′. Let S be a well-formed multiset with a ground +tree Tg ∈ S. Denote by +Lin(S) = +� +� +�Ψ(S) + +� +T ′ +i ∈S\{Tg} +ti · Ψ(T ′ +i) +����� +ti ∈ N +� +� +� . +(1) +The set Lin(S) is linear by the definition. Since there are only finitely many different sets S′ +w, +the set Ψ(L(G)) is a finite union of linear sets Lin(S′ +w), so it is semilinear by the definition. +The scheme of the proof is depicted in Fig. 2 +L(G) +Ψ(L(G)) +{Tw : w ∈ L(G)} +{Sw : w ∈ L(G)} +{S′ +w : w ∈ L(G)} +{Lin(S′ +w) : w ∈ L(G)} +Ψ +∪ +Figure 2 Scheme of the proof +3.1 +Auxiliary Lemmas +▶ Lemma 5. Let S be a well-formed multiset of minimal trees and S′′ be a multiset. Denote +by ⃗v and ⃗v′′ the corresponding vectors of S and S′′, respectively. Suppose ⃗vi > 0 iff ⃗v′′ +i > 0; +then S′′ is a well-formed multiset as well. +Proof. The statement is equivalent to the conjunction of two claims. The first one: if ⃗vi > 0 +and ⃗v′′ +i = ⃗vi + 1 while ⃗v′′ +j = ⃗vj for j ̸= i, then S′′ is well-formed. The second one: if ⃗vi > 1 +and ⃗v′′ +i = ⃗vi − 1 while ⃗v′′ +j = ⃗vj for j ̸= i, then S′′ is well-formed. Starting with the vector ⃗v +and subsequently increasing or decreasing its components by 1 we can obtain any vector ⃗v′′ +satisfying the conditions of the lemma. Since at each step the condition of one of the claims +hold, we obtain that the condition of the lemma holds in the result. +Recall Definitions 2 and 3 of trees and multiset derivations. Denote by Ti = (A +∗=⇒ uAv) +the i-th block-tree. Fix a derivation tree T such that S ⊢ +∗ {T} with a derivation of the tree +T as well. +Proof of the first claim. Due to the form of Ti ∈ S, the tree T contains the non-terminal A. +So we can glue Ti into the place of some occurrence of the non-terminal A and obtain the +tree T ′ as the result. S′′ is well-formed, since T ′ is a derivation tree of G by the construction. +Proof of the second claim. If Ti is a subtree of T, than it can be removed from T (as +in Fig. 1) and the resulting tree T ′ would also be a derivation tree of G. Otherwise, let us +consider the steps of the fixed derivation of T. We change this derivation as follows. Recalling +that ⃗vi > 1, fix two copies T 1 +i and T 2 +i of the tree Ti in the multiset S such that T 1 +i was used + +A. Rubtsov +5 +earlier than T 2 +i in the derivation of T. When T 2 +i must be composed with the ground tree, we +skip this step. If at some step a tree Tj = (B +∗=⇒ u′Bv′) is glued into a node B of T 2 +i , then +we glue it into the corresponding node B of T 1 +i (that is already in the ground tree). So, this +modification yields a derivation S ⊢ +∗ {T ′, T 2 +i }, where T ′ is some ground tree. But then we +could do the same starting from S′′ and obtain S′′ ⊢ +∗ {T ′}. Therefore S′′ is well-formed. +◀ +▶ Corollary 6. For any w ∈ L(G) : Lin(S′ +w) ⊆ Ψ(L(G)). +▶ Lemma 7. For any w ∈ L(G): +Ψ(w) ∈ Lin(S′ +w). +Proof. From the definitions it follows that Ψ(w) = Ψ(Sw) ∈ Lin(Sw). +We prove that +Lin(Sw) ⊆ Lin(S′ +w). +Note that Ψ(Sw) = Ψ(S′ +w) + �m +i=1(⃗vi − 1)Ψ(Ti), where ⃗v is the +corresponding vector of Sw and Ti is the i-th minimal tree (in the enumeration above). By +the definition a vector ⃗u ∈ Lin(Sw) has the form +⃗u = Ψ(Sw) + +� +T ′ +j∈Sw\{Tg} +t′ +j · Ψ(T ′ +j) = Ψ(Sw) + +n +� +i=1 +ti · Ψ(Ti), +where in the second equality we put together all T ′ +j’s that are copies of the same tree Ti. +Note that each T ′ +j is a minimal block tree, so T ′ +j = Ti for some i. Thus, +⃗u = +� +�Ψ(S′ +w) + +� +i:⃗vi>0 +(ti + ⃗vi − 1)Ψ(Ti) +� +� ∈ Lin(S′ +w) +◀ +3.2 +Proof of Parikh’s theorem +Let G be a context-free grammar generating L. For a word w ∈ L denote by T(w) the set of +all derivation trees Tw. Note that T(w) ̸= ∅ and it can be even infinite if there are ε-rules +in G. Denote by S(w) = {Sw | ∃Tw ∈ T(w) : Sw ⊢ +∗ {Tw}} where Sw is a multiset consisting +only of minimal trees (as before). Finally S′(w) = {S′ +w | Sw ∈ S(w)}; recall that S′ +w is +the set obtained from Sw by deleting the duplicates. Now we show that S′(w) is finite for +each w and moreover the union ∪w∈LS′(w) = S′(L) is finite as well. Recall that there are +finitely many minimal trees, and we denote them by T1, . . . , Tm. Therefore, each Sw has a +corresponding m-dimensional vector ⃗v such that +Ψ(w) = Ψ(Sw) = +m +� +i=1 +⃗vi · Ψ(Ti). +The corresponding vector ⃗v′ of S′ +w is a 0-1 vector such that ⃗v′ +i = 1 iff ⃗vi > 0. So since there +are only finitely many 0-1 vectors of length m and each S′ +w ∈ S′(L) has a corresponding 0-1 +vector ⃗v′, then the set S′(L) is finite as well. +Putting everything together, by Lemma 7 +Ψ(L) ⊆ +� +S′∈S′(L) +Lin(S′) +and by Corollary 6 +� +S′∈S′(L) +Lin(S′) ⊆ Ψ(L). + +6 +The Simplest Proof of Parikh’s Theorem via Derivation Trees +Since the set S′(L) is finite and each set Lin(S′) is linear by Eq. (1), the equality +Ψ(L) = +� +S′∈S′(L) +Lin(S′) +proves Parikh’s theorem. +◀ +References +1 +Javier Esparza, Pierre Ganty, Stefan Kiefer, and Michael Luttenberger. Parikh’s theorem: A +simple and direct automaton construction. Information Processing Letters, 111(12):614–619, +2011. +URL: https://www.sciencedirect.com/science/article/pii/S0020019011000822, +doi:https://doi.org/10.1016/j.ipl.2011.03.019. +2 +Jonathan Goldstine. A simplified proof of Parikh’s theorem. Discrete Mathematics, 19(3):235– +239, 1977. +3 +John E Hopcroft, Rajeev Motwani, and Jeffrey D Ullman. Introduction to automata theory, +languages, and computation. Acm Sigact News, 32(1):60–65, 2001. +4 +Caleb Koch. A friendly tour of Parikh’s theorem. 2018. URL: http://cakoch10.github.io/ +papers/Kleene_Algebra.pdf. +5 +Toshihiro Koga. A proof of Parikh’s theorem via Dickson’s lemma. International Journal of +Foundations of Computer Science, 32(02):163–173, 2021. doi:10.1142/S012905412150009X. +6 +Rohit J. Parikh. +On Context-Free Languages. +J. ACM, 13(4):570–581, oct 1966. +doi: +10.1145/321356.321364. +7 +Jeffrey O. Shallit. A Second Course in Formal Languages and Automata Theory. Cambridge +University Press, 2008. +8 +Ryoma Sin’ya. Simple proof of Parikh’s theorem a la Takahashi. CoRR, abs/1909.09393, 2019. + diff --git a/6tAyT4oBgHgl3EQfQfaP/content/tmp_files/load_file.txt b/6tAyT4oBgHgl3EQfQfaP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..84c0040ec2469054597f776a477d4f6059a98352 --- /dev/null +++ b/6tAyT4oBgHgl3EQfQfaP/content/tmp_files/load_file.txt @@ -0,0 +1,215 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf,len=214 +page_content='The Simplest Proof of Parikh’s Theorem via Derivation Trees Alexander Rubtsov � � National Research University Higher School of Economics Moscow Institute of Physics and Technology Abstract Parikh’s theorem is a fundamental result of the formal language’s theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' There had been published many proofs and many papers claimed to provide a simplified proof, but most of them are long and still complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We provide the proof that is really short, simple and discloses the nature of this fundamental result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We follow the technique closed to the original Parikh’s paper and our proof is similar to the proof by Ryoma Sin’ya 2019, but we provide more detailed exposition and pretend to more simplicity as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We achieve the simplicity via nonconstructivenes that allows us avoiding many difficulties met by other proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' 2012 ACM Subject Classification Theory of computation → Grammars and context-free languages Keywords and phrases Formal Languages, Context-Free Languages, Parikh’s Theorem Funding The paper was supported by RFBR grant 20-01-00645 Acknowledgements I want to thank Dmitry Chistikov for the helpfull discussion of the paper 1 Introduction Parikh’s theorem [6] is a fundamental theorem of the formal language’s theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' There had been published many proofs (see [1], [4] for the survey of different proofs and detailed exposition on the topic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Many papers claimed to provide a simplified proof ([2] looks to be the most known), but most of them are long and still complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Despite really short and simple proofs are already known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=', [7]), papers with another proofs continues to be published [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We provide the proof that is really short, simple and discloses the nature of this fundamental result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Our proof is based on derivations trees so as the original Parikh’s proof [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' As Parikh, we decompose derivation trees into small ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' we consider similar kinds of trees: ordinary derivation trees and auxiliary ones (with only nonterminal in the crown that is the same as the root).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We get rid of duplicates in auxiliary trees and pump them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' While our technique is similar to the original proof in general, it is simpler since we do not have restrictions on grammar and other technical issues (like considering derivation trees that contains nonterminals from a fixed subset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' The simplicity of our construction is based on nonconstructivenes that allows us avoiding many technical issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Also trees (in the decomposition) in our construction have linear height (in the number of nonterminals) while in the Parikh’s construction trees have quadratic height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' In our proof we generalize the idea of derivation from words to trees and translate the idea of the pumping lemma to the trees as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' It makes our construction clear enough to explain students during a lecture in a basic course of formal languages and automata theory, and so we hope that the detailed version will help to spread it in the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Our technique is similar to [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Unlike concise exposition in [8], we provide more detailed exposition and explain the intuition of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' And we do not use auxiliary results for our proof (thanks to nonconstructivenes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='00047v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='FL] 30 Dec 2022 2 The Simplest Proof of Parikh’s Theorem via Derivation Trees 2 Definitions We denote by N non-negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Let Σk = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' , ak} be an alphabet, k ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We denote by Ψ : Σ∗ k → Nk the Parikh mapping that maps a word w to its Parikh’s image the vector (|w|a1, |w|a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' , |w|ak), where |w|aj is the number of letters aj in w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We denote the Parikh’s image of a language L ⊆ Σ∗ k in the natural way: Ψ(L) = {Ψ(w) | w ∈ L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' A set S ⊆ Nk is linear if there exist vectors v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' , vm ∈ Nk such that S = {v0 + v1t1 + v2t2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' vmtm | t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' tm ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' A set is semi-linear if it is a union of finitely many linear sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' ▶ Theorem 1 (Parikh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' For each context-free language L the set Ψ(L) is semi-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We use classical notation for context-free grammars [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We denote by N the set of nonterminals, and denote nonterminals by capital letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Small letters from the end of the alphabet denote words and Greek letters denote words over the alphabet Σ ∪ N, called sentential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Our proof uses derivation trees and exploits the idea similar to the classical proof of the pumping lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We use derivation trees not only for words, but also for sentential forms of a special kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We call a tree that corresponds to a derivation of the form A ∗=⇒ uAv a block tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We call derivation trees for words ground trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We say that a tree T (a ground or a block one) is minimal if it does not have a block tree as a subtree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' T does not have the form (S ∗=⇒ xAz ∗=⇒ xuAvz ∗=⇒ xuβvz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Hereinafter, we describe a tree by any derivation corresponding to the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' S A A x u y v z S A y x z A A u v Figure 1 Decomposition of tree T into the pair of trees (T1, T2) A non-minimal ground tree has the form S ∗=⇒ xAz ∗=⇒ xuAvz ∗=⇒ xuyvz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' It can be decomposed into a pair of the ground tree S ∗=⇒ xAz ∗=⇒ xyz and the block tree A ∗=⇒ uAv, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Formally, we say that a tree T is decomposed to the pair of trees (T1, T2) if for some B ∈ N : T = A ∗=⇒ xBz ∗=⇒ xuBvz ∗=⇒ xuβvz, T1 = A ∗=⇒ xBz ∗=⇒ xβz and T2 = B ∗=⇒ uBv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' here β is either A if T is a block tree or β ∈ Σ∗ if T is a ground tree (and in this case A is the axiom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We say that T can be composed from T1 and T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Note that T1 and T2 can be composed in several ways if T1 has several B nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We denote T1 ◦ T2 = {T | T can be decomposed into (T1, T2)} A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Rubtsov 3 In addition to derivation of words we consider derivation of trees as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' ▶ Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' A derivation of a tree starts with a minimal ground tree and each derivation step leads to a ground tree as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' At a derivation step we choose a node A and a minimal block tree A ∗=⇒ xAy, replace the chosen node by the block tree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' the subtree of the chosen node is glued into A from the crown of the block tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' So in the trees derivation a minimal ground tree has the role of the axiom (there can be several ones), and a replacement A → Ti where Ti = (A ∗=⇒ xAz) has the role of the production rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' ▶ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Let S be a multiset of trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We say that S′ is derived from S if S′ is obtained from S by the replacement of two trees T1 and T2 by a tree T ∈ T1 ◦ T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We denote it as S ⊢ S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We say that a multiset of trees S is well-formed such that S ⊢ ∗ {T} for some ground tree T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' It is easy to see, that if S contains only minimal trees and S ⊢ ∗ {T}, then there exists a derivation of T such that S = {T0, T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' , Tn} where T0 is the ground tree and Ti, i ⩾ 1 is the block tree that was chosen at the i-th derivation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We define the Parikh image for the trees in a natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Ψ(T) = Ψ(w) if T = (S ∗=⇒ w) and Ψ(T) = Ψ(xz) if T = (A ∗=⇒ xAz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' The following lemma directly follows from the definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' ▶ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' If T, T ′ ∈ T1 ◦ T2, then Ψ(T) = Ψ(T ′) = Ψ(T1) + Ψ(T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' If S ⊢ ∗ {T} and S ⊢ ∗ {T ′} then Ψ(T) = Ψ(T ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Lemma 4 implies that we can extend the definition of the Parikh map to well-formed multisets: Ψ(S) = � T ∈S Ψ(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' By the pigeonhole principle each minimal tree has depth at most |N| − 1 (otherwise on the longest path from the root to a leaf there would be a repetition of nonterminals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Since (for a fixed grammar) each node of a tree has a bounded degree, there is only finitely many minimal trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Let us enumerate them and denote the number of minimal trees by m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' So, each multiset of minimal trees S has a corresponding vector ⃗v ∈ Nm where ⃗vi is the number of occurrences of Ti in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' 3 Proof We begin with the proof idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Any word w derived from a grammar G has some derivation tree Tw and Ψ(Tw) = Ψ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We decompose Tw into a multiset Sw of minimal trees (Sw ⊢ ∗ Tw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Denote by Tg the ground tree from Sw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Obtain the set S′ w from Sw by removing repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Note that S′ w ⊆ MG where MG is the finite set of minimal trees of grammar G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' So we have defined a mapping w �→ S′ w with a finite codomain 2MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' For any multiset S obtained from S′ w by repetition of non-ground trees, there exists a derivation tree T (of G) such that S ⊢ ∗ T (S is well-formed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Therefore Ψ(T) ∈ Ψ(L(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' In other words, � �Ψ(S′ w) + � T ′∈S′w\\{Tg} tT ′ · Ψ(T ′) � � ∈ Ψ(L(G)) 4 The Simplest Proof of Parikh’s Theorem via Derivation Trees no matter how we choose tT ′ ∈ N for each T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Let S be a well-formed multiset with a ground tree Tg ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Denote by Lin(S) = � � �Ψ(S) + � T ′ i ∈S\\{Tg} ti · Ψ(T ′ i) ����� ti ∈ N � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' (1) The set Lin(S) is linear by the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Since there are only finitely many different sets S′ w, the set Ψ(L(G)) is a finite union of linear sets Lin(S′ w), so it is semilinear by the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' The scheme of the proof is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' 2 L(G) Ψ(L(G)) {Tw : w ∈ L(G)} {Sw : w ∈ L(G)} {S′ w : w ∈ L(G)} {Lin(S′ w) : w ∈ L(G)} Ψ ∪ Figure 2 Scheme of the proof 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='1 Auxiliary Lemmas ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Let S be a well-formed multiset of minimal trees and S′′ be a multiset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Denote by ⃗v and ⃗v′′ the corresponding vectors of S and S′′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Suppose ⃗vi > 0 iff ⃗v′′ i > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' then S′′ is a well-formed multiset as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' The statement is equivalent to the conjunction of two claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' The first one: if ⃗vi > 0 and ⃗v′′ i = ⃗vi + 1 while ⃗v′′ j = ⃗vj for j ̸= i, then S′′ is well-formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' The second one: if ⃗vi > 1 and ⃗v′′ i = ⃗vi − 1 while ⃗v′′ j = ⃗vj for j ̸= i, then S′′ is well-formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Starting with the vector ⃗v and subsequently increasing or decreasing its components by 1 we can obtain any vector ⃗v′′ satisfying the conditions of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Since at each step the condition of one of the claims hold, we obtain that the condition of the lemma holds in the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Recall Definitions 2 and 3 of trees and multiset derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Denote by Ti = (A ∗=⇒ uAv) the i-th block-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Fix a derivation tree T such that S ⊢ ∗ {T} with a derivation of the tree T as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Proof of the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Due to the form of Ti ∈ S, the tree T contains the non-terminal A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' So we can glue Ti into the place of some occurrence of the non-terminal A and obtain the tree T ′ as the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' S′′ is well-formed, since T ′ is a derivation tree of G by the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Proof of the second claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' If Ti is a subtree of T, than it can be removed from T (as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' 1) and the resulting tree T ′ would also be a derivation tree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Otherwise, let us consider the steps of the fixed derivation of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We change this derivation as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Recalling that ⃗vi > 1, fix two copies T 1 i and T 2 i of the tree Ti in the multiset S such that T 1 i was used A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Rubtsov 5 earlier than T 2 i in the derivation of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' When T 2 i must be composed with the ground tree, we skip this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' If at some step a tree Tj = (B ∗=⇒ u′Bv′) is glued into a node B of T 2 i , then we glue it into the corresponding node B of T 1 i (that is already in the ground tree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' So, this modification yields a derivation S ⊢ ∗ {T ′, T 2 i }, where T ′ is some ground tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' But then we could do the same starting from S′′ and obtain S′′ ⊢ ∗ {T ′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Therefore S′′ is well-formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' ◀ ▶ Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' For any w ∈ L(G) : Lin(S′ w) ⊆ Ψ(L(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' ▶ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' For any w ∈ L(G): Ψ(w) ∈ Lin(S′ w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' From the definitions it follows that Ψ(w) = Ψ(Sw) ∈ Lin(Sw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' We prove that Lin(Sw) ⊆ Lin(S′ w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Note that Ψ(Sw) = Ψ(S′ w) + �m i=1(⃗vi − 1)Ψ(Ti), where ⃗v is the corresponding vector of Sw and Ti is the i-th minimal tree (in the enumeration above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' By the definition a vector ⃗u ∈ Lin(Sw) has the form ⃗u = Ψ(Sw) + � T ′ j∈Sw\\{Tg} t′ j · Ψ(T ′ j) = Ψ(Sw) + n � i=1 ti · Ψ(Ti), where in the second equality we put together all T ′ j’s that are copies of the same tree Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Note that each T ′ j is a minimal block tree, so T ′ j = Ti for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Thus, ⃗u = � �Ψ(S′ w) + � i:⃗vi>0 (ti + ⃗vi − 1)Ψ(Ti) � � ∈ Lin(S′ w) ◀ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='2 Proof of Parikh’s theorem Let G be a context-free grammar generating L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' For a word w ∈ L denote by T(w) the set of all derivation trees Tw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Note that T(w) ̸= ∅ and it can be even infinite if there are ε-rules in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Denote by S(w) = {Sw | ∃Tw ∈ T(w) : Sw ⊢ ∗ {Tw}} where Sw is a multiset consisting only of minimal trees (as before).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Finally S′(w) = {S′ w | Sw ∈ S(w)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' recall that S′ w is the set obtained from Sw by deleting the duplicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Now we show that S′(w) is finite for each w and moreover the union ∪w∈LS′(w) = S′(L) is finite as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Recall that there are finitely many minimal trees, and we denote them by T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' , Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Therefore, each Sw has a corresponding m-dimensional vector ⃗v such that Ψ(w) = Ψ(Sw) = m � i=1 ⃗vi · Ψ(Ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' The corresponding vector ⃗v′ of S′ w is a 0-1 vector such that ⃗v′ i = 1 iff ⃗vi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' So since there are only finitely many 0-1 vectors of length m and each S′ w ∈ S′(L) has a corresponding 0-1 vector ⃗v′, then the set S′(L) is finite as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Putting everything together, by Lemma 7 Ψ(L) ⊆ � S′∈S′(L) Lin(S′) and by Corollary 6 � S′∈S′(L) Lin(S′) ⊆ Ψ(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' 6 The Simplest Proof of Parikh’s Theorem via Derivation Trees Since the set S′(L) is finite and each set Lin(S′) is linear by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' (1), the equality Ψ(L) = � S′∈S′(L) Lin(S′) proves Parikh’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' ◀ References 1 Javier Esparza, Pierre Ganty, Stefan Kiefer, and Michael Luttenberger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Parikh’s theorem: A simple and direct automaton construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Information Processing Letters, 111(12):614–619, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' URL: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='com/science/article/pii/S0020019011000822, doi:https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='ipl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' 2 Jonathan Goldstine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' A simplified proof of Parikh’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Discrete Mathematics, 19(3):235– 239, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' 3 John E Hopcroft, Rajeev Motwani, and Jeffrey D Ullman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Introduction to automata theory, languages, and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Acm Sigact News, 32(1):60–65, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' 4 Caleb Koch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' A friendly tour of Parikh’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' URL: http://cakoch10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='io/ papers/Kleene_Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' 5 Toshihiro Koga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' A proof of Parikh’s theorem via Dickson’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' International Journal of Foundations of Computer Science, 32(02):163–173, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='1142/S012905412150009X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' 6 Rohit J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Parikh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' On Context-Free Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' ACM, 13(4):570–581, oct 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='1145/321356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='321364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' 7 Jeffrey O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Shallit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' A Second Course in Formal Languages and Automata Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Cambridge University Press, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' 8 Ryoma Sin’ya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' Simple proof of Parikh’s theorem a la Takahashi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content=' CoRR, abs/1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} +page_content='09393, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfQfaP/content/2301.00047v1.pdf'} diff --git a/6tE0T4oBgHgl3EQfwAE6/content/tmp_files/2301.02625v1.pdf.txt b/6tE0T4oBgHgl3EQfwAE6/content/tmp_files/2301.02625v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..43ae0d1a0808b8bf5d141ff2ecaa1239e8036172 --- /dev/null +++ b/6tE0T4oBgHgl3EQfwAE6/content/tmp_files/2301.02625v1.pdf.txt @@ -0,0 +1,4228 @@ +arXiv:2301.02625v1 [math.PR] 6 Jan 2023 +STRONG SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH +DISCONTINUOUS AND UNBOUNDED COEFFICIENTS +YAOZHONG HU AND QUN SHI +ABSTRACT. In this paper we study the existence and uniqueness of the strong solution +of following d-dimensional stochastic differential equation (SDE) driven by Brownian mo- +tion: +dXt = b(t,Xt)dt +σ(t,Xt)dBt, X0 = x, +where B is a d-dimensional standard Brownian motion; the diffusion coefficient σ is a +H¨older continuous and uniformly non-degenerate d × d matrix-valued function and the +drift coefficient b may be discontinuous and unbounded, not necessarily in Lq +p, extending +the previous works to discontinuous and unbounded drift coefficient situation. The idea +is to combine the Zvonkin’s transformation with the Lyapunov function approach. To this +end, we need to establish a local version of the connection between the solutions of the +SDE up to the exit time of a bounded connected open set D and the associated partial +differential equation on this domain. As an interesting by-product, we establish a localized +version of the Krylov estimates (Theorem 4.1) and a localized version of the stability result +of the stochastic differential equations of discontinuous coefficients (Theorem 4.5). +Keywords: Discontinuity; localized Krylov estimate; local Zvonkin’s transformation; +localized stability; Lyapunov function; strong solution; pathwise uniqueness. +1. Introduction +Let (Ω,F,P,(Ft)t≥0) be a complete filtered probability space with a filtration (Ft)t≥0 +satisfying the usual conditions and let (Bt)t≥0 be a d-dimensional Ft-adapted standard +Brownian motion. +In this work we study the following stochastic differential equation (SDE) driven by +multi-dimensional Brownian motion: +dXt = b(t,Xt)dt + σ(t,Xt)dBt, +X0 = x ∈ Rd, +(1.1) +where the diffusion coefficient σ : R+ × Rd → Rd ⊗ Rd is uniformly elliptic and H¨older +continuous with respect to spatial variable in any bounded domain D and the drift coeffi- +cient b : R+ × Rd → Rd is integrable in any bounded domain. More precisely, we make +the following assumptions. +(Hσ): For any bounded domain D ⊆ Rd there exist constants α ∈ (0,1], κ = κα,D > 1, +such that for all (t,x) ∈ [0,T]× D, +κ−1|ξ|2 ≤ |σt(t,x)ξ|2 ≤ κ|ξ|2, ∀ξ ∈ Rd , +Y.Hu was supported by the NSERC discovery fund and a centennial fund of University of Alberta, National +Natural Science Foundation of China (12261046). +Q.Shi was supported by China Overseas Education Fund Committee, National Natural Science Foundation of +China (11901257, 12261046), and an NSERC discovery fund. +AMS Mathematics Subject Classification (2010): 60G15; 60H07; 60H10; 65C30 . +1 + +2 +YAOZHONG HU AND QUN SHI +and +∥σ(t,x)− σ(t,y)∥HS ≤ κ|x− y|α, +where σt stands for the transpose of the matrix σ and where for a matrix A, +∥A∥HS = tr(ATA) stands for its Hilbert-Schmidt norm. We also assume +∇σ ∈ Lq +p([0,T]× D) +for certain p and q satisfying d +p + 2 +q < 1, +(1.2) +where Lq +p([0,T]× D) is defined by (2.1) in the next section. +For the drift coefficient we make the following assumptions. +(Hb): b is uniformly locally integrable on any bounded domain of Rd, namely, +b ∈ Lq +p([0,T]× D) < ∞, +∀ bounded domain D, +for p and q appeared in (1.2). +(HL): There exists a non-negative (Lyapunov) function V ∈ C1,2([0,T]× Rd) [the set of +all functions { f(t,x),t ∈ [0,T],x ∈ Rd} which are continuously differentiable in t +and have continuous derivatives with respect to the spatial variable x up to second +order] satisfying +lim +R→∞ +inf +0≤t≤T,|x|≤RV(t,x) = ∞ +and +LtV(t,x) ≤ CV(t,x), ∀t ∈ [0,T], x ∈ Rd , +(1.3) +for some constant C > 0, where Lt is the differential operator associated with (1.1): +Lt := ∂ +∂t + Lσ,b = ∂ +∂t + +d +∑ +i=1 +bi(t,x) ∂ +∂xi ++ 1 +2 +d +∑ +i,j=1 +d +∑ +k=1 +(σikσ jk)(t,x) +∂ 2 +∂xi∂xj +. +(1.4) +The objective of this work is to show that under the above assumptions (Hσ), (Hb) and +(HL), the equation (1.1) has a unique strong solution (e.g. Theorem 5.2). +For stochastic differential equations with discontinuous (yet bounded) coefficients there +are rather complete general results about the weak solution and we refer to the classi- +cal work [28] and references therein. For the strong solutions there have been also some +important progresses. Among them let us mention the works [18, 33, 35, 38]. Let us em- +phasize an important point that in these works, the assumptions that σ is uniformly elliptic +on the whole space Rd and b ∈ Lq +p = Lq +p(Rd) are usually needed, which generally require +that σ and b are bounded when |x| → ∞. In [34], the authors show the weak differentia- +bility of the unique strong solution with respect to the starting point x as well as Bismut- +Elworthy-Li’s derivative formula for SDE (1.1) under the condition that 1) σ is bounded, +uniformly continuous, and nondegenerate, 2) b ∈ �Lq1 +p1, ∇σ ∈ �Lq2 +p2 for some p1,q1 ∈ [2,∞) +with d +pi + 2 +qi < 1, i = 1,2, where �Lqipi are some localized spaces of Lqipi. But unbounded +drift coefficients may still not be contained in this space because if b is unbounded, then +|||b|||�Lqipi := supz∈R ||b · χz +r||Lqipi = ∞ with χ ∈ C∞ +c (Rd) and χz +r(x) := χr(x − z) := χ( x−z +r ), +r > 0. The idea in all of these works is to use the Zvonkin transformation to transform +equation (1.1) with discontinuous coefficients to an equation without drift and hence the +problem of discontinuity of the drift coefficient disappears. +Recently, the more and more interest in studying SDE (1.1) with discontinuous coeffi- +cients is partly due to its more and more important role played in applications. For example, +the threshold Ornsten-Uhlenbeck processes (e.g. [11] and the references therein) widely +used in application contain discontinuous but piecewise linear drift; the work of Flandoli, + +WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT +3 +Gubinelli and Priola [6] discovers that noises can prevent the singularity for linear transport +equations. In these equations the drift coefficient b is not in Lq +p. +In fact, for an SDE with discontinuous and unbounded coefficients there are also some +study although the number of works is limited. Let us mention a recent work [36] in which +Zhang et al show the existence and uniqueness of the martingale solution (weak solution) +to the above homogeneous (time independent) SDE (1.1) under the following assumptions: +(1) the drift coefficient b is decomposed into b = b1 + b2, where +⟨x,b1(x)⟩ +√ +1 + x2 ≤ −κ0|x|ϑ + κ1, |b1(x)| ≤ κ2(1 + |x|ϑ) +(1.5) +for some ϑ ≥ 0 and κ0,κ1,κ2 > 0, +b2 ∈ H−α +p +for some α ∈ (0,1/2] and p ∈ +� +d +1 − α ,∞ +� +. +(2) The diffusion coefficient σ is uniformly elliptic and +||(−∆)β/2σ||Lr < ∞ for some β ∈ [α,1], r ∈ (d/β,∞). +They obtain sharp two-sided as well as the gradient estimates of the heat kernel associated +to the above SDE. Moreover, they study the ergodicity and global regularity of the invariant +measures of the associated semigroup. However, there is no study on the strong solution. +If b and σ are locally Lipschitzian on any bounded domain, then it is well-known that +the assumption (HL) is the famous Lyapunov type condition to guarantee non explosion +of the equation. So, in some sense our condition is to combine the Lyapunov condition +and the integrability condition. Our new set of conditions can be applied to some new +situations. We give only some very special examples as follows. +Example 1.1. ([11]) One example is the threshold Ornstein-Ulenbeck processes: +dXt = +n +∑ +i=1 +(βi − αiXt)I{θi−1≤Xt<θi}dt + σdBt , +(1.6) +where β1,··· ,βn,α1,··· ,αn,−∞ = θ0 < θ1 < ··· < θn−1 < θn = ∞ are constants. It is clear +that for all parameters βi,αi,θi, the drift coefficient b(x) = ∑n +i=1(βi − αix)I{θi−1≤x<θi} and +the diffusion coefficient σ(x) = σ satisfy (1.3) with V(x) = |x|2 and with some constant C. +But it does not satisfy (1.5) unless α1,··· ,αn are positive. +Example 1.2. Another interesting example is +dXt = +n +∑ +k=1 +� mk +∑ +j=1 +βk,jX j +t +� +I{θk−1≤Xt<θk}dt + σdBt , +(1.7) +where βk,j are constants and βk,mk ̸= 0 and −∞ = θ0 < θ1 < ··· < θn−1 < θn = ∞ are con- +stants. It is clear that if m1,mn are odd, and if β1,n1,βn,nn are negative, then (1.3) is satisfied +with V(x) = |x|2 and with some constant C. But (1.5) will generally not be satisfied. +In the above assumption (Hb), b may be unbounded in Rd so we cannot use the existing +theory to solve (1.1). To study the well-posedness of this equation, our strategy is to +consider first the solvability of the associated parabolic differential equation with Cauchy- +Dirichlet problem on bounded domain + + + +∂tu + Lσu + b ·∇u + f = 0, (t,x) ∈ (0,T)× D, +u(T,x) = 0, x ∈ D, +u(t,x) = g(t,x), (t,x) ∈ (0,T)× ∂D, +(1.8) + +4 +YAOZHONG HU AND QUN SHI +where D is bounded nonempty domain (a connected open subset) of Rd so that ∂D ∈ +C2, g is integrable functions on (0,T) × ∂D. Then, with the aid of the solution to the +above equation we find a C1-diffeomorphism Φ to transform (1.1) to another equation with +only diffusion term. This transformed equation has a unique strong solution in bounded +nonempty domain. Finally, by a stopping time argument combined with the Lyapunov +function V (given in assumption (HL)) we obtain the unique strong solution in Rd of SDE +(1.1). +There are new challenges in each of our above steps. First, when the coefficients are +nice equation (1.8) is a classical Cauchy-Dirichlet problem in bounded domain. However, +when the coefficient is only locally integrable, it seems there is no study on such equa- +tion, to our best knowledge. Relevant works we found is the parabolic problem for whole +space Rd (which was used in the works [18, 33, 35, 38]) and elliptic problem for general +(including bounded) domain). These works can be found for example in [15], where are +also mentioned that the results on elliptic problem on general domain can be extended to +parabolic problem. Since we need to know what exact we can cite, we present a detail +study on this problem in Section 3. +In the previous works on the strong solution with integrable drift coefficients, an impor- +tant technique is the so-called the Zvonvin transformation which reduced the original SDE +to an SDE without drift terms. Now we need study SDE on a bounded domain. Since the +solution will stay in the boundary only in some finite stopping time, we need to study the +solution with a finite (random) life time for the equation. Not much study on the stochastic +differential equation is available. To obtain the strong solution of an SDE on the boundary +domain, we extend the coefficients to the whole Euclidean space Rd. However, there is no +available extension theorem we can immediately use. We need to modify some existing +results so that they are applicable to our situation. Some of these are given in Section 2 +and some are given in Appendix (Section 6). The stochastic differential equations on a +bounded domain is studied in detail in Section 4. For the existence and uniqueness one +of the most important tools is the Krylov estimate. We deduce the Krylov type estimate +for stochastic differential equations on bounded domain, namely, a localized version of the +Krylov estimate, also in Section 4. In Section 5, we prove the main result of the existence +and uniqueness of strong solutions to SDE (1.1) under Lyapunov condition. In Section 2, +we recall some well-known results and give briefly some preliminaries about the Sobolev +differentiabilities of random vector fields. The last section contains some technical results +obtained and used in the paper. +Throughout this paper, we use the following convention: C with or without subscripts +will denote a positive constant, whose value may change in different places, and whose +dependence on the parameters can be traced from the calculations. +2. PRELIMIARIES +We first introduce some spaces and notations for later use. Let p,q ∈ [1,∞), T > 0 and +let D be a bounded,connected domain in Rd with C2 boundary. We denote by Lq +p((0,T)× +D) the space of all real-value Borel functions on [0,T]× D such that +∥ f∥Lq +p((0,T)×D) := +�� T +0 +�� +D |f(t,x)|pdx +�q/p +dt +�1/q +< ∞. +(2.1) +For p,q = ∞, +∥ f∥L∞∞((0,T)×D) := sup +t∈[0,T] +sup +x∈D +|f(t,x)| < ∞. + +WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT +5 +When q = p we denote Lp((0,T) × D) := Lp +p((0,T) × D). When D = Rd, we denote +Lq +p = Lq +p(0,T) := Lq +p((0,T)× Rd). +For any positive integer m and any p ≥ 1, W m +p (D) is used to denote the usual Sobolev +space over the domain D ⊆ Rd with norm +∥ f∥Wm +p (D) := +m +∑ +k=0 +∥∇k +x f∥Lp(D) < +∞, +where ∇k +x denotes the k-order gradient operator on spatial variable. When p = q and D = +Rd, we denote Lp = Lp +p(0,T) := Lp +p((0,T)×Rd). For β ∈ R, let Hβ +p := (I−∆)− β +2 (Lp) be +the usual Bessel potential space with norm (we refer to [27], [30]) +∥ f∥Hβ +p = ∥(I− ∆) +β +2 f∥p , +where ∥ ·∥p is the usual Lp-norm. Notice that for m ∈ N and p > 1, +∥ f∥Hmp ≍ ∥ f∥Wmp , +where and throughout the paper we use A ≲ B to denote that fact that there is a constant C +such that A ≤ CB and A ≍ B means A ≲ B and B ≲ A. For β ∈ [0,2) and p ∈ (1,∞), by +Mihlin’s multiplier theorem (see [29]), we know +∥ f∥Hβ +p ≍ ∥(I− ∆) +β +2 f∥p ≍ ∥ f∥p + ∥(−∆) +β +2 f∥p . +Let C(Rd) be the collection of all continuous functions in Rd, equipped with the norm +||f||C(Rd) := sup +x∈Rd |f(x)|. +Obviously, C(Rd) is a Banach space. Let k ∈ N. Then +Ck(Rd) := { f ∈ C(Rd) : Dα f ∈ C(Rd) if |α| ≤ k} +is a Banach space equipped with the norm +||f||Ck(Rd) = ∑ +|α|≤k +||Dα f||C(Rd). +Here, we use standard notations: α = (α1,··· ,αd) with αj ∈ N0 := {0,1,2,···} is a mul- +tiindex, |α| = ∑d +j=1 αj and +Dα f(x) = +∂ |α| f +∂ α1 +x1 ···∂ αd +xd +(x). +For any positive number 0 < δ < 1, let C δ be the usual H¨older space with finite norm +∥ f∥C δ (Rd) := sup +x∈Rd +|f(x)|+ +sup +x̸=y,x,y∈Rd +|f(x)− f(y)| +|x− y|δ +< ∞. +For any 0 < δ ̸∈ N0, we put +δ = [δ]+ {δ}, + +6 +YAOZHONG HU AND QUN SHI +where [δ] = max{k ∈ Z;,k ≤ δ} is the integer part of δ, 0 < {δ} < 1. Denote the H¨older +space (refer to [29, Chapter 1.2]) +C δ(Rd) : += +� +f ∈ C(Rd) : ∥ f∥C δ (Rd) = ||f||C[δ](Rd) ++ ∑ +|α|=[δ] +� +sup +x∈Rd |Dα f(x)|+ +sup +x̸=y,x,y∈Rd +|Dα f(x)− Dα f(y)| +|x− y|δ−[δ] +�� +. (2.2) +By Sobolev’s embedding theorem ([3, Corollary 7.11]), we have +∥ f∥C δ (D) ≤ C∥ f∥Hβ +p(D), β − δ > d +p, δ ≥ 0. +(2.3) +In particular, we take δ = 0 to obtain ∥ f∥∞ ≤ C∥ f∥Hβ +p , as β > d +p. +Furthermore, we also need the following Sobolev space: for p,q ∈ [1,∞),m ≥ 0 we +denote by Wm,q +p ((0,T)×D) the set of all Borel functions +� +f(t,x),0 ≤ t ≤ T,x ∈ Rd� +such +that +∥ f∥Wm,q +p +((0,T)×D) := +�� T +0 +� +∥∂t f(t)∥q +Lp(D) + +m +∑ +k=0 +∥∇k f(t)∥q +Lp(D) +� +dt +�1/q +< ∞, +where ∇k is the gradient with respect to x only. By ˚W1,q +p ((0,T) × D) we mean the subset +of W1,q +p ((0,T) × D) consisting of all functions vanishing on the boundary ∂D. For k ∈ +{2,3,...} let +˚Wk,q +p ((0,T)× D) = ˚W1,q +p ((0,T)× D)∩Wk,q +p ((0,T)× D). +The norm in ˚Wk,q +p ((0,T)×D), k ∈ {1,2,...}, is taken to be the same as in Wk,q +p ((0,T)×D). +The introduction of ˚Wk,q +p ((0,T)×D) is to express the Dirichlet boundary condition. When +we say u = g on ∂D we understand the following condition +u − g ∈ ˚Wk,q +p ((0,T)× D). +We can also introduce the function class Wk,q +p ((0,T)×∂D), k = 0,1,2,... By [15, Theorem +13.7.2], a function g in Wk,q +p ((0,T)× ∂D) can be extended to a function on D so that it is +in Wk,q +p ((0,T)×D). We shall use this property in the future instead of giving the definition +of Wk,q +p ((0,T)× ∂D). +Let f be a locally integrable function on Rd . The classical Hardy-Littlewood maximal +function is defined by +M f(x) := sup +0 1. We also need to use a similar property of this operator on a domain D, which +is given in Appendix. + +WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT +7 +3. Solvability of parabolic differential equations with Dirichlet boundary +Denote +Lσ := 1 +2aij(t,x) +∂ 2 +∂xi∂xj , +where +aij(t,x) = +d +∑ +k=1 +(σikσ jk)(t,x). +(3.1) +This section is devoted to study the following Cauchy-Dirichlet problem in bounded do- +main D for non-smooth coefficient parabolic equation: + + + +∂tu + Lσu + b ·∇u + f = 0, (t,x) ∈ (0,T)× D, +u(T,x) = 0, x ∈ D, +u(t,x) = g(t,x), (t,x) ∈ (0,T)× ∂D, +(3.2) +where D is a bounded nonempty domain (connected open subset) of Rd so that ∂D ∈ C2 +and g is integrable functions on (0,T) × ∂D. We assume that the coefficients σ and b +satisfy (Hσ), (Hb) on D. We mentioned a few time that ∂D ∈ C2. We recall this concept in +the following definition. +Definition 3.1. Let k ∈ {1,2,···} and let D be a domain in Rd. We say that the domain D +is of class Ck (denoted by D ∈ Ck (or ∂D ∈ Ck)), if there are numbers κ,ρ0 such that for +any point z ∈ ∂D there exists a diffeomorphism ψ = ψz (we omit the dependence of ψ on +z) from a ball Bρ0(z) of center z with radius ρ0 onto a domain Dz ⊆ Rd such that +(i) +Dz ++ := ψ(Bρ0(z)∩D) ⊆ Rd ++ := +� +x = (x1,x2,··· ,xd) ∈ Rd,x1 > 0 +� +and ψ(z) = 0 ∈ Rd, +(ii) +ψ(Bρ0(z)∩∂D) = Dz ∩∂Rd ++, +(iii) +ψ ∈ Ck(Bρ0(z)),ψ−1 ∈ Ck(Dz), +and |ψ|Ck(Bρ0(z)) + |ψ−1|Ck(Dz) ≤ κ. +(3.3) +Such diffeomorphismψ is said to straighten or flatten the boundary of D near z. Through- +out this paper we fix a domain D ∈ ∂C2. +3.1. Operators in a neighborhood of a boundary point. For the moment we fix z ∈ ∂D. +To ease our notation, for functions v and �v defined in (0,T)×(Bρ0(z)∩D) and (0,T)×Dz ++, +respectively, we write +v(t,x) = �v(t,y) +(3.4) +if this equality holds for +y = ψ(x) = ψz(x) = (ψ1(x),··· ,ψd(x)). +(3.5) +This equality is also used to introduce a function �v on (0,T) × Dz ++ if we are given a +function v on (0,T) × (Bρ0(z) ∩ D), and vice versa. In this subsection if in a formula we +have both x and y, we will always assume that they are related by (3.5). +Let v ∈ C1,2((0,T) × (Bρ0(z) ∩ D)). In the domain Dz ++ define the function �v(t,y) by +relationship (3.4). Obviously, in (0,T)× (Bρ0(z)∩D), +∂tv(t,x) += +∂t�v(t,y), +vxi(t,x) += +�vyr(t,y)ψr +xi(x), +vxixj(t,x) += +�vyr(t,y)ψr +xixj(x)+ �vyryl(t,y)ψr +xi(x)ψl +xj(x). +(3.6) +In fact, the above formulas are also true if v ∈ W2,q +p ((0,T) × (Bρ0(z) ∩ D)) by a limiting +argument. + +8 +YAOZHONG HU AND QUN SHI +Lemma 3.2. For any function ζ ∈ C∞ +0 such that ζ(t,x) = 0 for |x| ≥ ρ0/2, t ∈ (0,T) and +0 ≤ ζ ≤ 1 everywhere, where ρ0 is the one appeared in Definition 3.1 set +ζ z(t,x) = ζ(t,x− z), +�ζ z(t,y) = ζ z(t,x). +(3.7) +Let z ∈ ∂D and let v, �v be functions on (0,T)×(Bρ0(z)∩D) and (0,T)×Dz ++, respectively, +related by (3.5). Then +ζ zv ∈ ˚Wk,q +p ((0,T)× (Bρ0(z)∩D)) ⇐⇒ �ζ z�v ∈ ˚Wk,q +p ((0,T)× Dz ++). +(3.8) +In addition, for r = 0,1,...,k, +1 +C∥ζ zv∥Wr,q +p ((0,T)×(Bρ0(z)∩D)) ≤ ∥�ζ z�v∥Wr,q +p ((0,T)×Dz ++) ≤ C∥ζ zv∥Wr,q +p ((0,T)×(Bρ0(z)∩D)) , (3.9) +where C = C(d, p,q,k,κ,ζ). Furthermore, the assertion (3.8) is also true if we replace +˚Wk,q +p ((0,T)× (Bρ0(z)∩D)) by Wk,q +p ((0,T)× (Bρ0(z)∩D)). +Proof +By relationships (3.4), (3.5) and (3.7), we have +� T +0 +�� +Bρ0(z)∩D |ζ zv|p(t,x)dx +�q/p +dt = +� T +0 +�� +Dz ++ +|�ζ z�v|p(t,y)|J|dy +�q/p +dt , +where J is the Jacobian of the map ψ−1, i.e. J := ∂ψ−1 +∂y . This shows (3.9) when r = 0. +Without loss of generality, we assume v ∈ C1,1((0,T)× (Bρ0(z)∩D)). Since +(ζ zv)xi(t,x) += +v(t,x)ζ z +xi(t,x)+ ζ z(t,x)vxi(t,x) += +v(t,x)�ζ z +yr(t,y)ψr +xi(x)+ ζ z(t,x)�vyr(t,y)ψr +xi(x) += +(�ζ z�v)yr(t,y)ψr +xi(x), +we have +� T +0 +�� +Bρ0(z)∩D |ζ zv|p +xi(t,x)dx +�q/p +dt ≤ Kq +1 +� T +0 +�� +Dz ++ +|�ζ z�v|p +yr(t,y)|J|dy +�q/p +dt. +This shows the first inequality in (3.9) when r = 1. +For general k, we have +� T +0 +�� +Bρ0(z)∩D |∂ k(ζ zv)|p(t,x)dx +�q/p +dt ≤ Cκ,q +� T +0 +�� +Dz ++ +k +∑ +i=1 +|∂ i(�ζ z�v)|p(t,y)|J|dy +�q/p +dt. +Owing to ∂t(ζ zv)(t,x) = ∂t(�ζ z�v)(t,x), we see +� T +0 +�� +Bρ0(z)∩D |∂t(ζ zv)|p(t,x)dx +�q/p +dt = +� T +0 +�� +Dz ++ +|∂t(�ζ z�v)|p(t,y)|J|dy +�q/p +dt. +Thus, we have +∥ζ zv∥Wk,q +p ((0,T)×(Bρ0(z)∩D)) ≤ C∥�ζ z�v∥Wk,q +p ((0,T)×Dz ++). +This proves the first inequality in (3.9). As for the second inequality in (3.9), we just need +to make the change of variables by using ψ instead of ψ−1. +If (ζ zv)(t,x) = 0 for x ∈ ∂(Bρ0(z)∩D), then (�ζ z�v)(t,y) = (ζ zv)(t,x) = 0 for y ∈ ∂Dz ++. +This shows that the assertion (3.8) is also true if we replace ˚Wk,q +p ((0,T)×(Bρ0(z)∩D)) by +Wk,q +p ((0,T)× (Bρ0(z)∩D)). + +WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT +9 +Let v ∈ C1,2((0,T)× (Bρ0(z)∩D)). Denote +L0 +t v(t,x) := ∂tv(t,x)+ 1 +2 +d +∑ +i,j=1 +aij(t,x)vxixj(t,x) +(3.10) +and +�Lz +t �v(t,y) := ∂t�v(t,y)+ 1 +2 +d +∑ +r,l=1 +�arl(t,y)�vyryl(t,y)+ +d +∑ +r=1 +�br(t,y)�vyr(t,y), +(3.11) +where +� +�arl(t,y) := ∑d +i,j=1 aij(t,x)ψr +xi(x)ψl +xj(x); +�br(t,y) := ∑d +i,j=1 aij(t,x)ψr +xixj(x) +(3.12) +and we recall that x is related to y by (3.5). +We are going to discuss the equation +L0 +t v(t,x) = f(t,x) +(3.13) +for some function f or +˜Lt�v(t,y) = �f (t,y), +(3.14) +where �f (t,y) = f(t,x). +Lemma 3.3. Let σ satisfy (Hσ) on the domain D. +(i) For any ψ and Dz defined in Definition 3.1 and for any y,y1,y2 ∈ Dz, we have +|�arl(t,y)| ≤ C, |�arl(t,y1)− �arl(t,y2)| ≤ C|y1 − y2|α , +where C = C(d,κ). +(ii) A function v ∈ W2,q +p ((0,T)×(Bρ0(z)∩D)) b satisfies (3.13) in (0,T)×(Bρ0(z)∩D) +if and only if �v ∈ W2,q +p ((0,T)× Dz ++) satisfies (3.14) in (0,T)× Dz ++. +(iii) The operator �Lt is parabolic in (0,T)× Dz, that is, there is a �κ > 0 such that +�arl(t,y)θ rθ l ≥ �κ|θ|2, ∀θ ∈ Rd, t ∈ (0,T), y ∈ Dz. +Proof +(i) The first assertion is obvious. +(ii) Using the computations in (3.6), we see that if v satisfies (3.13), then ˜v satisfies (3.14). +The reverse part is similar. +(iii) Notice that +�arl(t,y)θ rθ l = arl(t,x)ψr +xi(x)ψl +xj(x)θ rθ l += arl(t,x)(θψ)r +xi(x)(θψ)l +xj(x) ≥ κ|(θ ·ψ)x(x)|2 . +Letting φ := ψ−1 we have φ(y) = x and +ψr +xi(x)φi +yj(y) = δr j = +� 0, if r ̸= j, +1, if r = j, +(θ ·ψ)r +xi(x)φi +yj(y) = θ rδr j = θ j . +Thus, we have +|θ|2 ≤ Cκ2|(θψ)x|2. + +10 +YAOZHONG HU AND QUN SHI +This yields +�arl(t,y)θ rθ l ≥ �κ|θ|2 +with �κ depending on d,κ. +Next, we fix a function η ∈ C∞ +0 so that +η(t,x) = +� +1, +for t ∈ (0,T),|x| ≤ ρ0/2, +0, +for t ∈ (0,T),|x| ≥ 3ρ0/4, +and 0 ≤ η ≤ 1 everywhere. Define +ηz(t,x) := η(t,x− z), +�ηz(t,y) := ηz(t,x), +�Lz +t (y) := �ηz(t,y)�Lz +t (y)+ (1 − �ηz(t,y))(∆+ ∂t). +(3.15) +Let ζ z be the function introduced in Lemma 3.2. It is easy to verify +�Lz +t (ζ z·) = �Lz +t (ζ z·). +(3.16) +Now, we first deal with the case that b is identically equal to zero in operator Lt and +recall that +L0 +t := ∂t + 1 +2 +d +∑ +i,j=1 +d +∑ +k=1 +(σikσ jk)(t,x) +∂ 2 +∂xi∂xj +, +where the coefficient σ is defined on D and satisfies the condition (Hσ) on D. By Propo- +sition 6.1 we can extend σ a function on Rd satisfying (Hσ) on the whole space Rd. The +relation (3.16) and [15, Theorem 7.2.8 with b = 0] allow us to find a λ0 ≥ 1 depending +only on κ,ρ0, p,q,d,α such that for λ ≥ λ0, the operator +λ − �Lz +t = λI − �Lz +t : W2,q +p ((0,T)× Rd) → Lq +p((0,T)× Rd) +is invertible, where I is the identity operator. On the other hand, applying a time dependent +version of [14, Lemma 8.2.1] yields that for λ ≥ λ0 the operator +λ − �Lz +t : ˚W2,q +p ((0,T)× Rd ++) → Lq +p((0,T)× Rd ++) +(3.17) +is invertible. We denote this inverse by +� +Rz +λ,t := +� +λ − �Lz +t +�−1 +. +Define +Ψ +: +w = w(t,y) → Ψw(t,x) = w(t,ψ(x)), +Ψ−1 +: +v = v(t,x) → Ψ−1v(t,y) = v(t,ψ−1(y)), +Rz +λ,t +: +f = f(t,x) → Rz +λ,t f(t,x) = Ψ � +Rz +λ,tΨ−1[ηz f](t,x). +(3.18) +From the definitions of Ψ and Ψ−1 it follows +�v = Ψ−1v, v = Ψ�v +and the identities (3.11) and (3.16) imply that +�Lz +t Ψ−1(ζ zv) = �Lz +t Ψ−1(ζ zv) = Ψ−1Lt(ζ zv), +Ψ�Lz +t Ψ−1(ζ zv) = Lt(ζ zv), �Lz +t (�ζ z�v) = Ψ−1LtΨ(�ζ z�v). +(3.19) + +WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT +11 +Theorem 3.4. (i) If ζ zv ∈ ˚W2,q +p ((0,T)× D), then for any λ ≥ λ0, we have +ζ zv = Rz +λ,t(λ − L0 +t )(ζ zv) +on (0,T)× (Bρ0(z)∩D). +(ii) There is a constant C depending only on κ,ρ0, p,q,d,α such that for λ ≥ λ0 and +f ∈ Lq +p((0,T)× D), we have +ζ zRz +λ,t f ∈ ˚W2,q +p ((0,T)× (Bρ0(z)∩D)), +and +λ∥ζ zRz +λ,t f∥Lq +p((0,T)×D) + λ 1/2∥ζ zRz +λ,t f∥W1,q +p ((0,T)×D) + ∥ζ zRz +λ,t f∥W2,q +p ((0,T)×D) ++∥∂t(ζ zRz +λ,t f)∥Lq +p((0,T)×D) ≤ C∥ f∥Lq +p((0,T)×(Bρ0(z)∩D)). +(3.20) +Proof +(i) Define w := ζ zv, f := λw − L0 +t w. Then by Lemmas 3.2 and 3.3 we have �w ∈ +˚W2,q +p ((0,T)× Dz ++). Since ηz f = f in ((0,T)× (Bρ0/2(z))∩D), we see +�f = Ψ−1 f = Ψ−1[ηz f]. +This yields +�w(t,y) = (λ − �Lz +t )−1 �f(t,y) = � +Rz +λ,t �f(t,y) = � +Rz +λ,tΨ−1[ηz f](t,y), +and +w(t,x) = Ψ�w(t,x) = Ψ � +Rz +λ,tΨ−1[ηz f](t,x) = Rz +λ,t f(t,x) = Rz +λ,t(λ − L0 +t )(ζ zv) +for y ∈ Dz ++ and x = ψ−1(y) ∈ Bρ0/2(z))∩D. +(ii) By Lemma 3.2 and equation (3.17), we have for λ ≥ λ0 ≥ 1, +λ∥ζ zRz +λ,t f∥Lq +p((0,T)×D) + λ 1/2∥ζ zRz +λ,t f∥W1,q +p ((0,T)×D) ++∥ζ zRz +λ,t f∥W2,q +p ((0,T)×D) + ∥∂t(ζ zRz +λ,t f)∥Lq +p((0,T)×D) +≤ C +� +λ∥�ζ z � +Rz +λ,t �f ∥Lq +p((0,T)×Rd+) + λ 1/2∥�ζ z � +Rz +λ,t �f ∥W1,q +p ((0,T)×Rd+) ++∥�ζ z � +Rz +λ,t �f∥W2,q +p ((0,T)×Rd ++) + ∥∂t(�ζ z � +Rz +λ,t �f)∥Lq +p((0,T)×Rd+) +� +≤ C +� +λ∥ � +Rz +λ,tΨ−1[ηz f]∥Lq +p((0,T)×Rd+) + λ 1/2∥ � +Rz +λ,tΨ−1[ηz f]∥W1,q +p ((0,T)×Rd+) ++∥ � +Rz +λ,tΨ−1[ηz f]∥W2,q +p ((0,T)×Rd+) + ∥∂t( � +Rz +λ,tΨ−1[ηz f])∥Lq +p((0,T)×Rd+) +� +≤ C +� +λ∥ � +Rz +λ,tΨ−1[ηz f]∥Lq +p((0,T)×Rd ++) + λ 1/2∥D � +Rz +λ,tΨ−1[ηz f]∥Lq +p((0,T)×Rd ++) ++∥D2 � +Rz +λ,tΨ−1[ηz f]∥Lq +p((0,T)×Rd+) + ∥∂t( � +Rz +λ,tΨ−1[ηz f])∥Lq +p((0,T)×Rd+) +� +≤ C∥Ψ−1[ηz f]∥Lq +p((0,T)×Rd+) ≤ C∥ηz f∥Lq +p((0,T)×(Bρ0(z)∩D)) +≤ C∥ f∥Lq +p((0,T)×(Bρ0(z)∩D)). +The theorem is then proved. +Lemma 3.5. If f ∈ Lq +p((0,T)× (Bρ0(z)∩D)), then +(λ − L0 +t )Rz +λ,t f = f +on (0,T)× (Bρ0/2(z)∩D). + +12 +YAOZHONG HU AND QUN SHI +Proof +Denote f1 := Rz +λ,t f, and f2 := (λ − L0 +t )f1. It suffices to prove f = f2 in (0,T)× +(Bρ0/2(z)∩D). Since +f ∈ Lq +p((0,T)× (Bρ0(z)∩D)) ⇒ f1 ∈ W2,q +p ((0,T)× (Bρ0(z)∩D)) +we have by Lemma 3.3 (ii) +�f1 ∈ W2,q +p ((0,T)× Dz ++) +and +�f2 = (λ − �Lz +t ) �f1 +in (0,T)× Dz ++. +Therefore, we have �f1 = (λ − �Lz +t )−1 �f2 = � +Rz +λ,t �f2 in (0,T) × Dz ++. On the other hand, by +definition of ηz, we see ηz f = f in (0,T)× Bρ0/2(z), and then we have +�f1 += +Ψ−1 f1 = Ψ−1Rz +λ,t f = Ψ−1(Ψ � +Rz +λ,tΨ−1[ηz f]) += +� +Rz +λ,tΨ−1[ηz f] = � +Rz +λ,tΨ−1 f = � +Rz +λ,t �f . +This yields +� +Rz +λ,t �f2 = � +Rz +λ,t �f . +Since � +Rz +λ,t and Ψ−1 are linear operators, we conclude f = f2 in (0,T)×(Bρ0/2(z)∩D). +3.2. Piece together the neighborhoods of boundary points. Take a function ξ ∈C∞ +0 (Rd) +such that 0 ≤ ξ ≤ 1, and +ξ(x) = +� +0 +when |x| ≥ ρ0/2, +1 +when |x| ≤ ρ0/4. +Next, take points z1,z2,...,zn ∈ ∂D so that +� +|zi − zj| ≥ ρ0/8 +for i ̸= j and +the boundary ∂D is covered by Bρ0/8(zi) . +This is possible because we assume that D is bounded so that the number n of the points zi +can be chosen to be finite. Observe that n can be estimated through ρ0,d and diam(D). +Denote +ξ i(x) = ξ(x− zi), i = 1,2,...,n. +(3.21) +These function are used to smooth a function near the boundary ∂D of the domain D. To +smooth a function inside the domain we introduce another function ξ 0 ∈ C∞ +0 (Rd) such that +0 ≤ ξ 0 ≤ 1, +� ξ 0(x) = 0 for x ∈ D with dist(x,∂D) ≤ ρ0/16, +ξ 0(x) = 1 for x ∈ D with dist(x,∂D) ≥ ρ0/8. +(3.22) +This is possible, for instance, by mollifying the indicate function of +D\ {x : dist(x,∂D) ≤ 3ρ0/32}. +Notice that +n +∑ +i=1 +(ξ i(x))2 ≥ 1 if x ∈ D and dist(x,∂D) ≤ ρ0/8. +Therefore, +ξ := +n +∑ +i=0 +(ξ i(x))2 ≥ 1 in D. + +WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT +13 +Moreover, ξ and its derivatives of any order are bounded in D by a constant depending +only on d,n,ρ0, and the order of the derivative. Finally, define +ζ i(x) := ξ iξ +−1/2, i = 0,1,...,n. +(3.23) +It is easy to see that all ζ i are infinitely differentiable and +n +∑ +i=0 +(ζ i(x))2 = 1, +on (0,T)× D. +Denote +R(i) +λ,t = Rzi +λ,t, i = 1,...,n +for λ ≥ λ0 ≥ 1 and zi ∈ ∂D. Let R(0) +λ,t be the inverse operator of (λ − L0 +t ) : W2,q +p ((0,T) × +Rd) −→ Lq +p((0,T)× Rd). +We may increase λ0 if needed so that [15, Theorem 7.2.8] can be applied (with b = 0) +to yield that for λ ≥ λ0 +λ∥R(0) +λ,t f∥Lq +p((0,T)×Rd) + λ 1/2∥R(0) +λ,t f∥W1,q +p ((0,T)×Rd) + ∥R(0) +λ,t f∥W2,q +p ((0,T)×Rd) ++ ∥∂tR(0) +λ,t f∥Lq +p((0,T)×Rd) ≤ C∥ f∥Lq +p((0,T)×Rd) . +(3.24) +Lemma 3.6. Let u ∈ ˚W2,q +p ((0,T)×D) and set f = λu−L0 +t u. If λ ≥ λ0, then in (0,T)×D +u = ∑ +0≤k≤n +ζ kR(k) +λ,t(ζ k f − Lt,ku), +(3.25) +where +Lt,ku := aijζ k +xixju + 2aij∂iζ k∂ju. +(3.26) +Proof +Since ζ 0u ∈ W2,q +p ((0,T)× Rd) and by the definition of R(0) +λ,t, we have +ζ 0u = R(0) +λ,t(λ(ζ 0u)− L0 +t (ζ 0u)). +Thus by Theorem 3.4 (i) we have +u += +u +n +∑ +k=0 +(ζ k(x))2 = +n +∑ +k=0 +ζ k(ζ ku) = +n +∑ +k=0 +ζ kR(k) +λ,t((λ − L0 +t )ζ ku) += +n +∑ +k=0 +ζ kR(k) +λ,t(λ(ζ ku)− ζ kL0 +t u − Lt,ku) = +n +∑ +k=0 +ζ kR(k) +λ,t(ζ k f − Lt,ku). +This completes the proof. +3.3. Zero drift term and zero Dirichlet boundary condition. Now we turn to study +λu − L0 +t u = f +(3.27) +in ˚W2,q +p ((0,T) × D). Instead of solving this equation directly, we shall solve equation +(3.25), which is equivalent to (3.27). In fact, by Lemma 3.6 we see that if u satisfies (3.27), +then it satisfies (3.25). In next lemma we prove the opposite part of Lemma 3.6, namely, +any solution of equation (3.25) is indeed a solution of (3.27). +Lemma 3.7. Let λ ≥ λ0 and let f ∈ Lq +p((0,T)× D), where λ0 is the one in [15, Theorem +7.2.8]. If u ∈ ˚W1,q +p ((0,T)× D) is a solution of (3.25), then u ∈ ˚W2,q +p ((0,T)× D). Further- +more, there exists a constant λ1 ≥ (1∨λ0), depending only on d, p,q,α,κ,ρ0 and diam(D), +such that for all λ ≥ λ1, the solution u of (3.25) satisfies (3.27) in (0,T)× D. + +14 +YAOZHONG HU AND QUN SHI +Proof +Let u ∈ ˚W1,q +p ((0,T) × D) be a solution of (3.25). Owing to the fact that Lt,i, +(i = 1,...,n) defined by (3.26) are first order differentiable operators, we have +ζ i f − Lt,iu ∈ Lq +p((0,T)× D). +Hence by Lemma 3.5 +ζ iR(i) +λ,t(ζ i f − Lt,iu) ∈ ˚W2,q +p ((0,T)× D). +From the expression (3.25) for u it is easy to see that u ∈ ˚W2,q +p ((0,T)× D). +Next, denote h := λu − L0 +t u. To complete the proof, we only need to show that as long +as λ is sufficiently large, then f = h. By Lemma 3.6, u = ∑0≤i≤nζ iR(i) +λ,t(ζ ih − Lt,iu). On +the other hand, u is a solution of (3.25), ie, u = ∑0≤i≤nζ iR(i) +λ,t(ζ i f − Lt,iu). +Thus, we have +0 = ∑ +0≤i≤n +ζ iR(i) +λ,t(ζ i f − Lt,iu)− ∑ +0≤i≤n +ζ iR(i) +λ,t(ζ ih − Lt,iu) = ∑ +0≤i≤n +ζ iR(i) +λ,t(ζ i(f − h)). +This means that Rλ,t(f − h) = 0, where +Rλ,t f = ∑ +0≤i≤n +ζ iR(i) +λ,t(ζ i f), +which is an operator associated with the operator λ − L0 +t (It is a regularizer of the inverse +of λ − L0 +t ). Thus, we only need to show the following: +If ν ∈ Lq +p((0,T)× D) and if Rλ,t(ν) = 0 for λ ≥ λ0, +then ν = 0 in (0,T)× D . +(3.28) +This is the objective of the following paragraphs. By Lemma 3.5 we have +(λ − L0 +t )R(i) +λ,t(ζ i) = ζ i +on the set in (0,T) × D, where ζ i is defined by (3.23). Therefore, if Rλ,t(ν) = 0 then we +have +0 += +(λ − L0 +t )Rλ,t(ν) = λRλ,t(ν)− ∑ +0≤i≤n +L0 +t (ζ iR(i) +λ,t(ζ i(ν))) += +∑ +0≤i≤n +λ(ζ iR(i) +λ,t(ζ i(ν)))− +� +∑ +0≤i≤n +ζ i(L0 +t R(i) +λ,t(ζ i(ν)))+ ∑ +0≤i≤n +Lt,iR(i) +λ,t(ζ i(ν)) +� += +∑ +0≤i≤n +ζ i(λ − L0 +t )R(i) +λ,t(ζ i(ν))− ∑ +0≤i≤n +Lt,iR(i) +λ,t(ζ i(ν)) += +∑ +0≤i≤n +(ζ i)2(ν)− ∑ +0≤i≤n +Lt,iR(i) +λ,t(ζ i(ν)) := ν− Tλ(ν), +where +Tλ(ν) := ∑ +0≤i≤n +Lt,iR(i) +λ,t(ζ i(ν)). +To finish the proof of (3.28), it suffices to show that for sufficiently large λ the operator Tλ +is a contraction operator in Lq +p((0,T)× D). In fact, from +Lt,ku := aijζ k +xixju + 2aij∂iζ k∂ju + +WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT +15 +and from (3.24) and Theorem 3.4 (ii) it follows that +∥Tλν∥Lq +p((0,T)×D) +≤ +Cλ −1 +n +∑ +i=1 +∥ζ i(ν)∥Lq +p((0,T)×(Bρ0(zi)∩D)) ++ +Cλ −1∥ζ 0(ν)∥Lq +p((0,T)×D) +≤ +Cλ −1∥ν∥Lq +p((0,T)×D) . +Since the above constant C does not depend on λ, we may choose λ sufficiently large so +that Cλ −1 < 1. This means that Tλ is a contraction since Tλ is a linear operator. Thus ν = 0 +in (0,T)× D and (3.28) is proved. The proof of lemma is then completed. +The following lemma will show that the equation (3.25) has a unique solution satisfying +some desired estimate. +Lemma 3.8. There exists a constant λ1 ≥ 1, depending only on d, p,q,α,κ,ρ0 and diam(D), +such that for any f ∈ Lq +p((0,T) × D), there exists a unique solution u ∈ ˚W1,q +p ((0,T) × D) +of equation (3.25) (for any fixed λ ≥ λ1). Furthermore, this solution satisfies +λ∥u∥Lq +p((0,T)×D) + λ 1/2∥u∥W1,q +p ((0,T)×D) + ∥∂tu∥Lq +p((0,T)×D) ≤ C∥ f∥Lq +p((0,T)×D) , +(3.29) +where C depending only on d, p,q,α,κ,ρ0 and diam(D). +Proof +Define +F(v) := ∑ +0≤i≤n +ζ iR(i) +λ,t(ζ i f − Lt,iv) +and let u = F(v) for v ∈ ˚W1,q +p ((0,T)× D). +Thus we have u ∈ ˚W1,q +p ((0,T)× D). Owing to Theorem 3.4 (ii) we have +λ∥u∥Lq +p((0,T)×D) +≤ +∑ +0≤i≤n +λ∥ζ iR(i) +λ,t(ζ i f − Lt,iv)∥Lq +p((0,T)×D) +≤ +C ∑ +0≤i≤n +∥ζ i f − Lt,iv∥Lq +p((0,T)×D) +≤ +C +� +∥ f∥Lq +p((0,T)×D) + ∥v∥W1,q +p ((0,T)×D) +� +. +(3.30) +In the same way we can obtain +λ 1/2∥u∥W1,q +p ((0,T)×D) ≤ C +� +∥ f∥Lq +p((0,T)×D) + ∥v∥W1,q +p ((0,T)×D) +� +. +(3.31) +Furthermore, we have also +∥∂tu∥Lq +p((0,T)×D) = +���∂t +� +∑ +0≤i≤n +ζ iR(i) +λ,t(ζ i f − Lt,iv) +���� +Lq +p((0,T)×D) +≤ ∑ +1≤i≤n +∥∂t(ζ iR(i) +λ,t(ζ i f − Lt,iv)∥Lq +p((0,T)×D) ++∥ζ 0∂t(R(0) +λ,t(ζ 0 f − Lt,0v))∥Lq +p((0,T)×D) ++∥(∂tζ 0)R(0) +λ,t(ζ 0 f − Lt,0v)∥Lq +p((0,T)×D) +≤ C ∑ +0≤i≤n +∥ζ i f − Lt,iv∥Lq +p((0,T)×D) +≤ C +� +∥ f∥Lq +p((0,T)×D) + ∥v∥W1,q +p ((0,T)×D) +� +. +(3.32) + +16 +YAOZHONG HU AND QUN SHI +These three estimates (3.30)-(3.32) yield +λ∥u∥Lq +p((0,T)×D) + λ 1/2∥u∥W1,q +p ((0,T)×D) + ∥∂tu∥Lq +p((0,T)×D) +≤ C +� +∥ f∥Lq +p((0,T)×D) + ∥v∥W1,q +p ((0,T)×D) +� +. +(3.33) +Let u1 = F(v1) and u2 = F(v2). By the Affinity of F and the above inequality (3.33) we +have +λ∥F(v1 − v2)∥Lq +p((0,T)×D) + λ 1/2∥F(v1 − v2)∥W1,q +p ((0,T)×D) ++ ∥∂tF(v1 − v2)∥Lq +p((0,T)×D) ≤ C∥v1 − v2∥W1,q +p ((0,T)×D). +Take λ ≥ λ1 large enough so that Cλ −1 < 1 and we see that F(v) is a contraction. Thus +F(u) = u has a unique fixed point and this implies that equation (3.25) admits a unique +solution u ∈ ˚W1,q +p ((0,T) × D). The inequality (3.29) follows straightforward from the +inequality (3.33) with v replaced by u. +3.4. Zero drift term and general Dirichlet boundary condition. +Theorem 3.9. Let λ1 be sufficiently large and let ∂D ∈ C2. Assume that σ satisfies (Hσ). +Then for any λ ≥ λ1, f ∈ Lq +p((0,T)× D) and g ∈ W2,q +p ((0,T)× D), there exists a unique +solution u0 ∈ W2,q +p ((0,T)× D) to the equation +λu0 − L0 +t u0 = f +(3.34) +in (0,T)× D such that u0 = g on (0,T)× ∂D, and +λ∥u0 − g∥Lq +p((0,T)×D) + λ 1/2∥u0 − g∥W2,q +p ((0,T)×D) + ∥∂t(u0 − g)∥Lq +p((0,T)×D) +≤ C +� +∥ f∥Lq +p((0,T)×D) +Cλ∥g∥W2,q +p ((0,T)×D) +� +, +(3.35) +where C depending only on d, p,q,α,κ,ρ0 and diam(D) and is independent of λ, and Cλ +depends only on λ. +Proof +Let us recall that u0 = g on (0,T) × ∂D means that u0 − g ∈ ˚W2,q +p ((0,T) × D). +Choose λ1 to be sufficiently large such that both Lemmas 3.7 and 3.8 hold true. We shall +apply Lemma 3.8 to u0 − g. Since u0 − g ∈ ˚W2,q +p ((0,T)× D) and +λ(u0 − g)− L0 +t (u0 − g) = f + L0 +t g − λg, +by inequality (3.29), we see +λ∥u0 − g∥Lq +p((0,T)×D) ++ +λ 1/2∥u0 − g∥W2,q +p ((0,T)×D) + ∥∂t(u0 − g)∥Lq +p((0,T)×D) +≤ C∥ f + L0 +t g − λg∥Lq +p((0,T)×D) +≤ C +� +∥ f∥Lq +p((0,T)×D) +Cκ,λ∥g∥W2,q +p ((0,T)×D)). +The theorem is proved. +In the above theorem, we need to assume λ ≥ λ1 for some λ1 > 0. We can improve +this condition to λ ≥ 0 by observing the following fact: u0 satisfies (3.34) if and only if +vλ(t,x) := e−λtu0(t,x) satisfies +∂tv+ Lσv+ e−λt f = 0 in (0,T)× D +and vλ = e−λtg on (0,T)× ∂D for λ ≥ λ1. + +WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT +17 +Theorem 3.10. Let ∂D ∈ C2. Assume that b = 0 and σ satisfies (Hσ) on the domain D. +Then for any f ∈ Lq +p((0,T)× D) and g ∈ W2,q +p ((0,T)× D), there exists a unique solution +u0 ∈ W2,q +p ((0,T)× D) to the equation +∂tu0 + Lσu0 = f +in (0,T)× D such that u0 = g on (0,T)× ∂D, and +∥u0 − g∥W2,q +p ((0,T)×D) ≤ C +� +∥ f∥Lq +p((0,T)×D) + ∥g∥W2,q +p ((0,T)×D) +� +, +(3.36) +where C depending only on p,q,α,κ,ρ0 and diam(D). +Furthermore, if p,q ∈ (1,∞), f ∈ Lq +p((0,T) × D), g ∈ W2,q +p ((0,T) × D) , then for any +β ∈ [0,2) and γ > 1 with d +p + 2 +q < 2 − β + d +γ , we have +||u0(t)||Hβ +γ (D) ≤ C(T −t)(2−β)/2−d/2p−1/q+d/2γ × +� +||f||Lq +p((t,T)×D) + ||g||Lq +p((t,T)×D) +� +, +(3.37) +where C = C(d,κ, p,q,β,γ,α) is a positive constant independent of t. +Proof +We only need to prove estimate (3.37) and we shall do this by using mollifying +technique and weak convergence argument. Let ρ be a nonnegative smooth function on +(0,T)×Rd with support in {(t,x) ∈ (0,T)×Rd : t +|x|2 ≤ 1} and +� +(0,T)×Rd ρ(t,x)dxdt = 1. +Set ρn(t,x) := nd+1ρ(nt,nx). Define +σn := σ ∗ ρn, fn := f ∗ ρn, gn := g ∗ ρn. +It is a classical fact (e.g. [5, Chapter I]) that for a bounded domain D of Rd, there exists the +parabolic Dirichlet-Green function for the operator ∂t +Lσn, restricted on D with boundary +value g, denoted by GD,g,n(s,x;t,y). +Define +T n +s,t f(t,x) := +� +D f(t,y)GD,g,n(s,x;t,y)dy, +∀ f ∈ C1,2([0,T]× D). +The function T n +s,t fn(t,x) satisfies that for all (t,x) ∈ [0,T]× D, +∂sT n +s,t fn(t,x)+ LσnT n +s,t fn(t,x) = 0, +lim +s↑t T n +s,t fn(t,x) = fn(t,x). +(3.38) +Furthermore, for all x,y ∈ D and 0 ≤ s ≤ t ≤ T, using (1.4.9) with µ = d/2, (1.4.15) and +Lemma 1.4.3 in [5, Chapter I], we have for any 0 < κ∗ < κ, +|GD,g,n(s,x;t,y)| ≤ Cd,α,κ∗(t − s)− d +2 exp +� +− κ∗|x− y|2 +2(t − s) +� +. +(3.39) +Using (1.4.10) with µ = (d + 1)/2, (1.4.15) and Lemma 1.4.3 in [5, Chapter I], we have +for any 0 < κ∗ < κ, +|∇xGD,g,n(s,x;t,y)| ≤ Cd,α,κ∗(t − s)− d+1 +2 exp +� +− κ∗|x− y|2 +2(t − s) +� +. +(3.40) +Using (1.4.11) with µ = (d + 2)/2, (1.4.15) and Lemma 1.4.3 in [5, Chapter I], we have +for any 0 < κ∗ < κ, +|∇2 +xGD,g,n(s,x;t,y)| ≤ Cd,α,κ∗(t − s)− d+2 +2 exp +� +− κ∗|x− y|2 +2(t − s) +� +. +(3.41) +In the above inequalities, the constant Cd,α,κ∗ depends on α,κ,d and presumably also on n. +However, it depends on n via the maximum values of σn and their derivatives on D, which + +18 +YAOZHONG HU AND QUN SHI +are again by the property of the mollifying operator, bounded by the maximum values of σ +and their derivatives on D. So the constant Cd,α,κ∗ can be chosen so that it does not depend +on n. In fact, from [5, Chapter 9] if we take the fundamental solution (Green’s function) +of parabolic equations with coefficients depending on t by freezing the original equation +only on the spatial point (not at the time point) we can obtain all the above estimates even +if σn is not uniformly H¨older continuous in t with Cd,α,κ∗ depending only on α,κ,d and +the values of σ on D. +By the gradient estimates (3.39),(3.40) and (3.41), we have for all p ∈ [1,∞], +||∇j +xT n +s,t fn(t,·)||Lp(D) ≤ C(t − s)− j/2||fn(t,·)||Lp(D), j = 0,1,2. +Moreover, by Gagliardo-Nirenberg’s inequality (e.g. [31, Theorem 2.15]), complex inter- +polation inequality e.g. [31, Theorem 2.1]), and the gradient estimates, we have for any +p,γ > 1 and β ∈ [0,2), there exists a constant C such that +∥T n +s,t fn(t,·)∥Hβ +γ (D) +≤ +∥(−∆)β/2T n +s,t fn(t,·)||Lγ (D) + ∥T n +s,t fn(t,·)∥Lγ(D) +≤ +C||∇T n +s,t fn(t,·)||β/2 +L βγ +2 +(D) ·||T n +s,t fn(t,·)||1−β/2 +L∞(D) + ∥T n +s,t fn(t,·)∥Lγ(D) +≤ +C||∇2T n +s,t fn(t,·)||β/2+d/(2p)−d/(2γ) +Lp(D) +·||T n +s,t fn(t,·)||(2−β)/2−d/(2p)+d/(2γ) +Lp(D) ++ ∥T n +s,t fn(t,·)∥Lp(D) +≤ +C(t − s)−β/2−d/(2p)+d/(2γ)||f(t,·)||Lp(D). +(3.42) +Define +J n +s,th(t,x) := +� +∂D h(t,ξ)GD,g,n(s,x;t,ξ)dS(ξ), +∀ h ∈ C1,2([0,T]× D), +where dS(ξ) is the surface ”area” element on ∂D. +Therefore, by the gradient estimates (3.39),(3.40) and (3.41), we have for all p ∈ [1,∞], +||∇j +xJ n +s,tgn(t,·)||p +Lp(D) += +� +D |∇j +xJ n +s,tgn(t,x)|pdx +≤ +C +� +D +� +∂D |gn|p(t,ξ)× |∇j +xGD,g,n(s,x;t,ξ)|pdS(ξ)dx +≤ +C(t − s)− jp/2 +� +∂D |gn|p(t,ξ)dS(ξ) +≤ +C(t − s)− jp/2∥gn∥p +Lp(D) +≤ +C(t − s)− jp/2∥g(t,·)∥p +Lp(D), j = 0,1,2, +(3.43) +where C depends on α,κ,d, and the values of σ on D. +Using the same method and inequalities for T n +s,t fn(t,x), we can get +∥J n +s,tgn(t,·)∥Hβ +γ (D) +≤ +C(t − s)−β/2−d/(2p)+d/(2γ)||g(t,·)||Lp(D). +(3.44) +Now let us consider the function +un,1(s,x) := +� T +s +� +D fn(t,y)GD,g,n(s,x;t,y)dydt . + +WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT +19 +Then un,1 satisfies +� +∂sun,1 + Lσnun,1 + fn = 0, (s,x) ∈ (0,T)× D, +un,1(T,x) = 0, x ∈ D, +(3.45) +and +∥un,1∥Lq +p((t,T)×D) ≤ C∥ fn∥Lq +p((t,T)×D) +(3.46) +by Theorem 1.6.9 in [5]. +On the other hand, we can also consider the function +un,2(s,x) := +� T +s +� +∂D GD,g,n(s,x;t,ξ)(gn(t,ξ)− un,1(t,ξ))dS(ξ)dt . +This function un,2 satisfies + + + +∂sun,2 + Lσnun,2 = 0, (s,x) ∈ (0,T)× D, +un,2(T,x) = 0, x ∈ D +un,2(s,x) = gn(s,x)− un,1(s,x), (s,x) ∈ (0,T)× ∂D. +(3.47) +Now define +un(s,x) := un,1(s,x)+ un,2(s,x), s ∈ [0,T], x ∈ D. +From the expressions of un,1(s,x) and un,2(s,x) we see un ∈ W2,q +p ((0,T) × D). By (3.46) +and (3.47), it is easy to see that un satisfies + + + +∂sun + Lσnun + fn = 0, (s,x) ∈ (0,T)× D, +un(T,x) = 0, x ∈ D, +un(s,x)|x∈∂D = gn(s,x), s ∈ (0,T). +(3.48) +In fact, we can apply (3.36) to obtain +∥un∥Lq +p((0,T)×D) + ∥∂tun∥Lq +p((0,T)×D) + ∥∇2 +xun∥Lq +p((0,T)×D) +≤ C(∥ fn∥Lq +p((0,T)×D) + ∥gn∥Lq +p((0,T)×D)) +≤ C(∥ f∥Lq +p((0,T)×D) + ∥g∥Lq +p((0,T)×D)), +(3.49) +where the constant C is independent of n and the above last inequality follows from the +property of mollifying operator. This proves that un is a bounded sequence in W2,q +p ((0,T)× +D). +By the weak compactness of W2,q +p ((0,T) × D) (we refer to [20, Page 347, The- +orem 11.65] for detail), there exists a subsequence still denoted by un and a function +u0 ∈ W2,q +p ((0,T) × D) with u0(T) = 0 such that un converges weakly to u0. In fact, for +any ϕ ∈ C∞ +0 ((0,T)× D), we have +⟨Lσnun − Lσu0,ϕ⟩(0,T)×D += +� T +0 +� +D(Lσnun − Lσu0)(t,x)ϕ(t,x)dxdt += +� T +0 +� +D(Lσnun − Lσun)(t,x)ϕ(t,x)dxdt ++ +� T +0 +� +D(Lσun − Lσu0)(t,x)ϕ(t,x)dxdt +:= +I1 + I2. + +20 +YAOZHONG HU AND QUN SHI +By the boundedness of ϕ and H¨older’s inequality, we have +I1 +≤ +C +�� T +0 (∥σn(t)− σ(t)∥L∞(D) ·∥∇2 +xun(t)∥Lp(D)dt +� +≤ +C +�� T +0 (∥σn(t)− σ(t)∥L∞(D)) +q +q−1 dt +� q−1 +q +·∥∇2 +xun∥Lq +p((0,T)×D). +Letting n → ∞, we have by (3.49) +lim +n→∞I1 = 0. +Since σ is bounded and since un converges weakly to u0, we have +lim +n→∞I2 ≤ Cκ lim +n→∞ +�� T +0 +� +D +��un(t,x)− u0(t,x))∇2 +xϕ(t,x) +��dxdt +� += 0. +Hence, +lim +n→∞ +� T +0 +� +D(Lσnun − Lσu0)(t,x)ϕ(t,x)dxdt = 0. +Similarly, +lim +n→∞ +� T +0 +� +D(∂tun − ∂tu0)(t,x)ϕ(t,x)dxdt = − lim +n→∞ +� T +0 +� +D(un − u0)(t,x)∂tϕ(t,x)dxdt = 0, +since un converges weakly to u0. +Next, we will show the convergence of boundary function, that is: u0(s,x)|∂D = g(s,x) +for all s ∈ (0,T). Denote t = xd+1, ai(d+1) = a(d+1)i = 0 for all i = 1,2,··· ,d +1. Then the +parabolic equation (3.48) with i = 1,2,··· ,d + 1, x = (x1,··· ,xd+1) can be considered as +an elliptic equation on (0,T)×D in place of D. The corresponding boundary of (0,T)×D +∂(D× (0,T)) = ({t = 0} × D)∪({t = T} × D)∪((0,T)× ∂D) +is Lipschitz continuous. Thus, we can use the compactness of trace operator (see [20, Page +592-594,Theorem18.1 and Corollary 18.6]) to obtain u0(s,x)|∂D = g(s,x) for all s ∈ (0,T). + +WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT +21 +Moreover, for any p,q,γ ∈ (1,∞) and β ∈ [0,2) with d +p + 2 +q < 2 − β + d +γ , by (3.42), +(3.44) and H¨older’s inequality , we have +||un(t)||Hβ +γ (D) +≤ +� T +t +||T n +t,r fn(r,·)||Hβ +γ (D)dr + +� T +t +||J n +t,rgn(r,·)||Hβ +γ (D)dr ++ +� T +t +||J n +t,run,1(r,·)||Hβ +γ (D)dr +≤ +C +� T +t (r −t)−β/2−d/(2p)+d/2γ� +||fn(r,·)||Lp(D) ++||gn(r,·)||Lp(D) + ||un,1(r,·)||Lp(D) +� +dr +≤ +C +�� T +t (r −t)−βq∗/2−dq∗/(2p)+dq∗/2γdr +�1/q∗ +× +� +||fn||Lq +p((t,T)×D) + ||gn||Lq +p((t,T)×D) + ||un,1||Lq +p((t,T)×D) +� +≤ +C(T −t)(2−β)/2−d/(2p)−1/q+d/2γ +× +� +||fn||Lq +p((t,T)×D) + ||gn||Lq +p((t,T)×D) + ||un,1||Lq +p((t,T)×D) +� +≤ +C(T −t)(2−β)/2−d/(2p)−1/q+d/2γ +× +� +2||fn||Lq +p((t,T)×D) + ||gn||Lq +p((t,T)×D) +� +≤ +C(T −t)(2−β)/2−d/(2p)−1/q+d/2γ +× +� +2||f||Lq +p((t,T)×D) + ||g||Lq +p((t,T)×D) +� +, +(3.50) +where q∗ = +q +q−1 is the conjugate of q and C = C(d,κ, p,q,α,β,γ) is a positive constant +and where in the last inequality we used the fact that +lim +n→∞∥ fn − f∥Lq +p((0,T)×D) = 0. +Letting n → ∞ yields +||u0(t)||Hβ +γ (D) ≤ C(T −t)(2−β)/2−d/(2p)−1/q+d/2γ � +||f||Lq +p((t,T)×D) + ||g||Lq +p((t,T)×D) +� +, +where C = C(d,κ, p,q,α,β,γ) > 0. This shows the theorem. +3.5. General drift term and general Dirichlet boundary condition. Before stating the +following main result of this section, Let us recall [15, Theorem 13.7.2] that a function +in ∩∞ +k=1Wk,q +p ((0,T) × ∂D) can be continuously extended to D as a ∩∞ +k=1Wk,q +p ((0,T) × D) +function. Here is the main result in this section. +Theorem 3.11. Let ∂D ∈ C2. Assume that σ satisfies (Hσ) and b satisfies (Hb) on the +domain D. Then for any f ∈ Lq +p((0,T) × D) and g ∈ W2,q +p ((0,T) × D), there exists a +unique solution u ∈ W2,q +p ((0,T)× D) to the equation +∂tu + Lσu + b ·∇u = f in (0,T)× D +(3.51) + +22 +YAOZHONG HU AND QUN SHI +such that u = g on (0,T) × ∂D. Moreover, there exists a constant C depending only on +d, p,κ,diam(D),and ρ0 such that +||u − g||W2,q +p ((0,T)×D) +≤ +C +� +∥b∥Lq +p((0,T)×D) exp{CT qδ/3∥b∥q +Lq +p((0,T)×D)} + 1 +� +× +� +∥ f∥Lq +p((0,T)×D) + ∥g∥Lq +p((0,T)×D) +� ++C∥g∥W2,q +p ((0,T)×D) . +(3.52) +Furthermore, if p,q ∈ (1,∞), f ∈ Lq +p((0,T)× D), g ∈ W2,q +p ((0,T)× D), then for any β ∈ +[0,2) and γ > 1 with d +p + 2 +q < 2 − β + d +γ , we have +||∇u(t)||C δ/2(D) +≤ +C(T −t)δ/3exp{C1(T −t)qδ/3∥b∥q +Lq +p((0,T)×D)} +× +� +∥ f∥q +Lq +p((0,T)×D) + ∥g∥q +Lq +p((0,T)×D) +� +, +(3.53) +where δ > 0,C = C(d, p,q,β,γ,α) is a positive constant independent of t. +Proof +By standard continuity argument, we only need to prove the a priori estimate (3.52) +and (3.53). Letting δ := 1 +2 − d +2p − 1 +q > 0, by (2.3), (3.37) with suitable choices of β and γ +such that β − δ+1 +2 +> d +γ , we have +∥∇u∥q +L∞∞((0,T)×D) +≤ +∥∇u(t)∥q +C δ/2(D) ≤ C∥u(t)∥q +Hβ +γ (D) +≤ +C(T −t)qδ/3 · +� T +t +∥(b ·∇u)(s)+ f(s)∥q +Lp(D) + ∥g(s)∥q +Lp(D)ds +≤ +C(T −t)qδ/3 +� T +t +∥b(s)∥q +Lp(D) ·∥∇u(s)∥q +L∞(D)ds ++C(T −t)qδ/3� +∥ f∥q +Lq +p((0,T)×D) + ∥g∥q +Lq +p((0,T)×D) +� +≤ +C(T −t)qδ/3 +� T +t +∥b(s)∥q +Lp(D) ·∥∇u(s)∥q +C δ/2(D)ds ++C(T −t)qδ/3� +∥ f∥q +Lq +p((0,T)×D) + ∥g∥q +Lq +p((0,T)×D) +� +. +Now Gronwall’s inequality implies +∥∇u(t)∥C δ/2(D) +≤ +C1(T −t)δ/3exp{C1(T −t)qδ/3∥b∥q +Lq +p((0,T)×D)} +× +� +∥ f∥q +Lq +p((0,T)×D) + ∥g∥q +Lq +p((0,T)×D) +� +. +(3.54) +This is (3.53). Since +∂tu + Lσu = f − b ·∇u, + +WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT +23 +we have by (3.36) and (3.54) +∥u − g∥W2,q +p ((0,T)×D) +≤ C +� +∥ f − b ·∇u∥Lq +p((0,T)×D) + ∥g∥W2,q +p ((0,T)×D) +� +≤ C +� +∥ f∥Lq +p((0,T)×D) + ∥b∥Lq +p((0,T)×D) ·∥∇u∥L∞∞((0,T)×D) +� ++C∥g∥W2,q +p ((0,T)×D) +≤ C +� +∥b∥Lq +p((0,T)×D) exp{CT qδ/3∥b∥q +Lq +p(D)} + 1 +� +× +� +∥ f∥Lq +p((0,T)×D) + ∥g∥Lq +p((0,T)×D) +� ++C∥g∥W2,q +p ((0,T)×D). +This is (3.52). +4. Krylov-type estimates +Our first result in this section is a localized version of the Krylov type estimate for the +solution of a stochastic differential equation on a domain D (before its first exit time from +this domain). +Theorem 4.1. Suppose that σ,b satisfy (Hσ) and (Hb) on the domain D, respectively. +Suppose that (Xt,0 ≤ t ≤ τD) is the weak solution to the following stochastic differential +equation: +dXt = b(t,Xt)dt + σ(t,Xt)dBt, +X0 = x ∈ D, +(4.1) +where τD be the first exit time of Xt from D. Assume p,q ∈ (1,∞) with d +p + 2 +q < 2. For any +δ ∈ (0,1 − d +2p − 1 +q), there exists a positive constant C = C(κ,α, p,q,d,δ), such that for +any f ∈ Lq +p((0,T)× D), we have +Ex +�� s∧τD +r∧τD +��f(η,Xη) +��dη | Fr∧τD +� +≤ C(s−r)δ ||f||Lq +p((0,T)×D) , +∀ 0 ≤ r ≤ s ≤ T,x ∈ D. +(4.2) +Proof +Let p′ = d +1. Since Lp′ +p′((0,T)×D)∩Lq +p((0,T)×D) is dense in Lq +p((0,T)×D), +it suffices to show (4.2) for +f ∈ Lp′ +p′((0,T)× D)∩Lq +p((0,T)× D). +Fix s ∈ [0,T]. By Theorem 3.11, there exists a unique solution u ∈ W2,p′ +p′ ((0,s) × D) ∩ +W2,q +p ((0,s)×D) to the following parabolic partial differential equation on [0,s] with termi- +nal condition + + + +∂tu + Lσu + b ·∇u + f = 0, in (0,s)× D, +u = 0, on (0,s)× ∂D, +u(s,x) = 0. +(4.3) +Moreover, there exists some constant C = C(K,α, p,q,d) such that for all r ∈ [0,s], +||∂tu||Lp′ +p′((r,s)×D) + ||u||Lp′ +p′((r,s)×D) + ||∇2u||Lp′ +p′((r,s)×D) ≤ C||f||Lp′ +p′((r,s)×D) +(4.4) +and +||∂tu||Lq +p((r,s)×D) + ||u||Lq +p((r,s)×D) + ||∇2u||Lq +p((r,s)×D) ≤ C||f||Lq +p((r,s)×D). + +24 +YAOZHONG HU AND QUN SHI +In particular, by (2.3) and (3.53) with vanishing boundary, for any δ ∈ (0,1− d +2p − 1 +q), with +suitable choices of β and γ such that β − δ+1 +2 +> d +γ we have +sup +t∈(r,s) +||u(t,·)||L∞(D) ≤ sup +t∈(r,s) +||u(t,·)||Hβ +p(D) ≤ C(s− r)δ||f||Lq +p((r,s)×D). +(4.5) +Now we introduce a mollifier with both space and time. Let ρ be a nonnegative smooth +function in Rd+1 ++ +with support in {(t,x) ∈ Rd+1 ++ +: (t,x) ∈ (0,T)×D} and +� +Rd+1 ++ +ρ(t,x)dtdx = +1. Set ρn = nd+1ρ(nt,nx) and define +un(t,x) := u ∗ ρn(t,x) +(4.6) +and +fn(t,x) := −(∂tun(t,x)+ Lσun(t,x)+ b ·∇un(t,x)). +(4.7) +[We use the same notations as previous section without confusion.] Then by the property +of convolutions, we have +||fn − f||Lp′ +p′((0,T)×D) +≤ +||∂t(un − u)||Lp′ +p′((0,T)×D) + ||Lσun − Lσu||Lp′ +p′((0,T)×D) + ||b ·∇(un− u)||Lp′ +p′((0,T)×D) +≤ +||∂t(un − u)||Lp′ +p′((0,T)×D) + κ||∇2(un − u)||Lp′ +p′((0,T)×D) ++||b||Lp′ +p′((0,T)×D)||∇(un − u)||L∞∞((0,T)×D) +≤ +||∂tu ∗ ρn − ∂tu||Lp′ +p′((0,T)×D) + κ||∇2u ∗ ρn − ∇2u||Lp′ +p′((0,T)×D) ++||b||Lp′ +p′((0,T)×D)||∇u ∗ ρn − ∇u||L∞∞((0,T)×D) +≤ +||f ∗ ρn − f||Lp′ +p′((0,T)×D) + 2κ||∇2u ∗ ρn − ∇2u||Lp′ +p′((0,T)×D) ++2||b||Lp′ +p′((0,T)×D)||∇u ∗ ρn − ∇u||L∞∞((0,T)×D) +→ +0 as n → ∞. +Now we use the following the classical Krylov’s estimate [14, Theorem 4, Page 54] which +states +E +�� s∧τD +0 +f(η,Xη)dη +� +≤ C||f||Lp +p((0,T)×D) +for any p ≥ d, where the constant C = C(d, p,κ,diam(D)) depends on d, p,κ,diam(D) +since we assume f ∈ Lp′ +p′((0,T)× D)∩Lq +p((0,T)× D). +Thus +lim +n→∞E +�� s∧τD +r∧τD +|fn(η,Xη)− f(η,Xη)|dη +� +≤ C lim +n→∞||fn − f||Lp′ +p′((0,T)×D) = 0. +(4.8) +Now applying Itˆo’s formula to un(t,Xt), 0 < t < T, and by the definition (4.7) of fn we +have +un(t,Xt) = un(0,X0)+ +� t +0 fn(s,Xs)ds+ +d +∑ +k=1 +� t +0 ∂iun(s,Xs)σik(s,Xs)dBk +s . + +WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT +25 +In view of (4.5), we have +sup +η∈(r,s),x∈D +|∂iun(η,x)| +≤ +n2d+1 +sup +θ∈(0,η− T +n ),y∈ D +n +x +|u(η,y)| +sup +η∈(r,s),x∈D +� η− T +n +0 +� +D +n +x(∂iρ)(n(η − θ),n(x− y))dydθ +≤ +Cn +for some constant Cn which may depend on n. Thus, we can apply Doob’s optional sam- +pling theorem to obtain +E +�� s∧τD +r∧τD +∂iun(η,Xη)σik(η,Xη)dBk +η | Fr∧τD +� += 0. +Hence, we have +����E +�� s∧τD +r∧τD +fn(η,Xη)dη | Fr∧τD +����� += +|E(un(s∧τD,Xs∧τD)− un(r ∧τD,Xr∧τD) | Fr∧τD)| +≤ +2 +sup +η∈(r,s),x∈D +|un(η,x)| ≤ 2 +sup +η∈(r,s),x∈D +|u(η,x)| +≤ +C(s− r)δ||f||Lq +p((r,s)×D), +where the last inequality follows from (4.5). Finally, letting n → ∞ and using (4.8) we see +����E +�� s∧τD +r∧τD +f(η,Xη)dη | Fr∧τD +����� ≤ C(s− r)δ||f||Lq +p((0,T)×D). +Since f ∈ Lq +p((0,T)× D) implies |f| ∈ Lq +p((0,T)× D) this proves the theorem. +Next, we give a local stability result for the solutions to equation (4.1). Namely, we +want to know how the solution depends on the coefficients of the equation. Consider two +stochastic differential equations +� +dXt =b(t,Xt)dt + σ(t,Xt)dBt, +X0 = x; +dX′ +t =b′(t,X′ +t )dt + σ′(t,X′ +t )dBt, +X′ +0 = x. +(4.9) +Definition 4.2. Let b,b′ satisfy (Hb) on D and σ,σ′ satisfy (Hσ) on D. We say that +that the two equations have tied weak solutions if one can find a common probability +space (Ω,F,P) and a common Brownian motion (Bt,t ≥ 0), and two stochastic processes +(Xt,0 ≤ t ≤ τD) and (X′ +t ,0 ≤ t ≤ τ′ +D) such that + + + + + + + +Xt∧τD =x+ +� t∧τD +0 +b(s,Xs)ds+ +� t∧τD +0 +σ(s,Xs)dBs ; +X′ +t∧τ′ +D =x′ + +� t∧τ′ +D +0 +b′(s,X′ +s)ds+ +� t∧τD +0 +σ′(s,X′ +s)dBs . +(4.10) +First, we will give a local stability result for the solutions to the equation without drift co- +efficients. Assume that Xσ and Xσ′ are two tied weak solutions to (4.1) with drift term b′ = +b = 0 and with two different coefficients σ and σ′. We want to bound the difference of the +solutions Xσ −Xσ′ by the Sobolev norm of σ −σ′. We cannot no longer use Burkh¨older’s +inequality since when 0 < p0 < 1 we can only bound E +� +sups∈[0,T] |Xσ +s (x)− Xσ′ +s (x)|p0 +� +by +E +�� t +0 ||σ(r,Xσ +r )− σ′(r,Xσ′ +r )||2dr +�p0/2 +which cannot be bounded by ||σ − σ′||p0 +Lq +p((0,T)×D). +Instead, we shall use Lemma 6.4 in Appendix for positive constant p0 in place of Burkh¨older’s +inequality. + +26 +YAOZHONG HU AND QUN SHI +Lemma 4.3. Assume that σ,σ′ satisfy (Hσ) and let p,q ∈ (1,∞) with d +p + 2 +q < 1. Let Xσ, +Xσ′ be two tied weak solutions to equation (4.1) with b′ = b = 0, with the same initial +condition, and with diffusion coefficients σ, σ′, respectively. Let +τD = inf +� +t ≥ 0; +Xt ̸∈ D +or +X′ +t ̸∈ D +� +. +Then, for any p0 > 0, there exists a constant C = C(p0, p,q,d,κ,α) > 0 such that +sup +x∈D +E +� +sup +s∈[0,T∧τD] +|Xσ +s (x)− Xσ′ +s (x)|p0 +� +≤ C||σ − σ′||p0 +Lq +p((0,T)×D). +Proof +Set Zt = Xσ +t − Xσ′ +t . Then +Zt = +� t +0 +� +σ(r,Xσ +r )− σ′(r,Xσ′ +r ) +� +dBr. +By Itˆo’s formula, we have +|Zt|2 += +� t +0 ||σ(r,Xσ +r )− σ′(r,Xσ′ +r )||2dr + 2 +� t +0 +� +σ(r,Xσ +r )− σ′(r,Xσ′ +r ) +�t +ZrdBr += +� t +0 ξ(r)dr + +� t +0 η(r)dBr + +� t +0 |Zr|2β(r)dr + +� t +0 |Zr|2α(r)dBr, +where +ξ(r) +:= +||σ(r,Xσ +r )− σ′(r,Xσ′ +r )||2 − ||σ(r,Xσ +r )− σ(r,Xσ′ +r )||2, +η(r) +:= +2(σ(r,Xσ′ +r )− σ′(r,Xσ′ +r ))tZr, +β(r) +:= +||σ(r,Xσ +r )− σ(r,Xσ′ +r )||2/|Zr|2, +α(r) +:= +2(σ(r,Xσ +r )− σ(r,Xσ′ +r ))tZr/|Zr|2. +(4.11) +Here, we have used the convention 0 +0 := 0, that is, if β(r) = α(r) = 0, then |Zr| = 0. +By the inequality (4.2) and by the fact that the local maximal operator MD is bounded +on Lp +q((0,T)× D) for any p,q > 1, we have that for any 0 ≤ s < t ≤ T, +E +�� t∧τD +s∧τD +(|β(r)|+ |α(r)|2)dr | Fs∧τD +� +≤ CE +�� t∧τD +s∧τD +� +MD|∇σ|2(Xσ +r )+ MD|∇σ|2(Xσ′ +r ) +� +dr | Fs∧τD +� +≤ C(t − s)δ||MD|∇σ|2||Lq/2 +p/2((0,T)×D) ≤ C(t − s)δ|||∇σ|2||Lq/2 +p/2((0,T)×D) +≤ C(t − s)δ||∇σ||2 +Lq +p((0,T)×D) , +where MD is the local maximal operator (e.g.(6.1)). Now we want to bound ξ +. For any +γ ∈ (1 ∨ p0,1/(2/q + d/p)∨ p0), we have by Theorem 4.1 +E +�� t∧τD +s∧τD +||σ(r,Xσ′ +r )− σ′(r,Xσ′ +r )||2dr +�γ +≤ E +�� t∧τD +s∧τD +||σ(r,Xσ′ +r )− σ′(r,Xσ′ +r )||2γdr +� +≤ C(t − s)δ||||σ − σ′||2γ||Lq/(2γ) +p/(2γ)((0,T)×D) += C(t − s)δ||σ − σ′||2γ +Lq +p((0,T)×D) . + +WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT +27 +This implies +E +�� T∧τ +0 +ξ +(r)dr +�γ +≤ ||σ − σ′||2γ +Lq +p((0,T)×D) . +(4.12) +Using inequality (6.8) in Appendix (Lemma 6.4) with p0 > 0,γ2 = γ/p0 and γ3 = +2γ2 +γ2+1 we +obtain +E +� +sup +s∈[0,T∧τD] +|Zs|2p0 +� +≤ +C + + +���� +���� +�� T∧τD +0 +ξ +(r)dr +�p0���� +���� +Lγ2(Ω) ++ +����� +����� +�� T∧τD +0 +|η(r)|2dr +�p0/2����� +����� +Lγ3(Ω) + + +≤ +C +����� +����� +�� T∧τD +0 +|Zr|2||σ(r,Xσ′ +r )− σ′(r,Xσ′ +r )||2dr +�p0/2����� +����� +Lγ3(Ω) ++C||σ − σ′||2p0 +Lq +p((0,T)×D) . +(4.13) +Now using H¨older’s inequality, we have +E +� +sup +s∈[0,T∧τD] +|Zs|2p0 +� +≤ +C +� +E +sup +s∈[0,T∧τD] +|Zs|2p0 +�1/2 +× +����� +����� +�� T∧τD +0 +||σ(r,Xσ′ +r )− σ′(r,Xσ′ +r )||2dr +�p0/2����� +����� +Lγ2(Ω) ++C||σ − σ′||2p0 +Lq +p((0,T)×D) +≤ +C +� +E +sup +s∈[0,T∧τD] +|Zs|2p0 +�1/2 +× +� +||σ − σ′||2p0 +Lq +p((0,T)×D) +�1/2 ++C||σ − σ′||2p0 +Lq +p((0,T)×D) +≤ +1 +2 +� +E +sup +s∈[0,T∧τD] +|Zs|2p0 +� ++C||σ − σ′||2p0 +Lq +p((0,T)×D) . +(4.14) +Thus, we obtain for any p0 > 0, +E +� +sup +s∈[0,T∧τD] +|Zs|2p0 +� +≤ 2C||σ − σ′||2p0 +Lq +p((0,T)×D). +This proves the lemma. +The next theorem is about the stability on the drift coefficients of (4.1) on domain D. +Theorem 4.4. Assume that b,b′ satisfy (Hb) on the domain D with p,q satisfying condition +d +p + 2 +q < 1, and assume that σ satisfies (Hσ) on D. Let Xb,σ +t +and Xb′,σ +t +be two tied weak +solutions to (4.1) associated with coefficient pairs (b,σ), (b′,σ), respectively. Then for +any p0 > 0, +sup +x∈D +E +� +sup +s∈[0,T∧τD] +|Xb′,σ +s +(x)− Xb,σ +s +(x)|p0 +� +≤ C||b − b′||p0 +Lq +p((0,T)×D). +(4.15) + +28 +YAOZHONG HU AND QUN SHI +Proof +Since b(t,x),b′(t,x) ∈ Lq +p((0,T)× D), Theorem 3.11 implies that for any f, f ′ ∈ +Lq +p((0,T)×D) and g,g′ ∈ W2,q +p ((0,T)×∂D), there exist unique solutions ub,ub′ ∈ W2,q +p ((0,T)× +D) to the equations +∂tub + Lσub + b ·∇ub + f = 0, ub(T,x) = 0, +and +∂tub′ + Lσub + b′ ·∇ub′ + f ′ = 0, ub′(T,x) = 0, +such that ub − g ∈ ˚W2,q +p ((0,T)× D), ub′ − g′ ∈ ˚W2,q +p ((0,T)× D), respectively. +In the above equations we take f = bℓ, f ′ = b′ℓ for ℓ = 1,2,...,d. The corresponding so- +lutions are denoted by ub(t,x) := (ub,1(t,x),...,ub,d(t,x)), ub′(t,x) := (ub′,1(t,x),...,ub′,d(t,x)). +Set +Φb(t,x) := x+ ub(t,x), Φb′(t,x) := x+ ub′(t,x), in (0,T)× D. +Let δ := 1 +2 − d +2p − 1 +q > 0. By (3.53), there is a C = C(κ,α, p,q,d) > 0 such that for all +t ∈ [s0,t0] ⊆ [0,T], +||∇ub(t,·)||C δ (D) +≤ +C(t0 − s0)δ/3 exp{C1(T −t)qδ/3∥b∥q +Lq +p((0,T)×D)} +× +� +∥ f∥q +Lq +p((0,T)×D) + ∥g∥q +Lq +p((0,T)×D) +� +. +For given positive constant M, let us choose ε = ε(δ, p,q,C,M) > 0 small enough so that +for all t0 − s0 ≤ ε and ||b||Lq +p(s0,t0)×D) ≤ M, +sup +s0≤t≤t0 +||∇ub(t,·)||C δ (D) ≤ 1 +2. +In particular, we have +|ub(t,x)− ub(t,y)| ≤ |x− y| +2 +, +s0 ≤ t ≤ t0. +Thus by definition of Φb(t,x) = x+ ub(t,x) we have +1 +2|x− y| ≤ |Φb(t,x)− Φb(t,y)| ≤ 3 +2|x− y| +(4.16) +for all t0−s0 ≤ ε, any x,y ∈ D and ||b||Lq +p(s0,t0)×D) ≤ M. In the same way, for all t0 −s0 ≤ ε, +any x,y ∈ D and ||b′||Lq +p(s0,t0)×D) ≤ M, we also have +1 +2|x− y| ≤ |Φb′(t,x)− Φb′(t,y)| ≤ 3 +2|x− y|. +Next, we shall verify the following two estimates: +∇(∇Φb ·σ)(t,x) ∈ Lq +p((s0,t0)× D) +(4.17) +and +||Φb′ − Φb||L∞∞(s0,t0)×D) + ||∇Φb − ∇Φb′||Lq +p((s0,t0)×D) ≤ C||b′ − b||Lq +p(s0,t0)×D). +(4.18) +Indeed, +||∇(∇Φb ·σ)||Lq +p((s0,t0)×D) +≤ κ||∇2Φb||Lq +p((s0,t0)×D) + ||∇Φb||L∞∞((s0,t0)×D) ·||∇σ||Lq +p((s0,t0)×D) +≤ κ||∇2ub||Lq +p((s0,t0)×D) + ||∇Φb||L∞∞((s0,t0)×D) ·||∇σ||Lq +p((s0,t0)×D) +< ∞. + +WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT +29 +Thus (4.17) holds true. On the other hand, if we denote w := wb,b′ := ub − ub′, then w +satisfies the following equation + + + +∂tw+ Lσw+ b ·∇w = f − f ′ + (b′ − b)·∇ub′ in (0,T)× D, +w(T,x) = 0, +w(t,x) = 0, t ∈ (0,T), x ∈ ∂D. +(4.19) +As above, using the definition of C δ ((2.2)), and (2.3), (3.53), choosing suitable δ,β such +that β − (δ + 1) > d +p, we have +||∇ub′||L∞∞((s0,t0)×D) += +sup +t∈(s0,t0) +||∇ub′(t)||L∞(D) ≤ +sup +t∈(s0,t0) +||ub′(t)||C δ+1(D) +≤ +sup +t∈(s0,t0) +||ub′(t)||Hβ +p(D) +≤ +C(t0 − s0)δ� +||f ′||Lq +p((s0,t0)×D) + ||g′||Lq +p((s0,t0)×D) +� +< +∞. +(4.20) +Therefore, by inequalities (3.52) in Theorem 3.11 and (4.20), we get +||∇Φb − ∇Φb′||Lq +p((s0,t0)×D) += +||∇w||Lq +p((s0,t0)×D) ≤ ||w||W2,q +p ((s0,t0)×D) +≤ +C||f − f ′||Lq +p((s0,t0)×D) ++ +C||∇ub′||L∞∞((s0,t0)×D) × ||b − b′||Lq +p((s0,t0)×D) +≤ +C||b − b′||Lq +p((s0,t0)×D) . +(4.21) +On the other hand, from the inequalities (2.3), (3.53) in Theorem 3.11 and (4.20) it follows +||Φb′ − Φb||L∞∞(s0,t0)×D) += +||w||L∞∞(s0,t0)×D) = +sup +t∈(s0,t0) +||w(t)||L∞(D) +≤ +sup +t∈(s0,t0) +||w(t)||C δ (D) ≤ C sup +t∈(s0,t0) +||w(t)||Hβ +p(D) +≤ +C||f − f ′ + (b′ − b)·∇ub′||Lq +p((s0,t0)×D) +≤ +C||b − b′||Lq +p((s0,t0)×D) × (1 + ||∇ub′||L∞∞((s0,t0)×D)) +≤ +C||b − b′||Lq +p((s0,t0)×D). +Combining this estimate with (4.21) yields (4.18). +By the generalized Itˆo formula (e.g. [37, Lemma 4.3]), Xb,σ +t +(x) solves SDE (4.1) on +[0,T ∧τD] with initial value x ∈ D if and only if Y b +t (y) := Φb(t,Xb,σ +t +(x)) solves the follow- +ing SDE on [0,T ∧ τD] with initial value in D, where D is also the image of the function +Φb(t,x) by Lemma 6.2, +dY b +t = (∇Φb ·σ)◦ (t,Φb,−1(t,Y b +t ))dBt := Θb(t,Yt)dBt , +(4.22) +where +Θb(t,y) := (∇Φb ·σ)◦ (t,Φb,−1(t,y)) +(4.23) +and Φb,−1(t,y) is the inverse of Φb(t,·) : D → D with respect to spatial variable. Let Θb′ +be defined as above (4.23) with b replaced by b′, and Φb′,−1(t,y) is the inverse of Φb′(t,y) +with respect to spatial variable. To study the stability of our original equation we now + +30 +YAOZHONG HU AND QUN SHI +reduce it to the stability of the equation of (4.22) and to study the later equation, we need +to know how Θ depends on b. First, we have +Θb(t,y)− Θb′(t,y) += +(∇Φb ·σ)◦ (t,Φb,−1(t,y))− (∇Φb ·σ)◦ (t,Φb′,−1(t,y)) ++((∇Φb − ∇Φb′)·σ)◦ (t,Φb′,−1(t,y)) := I1(t,y)+ I2(t,y). +For the above first term I1(t,y), by [35, the inequality (2.2)] we have +|I1(t,y)| +≤ +C|Φb,−1(t,y)− Φb′,−1(t,y)| +× +� +MD|∇(∇Φb ·σ)|(t,Φb,−1(t,y))+ MD|∇(∇Φb ·σ)|(t,Φb′,−1(t,y)) +� +, +where MD is the Hardy-Littlewood maximal operator which is bounded operator in any +Lq +p((0,T)× D) space. Noticing that +sup +y∈D +|Φb,−1(t,y)− Φb′,−1(t,y)| += +sup +x∈D +|x− Φb′,−1 ◦ (t,Φb(t,x))| += +sup +x∈D +|Φb′,−1 ◦ (t,Φb′(t,x))− Φb′,−1 ◦ (t,Φb(t,x))| +≤ +||∇Φb′,−1(t,·)||L∞(D) × ||Φb′(t,·)− Φb(t,·)||L∞(D) +and by the change of variables, [35, (2.3)], (4.17) and (4.18) we have +||I1||Lq +p((s0,t0)×D) +≤ +C||MD|∇(∇Φb ·σ)|(·,Φb,−1)+ MD|∇(∇Φb ·σ)|(·,Φb′,−1)||Lq +p((s0,t0)×D) +×||Φb,−1 − Φb′,−1||L∞∞((s0,t0)×D) +≤ +C||∇Φb′,−1||L∞∞(s0,t0)×D) × ||Φb′ − Φb||L∞∞(s0,t0)×D) +×||MD|∇(∇Φb ·σ)|||Lq +p((s0,t0)×D) +≤ +C||∇(∇Φb ·σ)||Lq +p((s0,t0)×D) × ||Φb′ − Φb||L∞∞(s0,t0)×D) +≤ +C||b′ − b||Lq +p(s0,t0)×D). +(4.24) +For the second term I2(t,y), by the change of variables, boundedness of σ and (4.18), we +have +||I2||Lq +p((s0,t0)×D) += +||(∇Φb − ∇Φb′)·σ||Lq +p((s0,t0)×D) +≤ +κ||∇Φb − ∇Φb′||Lq +p((s0,t0)×D) +≤ +C||b′ − b||Lq +p(s0,t0)×D). +(4.25) +Combining (4.24) and (4.25) yields +||Θb − Θb′||Lq +p((s0,t0)×D) ≤ C||b − b′||Lq +p(s0,t0)×D). +(4.26) +Dividing [0,T] into subintervals, applying the above estimates on each interval and piecing +them together, we then obtain +||Θb − Θb′||Lq +p((0,T)×D) ≤ C||b − b′||Lq +p(0,T)×D). +(4.27) + +WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT +31 +Finally, combining the above estimate (4.27) with Lemma 4.3 gives +sup +x∈D +E +� +sup +t∈[0,T∧τD] +|Xb,σ +t +(x)− Xb′σ +t +(x)|p0 +� +≤ +sup +y∈D +E +� +sup +t∈[0,T∧τD] +|Y b +t (y)−Yb′ +t (y)|p0 +� +≤ +C||Θb − Θb′||p0 +Lq +p((0,T)×D) +≤ +C||b − b′||p0 +Lq +p((0,T)×D). +This completes the proof of the theorem. +Finally, we concluded this subsection with the following theorem: +Theorem 4.5. Assume that on the domain D, the coefficients b,b′ satisfy (Hb) and that +coefficients σ,σ′ satisfy (Hσ). Let Xb,σ +t +and Xb′,σ′ +t +be the tied weak solutions to (4.1) +associated with coefficients pairs (b,σ) and (b′,σ′), respectively. Then for all p0 > 0, +there is a constant C = Cp0,p,q,κ,α depending on p0, p,q,κ,α, and the constants appeared +in assumptions (Hb) and (Hσ), such that +sup +x∈D +E +� +sup +t∈[0,T∧τD] +|Xb,σ +t +(x)− Xb′,σ′ +t +(x)|p0 +� +≤C +� +||b − b′||p0 +Lq +p((0,T)×D) + ||σ − σ′||p0 +Lq +p((0,T)×D) +� +. +(4.28) +Proof +Since +E +� +sup +t∈[0,T∧τD] +|Xb,σ +t +(x)− Xb′,σ′ +t +(x)|p0 +� +≤ +2p0−1E +� +sup +t∈[0,T∧τD] +|Xb,σ +t +(x)− Xb′,σ +t +(x)|p0 +� ++ +2p0−1E +� +sup +t∈[0,T∧τD] +|Xb′,σ +t +(x)− Xb′,σ′ +t +(x)|p0 +� +=: +J1(x)+ J2(x). +By Theorem 4.4, we have already that +sup +x∈D +J1 ≤ C||b − b′||p0 +Lq +p((0,T)×D). +Next, we devote our effort to deal with J2(x). We shall use the Zvonkin transformation as +in the proof of Theorem 4.4. For any f, f ′ ∈ Lq +p((0,T)×D) and g,g′ ∈ W2,q +p ((0,T)×∂D), +Theorem 3.11 implies that there exist unique solutions uσ,uσ′ ∈ W2,q +p ((0,T) × D) to the +following two equations: +∂tuσ + Lσuσ + b ·∇uσ + f = 0, uσ(T,x) = 0, +and +∂tuσ′ + Lσ′uσ′ + b ·∇uσ′ + f ′ = 0, uσ′(T,x) = 0, +such that uσ − g ∈ ˚W2,q +p ((0,T)× D), uσ′ − g′ ∈ ˚W2,q +p ((0,T)× D), respectively. +Taking f = f ′ = bℓ for ℓ = 1,2,...,d and letting +uσ(t,x) := (uσ,1(t,x),...,uσ,d(t,x)), +uσ′(t,x) := (uσ′,1(t,x),...,uσ′,d(t,x)) +and +Φσ(t,x) := x+ uσ(t,x), Φσ′(t,x) := x+ uσ′(t,x), in (0,T)× D. + +32 +YAOZHONG HU AND QUN SHI +Applying the same procedure as that in the proof of Theorem 4.4 we have for all t0−s0 ≤ ε, +any x,y ∈ D and ||σ||Lq +p(s0,t0)×D),||σ′||Lq +p(s0,t0)×D) ≤ M, +1 +2|x− y| ≤ |Φσ(t,x)− Φσ(t,y)| ≤ 3 +2|x− y| +and +1 +2|x− y| ≤ |Φσ′(t,x)− Φσ′(t,y)| ≤ 3 +2|x− y|. +As in the proof of Theorem 4.4 we need to verify +∇(∇Φσ ·σ)(t,x) ∈ Lq +p((0,T)× D) +(4.29) +and +||Φσ′ − Φσ||L∞∞(s0,t0)×D) + ||∇Φσ − ∇Φσ′||Lq +p((s0,t0)×D) ≤ C||σ − σ′||Lq +p(s0,t0)×D). +(4.30) +Indeed, +||∇(∇Φσ ·σ)||Lq +p((s0,t0)×D) +≤ +κ||∇2Φσ||Lq +p((s0,t0)×D) ++||∇Φσ||L∞∞((s0,t0)×D) ·||∇σ||Lq +p((s0,t0)×D) +≤ +κ||∇2uσ||Lq +p((s0,t0)×D) ++||∇Φσ||L∞∞((s0,t0)×D) ·||∇σ||Lq +p((s0,t0)×D) +< +∞. +On the other hand, if we denote v := vσ,σ′ := uσ − uσ′, then v satisfies following equation + + + +∂tv+ Lσv+ b ·∇v = (Lσ′ − Lσ)·uσ′ in (0,T)× D, +v(T,x) = 0, +v(t,x) = 0, t ∈ (0,T), x ∈ ∂D. +(4.31) +As above, by the definition of C δ ((2.2)), and (2.3), (3.53), choosing suitable δ,β such +that β − (δ + 2) > d +γ , we have +||∇2uσ′||L∞∞((s0,t0)×D) += +sup +t∈(s0,t0) +||∇2uσ′(t)||L∞(D) ≤ +sup +t∈(s0,t0) +||uσ′(t)||C δ+2(D) +≤ +sup +t∈(s0,t0) +||uσ′(t)||Hβ +γ (D) +≤ +C(t0 − s0)δ� +||f ′||Lq +p((s0,t0)×D) + ||g′||Lq +p((s0,t0)×D) +� +< +∞. +(4.32) +Therefore, by inequalities (3.52) in Theorem 3.11 and (4.32), we get +||∇Φσ − ∇Φσ′||Lq +p((s0,t0)×D) += +||∇v||Lq +p((s0,t0)×D) ≤ ||v||W2,q +p ((s0,t0)×D) +≤ +C||(Lσ′ − Lσ)·uσ′||Lq +p((s0,t0)×D) +≤ +C||(σ′ − σ)·∇2uσ′||Lq +p((s0,t0)×D) +≤ +C||σ − σ′||Lq +p((s0,t0)×D) ·||∇2uσ′||L∞∞((s0,t0)×D) +≤ +C||σ − σ′||Lq +p((s0,t0)×D) . +(4.33) + +WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT +33 +On the other hand, by the inequalities (2.3), (3.53) in Theorem 3.11 and (4.32), we obtain +||Φσ′ − Φσ||L∞∞(s0,t0)×D) += +||v||L∞∞(s0,t0)×D) = +sup +t∈(s0,t0) +||v(t)||L∞(D) +≤ +sup +t∈(s0,t0) +||v(t)||C δ (D) ≤ C sup +t∈(s0,t0) +||v(t)||Hβ +p(D) +≤ +C||(Lσ′ − Lσ)·uσ′||Lq +p((s0,t0)×D) +≤ +C||σ − σ′||Lq +p((s0,t0)×D). +Combining this estimate with (4.33), we get (4.30). +Then dividing [0,T] into subintervals and applying the last estimates on each interval +and piecing them together, we see that (4.29) is verified. +By generalized Itˆo’s formula (e.g. [37, Lemma 4.3]), Xb,σ +t +(x) satisfies SDE (4.1) on +[0,T ∧ τD] with initial value x ∈ D if and only if Y σ +t (y) := Φσ(t,Xb,σ +t +(x)) satisfies the +following SDE on [0,T ∧τD] with initial value in D, where D is also the image domain of +function Φσ(t,y) by Lemma 6.2, +dY σ +t = (∇Φσ ·σ)◦ (t,Φσ,−1(t,Y σ +t ))dBt := Θσ(t,Y σ +t )dBt, +(4.34) +where +Θσ(t,y) = (∇Φσ ·σ)◦ (t,Φσ,−1(t,y)). +(4.35) +Let Θσ′ be defined as in (4.35) via σ replaced by σ′. We have +Θσ(t,y)− Θσ′(t,y) += +(∇Φσ ·σ)◦ (t,Φσ,−1(t,y))− (∇Φσ ·σ)◦ (t,Φσ′,−1(t,y)) ++(∇Φσ ·σ − ∇Φσ′ ·σ′)◦ (t,Φσ′,−1(t,y)) +:= +J21(t,y)+ J22(t,y). +We are going to bound J21(t,y) and J22(t,y) separately. First, we bound J21(t,y). Since +MD is bounded operator in any integrable functions, we have +|J21| +≤ +C|Φσ,−1(t,y)− Φσ′,−1(t,y)| +× +� +MD|∇(∇Φσ ·σ)|(t,Φσ,−1(t,y))+ MD|∇(∇Φσ ·σ)|(t,Φσ′,−1(t,y)) +� +. +Noticing that +sup +y∈D +|Φσ,−1(t,y)− Φσ′,−1(t,y)| += +sup +x∈D +|x− Φσ′,−1 ◦ (t,Φσ(t,x))| +≤ +||∇Φσ′,−1 ◦ (t,·)||L∞(D) × ||Φσ′(t,·)− Φσ(t,·)||L∞(D) +and by the change of variables, [35, (2.3)], (4.29), and (4.30) we see +||J21||Lq +p((s0,t0)×D) +≤ +C||MD|∇(∇Φσ ·σ)|(·,Φσ,−1)+ MD|∇(∇Φσ ·σ)|(·,Φσ′,−1)||Lq +p((s0,t0)×D) +×||Φσ,−1 − Φσ′,−1||L∞∞((s0,t0)×D) +≤ +C||∇Φσ′,−1||L∞∞(s0,t0)×D) × ||Φσ′ − Φσ||L∞∞(s0,t0)×D) +×||MD|∇(∇Φσ ·σ)|||Lq +p((s0,t0)×D) +≤ +C||∇(∇Φσ ·σ)||Lq +p((s0,t0)×D) × ||Φσ′ − Φσ||L∞∞(s0,t0)×D) +≤ +C||σ′ − σ||Lq +p(s0,t0)×D). + +34 +YAOZHONG HU AND QUN SHI +For J22(t,y), by the change of variables, boundedness of σ and (4.30), we have +||J22||Lq +p((s0,t0)×D) += +||∇Φσ ·σ − ∇Φσ′ ·σ′||Lq +p((s0,t0)×D) +≤ +||(∇Φσ − ∇Φσ′)·σ||Lq +p((s0,t0)×D) + ||∇Φσ′(σ − σ′)||Lq +p((s0,t0)×D) +≤ +κ||∇Φσ − ∇Φσ′||Lq +p((s0,t0)×D) ++ +||∇Φσ′||L∞∞(s0,t0)×D) × ||σ − σ′||Lq +p((s0,t0)×D) +≤ +C||σ − σ′||Lq +p(s0,t0)×D). +Thus, we arrive at +||Θσ − Θσ′||Lq +p((s0,t0)×D) ≤ C||σ − σ′||Lq +p(s0,t0)×D). +(4.36) +Dividing [0,T] into subintervals and applying the estimate (4.36) on each interval and +piecing them together, we obtain +||Θσ − Θσ′||Lq +p((0,T)×D) ≤ C||σ − σ′||Lq +p(0,T)×D). +(4.37) +Finally, combining the above inequality (4.37) and Lemma 4.3, we get +sup +x∈D +E +� +sup +t∈[0,T∧τD] +|Xb′,σ +t +(x)− Xb′σ′ +t +(x)|p0 +� +≤ +sup +y∈D +E +� +sup +t∈[0,T∧τD] +|Y σ +t (y)−Yσ′ +t (y)|p0 +� +≤ +C||Θσ − Θσ′||p0 +Lq +p((0,T)×D) +≤ +C||σ − σ′||p0 +Lq +p((0,T)×D). +This combined with Theorem 4.4 completes the proof of (4.28). +5. Existence and uniqueness of strong solution +In this section, we shall study the strong solution to a stochastic differential equation +whose coefficients satisfy the conditions (Hb), (Hσ) and (HL). +Before doing this we need to study the solution of a stochastic differential equation on +bounded domain D, which is of very interesting on its own. Assume that b : [0,T]× D → +Rd,σ : [0,T] × D → Rd ⊗ Rd are measurable and satisfy the conditions (Hb), (Hσ) on +the domain D. We first study the strong solution of the following stochastic differential +equation on D: +Xt = x+ +� t +0 b(s,Xs)ds+ +� t +0 σ(s,Xs)dBs, t ∈ [0,T], x ∈ D, +(5.1) +Theorem 5.1. Let D be any nonempty bounded (connected) domain of Rd such that ∂D ∈ +C2. Assume that b,σ satisfy (Hb) and(Hσ) on the domain D. Then, the equation (5.1) +has a unique solution Xt up to positive stopping time τD, which is the first exit time of the +process Xt from the domain D. Namely, there are a stopping time τD and unique stochastic +process (Xt ,0 ≤ t ≤ τD) such that +Xt∧τD = x+ +� t∧τD +0 +b(s,Xs)ds+ +� t∧τD +0 +σ(s,Xs)dBs, ∀ t ∈ [0,T]. +(5.2) +Proof +It is obvious that b can be extended from D to Rd so that it satisfies the condition +(Hb) on the whole space Rd (in fact, we can require that the extension has compact support). +From Proposition 6.1 it follows that σ can also be extended from D to Rd so that it satisfies +the condition (Hσ) on the whole space Rd (in fact, we can require that the extension has + +WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT +35 +compact support). We denote these two extensions by ˜b and ˜σ. Thus, by [35, Theorem +5.1] the equation (5.1) on the whole space Rd +˜Xt = x+ +� t +0 +˜b(s, ˜Xs)ds+ +� t +0 +˜σ(s, ˜Xs)dBs, ∀ t ∈ [0,T]. +has a unique strong solution ( ˜Xt,0 ≤ t ≤ T). Define τD = inf +� +t > 0; ˜Xt ̸∈ D +� +. Then the +process +Xt = ˜Xt , +0 ≤ t ≤ τD +satisfies (5.2). +Since a strong solution is always a weak solution the uniqueness of the solution to the +equation (5.1) follows from Theorem 4.5. +It is obvious that τD appeared in the above context is also the life time of the process Xt +in the domain D +τD = inf{t > 0; Xt ̸∈ D} +and XτD ∈ ∂D due the continuity of the process Xt. +Theorem 5.2. Assume that σ and b satisfy conditions (Hσ), (Hb), on any bounded domain +D and assume (HL) holds true. Then the SDE (1.1) has a unique strong solution. Namely, +there is a unique process (X(t),0 ≤ t < ∞) such that +Xt = x+ +� t +0 b(s,Xs)ds+ +� t +0 σ(s,Xs)dBs , +∀ t ∈ [0,∞). +(5.3) +Proof +Let +DR = inf +� +x ∈ Rd ; |x| ≤ R +� +be the centered ball of radius R in Rd. Then by Theorem 5.1 equation (5.3) has a unique +solution XR +t on DR with the life time τR. +It is obvious that the stopping time τR is an (almost surely) increasing function of R and +then it converges to a stopping time τ := limR→∞ τR as R → ∞. By the uniqueness of the +above theorem (Theorem 5.1) we have that when R ≤ R′, then XR +t = XR′ +t +for all t ≤ τR. We +can define the process (Xt,0 ≤ t ≤ τ) so that Xt = XR +t if t ≤ τR without ambiguity. Then X +satisfies equation (5.3) when t ≤ τ: +Xt∧τ = x+ +� t∧τ +0 +b(s,Xs)ds+ +� t∧τ +0 +σ(s,Xs)dBs , +∀ t ≥ 0. +Such solution is clearly also unique. Thus, the proof of the theorem is completed if we can +prove τ = ∞ a.s. and this is the objective of the following. +Applying Itˆo’s formula to e−CtV(t,x) we obtain +de−CtV(t,Xt) += +(e−Ct∂tV(t,Xt)−Ce−CtV(t,Xt))dt + e−Ct∂xV(t,Xt)dXt ++e−Ct 1 +2∇2 +xV(t,Xt))d⟨X,X⟩t += +e−Ct(LtV(t,Xt)−CV(t,Xt)))dt + e−Ct⟨∇V(t,Xt),σ(t,Xt)⟩dBt. +Thus we have by the assumption (HL), +e−CτRV(τR,XτR) ≤ V(0,x)+ +� τR +0 +e−Cs⟨∇V(s,Xs),σ(s,Xs)⟩dBs . +Replacing the above τR by τR,N = τR ∧N for any positive integer N yields +e−CτR,NV(τR,N,XτR,N) ≤ V(0,x)+ +� τR,N +0 +e−Cs⟨∇V(s,Xs),σ(s,Xs)⟩dBs . + +36 +YAOZHONG HU AND QUN SHI +Since σ is uniformly bounded and ∇V(s,Xs)I{0≤s≤τR} ≤ C, taking the expectation yields +Ee−CτR,NV(τR,N,XτR,N) ≤ V(0,x) < ∞. +(5.4) +Denote AN := {τR ≤ N}. The above inequality (5.4) yields +e−CNE[V(τR,XτR)IAN] ≤ Ee−CτR,NV(τR,N,XτR,N) ≤ V(0,x) < ∞. +Thus, +P(AN) ≤ +eCNV(0,x0) +inf0≤t<∞,|x|=RV(t,x) . +Letting R → ∞, condition lim|x|→∞ inft V(t,x) = ∞ implies P(AN) = 0 for any integer N. +Thus limR↑∞τR = ∞,a.s. This completes the proof of the theorem. +6. Appendix +Let D be a domain in Rd and Dc = Rd\D. Let f be a measurable function from the +domain D to Rd. The local Hardy-Littlewood maximal operator MD is defined by +MD f(x) := +sup +0 0, denote the open cylinder +C(x,r,h) ≡ {y ∈ Rd ; +|x′ − y′| < r,|yd − xd| < h}. +Since ∂U is Lipschitzian, for each x ∈ ∂D, there exist (upon rotating and relabeling the +coordinate axes if necessary ) two numbers r,h > 0 and a Lipschitz function γ : Rd−1 → R +such that +D∩C(x,r,h) = {y ∈ Rd ; +|x′ − y′| < r, +γ(y′) < yd < xd + h}. + +WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT +37 +Fix x ∈ ∂D. With r,h,γ as above, write +C := C(x,r,h), C′ := C(x,r/2,h/2), D+ := C′ ∩D, D− := C′\D. +Set +� +f +(y) = f(y) +if y ∈ D+ , +f −(y) = f(y′,2γ(y′)− yd), +if y ∈ D− . +Note f + = f − = f on ∂D∩C′. +First, we assume that supp (f) ⊆ D∩C′(x,r,h). The extension of f is defined as follows +([4, see the proof of Theorem 4.4.1]): +Qf = + + + + + +f + +on D+, +f − +on D−, +0 +on Rd\(D+ ∪D−). +(6.2) +It is proved in [4, Theorem 4.4.1] that if f ∈ W 1 +p (D), then Qf ∈ W 1 +p (Rd). +We want to show that if f is H¨older continuous, +|f(y)− f(z)| ≤ C|y− z|α +∀ y,z ∈ D, +for some constant α ∈ (0,1) and positive constant C, then Qf is also H¨older continuous +with the same exponent α: +|Qf(y)− Qf(z)| ≤ C|y− z|α +∀ y,z ∈ Rd. +We shall prove the above by dividing our discussion into three cases. +Case I: When y,z ∈ D+, or y,z ∈ D−, or y,z ∈ Rd\(D+ ∪ D−), it is obvious that Qf is +H¨older continuous. +Case II: When y ∈ D+ and z ∈ D−, by the definitions of D+,D−, it holds yd > γ(y′) and +zd < γ(z′). By the H¨older continuity of f, we have +|Qf(y)− Qf(z)| += +|f(y′,yd)− f(z′,2γ(z′)− zd)| ≤ C +� +|y′ − z′|α + |yd + zd − 2γ(z′)|α� +≤ C +� +|y′ − z′|α + |yd − γ(y′)+ zd − γ(z′)|α + |γ(y′)− γ(z′)|α� +≤ C +� +|y′ − z′|α + |yd − γ(y′)+ zd − γ(z′)|α� +. +If yd − γ(y′) > |zd − γ(z′)|, then +|yd − γ(y′)+ zd − γ(z′)| += +yd − γ(y′)+ zd − γ(z′) ≤ |yd − γ(y′)− zd + γ(z′)| +≤ |yd − zd|+ |γ(y′)− γ(z′)| ≤ |yd − zd|+C|y′ − z′|. +If yd − γ(y′) ≤ |zd − γ(z′)|, then +|yd − γ(y′)+ zd − γ(z′)| += +−yd + γ(y′)− zd + γ(z′) ≤ |yd − γ(y′)− zd + γ(z′)| +≤ |yd − zd|+C|y′ − z′|. +Thus, we obtain +|Qf(y)− Qf(z)| ≤ C +� +|y′ − z′|α + |yd − zd|α� +≤ C|y− z|α. +Case III: When y ∈ D+, and z ∈ Rd\(D+∪D−), let u ∈ ∂D+ be the intersection of boundary +∂D+ and the line connecting the two points y,z. By the continuity of f we see f(u) = 0. +Then we have +|Qf(y)− Qf(z)| = |f(y)| = |f(y)− f(u)| ≤ C|y− u|α ≤ C|y− z|α. + +38 +YAOZHONG HU AND QUN SHI +Similarly, when y ∈ D−, and z ∈ Rd\(D+ ∪D−), it also holds +|Qf(y)− Qf(z)| ≤ C|y− z|α. +Thus, this proposition is proved in case f with support in C′ ∩D. +Now we remove the restriction supp (f) ⊆ C′ ∩D. Since ∂D is compact, we can cover +∂D with finitely many cylinders C′ +k = C(xk,rk/2,hk/2) with each xk ∈ ∂D, k = 1,2,··· ,N. +Let {ηk}N +k=0 be a sequence of smooth functions (see [25, Theorem 2.13] for a construction) +such that + + + + + +0 ≤ ηk ≤ 1 +and +supp (ηk) ⊆ C′ +k, (k = 1,2,··· ,N), +0 ≤ η0 ≤ 1 +and +supp (η0) ⊆ D, +∑N +k=0 ηk = 1, on D. +Define Q(ηk f) (k = 1,2,··· ,N) as above (6.2) and let Qf := ∑N +k=1 Q(ηk f)+η0 f. For any +y,z ∈ Rd, we have by the above argument +|Qf(y)− Qf(z)| +≤ +N +∑ +k=1 +|Q(ηk f)(y)− Q(ηk f)(z)|+ |η0(y)f(y)− η0(z)f(z)| +≤ +CN|y− z|α + |η0(y)f(y)− η0(z)f(z)|. +To show the H¨older continuity of η0 f, we also divide our discussion into three cases: Case +I: both y and z are in D, Case II: both y and z are not in D and Case III: y ∈ D and z ̸∈ D. +Case II is easy. +Case I: y,z ∈ D. In this case we have +|η0(y)f(y)− η0(z)f(z)| +≤ +|η0(y)(f(y)− f(z))|+ |(η0(y)− η0(z))f(z)| +≤ +|f(y)− f(z)|+ |η0(y)− η0(z)| ≤ C|y− z|α . +Case III: y ∈ D and z ̸∈ D. In this case we notice that η0(z) = 0 when z ̸∈ D and we have +then +|η0(y)f(y)− η0(z)f(z)| += +|η0(y)f(y)| = |η0(y)f(y)− η0(z)f(y)| +≤ +|f(y)||η0(y)− η0(z)| ≤ C|y− z|α . +Then, for any y,z ∈ Rd, we have +|Qf(y)− Qf(z)| ≤ C|y− z|α. +Finally, the proof of this proposition is completed. +Lemma 6.2. Assume D is a bounded, connected with C2 boundary domain in Rd. Assume +that σ and b satisfy conditions (Hσ), (Hb),respectively. Let Φ(t,x) := x + u(t,x) with +u(t,x) = +� +u1(t,x),u2(t,x),...,ud(t,x) +� +satisfying following PDE: +∂tuℓ + Lσuℓ + b ·∇uℓ + bℓ = 0, ℓ = 1,2,··· ,d +(6.3) +and u(t,x) = 0 for x ∈ ∂D, then the image of Φ(t,x) is D. +Proof +Since the image of Φ(t,x) has the same boundary as D, and D is a bounded, con- +nected with C2 boundary domain, therefore by the Jordan-Brouwer Separation Theorem +(see [9]),we have that the image of Φ(t,x) is D. +Lemma 6.3. Assume that {β(t),t ∈ [0,T]} is a nonnegative measurable process adapted +to a flow {Ft,t ∈ [0,T]} of σ-algebras on some probability space (Ω,F,P) (this means + +WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT +39 +that Fs ⊆ Ft ⊆ F for 0 < s < t < T), and the random variable β(t) is Ft measurable for +each t ∈ [0,T). Further, assume that for 0 < s < t < T, and any stopping time τ, +E +�� t∧τ +s∧τ β(r)dr|Fs∧τ +� +≤ ρ(s,t), +where ρ(s,t) is a nonrandom interval function satisfying the following conditions: +(i) +ρ(t1,t2) ≤ ρ(t3,t4) if (t1,t2) ⊆ (t3,t4). +(ii) +limh→0 sup0 N, +where t ∈ [0,T ∧ τ] and N is a positive number (N → ∞ later). Since βN(t) is a bounded +process, for all real λ, +Eexp +� +λ +� T∧τ +0 +βN(r)dr +� +≤ eλNT < ∞. +For 0 < s < t < T and for real λ let +ϕλ(s,t,N) := E +� +exp +� +λ +� t∧τ +s∧τ βN(r)dr +� +|Fs∧τ +� +. +From the identity +exp +� +λ +� t +s βN(r)dr +� += 1 + λ +� t +s βN(r)exp +� +λ +� t +r βN(θ)dθ +� +dr. +Taking conditional expectation on both sides , we get an equation for the function ϕλ(s,t,N) +ϕλ(s,t,N) = 1 + λE +�� t∧τ +s∧τ βN(r)exp +� +λ +� t∧τ +r∧τ βN(θ)dθ +� +dr|Fs∧τ +� +(6.4) += 1 + λE +� +E +�� t +s∧τ βN(r)1r≤τ exp +� +λ +� t∧τ +r∧τ βN(θ)dθ +� +|Fr∧τdr +� +|Fs∧τ +� += 1 + λE +��� t +s∧τ EβN(r ∧τ)1r≤τ exp +� +λ +� t∧τ +r∧τ βN(θ)dθ +� +|Fr∧τdr +� +|Fs∧τ +� += 1 + λE +��� t∧τ +s∧τ βN(r ∧τ)Eexp +� +λ +� t∧τ +r∧τ βN(θ)dθ +� +|Fr∧τdr +� +|Fs∧τ +� += 1 + λE +�� t∧τ +s∧τ βN(r)ϕλ(r,t,N)dr|Fs∧τ +� +. +Thus, by induction, +ϕλ(s,t,N) = +∞ +∑ +k=0 +ϕ(k) +λ (s,t,N), +where ϕ(0) +λ (s,t,N) ≡ 1, and +ϕ(k+1) +λ +(s,t,N) := λE +�� t∧τ +s∧τ βN(r)ϕ(k) +λ (r,t,N)dr|Fs∧τ +� +. + +40 +YAOZHONG HU AND QUN SHI +To estimate ϕ(k) +λ (s,t,N) uniformly with respect to N, we observe that +E +�� t∧τ +s∧τ βN(r)dr|Fs∧τ +� +≤ E +�� t∧τ +s∧τ β(r)dr|Fs∧τ +� +≤ ρ(s,t). +By induction on k, we now easily arrive at the inequalities by using condition (i) +ϕ(k) +λ (s,t,N) ≤ (λρ(s,t))k, k = 0,1,..., +It then follows from (ii) that for a fixed λ < κ−1 +0 +there is an h0 > 0 such that λρ(s,t) < 1 +if t − s < h0. Therefore, for the given λ < κ−1 +0 , we obtain +ϕλ(s,t,N) = +∞ +∑ +k=0 +ϕ(k) +λ (s,t,N) ≤ +∞ +∑ +k=0 +(λρ(s,t))k ≤ (1 − λρ(s,t))−1 < ∞ +(6.5) +for all 0 < s < t < T with t −s < h0. Now partition [0,T ∧τ] by points 0 = t0 ∧τ < t1 ∧τ < +... < tn ∧τ = T ∧τ, so that max1≤k≤n(tk −tk−1) < h0. An easy computation deduces that +tk ∧τ −tk−1 ∧τ ≤ tk −tk−1. Then by (6.5) +Eexp +� +λ +� T∧τ +0 +βN(r)dr +� +(6.6) += E +n−1 +∏ +i=0 +exp +� +λ +� ti+1∧τ +ti∧τ +βN(r)dr +� += E +� +E +� +n−1 +∏ +i=0 +exp +� +λ +� ti+1∧τ +ti∧τ +βN(r)dr +� +|Ftn−1∧τ +�� += E +� +n−2 +∏ +i=0 +exp +� +λ +� ti+1∧τ +ti∧τ +βN(r)dr +� +E +� +exp +� +λ +� tn∧τ +tn−1∧τ βN(r)dr +� +|Ftn−1∧τ +�� += E +� +n−2 +∏ +i=0 +exp +� +λ +� ti+1∧τ +ti∧τ +βN(r)dr +� +ϕλ(tn−1,tn,N) +� +≤ (1 − λρ(tn−1,tn))−1E +n−2 +∏ +i=0 +exp +� +λ +� ti+1∧τ +ti∧τ +βN(r)dr +� +≤ ... ≤ +n +∏ +i=1 +(1 − λρ(ti−1,ti))−1. +Finally, by dominated convergence theorem, we get +Eexp +� +λ +� T∧τ +0 +β(r)dr +� += +lim +N→∞Eexp +� +λ +� T∧τ +0 +βN(r)dr +� +≤ +n +∏ +i=1 +(1 − λρ(ti−1,ti))−1 < ∞. +The proof of the lemma is completed . +Lemma 6.4. Let (ξ(t))t∈[0,T], (ζ(t))t∈[0,T] and (β(t))t∈[0,T] be three real-valued measur- +able Ft-adapted processes. Let (η(t))t∈[0,T] and (α(t))t∈[0,T] be two Rd-valued measur- +able Ft-adapted processes. Suppose there exist c > 0 and δ ∈ (0,1) such that for any +0 < s ≤ t ≤ T and any stopping time τ, +E +�� t∧τ +s∧τ |β(r)|+ |α(r)|2dr|Fs∧τ +� +≤ c(t − s)δ +(6.7) + +WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT +41 +and suppose that +ξ(t) = +� t +0 ζ(r)dr + +� t +0 η(r)dBr + +� t +0 ξ(r)β(r)dr + +� t +0 ξ(r)α(r)dBr. +Then for any p > 0 and γ2,γ4 > 1, we have +E +� +sup +0≤t≤T∧τ +ξ p(t) +� +≤C + + +���� +���� +�� T∧τ +0 +ζ +(r)dr +�p���� +���� +Lγ4(Ω) ++ +����� +����� +�� T∧τ +0 +|η(r)|2dr +�p/2����� +����� +Lγ2(Ω) + + , +(6.8) +where C = C(c,δ, p,γ2,γ4) > 0. +Proof +Write +M(t) := exp +�� t +0 α(r)dBr − 1 +2 +� t +0 |α(r)|2dr + +� t +0 β(r)dr +� +. +Applying Itˆo’s formula to ξ(t)M−1(t), we can see that +d(ξ(t)M−1(t)) +(6.9) += M−1(t)dξ(t)− ξ(t)M−1(t)(α(t)dBt − |α(t)|2dt + β(t)dt)+ ⟨dξ,dM−1⟩t += M−1(t)η(t)dBt + M−1(t)(ζ(t)− ⟨α(t),η(t)⟩)dt +and then +ξ(t) = M(t) +�� t +0 M−1(r)η(r)dBr + +� t +0 M−1(r)(ζ(r)− ⟨α(r),η(r)⟩)dr +� +. +(6.10) +By (6.7) and Lemma 6.3, we have for any p ∈ R, +Eexp +� +p +� T∧τ +0 +|α(r)|2dr + p +� T∧τ +0 +|β(r)|dr +� +≤ C < ∞, +(6.11) +where the positive constant C depends only on c, p,δ. Furthermore, for any p ∈ R, by +(6.11), +exp +� +p +� t∧τ +0 +α(r)dBr − p2 +2 +� t∧τ +0 +|α(r)|2dr +� +is an uniformly integrable exponential martingale. Now we introduce some notations for +the sake of simplification below, +I1(t) := exp +� +p +� t +0 α(r)dBr − p2p1 +2 +� t +0 |α(r)|2dr +� +, +I2(t) := exp +� p2p1 − p +2 +� t +0 |α(r)2dr + p +� t +0 β(r)dr +� +with p1 > 1, p ∈ R, +I3(t) := +� t +0 M−1(r)η(r)dBr, +and +I4(t) := +� t +0 M−1(r)(ζ(r)− ⟨α(r),η(r)⟩)dr. + +42 +YAOZHONG HU AND QUN SHI +Thus, by H¨older’s inequality, Doob’s maximal inequality [32, Theorem B] and Lemma 6.3, +we have for any p ∈ R, +E( +sup +0≤t≤T∧τ +|M(t)|p) +(6.12) += E +sup +0≤t≤T∧τ +exp +� +p +� t +0 α(r)dBr − p +2 +� t +0 |α(r)|2dr + p +� t +0 β(r)dr +� += E( +sup +0≤t≤T∧τ +I1(t)I2(t)) +≤ +� +E( +sup +0≤t≤T∧τ +I1)p1 +�1/p1 +× +� +E( +sup +0≤t≤T∧τ +I2)p2 +�1/p2 +≤ +� +p1 +p1 − 1 +�� +Eexp +� +pp1 +� T∧τ +0 +α(r)dBr − p2p2 +1 +2 +� T∧τ +0 +|α(r)|2dr +��1/p1 +× +� +E( +sup +0≤t≤T∧τ +I2)p2 +�1/p2 +≤ +� +p1 +p1 − 1 +�� +E( +sup +0≤t≤T∧τ +I2)p2 +�1/p2 +< ∞, +where p2 > 1 is the conjugate of p1. Finally, for any p > 0, by (6.10), H¨older’s inequality, +(6.12) and Burkh¨older’s inequality [24, corollary 4.2, Chapter IV], +E( +sup +0≤t≤T∧τ +(ξ(t))p) +(6.13) += E( +sup +0≤t≤T∧τ +|M(t)|p(I3(t)+ I4(t))p) +≤ Cp(E( +sup +0≤t≤T∧τ +|M(t)|p)γ1)1/γ1 × (E( +sup +0≤t≤T∧τ +I p +3 (t))γ2)1/γ2 ++Cp(E( +sup +0≤t≤T∧τ +|M(t)|p)γ3)1/γ3 × (E( +sup +0≤t≤T∧τ +I p +4 (t))γ4)1/γ4 +≤ Cp,γ1,γ3,c,δ +�� +E( +sup +0≤t≤T∧τ +I p +3 (t))γ2 +�1/γ2 ++ +� +E( +sup +0≤t≤T∧τ +I p +4 (t))γ4 +�1/γ4� +≤ Cp,γ1,γ3,c,δ +� +E +�� T∧τ +0 +|η(r)|2dr +� pγ2 +2 +�1/γ2 ++Cp,γ1,γ3,c,δ +� +E( +sup +0≤t≤T∧τ +I p +4 (t))γ4 +�1/γ4 +≤ Cp,γ1,γ3,c,δ + + +����� +����� +�� T∧τ +0 +|η(r)|2dr +� p +2 +����� +����� +Lγ2(Ω) ++ +���� +���� +�� T∧τ +0 +ζ +(r)dr +�p���� +���� +Lγ4(Ω) + +, +where γ1,γ2,γ3,γ4 > 1, γ1,γ2 and γ3,γ4 are two pairs of conjugate numbers. The proof of +6.8 is completed. +REFERENCES +[1] S, Banach. Introduction to the theory of real functions. 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Mat. +Sb. 93(135) 129-149, 152 (1974). +DEPARTMENT OF MATHEMATICAL AND STATISTICAL SCIENCES, UNIVERSITY OF ALBERTA AT EDMON- +TON, EDMONTON, CANADA, T6G 2G1 +Email address: +yaozhong@ualberta.ca +SCHOOL OF MATHEMATICS AND STATISTICS, JIANGXI NORMAL UNIVERSITY, NANCHANG, JIANGXI +330022, P.R.CHINA AND DEPARTMENT OF MATHEMATICAL AND STATISTICAL SCIENCES, UNIVERSITY +OF ALBERTA AT EDMONTON +Email address: shiq3@mail2.sysu.edu.cn + diff --git a/6tE0T4oBgHgl3EQfwAE6/content/tmp_files/load_file.txt b/6tE0T4oBgHgl3EQfwAE6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7c812ba05b002ea6ee387a1e4faeffc3f8a668c0 --- /dev/null +++ b/6tE0T4oBgHgl3EQfwAE6/content/tmp_files/load_file.txt @@ -0,0 +1,1438 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf,len=1437 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='02625v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='PR] 6 Jan 2023 STRONG SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS AND UNBOUNDED COEFFICIENTS YAOZHONG HU AND QUN SHI ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' In this paper we study the existence and uniqueness of the strong solution of following d-dimensional stochastic differential equation (SDE) driven by Brownian mo- tion: dXt = b(t,Xt)dt +σ(t,Xt)dBt, X0 = x, where B is a d-dimensional standard Brownian motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' the diffusion coefficient σ is a H¨older continuous and uniformly non-degenerate d × d matrix-valued function and the drift coefficient b may be discontinuous and unbounded, not necessarily in Lq p, extending the previous works to discontinuous and unbounded drift coefficient situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' The idea is to combine the Zvonkin’s transformation with the Lyapunov function approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' To this end, we need to establish a local version of the connection between the solutions of the SDE up to the exit time of a bounded connected open set D and the associated partial differential equation on this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' As an interesting by-product, we establish a localized version of the Krylov estimates (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='1) and a localized version of the stability result of the stochastic differential equations of discontinuous coefficients (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Keywords: Discontinuity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' localized Krylov estimate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' local Zvonkin’s transformation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' localized stability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Lyapunov function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' strong solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' pathwise uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Introduction Let (Ω,F,P,(Ft)t≥0) be a complete filtered probability space with a filtration (Ft)t≥0 satisfying the usual conditions and let (Bt)t≥0 be a d-dimensional Ft-adapted standard Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' In this work we study the following stochastic differential equation (SDE) driven by multi-dimensional Brownian motion: dXt = b(t,Xt)dt + σ(t,Xt)dBt, X0 = x ∈ Rd, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='1) where the diffusion coefficient σ : R+ × Rd → Rd ⊗ Rd is uniformly elliptic and H¨older continuous with respect to spatial variable in any bounded domain D and the drift coeffi- cient b : R+ × Rd → Rd is integrable in any bounded domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' More precisely, we make the following assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' (Hσ): For any bounded domain D ⊆ Rd there exist constants α ∈ (0,1], κ = κα,D > 1, such that for all (t,x) ∈ [0,T]× D, κ−1|ξ|2 ≤ |σt(t,x)ξ|2 ≤ κ|ξ|2, ∀ξ ∈ Rd , Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='Hu was supported by the NSERC discovery fund and a centennial fund of University of Alberta, National Natural Science Foundation of China (12261046).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='Shi was supported by China Overseas Education Fund Committee, National Natural Science Foundation of China (11901257, 12261046), and an NSERC discovery fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' AMS Mathematics Subject Classification (2010): 60G15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' 60H07;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' 60H10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' 65C30 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' 1 2 YAOZHONG HU AND QUN SHI and ∥σ(t,x)− σ(t,y)∥HS ≤ κ|x− y|α, where σt stands for the transpose of the matrix σ and where for a matrix A, ∥A∥HS = tr(ATA) stands for its Hilbert-Schmidt norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' We also assume ∇σ ∈ Lq p([0,T]× D) for certain p and q satisfying d p + 2 q < 1, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='2) where Lq p([0,T]× D) is defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='1) in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' For the drift coefficient we make the following assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' (Hb): b is uniformly locally integrable on any bounded domain of Rd, namely, b ∈ Lq p([0,T]× D) < ∞, ∀ bounded domain D, for p and q appeared in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' (HL): There exists a non-negative (Lyapunov) function V ∈ C1,2([0,T]× Rd) [the set of all functions { f(t,x),t ∈ [0,T],x ∈ Rd} which are continuously differentiable in t and have continuous derivatives with respect to the spatial variable x up to second order] satisfying lim R→∞ inf 0≤t≤T,|x|≤RV(t,x) = ∞ and LtV(t,x) ≤ CV(t,x), ∀t ∈ [0,T], x ∈ Rd , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='3) for some constant C > 0, where Lt is the differential operator associated with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='1): Lt := ∂ ∂t + Lσ,b = ∂ ∂t + d ∑ i=1 bi(t,x) ∂ ∂xi + 1 2 d ∑ i,j=1 d ∑ k=1 (σikσ jk)(t,x) ∂ 2 ∂xi∂xj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='4) The objective of this work is to show that under the above assumptions (Hσ), (Hb) and (HL), the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='1) has a unique strong solution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' For stochastic differential equations with discontinuous (yet bounded) coefficients there are rather complete general results about the weak solution and we refer to the classi- cal work [28] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' For the strong solutions there have been also some important progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Among them let us mention the works [18, 33, 35, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Let us em- phasize an important point that in these works, the assumptions that σ is uniformly elliptic on the whole space Rd and b ∈ Lq p = Lq p(Rd) are usually needed, which generally require that σ and b are bounded when |x| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' In [34], the authors show the weak differentia- bility of the unique strong solution with respect to the starting point x as well as Bismut- Elworthy-Li’s derivative formula for SDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='1) under the condition that 1) σ is bounded, uniformly continuous, and nondegenerate, 2) b ∈ �Lq1 p1, ∇σ ∈ �Lq2 p2 for some p1,q1 ∈ [2,∞) with d pi + 2 qi < 1, i = 1,2, where �Lqipi are some localized spaces of Lqipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' But unbounded drift coefficients may still not be contained in this space because if b is unbounded, then |||b|||�Lqipi := supz∈R ||b · χz r||Lqipi = ∞ with χ ∈ C∞ c (Rd) and χz r(x) := χr(x − z) := χ( x−z r ), r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' The idea in all of these works is to use the Zvonkin transformation to transform equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='1) with discontinuous coefficients to an equation without drift and hence the problem of discontinuity of the drift coefficient disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Recently, the more and more interest in studying SDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='1) with discontinuous coeffi- cients is partly due to its more and more important role played in applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' For example, the threshold Ornsten-Uhlenbeck processes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' [11] and the references therein) widely used in application contain discontinuous but piecewise linear drift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' the work of Flandoli, WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT 3 Gubinelli and Priola [6] discovers that noises can prevent the singularity for linear transport equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' In these equations the drift coefficient b is not in Lq p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' In fact, for an SDE with discontinuous and unbounded coefficients there are also some study although the number of works is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Let us mention a recent work [36] in which Zhang et al show the existence and uniqueness of the martingale solution (weak solution) to the above homogeneous (time independent) SDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='1) under the following assumptions: (1) the drift coefficient b is decomposed into b = b1 + b2, where ⟨x,b1(x)⟩ √ 1 + x2 ≤ −κ0|x|ϑ + κ1, |b1(x)| ≤ κ2(1 + |x|ϑ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='5) for some ϑ ≥ 0 and κ0,κ1,κ2 > 0, b2 ∈ H−α p for some α ∈ (0,1/2] and p ∈ � d 1 − α ,∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' (2) The diffusion coefficient σ is uniformly elliptic and ||(−∆)β/2σ||Lr < ∞ for some β ∈ [α,1], r ∈ (d/β,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' They obtain sharp two-sided as well as the gradient estimates of the heat kernel associated to the above SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Moreover, they study the ergodicity and global regularity of the invariant measures of the associated semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' However, there is no study on the strong solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' If b and σ are locally Lipschitzian on any bounded domain, then it is well-known that the assumption (HL) is the famous Lyapunov type condition to guarantee non explosion of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' So, in some sense our condition is to combine the Lyapunov condition and the integrability condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Our new set of conditions can be applied to some new situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' We give only some very special examples as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' ([11]) One example is the threshold Ornstein-Ulenbeck processes: dXt = n ∑ i=1 (βi − αiXt)I{θi−1≤Xt<θi}dt + σdBt , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='6) where β1,··· ,βn,α1,··· ,αn,−∞ = θ0 < θ1 < ··· < θn−1 < θn = ∞ are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' It is clear that for all parameters βi,αi,θi, the drift coefficient b(x) = ∑n i=1(βi − αix)I{θi−1≤x<θi} and the diffusion coefficient σ(x) = σ satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='3) with V(x) = |x|2 and with some constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' But it does not satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='5) unless α1,··· ,αn are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Another interesting example is dXt = n ∑ k=1 � mk ∑ j=1 βk,jX j t � I{θk−1≤Xt<θk}dt + σdBt , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='7) where βk,j are constants and βk,mk ̸= 0 and −∞ = θ0 < θ1 < ··· < θn−1 < θn = ∞ are con- stants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' It is clear that if m1,mn are odd, and if β1,n1,βn,nn are negative, then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='3) is satisfied with V(x) = |x|2 and with some constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' But (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='5) will generally not be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' In the above assumption (Hb), b may be unbounded in Rd so we cannot use the existing theory to solve (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' To study the well-posedness of this equation, our strategy is to consider first the solvability of the associated parabolic differential equation with Cauchy- Dirichlet problem on bounded domain \uf8f1 \uf8f2 \uf8f3 ∂tu + Lσu + b ·∇u + f = 0, (t,x) ∈ (0,T)× D, u(T,x) = 0, x ∈ D, u(t,x) = g(t,x), (t,x) ∈ (0,T)× ∂D, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='8) 4 YAOZHONG HU AND QUN SHI where D is bounded nonempty domain (a connected open subset) of Rd so that ∂D ∈ C2, g is integrable functions on (0,T) × ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Then, with the aid of the solution to the above equation we find a C1-diffeomorphism Φ to transform (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='1) to another equation with only diffusion term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' This transformed equation has a unique strong solution in bounded nonempty domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Finally, by a stopping time argument combined with the Lyapunov function V (given in assumption (HL)) we obtain the unique strong solution in Rd of SDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' There are new challenges in each of our above steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' First, when the coefficients are nice equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='8) is a classical Cauchy-Dirichlet problem in bounded domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' However, when the coefficient is only locally integrable, it seems there is no study on such equa- tion, to our best knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Relevant works we found is the parabolic problem for whole space Rd (which was used in the works [18, 33, 35, 38]) and elliptic problem for general (including bounded) domain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' These works can be found for example in [15], where are also mentioned that the results on elliptic problem on general domain can be extended to parabolic problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Since we need to know what exact we can cite, we present a detail study on this problem in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' In the previous works on the strong solution with integrable drift coefficients, an impor- tant technique is the so-called the Zvonvin transformation which reduced the original SDE to an SDE without drift terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Now we need study SDE on a bounded domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Since the solution will stay in the boundary only in some finite stopping time, we need to study the solution with a finite (random) life time for the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Not much study on the stochastic differential equation is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' To obtain the strong solution of an SDE on the boundary domain, we extend the coefficients to the whole Euclidean space Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' However, there is no available extension theorem we can immediately use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' We need to modify some existing results so that they are applicable to our situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Some of these are given in Section 2 and some are given in Appendix (Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' The stochastic differential equations on a bounded domain is studied in detail in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' For the existence and uniqueness one of the most important tools is the Krylov estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' We deduce the Krylov type estimate for stochastic differential equations on bounded domain, namely, a localized version of the Krylov estimate, also in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' In Section 5, we prove the main result of the existence and uniqueness of strong solutions to SDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='1) under Lyapunov condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' In Section 2, we recall some well-known results and give briefly some preliminaries about the Sobolev differentiabilities of random vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' The last section contains some technical results obtained and used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Throughout this paper, we use the following convention: C with or without subscripts will denote a positive constant, whose value may change in different places, and whose dependence on the parameters can be traced from the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' PRELIMIARIES We first introduce some spaces and notations for later use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Let p,q ∈ [1,∞), T > 0 and let D be a bounded,connected domain in Rd with C2 boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' We denote by Lq p((0,T)× D) the space of all real-value Borel functions on [0,T]× D such that ∥ f∥Lq p((0,T)×D) := �� T 0 �� D |f(t,x)|pdx �q/p dt �1/q < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='1) For p,q = ∞, ∥ f∥L∞∞((0,T)×D) := sup t∈[0,T] sup x∈D |f(t,x)| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' WELL-POSEDNESS OF SDE WITH DISCONTINUOUS AND UNBOUNDED DRIFT 5 When q = p we denote Lp((0,T) × D) := Lp p((0,T) × D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' When D = Rd, we denote Lq p = Lq p(0,T) := Lq p((0,T)× Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' For any positive integer m and any p ≥ 1, W m p (D) is used to denote the usual Sobolev space over the domain D ⊆ Rd with norm ∥ f∥Wm p (D) := m ∑ k=0 ∥∇k x f∥Lp(D) < +∞, where ∇k x denotes the k-order gradient operator on spatial variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' When p = q and D = Rd, we denote Lp = Lp p(0,T) := Lp p((0,T)×Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' For β ∈ R, let Hβ p := (I−∆)− β 2 (Lp) be the usual Bessel potential space with norm (we refer to [27], [30]) ∥ f∥Hβ p = ∥(I− ∆) β 2 f∥p , where ∥ ·∥p is the usual Lp-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Notice that for m ∈ N and p > 1, ∥ f∥Hmp ≍ ∥ f∥Wmp , where and throughout the paper we use A ≲ B to denote that fact that there is a constant C such that A ≤ CB and A ≍ B means A ≲ B and B ≲ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' For β ∈ [0,2) and p ∈ (1,∞), by Mihlin’s multiplier theorem (see [29]), we know ∥ f∥Hβ p ≍ ∥(I− ∆) β 2 f∥p ≍ ∥ f∥p + ∥(−∆) β 2 f∥p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Let C(Rd) be the collection of all continuous functions in Rd, equipped with the norm ||f||C(Rd) := sup x∈Rd |f(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Obviously, C(Rd) is a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Let k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Then Ck(Rd) := { f ∈ C(Rd) : Dα f ∈ C(Rd) if |α| ≤ k} is a Banach space equipped with the norm ||f||Ck(Rd) = ∑ |α|≤k ||Dα f||C(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Here, we use standard notations: α = (α1,··· ,αd) with αj ∈ N0 := {0,1,2,···} is a mul- tiindex, |α| = ∑d j=1 αj and Dα f(x) = ∂ |α| f ∂ α1 x1 ···∂ αd xd (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' For any positive number 0 < δ < 1, let C δ be the usual H¨older space with finite norm ∥ f∥C δ (Rd) := sup x∈Rd |f(x)|+ sup x̸=y,x,y∈Rd |f(x)− f(y)| |x− y|δ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' For any 0 < δ ̸∈ N0, we put δ = [δ]+ {δ}, 6 YAOZHONG HU AND QUN SHI where [δ] = max{k ∈ Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=',k ≤ δ} is the integer part of δ, 0 < {δ} < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Denote the H¨older space (refer to [29, Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='2]) C δ(Rd) : = � f ∈ C(Rd) : ∥ f∥C δ (Rd) = ||f||C[δ](Rd) + ∑ |α|=[δ] � sup x∈Rd |Dα f(x)|+ sup x̸=y,x,y∈Rd |Dα f(x)− Dα f(y)| |x− y|δ−[δ] �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='2) By Sobolev’s embedding theorem ([3, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='11]), we have ∥ f∥C δ (D) ≤ C∥ f∥Hβ p(D), β − δ > d p, δ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='3) In particular, we take δ = 0 to obtain ∥ f∥∞ ≤ C∥ f∥Hβ p , as β > d p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Furthermore, we also need the following Sobolev space: for p,q ∈ [1,∞),m ≥ 0 we denote by Wm,q p ((0,T)×D) the set of all Borel functions � f(t,x),0 ≤ t ≤ T,x ∈ Rd� such that ∥ f∥Wm,q p ((0,T)×D) := �� T 0 � ∥∂t f(t)∥q Lp(D) + m ∑ k=0 ∥∇k f(t)∥q Lp(D) � dt �1/q < ∞, where ∇k is the gradient with respect to x only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' By ˚W1,q p ((0,T) × D) we mean the subset of W1,q p ((0,T) × D) consisting of all functions vanishing on the boundary ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' For k ∈ {2,3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='} let ˚Wk,q p ((0,T)× D) = ˚W1,q p ((0,T)× D)∩Wk,q p ((0,T)× D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' The norm in ˚Wk,q p ((0,T)×D), k ∈ {1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='}, is taken to be the same as in Wk,q p ((0,T)×D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' The introduction of ˚Wk,q p ((0,T)×D) is to express the Dirichlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' When we say u = g on ∂D we understand the following condition u − g ∈ ˚Wk,q p ((0,T)× D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' We can also introduce the function class Wk,q p ((0,T)×∂D), k = 0,1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' By [15, Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content='2], a function g in Wk,q p ((0,T)× ∂D) can be extended to a function on D so that it is in Wk,q p ((0,T)×D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' We shall use this property in the future instead of giving the definition of Wk,q p ((0,T)× ∂D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' Let f be a locally integrable function on Rd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE0T4oBgHgl3EQfwAE6/content/2301.02625v1.pdf'} +page_content=' The classical Hardy-Littlewood maximal function is defined by M f(x) := sup 0 0. +The super- +radiance conditon in Eq. (16) gives another constraint +2rgω < (ma + qQr+)/r+. Combining these two inequal- +ities, one can obtain, +0 < 2rgω2 − rgµ2 − qQω +< +�ma +r+ ++ qQ +� +ω − rgµ2 − qQω += ma +r+ +ω − rgµ2. +(27) +So there is no superradiant bound state if m ≤ 0. +It also shows that Reissner-Nordstr¨om BHs could not +hold bounded scalar clouds [55]. The minimum KNBH +spin a allowing superradiant instability is approximately +rgr+µ/m. +Next, we look at Eq. (7) in the small r limit. For BH +superradiance, the inner boundary is the outer horizon +r = r+. It is more convenient to write the radial function +in terms of z = (r − r+)/2b, +z(z + 1) d +dz +� +z(z + 1)dR +dz +� ++ U(z)R = 0, +(28) +where U(z) can be written as an expansion of z, +U(z) = p2 + z +�4rgr+ω +b +� +r+ω − am +2r+ +− Q2ω +2rg +� +− (Λlm + r2 ++µ2 + a2ω2) + qQ +b (am + r+qQ − a2ω − 3r2 ++ω) +� ++ z2(a2ω2 − Λlm + 2µ2a2 − 3µ2r2 ++ + 6r2 ++ω2 + 2Q2µ2 + q2Q2 − 6r+qQω) ++ 4z3b [rgµ2 + 2r+(ω2 − µ2) − qQω] + 4z4b2(ω2 − µ2), +(29) +in which, +p = (r2 ++ + a2) +2b +(ω − ωc). +(30) +Note that both p and rgωc scale as O(α0). +In the limit of small α, the Λlm has the expanded form +Λlm = l(l+1)+O(α4). At the LO of α, we get the radial +equation in limit (r − r+) ≪ max(1/ω, 1/µ), +z(z + 1) d +dz +� +z(z + 1)dR +dz +� ++ +� +p2 − l′(l′ + 1)z(1 + z) +� +R = 0. +(31) +In principle, the l′ should be replaced by l in this order. +Nonetheless, the ǫ in l′ plays the role of a regulator in +the intermediate steps. It will be set to zero at the end. +The general solution of Eq. (31) is a linear combination +of two associated Legendre functions, and the physical +solution is the one with the ingoing wave at r → r+. +After changing the variable back to r, the solution of the +radial function is, +R(r) = +�r − r+ +r − r− +�−ip +2F1 +� +−l′, l′ + 1; 1 − 2ip; −r − r+ +2b +� +, +(32) +up to an arbitrary normalization. +Next, we apply the matching method first proposed +in [6] and further developed recently in Ref. [21]. The +solution of Eq. (26) is only valid in r ≫ rg limit, while +the solution in Eq. (32) requires r ≪ rgα−2 from the ig- +norance of terms proportional to z3 and z4. They have +an overlapped region in the limit α ≪ 1. In this region, +the two solutions are expected to have the same behav- +ior. The behavior of Eq. (26) in the overlapped region is +obtained by looking at its small r limit, which is, +(2κ)l′Γ(−2l′ − 1) +Γ(−l′ − λ) +rl′ + (2κ)−l′−1Γ(2l′ + 1) +Γ(l′ + 1 − λ) +r−l′−1. (33) +On the other hand, the behavior of Eq. (32) in the over- +lapped region is obtained by looking at its large r limit, +which is, +(2b)−l′Γ(2l′ + 1) +Γ(l′ + 1)Γ(l′ + 1 − 2ip)rl′ + (2b)l′+1Γ(−2l′ − 1) +Γ(−l′ − 2ip)Γ(−l′) r−l′−1. +(34) +The ratio of the coefficients of the rl′ and r−l′−1 should be +the same for the two solutions in the overlap region. The +obtained equation is the eigenequation of ω. It can be + +5 +solved perturbatively by the observation that the second +term in the expression (33) must be suppressed at small +r, indicating l′+1−λ is very close to zero or some negative +integer, +l′ + 1 − λ = −n − δλ, +(35) +where |δλ| ≪ 1 and n is zero or a positive integer. +Following the convention in literature, we also define +¯n = n + l + 1. Then the above relation is re-expressed as +λ = ¯n + ǫ + δλ. At LO of α, it reduces to λ = ¯n + δλ. +Combining with the definition of λ in Eq. (22), the rgκ +scales as α2, which is important in power-counting. Since +|δλ| ≪ 1, one could solve for δλ perturbatively with ex- +pressions (33) and (34). +The LO calculation of δλ for Kerr BHs was completed +in Ref. [6], with the regulator ǫ set to zero from the be- +ginning. Recently, we have confirmed a missing factor +of 1/2 in that result [21], which was first identified in +Ref. [34]. The missing factor is conjectured to be from +mistreatments of Γ functions with negative integer argu- +ments. The correct formula is provided in the appendix of +Ref. [21]. This subtle calculation turns out to be straight- +forward with the regulator ǫ kept in the intermediate +steps. +More details could be found in Ref. [21]. +For +KNBHs, the first LO calculation of δλ was completed +in Ref. [56]. It followed the same steps in Ref. [6] and +missed the factor 1/2 as well. After the correction, the +LO result of δλ is, +δλ(0) = − ip (4κb)2l+1 (n + 2l + 1)!(l!)2 +n! [(2l)!(2l + 1)!]2 +l� +j=1 +(j2 + 4p2), +(36) +where the superscript (0) indicates that it is the LO re- +sult. It scales as O(α4l+2). +The eigen-frequency ω can be expressed in terms of δλ +with Eqs. (22) and (35). Defining ω = ω0 + ω1δλ(0) in +Eq. (22) and expanding it to the linear term of δλ(0), one +arrives at, +λ = rg(2ω2 +0 − µ2) − qQω0 +� +µ2 − ω2 +0 ++ rgω0ω1(3µ2 − 2ω2 +0) − qQµ2ω1 +(µ2 − ω2 +0)3/2 +δλ(0) + O +� +(δλ(0))2� +. +(37) +On the other hand, we have λ = ¯n + δλ(0) from Eq. (35). +Then it is straightforward to get, +rg(2ω2 +0 − µ2) − qQω0 +� +µ2 − ω2 +0 += ¯n, +(38a) +rgω0ω1(3µ2 − 2ω2 +0) − qQµ2ω1 +(µ2 − ω2 +0)3/2 += 1. +(38b) +Note that in getting Eq. (38a), we have ignored the ǫ +which could be traced back to the l′ in Eq. (35). This +omission leads to an error in rgω0 at the order of O(α5). +Solving ω0 perturbatively from Eq. (38a), one arrives at, +ω(0) +0 +µ += 1 − 1 +2 +�rgµ − qQ +¯n +�2 ++ O(α4). +(39) +Then the ω1 could be expressed in terms of ω0 from +Eq. (38b) and expanded in powers of α, +ω(0) +1 +µ += (rgµ − qQ)2 +¯n3 ++ O(α4). +(40) +Since both ω0 and ω1 are real, ω0 and ω1δλ(0) are the +leading terms of the real and imaginary parts of ω, re- +spectively. +Especially, the imaginary part of ω scales +as O(α4l+5). +B. +Next-to-leading-order Approximation +In a previous work, we have carefully studied the su- +perradiance of a real scalar field around a Kerr BH [21]. +The LO eigenfrequency ω obtained in Ref. [6] has an er- +ror as large as 160% compared to the numerical result. +After correcting the missing factor 1/2, the convergence +is improved, with the error ≲ 80%. Except for the large +discrepancy, the LO result also has some strange behav- +iors. Since the LO result is the leading term in the Taylor +series of the exact ω at α = 0, it is expected to converge +to the exact ω with α approaching zero. Nonetheless, the +relative error seems to be a nonzero constant for small α, +reaching as large as 30% at α = 0.07 for a = 0.99. This +discrepancy at small α puts the question on the power- +counting strategy. Moreover, the discrepancy at small α +increases quickly with the BH spin parameter a. +These problems are solved by adding the NLO correc- +tion of ω [21]. Below we follow the same steps for the +KNBHs. The key observation is that the first term in +the square bracket in Eq. (29), which scales as α2, is +enhanced by a factor of 1/b. For BHs with large spin +a and/or charge Q, this term can be as important as +the LO contribution. Other NLO contributions are also +added for consistency. +The first NLO correction appears as ǫ in the asymp- +totic radial wave function at large r, which is given in +Eq. (26). It can be calculated from the definition of l′ in +Eq. (19), +ǫ = −8r2 +gµ2 + Q2µ2 + 8rgqQµ − q2Q2 +2l + 1 ++ O(α4). +(41) +The second NLO contribution is from the asymptotic ra- +dial wave function at small r. +The potential U(z) in +Eq. (29) can be approximated by p2−l′(l′+1)z(1+z)+zd, +where d is defined as +d =(4rgµ − 2qQ)p − 2(4rg − r+)rgµ2 ++ 2µqQ(4rg − r+) − q2Q2 + O(α3). + +6 +Up to an arbitrary normalization, the corresponding ra- +dial function at the NLO is, +R(r) =(r − r−) +√ +d−p2 +(r − r+)ip +2F1 +� +− l′ − ip + +� +d − p2, +l′ + 1 − ip + +� +d − p2; 1 − 2ip; −r − r+ +2b +� +. +(42) +In the r → +∞ limit, the asymptotic behavior of this +function is, +(2b)−l′−ip+√ +d−p2Γ(2l′ + 1)Γ(1 − 2ip) +Γ(l′ + 1 − ip − +� +d − p2)Γ(l′ + 1 − ip + +� +d − p2) +rl′ ++ +(2b)l′+1−ip+√ +d−p2Γ(−2l′ − 1)Γ(1 − 2ip) +Γ(−l′ − ip − +� +d − p2)Γ(−l′ − ip + +� +d − p2) +r−l′−1. +(43) +Following similar matching steps above, the NLO contri- +bution of δλ could be obtained after some algebra, +δλ(1) = +� d +2ǫ − ǫ +2 − ip +� (4κb)2l′+1 Γ(n + 2l′ + 2)Γpd +n! [Γ(2l′ + 1)Γ(2l′ + 2)]2 +, +(44) +where the superscript (1) indicates it is the NLO result, +and the Γpd is defined as, +Γpd = +���Γ(l′ + 1 + ip + +� +d − p2)Γ(l′ + 1 + ip − +� +d − p2) +��� +2 +Γ(1 + 2ǫ)Γ(1 − 2ǫ) +Γ(1 − ip − +� +d − p2 − ǫ)Γ(1 + ip + +� +d − p2 + ǫ)Γ(1 − ip + +� +d − p2 − ǫ)Γ(1 + ip − +� +d − p2 + ǫ) +. +(45) +The last NLO contribution is from ω0 and ω1. Defining +ω = ω(1) +0 ++ ω(1) +1 δλ(1), the expansion of λ in Eq. (37) is +still valid, only with δλ(0) replaced by δλ(1). Combining +with λ = ¯n + ǫ + δλ(1), one could follow the same steps +as in the LO calculation and obtain, +ω(1) +0 +µ +=1 − 1 +2 +�rgµ − qQ +¯n +�2 ++ (rgµ − qQ)2 +8¯n4 +[3(rgµ − qQ)(5rgµ − qQ) + 8¯nǫ] ++ O(α6), +(46a) +ω(1) +1 +µ +=(rgµ − qQ)2 +¯n3 +− 3(rgµ − qQ)2 +2¯n5 +[(rgµ − qQ)(5rgµ − qQ) + 2¯nǫ] ++ O(α6). +(46b) +Finally, we discuss a subtle problem related to the ω +dependence in the definition of p. In the calculation of the +δλ(1), the ω in p should be replaced by ω(0) +0 , rather than +ω(1) +0 . Here we explain the reason. In deriving the small-r +asymptotic form of the radial function, we approximate +U(z) in Eq. (29) by p2 − l′(l′ + 1)z(z + z) + zd. The +coefficient of z and z2 are accurate at O(α2) and O(α0), +respectively. At z ∼ O(α), this two terms are at the same +order of O(α4). Consequently, we only need to keep the +terms in p2 up to O(α4), which then leads to ω = ω(0) +0 +in p. In comparison to the numerical calculation, this +choice of ω gives a satisfactory NLO result. Using ω(1) +0 +in p is not as satisfactory, due to partially including the +high-order contributions. +IV. +RESULTS +The eigenfrequency of the Kerr BH superradiance has +been studied in Refs. [8, 10, 21]. In comparison, the case +for Kerr-Newman BH has two more parameters, the BH +charge Q and the scalar charge q. In this section, we first +study the superradiance of a neutral scalar field, focusing +on the effect of Q. Then we consider the superradiance of +a charged scalar field. Comparisons with the numerical +calculations in the literature are also provided. +A. +Neutral Scalar Fields +In the following study of neutral scalar superradiance, +we adopt the NLO δλ(1) in Eq. (44), where the scalar +charge q is set to zero. The ω(1) +0 +and ω(1) +1 +in Eqs. (46) +are used. Then the NLO eigen-frequency is ω = ω(1) +0 ++ +ω(1) +1 δλ(1). +The BH charge Q cannot be chosen arbitrarily. In our +derivation, we have implicitly assumed the KNBH has + +7 +horizons, which requires |Q| ≤ +� +r2g − a2. In addition, +neutral scalars could not distinguish the sign of the BH +charge. Mathematically, it means the BH charge Q can +only appear in the formulas as Q2. So it is sufficient to +only consider positive Q. +The superradiance condition in Eq. (16) with q = 0 +has the same form as the Kerr BH. The effect of the BH +charge Q is hidden in r+ = rg + +� +r2g − a2 − Q2. Keeping +the BH mass M and spin a fixed, larger charge Q results +in a larger upper limit of Re(ω). Thus massive scalars +too heavy to be produced with Kerr BH superradiance +may exist in the superradiant region of KNBHs. +Q=0 +Q=0.1 +Q=0.2 +Q=0.3 +Q=0.4 +0.0 +0.5 +1.0 +1.5 +10-13 +10-11 +10-9 +10-7 +rg +� +Im ( +� +) / +� +l=m=1 +l=m=2 +l=m=3 +a=0.9 +Q=0 +Q=0.2 +Q=0.4 +Q=0.6 +Q=0.7 +0.0 +0.5 +1.0 +1.5 +10-13 +10-11 +10-9 +10-7 +rg +� +Im ( +� +) / +� +l=m=1 +l=m=2 +l=m=3 +a=0.7 +FIG. 1. The imaginary part of NLO eigenfrequency with q = +0 as a function of rgµ. Only the curves with n = 0 are shown. +In the top (bottom) panel, the BH spin a is 0.9 (0.7). +In +both panels, from left to right, the three bunches correspond +to l = m = 1, 2, 3, respectively. In each bunch, the curves +with different colors correspond to different values of the BH +charge Q. +Fig. 1 shows the imaginary part of ω as a function +of rgµ. +For comparison, the curves for Kerr BHs are +also shown, labeled with Q = 0. +All curves have the +same qualitative behavior. With an increasing value of +rgµ, they first increase, then drop rapidly to below zero +after reaching the maxima. +There are three effects of +the BH charge Q. Firstly, the superradiant region of rgµ +is enlarged with larger Q. Correspondingly, the peak of +Q=0.1 +Q=0.2 +Q=0.3 +Q=0.4 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +rg μ +s (Q) +a=0.9 +Q=0.2 +Q=0.4 +Q=0.6 +Q=0.7 +0.00 +0.05 +0.10 +0.15 +0.20 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +rg μ +s (Q) +a=0.7 +FIG. 2. Factor s(Q) with q = 0 as a function of rgµ for BH +spin a = 0.9 (upper panel) and a = 0.7 (lower panel). The +vertical dashed line in each panel labels the value of rgµ where +Im ω(Q = 0) reaches its maximum value for the corresponding +spin parameter a. +the curve moves to the right with increasing Q. +The +maximum rgµ with positive Im(ω) is quite accurately +determined by µ = ωc. Secondly, the maximum Im(ω) +increases with larger Q. Fixing the BH spin to be a = +0.9, the maximum values of rgIm(ω) with Q = 0 are +2.088 × 10−8, 2.427 × 10−9 and 1.029 × 10−10 for l = +m = 1, 2, 3, respectively. The numbers for Q = 0.43 are +1.476 × 10−7, 2.006 × 10−8 and 8.760 × 10−10, which are +larger than the Q = 0 cases by factors of 7.07, 8.26 and +8.51. For BHs with spin a = 0.7, the maximum Q is 0.71. +The enhancement factors are 90.29, 269.91, and 707.16, +for l = m = 1, 2, 3, respectively. Finally, in the ranges of +small rgµ before reaching the round peaks of the Q = 0 +curves, the charge Q turns out to impede the growth of +the scalar clouds. We define a factor s(Q) as +s(Q) = +Im ω(Q) +Im ω(Q = 0). +(47) +In Fig. 2, we show s(Q) as a function of rgµ, for two +different BH spins and several values of Q. Interestingly, +the suppression factor varies slowly with rgµ. It decreases +with increasing Q, reaching the minimum value ∼ 0.8 for + +8 +a = 0.9 and ∼ 0.5 for a = 0.7. +In Ref. [56], the authors claim that when a ≳ 0.997rg, +the maximum value of Im ω decreases as Q grows. We do +not observe the same behavior. For any spin parameter +a, the peak value of Im ω from the NLO approximation +increases monotonically with Q. +B. +Charged Scalar Fields +In this part, we study the superradiance of KNBHs +under charged scalar perturbation. The NLO eigenfre- +qency is given by ω = ω(1) +0 ++ ω(1) +1 δλ(1), with the NLO +δλ(1) in Eq. (44), and the ω(1) +0 +and ω(1) +1 +in Eqs. (46). +Note that the ω in p should take the form of ω(0) +0 +in +Eq. (39), as explained at the end of Sec. III B. We also +compare the NLO results to the LO ones. The latter is +given by ω = ω(0) +0 ++ ω(0) +1 δλ(0), with the expressions de- +fined in Eqs. (36), (39) and (40). The ω in p is replaced +by µ for consistency. +-60 +-40 +-20 +0 +20 +10-9 +10-8 +10-7 +q +Im (ω) +μ=0.1 +μ=0.2 +μ=0.3 +μ=0.41 +FIG. 3. Comparison of the numerical result and the analytic +approximations for n = 0, l = m = 1, a = 0.98, and Q = 0.01, +with rg chosen to be 1 for compacity. The imaginary part of +ω is plotted as a function of the scalar field charge q. The +dashed (solid) curves are the LO (NLO) approximations and +the scattered dots are numerical results taken from Fig. 6 +in Ref. [53]. The curves with different colors correspond to +different values of µ, labeled above the corresponding curves +with the same color. +Fig. 3 shows the comparison of the LO and NLO ap- +proximations to the numerical results taken from Fig. 6 +in Ref. [53]. The NLO approximation agrees much better +with the numerical results. In particular, the average per- +centage errors of the NLO results for the points in Fig. 3 +are 6.7%, 9.9%, 20.7% and 48.3% for rgµ = 0.1, 0.2, 0.3 +and 0.41, respectively. These numbers can be used as +estimates of the NLO results for different values of α. +Moreover, the convergence of NLO results is better for a +smaller value of rgµ, qualifying the power-counting strat- +egy. To the contrary, the LO results do not seem to con- +verge to the numerical result at small rgµ, which is also +observed for Kerr BHs [21]. The reason for the bad con- +vergence of the LO result is explained at the beginning of +Sec. III B. A caveat is that the curves for the LO approx- +imations in Fig. 3 are not the same as those in Ref. [53]. +The latter misses a factor of 1/2. +Table. I shows the comparison of the NLO results and +the numerical solutions for five more parameter sets in +the literature. They are the most unstable modes with +different parameters. The percentage uncertainty of the +NLO approximation varies from 14% to 29% compared +to the numerical results. +TABLE I. Comparison of the NLO approximations of Im(ω) +with the numerical results from Ref. [56] (cases A to D) and +from Ref. [53] (case E). All cases are with n = 0 and l = +m = 1. The numbers below assume rg = 1 for compacity. +The percentage error is calculated by taking the difference +between the approximation and the numerical result, then +dividing it by the numerical result. +Case A: a = 0.9, Q = 0.2 , q = −0.264, µ = 0.282; +Case B: a = 0.99, Q = 0.1105 , q = −0.6335, µ = 0.397; +Case C: a = 0.997, Q = 0.004 , q = −18.91, µ = 0.39822; +Case D: a = 0.997, Q = 0.0001, q = −756.68, µ = 0.39816; +Case E: a = 0.98, Q = 0.01, q = −8, µ = 0.35. +Case +Type +Im(ω) +% error +LO +5.623×10−9 +74.9% +A +NLO +2.882×10−8 +28.5% +Numerical +2.243×10−8 +- +LO +1.224×10−8 +92.9% +B +NLO +1.981×10−7 +14.1% +Numerical +1.736×10−7 +- +LO +1.264×10−8 +92.9% +C +NLO +2.041×10−7 +14.1% +Numerical +1.788×10−7 +- +LO +1.263×10−8 +92.9% +D +NLO +2.041×10−7 +14.1% +Numerical +1.788×10−7 +- +LO +1.27×10−8 +88.8% +E +NLO +1.39×10−7 +22.7% +Numerical +1.13×10−7 +- +Next, we analyze the effect of q. In the formulas, the +q and Q appears as qQ and Q2. +So it is sufficient to +consider the case with Q > 0, and with q being either +positive or negative. There are two constraints for the +existence of the superradiant bound states. The superra- +diance requires ω < ωc in Eq. (16). And the existence of +the bound states gives the second constraint λ > 0 from +Eq. (22), which is approximately rgµ − qQ > 0. +If the scalar and the KNBH at the center have opposite +charges, i.e. +qQ < 0, the scalar cloud is more tightly +bounded. +In this case, the second constraint above is +automatically satisfied. Fig. 4 shows the imaginary part +of ω as a function of rgµ in the n = 0, l = m = 1 +bound state, with BH spin a = 0.9 and charge Q = +0.01. +The scalar charge q varies from −45 to 0. +The +region of superradiance shrinks when q is more negative, +which is a consequence that ωc decreases with q for fixed + +9 +q=0 +q=-5 +q=-10 +q=-15 +q=-30 +q=-45 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +10-20 +10-17 +10-14 +10-11 +10-8 +rg μ +Im ( +� +) / μ +FIG. 4. The imaginary part of NLO eigenfrequency as a func- +tion of rgµ with different negative values of q. Other param- +eters are n = 0, l = m = 1, a = 0.9 and Q = 0.01. +TABLE II. The maximum value of Im(ω) obtained by varying +q, with a and Q fixed. +(a,Q) +q +Im(ω) +-2.5 +2.10313×10−8 +(0.9, 0.01) +-2.25 +2.10329×10−8 +-2.2 +2.10329×10−8 +-2 +2.10268×10−8 +-1.25 +2.10814×10−8 +(0.9, 0.02) +-1.1 +2.10831×10−8 +-1 +2.10815×10−8 +-0.75 +2.10682×10−8 +-3 +4.14247×10−10 +(0.7, 0.01) +-2.8 +4.14270×10−10 +-2.75 +4.14260×10−10 +-2.5 +4.14104×10−10 +-1.5 +4.14863×10−10 +(0.7, 0.02) +-1.4 +4.14888×10−10 +-1.25 +4.14726×10−10 +-1 +4.13927×10−10 +Q. The peak value of Im(ω) seems to be smaller with +decreasing q. Nonetheless, a more careful study shows +that the maximum Im(ω) happens at some small but +nonzero |q| (see Table. II). +If the charges of the scalar and the KNBH have the +same sign, i.e. qQ > 0, the scalar cloud is less bounded. +The second constraint above gives rgµ > qQ for the ex- +istence of bound states. Fig. 5 shows the imaginary part +of ω as a function of rgµ in the n = 0, l = m = 1 +bound state, with BH spin a = 0.9 and charge Q = 0.01. +With larger value of positive q, the superradiance region +shrinks and the peak is lower as well. +q=0 +q=5 +q=10 +q=15 +q=30 +0.0 +0.1 +0.2 +0.3 +0� � +0.5 +10-20 +10-17 +10-14 +10-11 +10-8 +rg μ +Im ( +� +) / μ +FIG. 5. The imaginary part of NLO eigenfrequency as a func- +tion of rgµ with different positive values of q. Other parame- +ters are n = 0, l = m = 1, a = 0.9 and Q = 0.01. +V. +CONCLUSION +In this work, we have studied the scalar superradi- +ant instability of the KNBH and obtained the LO and +NLO expressions of the superradiant rate in the regime +of α ≪ 1. +The calculation is based on the matching +method which is proposed by Detweiler for Kerr BHs in +Ref. [6] and developed in our previous work [21]. In this +manuscript, we further refine the power-counting strat- +egy and apply it to the KNBH. +The LO scalar superradiant rate for KNBH has been +calculated previously in Ref. [53]. +With our refined +power-counting strategy, a similar result is obtained but +with an extra overall factor of 1/2. We conjecture the +factor is from the mistreatment of the Γ functions with +negative integer arguments, similar to the case of Kerr +BHs. More analysis could be found in our previous work +[21]. +We compare the LO and NLO results with the existing +numerical calculations in the literature. The LO results +are smaller than the numerical solutions by an order of +magnitude. To the contrary, the percentage error of the +NLO result ranges from a few percent to about 50%, de- +pending on the value of α (see Fig. 3 and Table I). In +particular, the error of the NLO result decreases for a +smaller value of α, qualifying our power-counting strat- +egy. +The obtained NLO expression has a compact form +and can be straightforwardly applied to phenomenolog- +ical studies of the KNBH superradiance as well as the +ultralight scalars, either neutral or charged. Besides the +superradiance condition Re(ω) < mΩH as the Kerr BHs, +there is another condition rgµ > qQ for the existence +of bound states. For neutral scalars, larger BH charge +Q leads to a larger superradiant range of rgµ as well as +the maximum superradiant rate (see Fig. 1). Thus mas- +sive neutral scalars too heavy to be produced with Kerr + +10 +BH superradiance may exist in the superradiant region +of KNBHs. The situation is different for charged scalars. +For fixed BH spin a and charge Q, increasing the scalar +charge q always leads to narrower superradiant range of +rgµ (see Figs. 4 and 5). Interestingly, the maximum su- +perradiant rate happens at a small negative scalar charge +q (see Table II). We have no explanation for this obser- +vation. +ACKNOWLEDGMENTS +This work is supported in part by the National Nat- +ural Science Foundation of China (NSFC) under Grant +No. +12075136 and the Natural Science Foundation of +Shandong Province under Grant No. ZR2020MA094. +[1] A. Arvanitaki and S. Dubovsky, Phys. Rev. D 83, 044026 +(2011) [arXiv:1004.3558 [hep-th]]. +[2] H. Yoshino and H. Kodama, Prog. Theor. Phys. 128, +153-190 (2012) [arXiv:1203.5070 [gr-qc]]. +[3] M. Baryakhtar, M. Galanis, R. Lasenby and O. Simon, +Phys. Rev. D 103, no.9, 095019 (2021) [arXiv:2011.11646 +[hep-ph]]. +[4] R. Penrose, Riv. Nuovo Cim. 1, 252-276 (1969) +[5] C. W. Misner, Phys. Rev. Lett. 28, 994-997 (1972) +[6] S. L. Detweiler, Phys. Rev. D 22, 2323-2326 (1980) +[7] T. J. M. Zouros and D. M. Eardley, Annals Phys. 118, +139-155 (1979) +[8] V. Cardoso and S. 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J. 185, +649-674 (1973) + diff --git a/AdE4T4oBgHgl3EQf4w7g/content/tmp_files/load_file.txt b/AdE4T4oBgHgl3EQf4w7g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..19c1a916cf2cc00e03ddca4754e4aa3849b2f0de --- /dev/null +++ b/AdE4T4oBgHgl3EQf4w7g/content/tmp_files/load_file.txt @@ -0,0 +1,1133 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf,len=1132 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='05317v1 [gr-qc] 12 Jan 2023 Next-to-leading-order Solution to Kerr-Newman Black Hole Superradiance Shou-Shan Bao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' ∗ Qi-Xuan Xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' † and Hong Zhang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' ‡ 1Institute of Frontier and Interdisciplinary Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Key Laboratory of Particle Physics and Particle Irradiation (MOE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Shandong University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' QingDao 266237,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' China 2Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Blackett Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Imperial College,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' SW7 2AZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' United Kingdom (Dated: January 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 2023) The superradiant instabilities of Kerr-Newman black holes with charged or uncharged massive spin-0 fields are calculated analytically to the next-to-leading order in the limit of α ∼ rgµ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' A missing factor of 1/2 in the previous leading-order result is identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The next-to-leading order result has a compact form and is in good agreement with existing numerical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The percentage error increases with α, from a few percent for α ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1 to about 50% for α ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Massive neutral scalars too heavy to be produced with Kerr black hole superradiance may exist in the superradiant region of Kerr-Newman black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' INTRODUCTION Ultralight boson condensate could form around a rotat- ing black hole (BH) if the boson’s Compton wavelength is comparable to the size of the BH horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' With proper choice of parameters, such scalar condensate can con- tinuously extract energy and angular momentum from the BH until the BH spin is below some critical value and/or nonlinear effects become important[1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' This phenomenon is known as BH superradiance [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' There exist numerous works on various bosons, including spin- 0 [1, 6–26], spin-1 [25–38] and spin-2 [39, 40] fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In this work, we focus on the ultralight scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The super- radiance of other types of bosons could be found in the comprehensive review [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The scalar superradiance, especially with a Kerr BH, is important in phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Such BH-condensate sys- tems have been widely studied for constraining the scalar properties and for the possible observation of the GW emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It has been shown that the BH evolves along the Regge trajectories on the mass-spin plot if the su- perradiant effect is strong [1, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Consequently, there are “holes” on the Regge plot in which BHs cannot re- side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Combing with the observed BH spin distribution, favored and unfavored scalar mass ranges can be iden- tified [42–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' On the other hand, with the continuous GW generated by the BH-condensate, works has been done to study the possibility of resolving these systems from the backgrounds [1, 13–15, 18, 19, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The pos- itive frequency drift [13, 27] and the beat-like pattern [46] have been proposed to distinguish them from other monochromatic GW sources, such as neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The unresolved BH-condensate systems have also been care- fully studied as stochastic backgrounds for GW detectors [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The phenomenological study of BH superradiance de- pends on the accurate determination of the bound state’s ∗ ssbao@sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='cn † qixuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='xu22@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='uk ‡ hong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='zhang@sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='cn eigenfrequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For Kerr BHs, the numerical continued fraction method was first proposed by Leaver for mass- less scalars [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It is later developed for massive scalars in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [8] and further refined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In the small α ∼ rgµ limit, an analytic approximation was obtained by Detweiler [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Nonetheless, these two solutions are not consistent with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The problem is recently resolved in our previous work by including the next-to- leading-order (NLO) contribution to the analytic approx- imation [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' A power-counting strategy is also proposed which facilitates the NLO calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [48], Damour et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' have shown that the su- perradiance can also be realized with a charged massive scalar field in Kerr-Newman spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Comparably, it does not attract as much attention as that for Kerr BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It may be because the Kerr-Newman BH (KNBH) is unlikely to play important roles in astrophysics [49– 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Nonetheless, as pointed out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [52], the KNBH provides an ideal testing ground for studying the inter- play between gravity and electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In the pre- vious studies of scalar superradiance with KNBHs, De- tweiler’s method has been applied to obtain the leading- order (LO) analytic approximation at the α ≪ 1 limit [53–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The numerical solution has also been achieved using the 3-term continued fraction method [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The pa- rameter space of the KNBH superradiance is also probed by analyzing the existence of the potential well [57–59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In this work, we refine the power-counting strategy in our previous work and apply it to calculate the NLO con- tribution of the KNBH superradiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' A compact NLO expression for α ≪ 1 is obtained which could be straight- forwardly applied to phenomenological study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The scalar field can be either neutral or charged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' By comparing to the existing numerical results, the percentage error of the analytic approximation increases with α, from a few percent for α ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1 to about 50% for α ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' II, we briefly review the Klein-Gordon equation to be solved and ob- tain the superradiance condition from its solution at the outer horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Detweiler’s method is applied to derive the LO and NLO analytic expressions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' IV, the obtained analytic expressions are compared 2 to the existing numerical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Some effects rel- evant to phenomenology are also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Finally, we summarize our results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' SCALARS IN KERR-NEWMAN SPACETIME The spacetime around a KNBH with mass M, angular momentum J and charge Q can be expressed in Boyer- Lindquist coordinates [60], ds2 = − � 1 − 2rgr − Q2 Σ2 � dt2 + Σ2 ∆ dr2 + Σ2dθ2 + � (r2 + a2) + (2rgr − Q2)a2 sin2 θ Σ2 � sin2 θdϕ2 − 2(2rgr − Q2)a sin2 θ Σ2 dtdϕ, (1) with a = J/M, (2a) rg = GM, (2b) Σ2 = r2 + a2 cos2 θ, (2c) ∆ = r2 − 2rgr + a2 + Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (2d) The equation ∆ = 0 gives two event horizons at r± = rg ± b with b = � r2g − a2 − Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In this work, we only consider the KNBHs with r2 g − a2 − Q2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' To study the superradiance of a scalar field close to a BH, one needs to solve the combined Einstein and Klein- Gordon field equations, which is a very difficult task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Es- pecially, the existence of the scalar perturbs the space- time around the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Nonetheless, it has been shown that this perturbation could be safely ignored due to the tiny energy-stress tensor of the scalar cloud for Kerr BH [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' We assume the same situation happens for the KNBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' We further assume the self-interaction of the scalar field can also be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Then the problem reduces to solv- ing the Klein-Gordon equation on the stationary Kerr- Newman background, (∇α − iqAα)(∇α − iqAα)φ − µ2φ = 0, (3) where µ and q are the mass and electric charge of the scalar field, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The vector Aα is the back- ground electromagnetic potential, Aα = Qr Σ2 � −1, 0, 0, a sin2 θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (4) For complex scalars, φ can be written with the separation of variables, φ(t, r, θ, ϕ) = � l,m � dωRlm(r)Slm(θ)eimϕe−iωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (5) Inserting it into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (3), one obtains the angular equa- tion, 1 sin θ d dθ � sin θdSlm dθ � + � −a2(µ2 − ω2) cos2 θ − m2 sin2 θ + Λlm � Slm = 0, (6) where Λlm is the eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Its solution Slm(θ) is called the spheroidal harmonic function, whose properties can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The corresponding radial equation is [62], ∆ d dr � ∆dRlm dr � + U(r)Rlm = 0, (7) with U(r) =[ω(a2 + r2) − am − qQr]2 + ∆[2amω − µ2r2 − a2ω2 − Λlm].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (8) These are the equations for the complex scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For real scalars, one should set q = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (3) and choose only the real part on the right side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The obtained angular and radial equations are the same as Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (6) and (7) with q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In the rest of this paper, we focus on the equations for the complex scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The case for the real scalars can then be simply obtained by choosing q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' To obtain a constraint of parameters that allow super- radiance, we change to the tortoise coordinates, dr∗ = r2 + a2 ∆ dr, (9) with which the interesting region r ∈ (r+, +∞) corre- sponds to r∗ ∈ (−∞, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' We also define, R∗(r∗) = � r2 + a2R(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (10) Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (7) can be rewritten into a Schr¨odinger-like equation, d2R∗(r∗) dr2∗ − V (r)R∗(r∗) = 0, (11) where the effective potential is, V (r) = − � ω − am + qQr a2 + r2 �2 + ∆µ2 a2 + r2 − ∆ (a2 + r2)2 � 2amω − Λlm + a2(µ2 − ω2) � + ∆[∆ + 2r(r − rg)] (a2 + r2)3 − 3∆2r2 (a2 + r2)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (12) In the region close to the outer horizon r+, the potential has the asymptotic form, lim r→r+ V (r) = −(ω − ωc)2 + O(r − r+), (13) 3 where the critical frequency is defined as, ωc = ma + qQr+ r2 + + a2 = ma + qQr+ 2rgr+ − Q2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (14) Inserting this asymptotic expression of V (r) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (11), one gets the solution at the outer horizon, lim r∗→−∞R∗(r∗) = d1e−i(ω−ωc)r∗ + d2ei(ω−ωc)r∗, (15) where the first term is the wave falling into the outer hori- zon, and the second term is the wave escaping from the outer horizon, d1 and d2 are their amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Physically, nothing can escape from the horizon, indicating d2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The superradiance requires the phase velocity and the group velocity to be in opposite directions, which leads to the superradiance condition for a KNBH, Re(ω) < ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (16) With Q fixed, it is more relaxed (strict) compared to the superradiance condition of a Kerr BH if the charges of the scalar and the BH have the same sign (different signs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' ANALYTIC SOLUTION AT α ≪ 1 In the small α limit, the asymptotic matching method first proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [6] gives a reasonable approxima- tion of the complex eigenfrequency ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In a previous work, we have further calculated the NLO contribution for Kerr BH superradiance [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The NLO result has a much bet- ter agreement with the numerical solutions compared to the LO approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In the current work, we apply the method to KNBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In this section, we first repeat the LO approximation in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' A missing factor of 1/2 is identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' We then continue to calculate the NLO contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The calculation is valid for both real and complex scalar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For a real scalar field, one simply sets q = 0 throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Leading-order Approximation We first formally introduce the power-counting param- eter α ∼ rgµ for the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The scaling of other pa- rameters are Re ω ∼ µ ∼ q and a ∼ Q ∼ r+ ∼ r− ∼ rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Unlike some previous calculations in which α is defined to be rgµ, here we leave α as a power-counting parame- ter, which could be rgµ or any other quantity with the same scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In the limit r → +∞, the derivative term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (7) divided by ∆2 can be written into a familiar form, 1 ∆ d dr � ∆dR dr � ≈ d2R dr2 + 2 r dR dr = 1 r d2 dr2 (rR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (17) The second term on the left side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (7) divided by ∆2 can be expanded in powers of rg/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Keeping terms up to r2 g/r2, the radial function at large r limit (r ≫ rg) can be simplified as, d2 dr2 (rR) + � (ω2 − µ2) + 2(2rgω2 − rgµ2 − qQω) r − l′(l′ + 1) r2 + O(r−3) � rR = 0, (18) where l′(l′ + 1) =Λlm + 4r2 g(µ2 − 3ω2) + a2(ω2 − µ2) + Q2(2ω2 − q2 − µ2) + 8rgqQω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (19) The l′ is related to the orbital angular number by, l′ = l + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (20) Here ǫ ∼ O(α2) plays the role of a regulator and cannot be simply dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For a confined profile, the real part of ω is less than the boson mass µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The physical solution is the one that decays exponentially at large r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It is more convenient to define, κ = � µ2 − ω2, (21) λ = 2rgω2 − rgµ2 − qQω κ , (22) y = κr, (23) u(y) = yR �y κ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (24) Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (18) can be rewritten as, d2u(y) dy2 + � −1 + 2λ y − l′(l′ + 1) y2 � u(y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (25) The two solutions are Whittaker functions, and only one of them has the correct behavior at r → +∞ required by the bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The solution with the correct behavior can be further written in terms of confluent hypergeo- metric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Finally, the radial function at large r 4 is, R(r) = e−κr(2κr)l′U(l′ + 1 − λ, 2l′ + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 2κr), (26) up to an arbitrary normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The bound states only exist if λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The super- radiance conditon in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (16) gives another constraint 2rgω < (ma + qQr+)/r+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Combining these two inequal- ities, one can obtain, 0 < 2rgω2 − rgµ2 − qQω < �ma r+ + qQ � ω − rgµ2 − qQω = ma r+ ω − rgµ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (27) So there is no superradiant bound state if m ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It also shows that Reissner-Nordstr¨om BHs could not hold bounded scalar clouds [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The minimum KNBH spin a allowing superradiant instability is approximately rgr+µ/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Next, we look at Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (7) in the small r limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For BH superradiance, the inner boundary is the outer horizon r = r+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It is more convenient to write the radial function in terms of z = (r − r+)/2b, z(z + 1) d dz � z(z + 1)dR dz � + U(z)R = 0, (28) where U(z) can be written as an expansion of z, U(z) = p2 + z �4rgr+ω b � r+ω − am 2r+ − Q2ω 2rg � − (Λlm + r2 +µ2 + a2ω2) + qQ b (am + r+qQ − a2ω − 3r2 +ω) � + z2(a2ω2 − Λlm + 2µ2a2 − 3µ2r2 + + 6r2 +ω2 + 2Q2µ2 + q2Q2 − 6r+qQω) + 4z3b [rgµ2 + 2r+(ω2 − µ2) − qQω] + 4z4b2(ω2 − µ2), (29) in which, p = (r2 + + a2) 2b (ω − ωc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (30) Note that both p and rgωc scale as O(α0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In the limit of small α, the Λlm has the expanded form Λlm = l(l+1)+O(α4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' At the LO of α, we get the radial equation in limit (r − r+) ≪ max(1/ω, 1/µ), z(z + 1) d dz � z(z + 1)dR dz � + � p2 − l′(l′ + 1)z(1 + z) � R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (31) In principle, the l′ should be replaced by l in this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Nonetheless, the ǫ in l′ plays the role of a regulator in the intermediate steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It will be set to zero at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The general solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (31) is a linear combination of two associated Legendre functions, and the physical solution is the one with the ingoing wave at r → r+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' After changing the variable back to r, the solution of the radial function is, R(r) = �r − r+ r − r− �−ip 2F1 � −l′, l′ + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 1 − 2ip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' −r − r+ 2b � , (32) up to an arbitrary normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Next, we apply the matching method first proposed in [6] and further developed recently in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (26) is only valid in r ≫ rg limit, while the solution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (32) requires r ≪ rgα−2 from the ig- norance of terms proportional to z3 and z4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' They have an overlapped region in the limit α ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In this region, the two solutions are expected to have the same behav- ior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The behavior of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (26) in the overlapped region is obtained by looking at its small r limit, which is, (2κ)l′Γ(−2l′ − 1) Γ(−l′ − λ) rl′ + (2κ)−l′−1Γ(2l′ + 1) Γ(l′ + 1 − λ) r−l′−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (33) On the other hand, the behavior of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (32) in the over- lapped region is obtained by looking at its large r limit, which is, (2b)−l′Γ(2l′ + 1) Γ(l′ + 1)Γ(l′ + 1 − 2ip)rl′ + (2b)l′+1Γ(−2l′ − 1) Γ(−l′ − 2ip)Γ(−l′) r−l′−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (34) The ratio of the coefficients of the rl′ and r−l′−1 should be the same for the two solutions in the overlap region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The obtained equation is the eigenequation of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It can be 5 solved perturbatively by the observation that the second term in the expression (33) must be suppressed at small r, indicating l′+1−λ is very close to zero or some negative integer, l′ + 1 − λ = −n − δλ, (35) where |δλ| ≪ 1 and n is zero or a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Following the convention in literature, we also define ¯n = n + l + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Then the above relation is re-expressed as λ = ¯n + ǫ + δλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' At LO of α, it reduces to λ = ¯n + δλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Combining with the definition of λ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (22), the rgκ scales as α2, which is important in power-counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Since |δλ| ≪ 1, one could solve for δλ perturbatively with ex- pressions (33) and (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The LO calculation of δλ for Kerr BHs was completed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [6], with the regulator ǫ set to zero from the be- ginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Recently, we have confirmed a missing factor of 1/2 in that result [21], which was first identified in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The missing factor is conjectured to be from mistreatments of Γ functions with negative integer argu- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The correct formula is provided in the appendix of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' This subtle calculation turns out to be straight- forward with the regulator ǫ kept in the intermediate steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' More details could be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For KNBHs, the first LO calculation of δλ was completed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It followed the same steps in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [6] and missed the factor 1/2 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' After the correction, the LO result of δλ is, δλ(0) = − ip (4κb)2l+1 (n + 2l + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='(l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' )2 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [(2l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (2l + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' ]2 l� j=1 (j2 + 4p2), (36) where the superscript (0) indicates that it is the LO re- sult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It scales as O(α4l+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The eigen-frequency ω can be expressed in terms of δλ with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (22) and (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Defining ω = ω0 + ω1δλ(0) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (22) and expanding it to the linear term of δλ(0), one arrives at, λ = rg(2ω2 0 − µ2) − qQω0 � µ2 − ω2 0 + rgω0ω1(3µ2 − 2ω2 0) − qQµ2ω1 (µ2 − ω2 0)3/2 δλ(0) + O � (δλ(0))2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (37) On the other hand, we have λ = ¯n + δλ(0) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Then it is straightforward to get, rg(2ω2 0 − µ2) − qQω0 � µ2 − ω2 0 = ¯n, (38a) rgω0ω1(3µ2 − 2ω2 0) − qQµ2ω1 (µ2 − ω2 0)3/2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (38b) Note that in getting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (38a), we have ignored the ǫ which could be traced back to the l′ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' This omission leads to an error in rgω0 at the order of O(α5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Solving ω0 perturbatively from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (38a), one arrives at, ω(0) 0 µ = 1 − 1 2 �rgµ − qQ ¯n �2 + O(α4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (39) Then the ω1 could be expressed in terms of ω0 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (38b) and expanded in powers of α, ω(0) 1 µ = (rgµ − qQ)2 ¯n3 + O(α4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (40) Since both ω0 and ω1 are real, ω0 and ω1δλ(0) are the leading terms of the real and imaginary parts of ω, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Especially, the imaginary part of ω scales as O(α4l+5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Next-to-leading-order Approximation In a previous work, we have carefully studied the su- perradiance of a real scalar field around a Kerr BH [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The LO eigenfrequency ω obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [6] has an er- ror as large as 160% compared to the numerical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' After correcting the missing factor 1/2, the convergence is improved, with the error ≲ 80%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Except for the large discrepancy, the LO result also has some strange behav- iors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Since the LO result is the leading term in the Taylor series of the exact ω at α = 0, it is expected to converge to the exact ω with α approaching zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Nonetheless, the relative error seems to be a nonzero constant for small α, reaching as large as 30% at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='07 for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' This discrepancy at small α puts the question on the power- counting strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Moreover, the discrepancy at small α increases quickly with the BH spin parameter a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' These problems are solved by adding the NLO correc- tion of ω [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Below we follow the same steps for the KNBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The key observation is that the first term in the square bracket in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (29), which scales as α2, is enhanced by a factor of 1/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For BHs with large spin a and/or charge Q, this term can be as important as the LO contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Other NLO contributions are also added for consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The first NLO correction appears as ǫ in the asymp- totic radial wave function at large r, which is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It can be calculated from the definition of l′ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (19), ǫ = −8r2 gµ2 + Q2µ2 + 8rgqQµ − q2Q2 2l + 1 + O(α4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (41) The second NLO contribution is from the asymptotic ra- dial wave function at small r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The potential U(z) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (29) can be approximated by p2−l′(l′+1)z(1+z)+zd, where d is defined as d =(4rgµ − 2qQ)p − 2(4rg − r+)rgµ2 + 2µqQ(4rg − r+) − q2Q2 + O(α3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 6 Up to an arbitrary normalization, the corresponding ra- dial function at the NLO is, R(r) =(r − r−) √ d−p2 (r − r+)ip 2F1 � − l′ − ip + � d − p2, l′ + 1 − ip + � d − p2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 1 − 2ip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' −r − r+ 2b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (42) In the r → +∞ limit, the asymptotic behavior of this function is, (2b)−l′−ip+√ d−p2Γ(2l′ + 1)Γ(1 − 2ip) Γ(l′ + 1 − ip − � d − p2)Γ(l′ + 1 − ip + � d − p2) rl′ + (2b)l′+1−ip+√ d−p2Γ(−2l′ − 1)Γ(1 − 2ip) Γ(−l′ − ip − � d − p2)Γ(−l′ − ip + � d − p2) r−l′−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (43) Following similar matching steps above, the NLO contri- bution of δλ could be obtained after some algebra, δλ(1) = � d 2ǫ − ǫ 2 − ip � (4κb)2l′+1 Γ(n + 2l′ + 2)Γpd n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [Γ(2l′ + 1)Γ(2l′ + 2)]2 , (44) where the superscript (1) indicates it is the NLO result, and the Γpd is defined as, Γpd = ���Γ(l′ + 1 + ip + � d − p2)Γ(l′ + 1 + ip − � d − p2) ��� 2 Γ(1 + 2ǫ)Γ(1 − 2ǫ) Γ(1 − ip − � d − p2 − ǫ)Γ(1 + ip + � d − p2 + ǫ)Γ(1 − ip + � d − p2 − ǫ)Γ(1 + ip − � d − p2 + ǫ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (45) The last NLO contribution is from ω0 and ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Defining ω = ω(1) 0 + ω(1) 1 δλ(1), the expansion of λ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (37) is still valid, only with δλ(0) replaced by δλ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Combining with λ = ¯n + ǫ + δλ(1), one could follow the same steps as in the LO calculation and obtain, ω(1) 0 µ =1 − 1 2 �rgµ − qQ ¯n �2 + (rgµ − qQ)2 8¯n4 [3(rgµ − qQ)(5rgµ − qQ) + 8¯nǫ] + O(α6), (46a) ω(1) 1 µ =(rgµ − qQ)2 ¯n3 − 3(rgµ − qQ)2 2¯n5 [(rgµ − qQ)(5rgµ − qQ) + 2¯nǫ] + O(α6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (46b) Finally, we discuss a subtle problem related to the ω dependence in the definition of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In the calculation of the δλ(1), the ω in p should be replaced by ω(0) 0 , rather than ω(1) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Here we explain the reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In deriving the small-r asymptotic form of the radial function, we approximate U(z) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (29) by p2 − l′(l′ + 1)z(z + z) + zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The coefficient of z and z2 are accurate at O(α2) and O(α0), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' At z ∼ O(α), this two terms are at the same order of O(α4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Consequently, we only need to keep the terms in p2 up to O(α4), which then leads to ω = ω(0) 0 in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In comparison to the numerical calculation, this choice of ω gives a satisfactory NLO result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Using ω(1) 0 in p is not as satisfactory, due to partially including the high-order contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' RESULTS The eigenfrequency of the Kerr BH superradiance has been studied in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [8, 10, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In comparison, the case for Kerr-Newman BH has two more parameters, the BH charge Q and the scalar charge q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In this section, we first study the superradiance of a neutral scalar field, focusing on the effect of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Then we consider the superradiance of a charged scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Comparisons with the numerical calculations in the literature are also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Neutral Scalar Fields In the following study of neutral scalar superradiance, we adopt the NLO δλ(1) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (44), where the scalar charge q is set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The ω(1) 0 and ω(1) 1 in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (46) are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Then the NLO eigen-frequency is ω = ω(1) 0 + ω(1) 1 δλ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The BH charge Q cannot be chosen arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In our derivation, we have implicitly assumed the KNBH has 7 horizons, which requires |Q| ≤ � r2g − a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In addition, neutral scalars could not distinguish the sign of the BH charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Mathematically, it means the BH charge Q can only appear in the formulas as Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' So it is sufficient to only consider positive Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The superradiance condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (16) with q = 0 has the same form as the Kerr BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The effect of the BH charge Q is hidden in r+ = rg + � r2g − a2 − Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Keeping the BH mass M and spin a fixed, larger charge Q results in a larger upper limit of Re(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Thus massive scalars too heavy to be produced with Kerr BH superradiance may exist in the superradiant region of KNBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Q=0 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='3 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 10-13 10-11 10-9 10-7 rg � Im ( � ) / � l=m=1 l=m=2 l=m=3 a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9 Q=0 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='4 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='6 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 10-13 10-11 10-9 10-7 rg � Im ( � ) / � l=m=1 l=m=2 l=m=3 a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The imaginary part of NLO eigenfrequency with q = 0 as a function of rgµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Only the curves with n = 0 are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In the top (bottom) panel, the BH spin a is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In both panels, from left to right, the three bunches correspond to l = m = 1, 2, 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In each bunch, the curves with different colors correspond to different values of the BH charge Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 1 shows the imaginary part of ω as a function of rgµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For comparison, the curves for Kerr BHs are also shown, labeled with Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' All curves have the same qualitative behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' With an increasing value of rgµ, they first increase, then drop rapidly to below zero after reaching the maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' There are three effects of the BH charge Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Firstly, the superradiant region of rgµ is enlarged with larger Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Correspondingly, the peak of Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='3 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='3 rg μ s (Q) a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='4 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='6 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='0 rg μ s (Q) a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Factor s(Q) with q = 0 as a function of rgµ for BH spin a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9 (upper panel) and a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7 (lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The vertical dashed line in each panel labels the value of rgµ where Im ω(Q = 0) reaches its maximum value for the corresponding spin parameter a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' the curve moves to the right with increasing Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The maximum rgµ with positive Im(ω) is quite accurately determined by µ = ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Secondly, the maximum Im(ω) increases with larger Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Fixing the BH spin to be a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9, the maximum values of rgIm(ω) with Q = 0 are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='088 × 10−8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='427 × 10−9 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='029 × 10−10 for l = m = 1, 2, 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The numbers for Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='43 are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='476 × 10−7, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='006 × 10−8 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='760 × 10−10, which are larger than the Q = 0 cases by factors of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='07, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='26 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For BHs with spin a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7, the maximum Q is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The enhancement factors are 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='29, 269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='91, and 707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='16, for l = m = 1, 2, 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Finally, in the ranges of small rgµ before reaching the round peaks of the Q = 0 curves, the charge Q turns out to impede the growth of the scalar clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' We define a factor s(Q) as s(Q) = Im ω(Q) Im ω(Q = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (47) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 2, we show s(Q) as a function of rgµ, for two different BH spins and several values of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Interestingly, the suppression factor varies slowly with rgµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' It decreases with increasing Q, reaching the minimum value ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='8 for 8 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9 and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [56], the authors claim that when a ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='997rg, the maximum value of Im ω decreases as Q grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' We do not observe the same behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For any spin parameter a, the peak value of Im ω from the NLO approximation increases monotonically with Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Charged Scalar Fields In this part, we study the superradiance of KNBHs under charged scalar perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The NLO eigenfre- qency is given by ω = ω(1) 0 + ω(1) 1 δλ(1), with the NLO δλ(1) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (44), and the ω(1) 0 and ω(1) 1 in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Note that the ω in p should take the form of ω(0) 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (39), as explained at the end of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' We also compare the NLO results to the LO ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The latter is given by ω = ω(0) 0 + ω(0) 1 δλ(0), with the expressions de- fined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (36), (39) and (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The ω in p is replaced by µ for consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 60 40 20 0 20 10-9 10-8 10-7 q Im (ω) μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1 μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2 μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='3 μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='41 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Comparison of the numerical result and the analytic approximations for n = 0, l = m = 1, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='98, and Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='01, with rg chosen to be 1 for compacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The imaginary part of ω is plotted as a function of the scalar field charge q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The dashed (solid) curves are the LO (NLO) approximations and the scattered dots are numerical results taken from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 6 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The curves with different colors correspond to different values of µ, labeled above the corresponding curves with the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 3 shows the comparison of the LO and NLO ap- proximations to the numerical results taken from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 6 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The NLO approximation agrees much better with the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In particular, the average per- centage errors of the NLO results for the points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 3 are 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7%, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9%, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7% and 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='3% for rgµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='41, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' These numbers can be used as estimates of the NLO results for different values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Moreover, the convergence of NLO results is better for a smaller value of rgµ, qualifying the power-counting strat- egy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' To the contrary, the LO results do not seem to con- verge to the numerical result at small rgµ, which is also observed for Kerr BHs [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The reason for the bad con- vergence of the LO result is explained at the beginning of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' A caveat is that the curves for the LO approx- imations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 3 are not the same as those in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The latter misses a factor of 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' I shows the comparison of the NLO results and the numerical solutions for five more parameter sets in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' They are the most unstable modes with different parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The percentage uncertainty of the NLO approximation varies from 14% to 29% compared to the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Comparison of the NLO approximations of Im(ω) with the numerical results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [56] (cases A to D) and from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [53] (case E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' All cases are with n = 0 and l = m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The numbers below assume rg = 1 for compacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The percentage error is calculated by taking the difference between the approximation and the numerical result, then dividing it by the numerical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Case A: a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9, Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2 , q = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='264, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='282;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Case B: a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='99, Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1105 , q = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='6335, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='397;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Case C: a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='997, Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='004 , q = −18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='91, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='39822;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Case D: a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='997, Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='0001, q = −756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='68, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='39816;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Case E: a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='98, Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='01, q = −8, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Case Type Im(ω) % error LO 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='623×10−9 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9% A NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='882×10−8 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5% Numerical 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='243×10−8 LO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='224×10−8 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9% B NLO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='981×10−7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1% Numerical 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='736×10−7 LO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='264×10−8 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9% C NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='041×10−7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1% Numerical 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='788×10−7 LO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='263×10−8 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9% D NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='041×10−7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1% Numerical 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='788×10−7 LO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='27×10−8 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='8% E NLO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='39×10−7 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7% Numerical 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='13×10−7 Next, we analyze the effect of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In the formulas, the q and Q appears as qQ and Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' So it is sufficient to consider the case with Q > 0, and with q being either positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' There are two constraints for the existence of the superradiant bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The superra- diance requires ω < ωc in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' And the existence of the bound states gives the second constraint λ > 0 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (22), which is approximately rgµ − qQ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' If the scalar and the KNBH at the center have opposite charges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' qQ < 0, the scalar cloud is more tightly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In this case, the second constraint above is automatically satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 4 shows the imaginary part of ω as a function of rgµ in the n = 0, l = m = 1 bound state, with BH spin a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9 and charge Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The scalar charge q varies from −45 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The region of superradiance shrinks when q is more negative, which is a consequence that ωc decreases with q for fixed 9 q=0 q=-5 q=-10 q=-15 q=-30 q=-45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='35 10-20 10-17 10-14 10-11 10-8 rg μ Im ( � ) / μ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The imaginary part of NLO eigenfrequency as a func- tion of rgµ with different negative values of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Other param- eters are n = 0, l = m = 1, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9 and Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The maximum value of Im(ω) obtained by varying q, with a and Q fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' (a,Q) q Im(ω) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10313×10−8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='01) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10329×10−8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10329×10−8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10268×10−8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10814×10−8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='02) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10831×10−8 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10815×10−8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='10682×10−8 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='14247×10−10 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='01) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='14270×10−10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='14260×10−10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='14104×10−10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='14863×10−10 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='02) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='14888×10−10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='14726×10−10 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='13927×10−10 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The peak value of Im(ω) seems to be smaller with decreasing q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Nonetheless, a more careful study shows that the maximum Im(ω) happens at some small but nonzero |q| (see Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' If the charges of the scalar and the KNBH have the same sign, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' qQ > 0, the scalar cloud is less bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The second constraint above gives rgµ > qQ for the ex- istence of bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 5 shows the imaginary part of ω as a function of rgµ in the n = 0, l = m = 1 bound state, with BH spin a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9 and charge Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' With larger value of positive q, the superradiance region shrinks and the peak is lower as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' q=0 q=5 q=10 q=15 q=30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='3 0� � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='5 10-20 10-17 10-14 10-11 10-8 rg μ Im ( � ) / μ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The imaginary part of NLO eigenfrequency as a func- tion of rgµ with different positive values of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Other parame- ters are n = 0, l = m = 1, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='9 and Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' CONCLUSION In this work, we have studied the scalar superradi- ant instability of the KNBH and obtained the LO and NLO expressions of the superradiant rate in the regime of α ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The calculation is based on the matching method which is proposed by Detweiler for Kerr BHs in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [6] and developed in our previous work [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In this manuscript, we further refine the power-counting strat- egy and apply it to the KNBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The LO scalar superradiant rate for KNBH has been calculated previously in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' With our refined power-counting strategy, a similar result is obtained but with an extra overall factor of 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' We conjecture the factor is from the mistreatment of the Γ functions with negative integer arguments, similar to the case of Kerr BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' More analysis could be found in our previous work [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' We compare the LO and NLO results with the existing numerical calculations in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The LO results are smaller than the numerical solutions by an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' To the contrary, the percentage error of the NLO result ranges from a few percent to about 50%, de- pending on the value of α (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 3 and Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' In particular, the error of the NLO result decreases for a smaller value of α, qualifying our power-counting strat- egy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The obtained NLO expression has a compact form and can be straightforwardly applied to phenomenolog- ical studies of the KNBH superradiance as well as the ultralight scalars, either neutral or charged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Besides the superradiance condition Re(ω) < mΩH as the Kerr BHs, there is another condition rgµ > qQ for the existence of bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For neutral scalars, larger BH charge Q leads to a larger superradiant range of rgµ as well as the maximum superradiant rate (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Thus mas- sive neutral scalars too heavy to be produced with Kerr 10 BH superradiance may exist in the superradiant region of KNBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' The situation is different for charged scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' For fixed BH spin a and charge Q, increasing the scalar charge q always leads to narrower superradiant range of rgµ (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Interestingly, the maximum su- perradiant rate happens at a small negative scalar charge q (see Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' We have no explanation for this obser- vation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work is supported in part by the National Nat- ural Science Foundation of China (NSFC) under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 12075136 and the Natural Science Foundation of Shandong Province under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' ZR2020MA094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Arvanitaki and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Dubovsky, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Rev.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Cardoso and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Pani, Physics,” Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Notes Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' 906, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='1-237 (2015) 2020, ISBN 978-3- 319-18999-4, 978-3-319-19000-6, 978-3-030-46621-3, 978- 3-030-46622-0 [arXiv:1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='06570 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' [42] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Ng, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Hannuksela, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Vitale and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Li, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content=' D 103, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} +page_content='6, 063010 (2021) [arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE4T4oBgHgl3EQf4w7g/content/2301.05317v1.pdf'} 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Hannaa,b, Jos´e V. Aguadoa, Sebastien Comas-Cardonaa, Ramzi +Askrib, Domenico Borzacchielloa +aNantes Universit´e, Ecole Centrale Nantes, CNRS, GeM, UMR 6183, 1 Rue de la No¨e, +44300 Nantes, France +bNantes Universit´e, IRT Jules Verne, 44340 Bouguenais, France +Abstract +The paper’s goal is to provide a simple unified approach to perform sensitiv- +ity analysis using Physics-informed neural networks (PINN). The main idea lies +in adding a new term in the loss function that regularizes the solution in a small +neighborhood near the nominal value of the parameter of interest. The added +term represents the derivative of the residual with respect to the parameter of +interest. The result of this modification is a solution to the problem along with +the derivative of the solution with respect to the parameter of interest (the sen- +sitivity). We call the new technique to perform sensitivity analysis within this +context SA-PINN. We show the effectiveness of the technique using 3 examples: +the first one is a simple 1D advection-diffusion problem to show the methodol- +ogy, the second is a 2D Poisson’s problem with 9 parameters of interest and the +last one is a transient two-phase flow in porous media problem. +Keywords: +Physics-informed neural networks, sensitivity analysis, two-phase +flow in porous media +1. Introduction +Sensitivity analysis is a technique to measure the effect of uncertainties in +one or more input parameters on output quantities. Sensitivity can be regarded, +quantitatively, as the derivative of output quantities with respect to input pa- +rameters that might have some uncertainties. It has great importance in several +Preprint submitted to Computer Methods in Applied Mechanics and Engineering +arXiv:2301.02428v1 [math.NA] 6 Jan 2023 + +engineering applications. Examples of these applications include aerodynamic +optimization [1], shape optimization in solid mechanics [2], injection molding [3] +and biomedical applications [4, 5]. This is just to name few applications. +Several methods exist to calculate the sensitivities; the simplest one is the +finite difference approach. It includes solving the system repeatedly, using any +numerical technique, while varying the parameter of interest. Afterwards, the +derivatives can be approximated by calculating the finite differences. +This +method becomes impractical when the number of the parameters of interest +increases because the number of times the full system needs to be solved will +grow exponentially, thus getting the sensitivities will become computationally +intractable [6]. +The adjoint method is becoming the state of the art for performing sensi- +tivity analysis specially in CFD applications [7]. The method attempts to solve +an adjoint system of equations that is usually derived from the primal system. +By solving the adjoint system, one obtains the gradient values with respect to +parameters of interest. Despite the success and increasing widespread of the +method, some issues still exist. One of these issues is related to the differentia- +bility of the solution with discontinuities appearing in shock wave problems or +two-phase flow problems that leads to instabilities [8]. +Machine learning based techniques have been growing rapidly to solve prob- +lems governed by partial differential equations (PDEs). Physics-informed neu- +ral networks (PINN) is one of the fast growing fields that attempts to solve +forward and inverse problems governed by PDEs [9]. PINN gained wide in- +terest due to the lack of need to a big data set since the physics governed by +PDEs are used to regularize the model. PINN is powerful due to the use of feed +forward neural networks, that are universal approximators [10], as the approx- +imation space and the recent advances in automatic differentiation capabilities +[11] that facilitated the derivative calculations. The method has been applied +to several fields including solid mechanics [12], fluid mechanics [13, 14], addi- +tive manufacturing [15], two-phase flow in porous media [16] and many others +[17, 18, 19, 20, 21, 22, 23, 24]. +2 + +In this article, PINN is used as the base framework to develop the new sensi- +tivity analysis technique. The main objectives of the article can be summarized +as follows: +• Introducing a new technique to perform sensitivity analysis based on the +framework of PINN and we call it SA-PINN. +• Showing the ease and effectiveness of getting the sensitivity with respect +to multiple parameters of interest at once. +• Displaying the ability of SA-PINN to get the sensitivities for problems +with discontinuities such as moving boundaries or sharp gradients. +The paper is organized in the following manner. In chapter 2, a quick intro- +duction to PINN is given followed by explanation to SA-PINN. Chapter 3 intro- +duces three problems presented in the paper: 1D advection-diffusion problem, +2D Poisson’s problem with multiple parameters of interest and a 1D unsteady +two-phase flow in porous media problem. Chapter 4 gives the results to the +three problems which includes the calculation of some sensitivities of interest. +Chapter 5 offers a summary to the method and a conclusion. +2. Sensitivity Analysis-PINN (SA-PINN) +We consider a general partial differential equation of the form +ut + µL(u) = 0, +x ∈ Ω, t ∈ [0, T] +(1) +where ut is the time derivative, L a general differential operator and µ a ma- +terial parameter. Initial and boundary conditions for the problems are defined +as +u(0, x) = u0 +(2) +u(t, xD) = uD +(3) +B(u(t, xN)) = f(xN) +(4) +3 + +where B is a differential operator, xD the boundary where Dirichlet bound- +ary condition is enforced and xN the boundary where Neumann boundary con- +dition is applied. +2.1. PINN +The first step to solve this problem with PINN is to choose the approxima- +tion space. The choice is a feed forward neural network. Through automatic +differentiation, a combined loss is formed of the residual of the PDE defined +at spatio-temporal points called collocation points and the error in the ini- +tial/boundary conditions’ enforcement. +A solution can be obtained through +updating the weights and biases of the neural network by minimizing the loss +function using algorithms such as: gradient-descent, Adam, BFGS, etc. The +loss function can be written as: +Loss = λ0 loss0 + λD lossD + λN lossN + λ1 lossr +(5) +where λi are the weights for each loss term which play an important role in +the optimization process and: +loss0 = 1 +N0 +N0 +� +i=1 +r2 +0(ti +0, xi +0) = 1 +N0 +N0 +� +i=1 +||u(ti +0, xi +0) − ui +0||2 +(6) +lossD = +1 +ND +ND +� +i=1 +r2 +D(ti +D, xi +D) = +1 +ND +ND +� +i=1 +||u(ti +D, xi +D) − ui +D||2 +(7) +lossN = +1 +NN +NN +� +i=1 +r2 +N(ti +N, xi +N) = +1 +NN +NN +� +i=1 +||B(u(ti +N, xi +N)) − f i +N||2 +(8) +lossr = 1 +Nr +Nr +� +i=1 +||r(ti +r, xi +r)||2 = 1 +Nr +Nr +� +i=1 +||ut + µL(u)||2 +(ti +r,xi +r) +(9) +(10) +lossi respectively are the losses representing the initial condition, Dirichlet, +Neumann boundary conditions and the PDE residual. +4 + +2.2. SA-PINN +The main objective of PINN is to find a solution that minimizes the residual +of the PDE within the spatiotemporal domain that is represented by the collo- +cation points while respecting the initial and boundary conditions. The result +is a solution ˆu(t, x; ˆµ) to the PDE at a specific value ˆµ. To perform sensitivity +analysis with µ as the input parameter of interest, we would like to find not +only the solution ˆu but also the derivative of the solution with respect to µ (the +sensitivity) at a given nominal value ˆµ. +One way to obtain the sensitivity in PINN is to build a parametric model. +This is done by changing the structure of the neural network to accommodate for +another input which is the parameter of interest µ. Then, one adds collocation +points in the spatiotemporal-parametric space. Then, the residual is minimized +in the whole parametric domain while respecting the initial and boundary con- +ditions. +Afterwards, the derivative of the solution with respect to µ can be +easily obtained through automatic differentiation. The main issue of building +such parametric models is that the number of collocation points grows exponen- +tially with the number of parameters of interest. The problem can, then, easily +become computationally intractable if we have several parameters of interest +which is common in most engineering applications. +To overcome this issue, we thought that instead of only minimizing the +residual of the PDE, we can also minimize the derivative of the residual with +respect to the parameter of interest. First, the structure of the neural network +should accommodate for µ as an input. Then, the loss function is formed as the +sum of the residual and the derivative of the residual with respect to µ along +with the terms to respect the initial and boundary conditions. This way we +make sure that the solution is accurate within a small neighborhood of ˆµ, thus +sensitivity can be calculated. We call this technique SA-PINN. The technique +can be summarized in points as follows: +• Choose the neural network to have inputs related to space, time and pa- +rameter of interest. +5 + +• Sample the collocation points only in space and time, however the points +will be living in a higher dimension but without adding more points. +• Create the loss function having terms related to PDE residual, the residual +derivative with respect to the parameter of interest and the terms related +to the initial and boundary conditions. +The modified loss function will then be +Lossm = Loss + λ0µ loss0µ + λDµ lossDµ + λNµ lossNµ + λ1µ lossrµ +(11) +where +loss0µ = 1 +N0 +N0 +� +i=1 +���� +∂r0(ti +0, xi +0, µi +0) +∂µ +���� +(12) +lossDµ = +1 +ND +ND +� +i=1 +���� +∂rD(ti +D, xi +D, µi +D) +∂µ +���� +(13) +lossNµ = +1 +NN +NN +� +i=1 +���� +∂rN(ti +N, xi +N, µi +N) +∂µ +���� +(14) +lossrµ = 1 +Nr +Nr +� +i=1 +���� +∂r(ti +r, xi +r, µi +r) +∂µ +���� +(15) +(16) +Figure 1 shows a diagram that summarizes the methodology of SA-PINN. +The parts in orange are the added parts from classical PINN. The u − ˆu term +represents the mismatch of the solution from the initial and boundary condi- +tions. It must be noted that we sample the collocation points only in space and +time, but the points have another coordinate µ and all have a nominal value ˆµ. +6 + +Figure 1: Diagram explaining the methodology of SA-PINN. +3. Model problems +In this section, we introduce the models of the three examples that are used +to show the effectiveness of the technique. +3.1. 1D diffusion-advection equation +The first example is a steady one-dimensional diffusion-advection equation +where we would like to study the effect of perturbations in the diffusion term ϵ +on the solution. The strong form of the problem can be written as follows: +ϵ yxx − yx + 1 = 0, +x ∈ [0, 1], +y(0) = 1, +y(1) = 3 +(17) +The chosen nominal value for ϵ is 0.1. yxx and yx are respectively the second +and first order derivatives of the solution y. The weights for the different terms +in the loss function are set to 1 for the original PINN terms and 0.1 for the +added sensitivity terms. +7 + +X +dr +ne +t +NN +LosS +u +u-a +a(u-a) +ne3.2. 2D Poisson’s problem +The next example is a 2-dimensional Poisson’s problem where we have mul- +tiple parameters to study their effect on the solution. The domain is shown in +figure 2 where there exists 9 subdomains each having different diffusivity value. +k1 +k3 +k2 +k4 +k9 +k6 +k5 +k8 +k7 +Figure 2: 2D Poisson’s problem domain +The strong form of the problem can be written as: +k ∆u = −1, +in Ω, +u = 0, +on ∂Ω +(18) +where Ω is a square with unit sides and k is the diffusivity. The 9 subdomains +have equal areas. The nominal values for the diffusivity is 1; k1 = k2 = ... = +k9 = 1. +The main PINN terms weights are set to 1 and 0.1 for the added +sensitivity terms. +3.3. 1D two-phase flow in porous media +In this section, we introduce a 1D two-phase flow in porous media problem. +The problem is faced in Liquid Transfer Molding composite manufacturing pro- +cesses, where resin is injected in a mold that has prepositioned fibrous matrix. +The problem is shown in figure 3. At t = 0, the domain is initially saturated +with one fluid (fluid 1). Another fluid (fluid 2) is being injected from the left +end at constant pressure pin, while the pressure at the other end is fixed to pout. +8 + +pin +pout +Flow front +fluid 2 +fluid 1 +Figure 3: One-dimensional domain (filling problem) +The momentum equation can be approximated with Darcy’s law that can +be written in 1D as follow: +v = − k +φµpx +(19) +where v is the volume average Darcy’s velocity, µ the viscosity, and px the +pressure gradient, and φ the porosity. Both fluids are assumed to be incom- +pressible, therefore, the mass conservation equation reduces to +vx = 0 +(20) +Pressure boundary conditions can prescribed on the inlet and oulet: +p(xinlet, t) = pin, +p(xoutlet, t) = pout +(21) +To track the interface between the two fluids, the Volume Of Fluids (VOF) +technique is used; a fraction function c is introduced which takes a value 1 for +the resin and 0 for the air. The viscosity µ is redefined as +µ = cµ2 + (1 − c)µ1 +(22) +where µ2 and µ1 are the two fluids’ viscosities. c evolves with time according +to the following advection equation +ct + vcx = 0 +(23) +9 + +where ct and cx are the time and spatial derivative of the fraction function +c, respectively. +Initial and boundary conditions are defined to solve the advection of c. +c(x, t = 0) = c0(x), +c(xinlet, t) = 1 +(24) +To sum up, the strong form of the problem can be written as: +ct + v cx = 0, +x ∈ [0, l], +t ∈ [0, T], +v = − k +φµpx, +x ∈ [0, l], +t ∈ [0, T], +vx = 0, +x ∈ [0, l], +t ∈ [0, T], +µ = cµ2 + (1 − c)µ1 +p(0, t) = pin, +p(l, t) = pout +c(0, t) = 1, +c(x, 0) = 0 +(25) +The parameters of the problem are shown in table 1. +Table 1: Parameters used for the two-phase flow problem. +Parameter +Value +k +1 +µ1 +10−5 +µ2 +1 +pin +1 +pout +0 +l +1 +φ +1 +The main PINN terms weights are set to 1 and 0.01 for the added sensitivity +terms. The adaptivity algorithm presented in [16] is used to get a better sharper +solution. +10 + +4. Results +4.1. 1D diffusion-advection equation +The solution u using PINN and SA-PINN is shown in figure 4 along with +the analytical solution for ϵ = 0.1. +Figure 4: Solution u at ϵ = 0.1 using PINN and SA-PINN along with the analytical solution +of the 1D advection-diffusion problem. +From figure 4, we can see that PINN and SA-PINN acturetly captures the +analytical solution to the problem. The derivative of the solution with respect +to ϵ at to ϵ = 0.1 is shown in figure 5. +11 + +3.00 +2.75- +2.50 +2.25 - +u +2.00 +1.75 - +1.50 +Analytical +PINN +1.25- +SA-PINN +1.00 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +XFigure 5: +∂u +∂ϵ at ϵ = 0.1 using PINN and SA-PINN along with the finite difference solution of +the 1D advection-diffusion problem. +The reference finite difference solution in figure 5 is obtained by obtaining +different PINN solutions near ϵ = 0.1 and then calculating the derivative. We +can see that classical PINN fails to predict the derivative, while, SA-PINN +accurately predicts the derivative due to the added regularization term in the +loss function. The loss function for different values of ϵ is plotted in figure 6 for +PINN and SA-PINN. +12 + +4 +-2 +3e/ne +-4 +-6 +Finite difference +-8 - +PINN +SA-PINN +-10 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +XFigure 6: Loss function for different ϵ values using PINN and SA-PINN of the 1D advection- +diffusion problem. +As seen in figure 6, SA-PINN has the effect of greatly flattening the loss +curve in a neighborhood near the nominal value of ϵ = 0.1. This leads to better +solutions that PINN in the neighborhood and accurate derivative calculation at +ϵ = 0.1. +4.2. 2D Poisson’s problem +The PINN solution of the boundary value problem is shown in figure 7. The +solution appears to be accurate and agrees with the analytical solution of the +problem. +13 + +3.0 +PINN +SA-PINN +2.5 +2.0 +Loss funcion +1.5 +1.0 +0.5 +0.06 +0.08 +0.10 +0.12 +0.14Figure 7: PINN solution of the 2D Poisson’s boundary value problem. +The sensitivity terms +∂u +∂ki can then be plotted to see the effect of the diffu- +sivity on the solution. +14 + +1.0 +0.08 +0.07 +0.8 +0.06 +0.05 +0.6 +0.04 +0.4 - +0.03 +0.02 +0.2 - +0.01 ++0'0 +0.00 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +XFigure 8: Different derivatives of the solution with respect to ki of the 2D Poisson’s problem. +The computational time is plotted versus the number of parameters with +respect to which sensitivity terms are added in figure 9. +15 + +au/aki +au/ak2 +au/ak3 +1.0 - + 0.002 +1.0 +0.0025 +1.0 +0.002 + 0.000 +0.0000 +0.000 +0.8 +0.8 +0.0025 +0.8 +0.002 +0.002 +0.0050 +0.6 +0.004 +0.6 - +0.6 +0.004 +0.0075 +y +0.006 +y +y +0.006 +0.0100 +0.4 +0.008 +0.4 +0.4 +0.0125 +0.008 +0.010 +0.2 +0.2 +0.0150 +0.010 +0.2 +0.012 +0.0175 +0.012 +0.0 +0.014 +0.0 +0.0200 +0'0 +0.2 +0.4 +0.6 +8'0 +1.0 +0'0 +0.2 +0.4 +0.6 +8'0 +1.0 +0.0 + +0.2 +0.4 +0.8 +0.014 +0'0 +0.6 +1.0 +x +x +x +au/ak4 +au/aks +au/ak6 +1.0 +0.002 +1.0 +0.000 +1.0 +0.0000 ++0.000 +0.003 +0.0025 +80 +0.002 +0.8 +0.8 +0.006 +0.0050 +0.004 +0.6 +0.009 +0.6 +0.6 +0.0075 +0.006 +y +y +0.012 +y +0.0100 +0.008 +0.4 +0.4 +0.015 +0.4 +0.0125 +0.010 +0.018 +0.2 +0.0150 +0.2 - +0.012 +0.2 +0.014 +0.021 +0.0175 +0.0 + +0.016 +0.0 +0.2 +0.4 +0.6 +8'0 +1.0 +0.024 +0.0+ +0.4 +0.6 +80 +0.0200 +0.0 +0.2 +0.4 +9'0 +8:0 +1.0 +0.0 +0.0 +0.2 +1.0 +x +x +x +au/ak7 +au/akg +au/akg +1.0 +T 0.000 +1.0 +0.0000 +1.0 - +0.002 +0.0025 +0.002 +0.000 +0.8 +0.8 +80 +0.0050 +0.004 +0.002 +0.6 +0.6 +0.0075 +0.6 +0.004 +0.006 +y +y +0.0100 +y +0.006 +0.4 +0.008 +0.4 +0.0125 +0.4 +0.008 +0.010 +0.0150 +0.010 +0.2 +0.2 +0.2 - +0.012 +0.0175 +0.012 +0.0 + +0.014 +0.0 + +0.0200 +0.0+ +0.014 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +8'0 +1.0 +x +x +xFigure 9: Computational time vs. no of sensitivity parameters. +It can be seen from the figure that the computational time grows linearly +when increasing the number of parameters the sensitivity is calculated with +respect to. This happens because the number of collocation points is the same +when adding a new term to the loss function; the added cost is the same when +adding new sensitivity terms. +4.3. 1D transient two-phase flow in porous media +First, we plot the front location for three different values of k by taking the +0.5 level set of the fraction function c in figure 10. We compare SA-PINN with +classical PINN along with the analytical solution. +16 + +1000 +computation time (s) +800 +600 +400 +200 - +0 +2 +4 +6 +8 +no of parametersFigure 10: Flow front location vs. time for three different values of k (k = 1, 0.5 and 2) of +the transient two-phase flow in porous media problem. +We can notice that SA-PINN provides good results for values of k away from +the nominal value k = 1. Classical PINN accurately predicts the solution only +at the nominal values, however, away from that values, random solutions were +obtained which is clear from the two red lines. +In figure 11, we plot the time the flow front reaches x = 0.5 vs. k. We +compare the solution from SA-PINN with the analytical solution. +17 + +1.0 +1.0 +1.0 +0.8 +0.8 - +0.8 +0.6 +0.6 - +0.6 . +0.4 +0.4 +0.4 - +0.2 . +0.2 1/ +0.2 +Analytical +PINN +SA-PINN +0.0 + +0.0 +0.1 +0.2 +0 +0.4 +0.5 +0.1 +0.2 +0.3 +0.4 +0.5 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.0 +0.0 +TimeFigure 11: Time at which the flow front reaches x = 0.5 vs. k for the transient two-phase flow +in porous media problem. +We can see a good estimation of the filling time at different values of k using +SA-PINN. This result can be useful in applications of injection processes to +estimate the filling time as a function of a parameter of interest. +5. Conclusion +In the article, we presented a new method to perform sensitivity analysis +based on the paradigm of PINN. The method is easy to implement using any +of the machine learning libraries as TensorFlow or PyTorch. We show, through +the examples, that the technique is easy to use when sensitivity with respect to +multiple parameters of interest are studied at the same time. The computation +time grows linearly as the parameters increase which is an advantage to the +method. We also show through the last example that the method is working for +a problem where a discontinuity exists (flow front) and VOF method is used. +18 + +0.5 +Analytical +SA-PINN +0.4 - +0.3 +time +0.2 +0.1 +0.0 +0.6 +0.8 +1.0 +1.2 +1.4 +kAcknowledgements +This study was funded under the PERFORM Thesis program of IRT Jules +Verne. +References +[1] J. E. Peter, R. P. 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Rabczuk, Transfer learning +enhanced physics informed neural network for phase-field modeling of frac- +ture, Theoretical and Applied Fracture Mechanics 106 (2020) 102447. +22 + diff --git a/F9E0T4oBgHgl3EQfhQEq/content/tmp_files/load_file.txt b/F9E0T4oBgHgl3EQfhQEq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3ba5b79865ec9f9b9cead59bd0c46df589f6dc93 --- /dev/null +++ b/F9E0T4oBgHgl3EQfhQEq/content/tmp_files/load_file.txt @@ -0,0 +1,618 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf,len=617 +page_content='Sensitivity analysis using Physics-informed neural networks John M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Hannaa,b, Jos´e V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Aguadoa, Sebastien Comas-Cardonaa, Ramzi Askrib, Domenico Borzacchielloa aNantes Universit´e, Ecole Centrale Nantes, CNRS, GeM, UMR 6183, 1 Rue de la No¨e, 44300 Nantes, France bNantes Universit´e, IRT Jules Verne, 44340 Bouguenais, France Abstract The paper’s goal is to provide a simple unified approach to perform sensitiv- ity analysis using Physics-informed neural networks (PINN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The main idea lies in adding a new term in the loss function that regularizes the solution in a small neighborhood near the nominal value of the parameter of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The added term represents the derivative of the residual with respect to the parameter of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The result of this modification is a solution to the problem along with the derivative of the solution with respect to the parameter of interest (the sen- sitivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' We call the new technique to perform sensitivity analysis within this context SA-PINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' We show the effectiveness of the technique using 3 examples: the first one is a simple 1D advection-diffusion problem to show the methodol- ogy, the second is a 2D Poisson’s problem with 9 parameters of interest and the last one is a transient two-phase flow in porous media problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Keywords: Physics-informed neural networks, sensitivity analysis, two-phase flow in porous media 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Introduction Sensitivity analysis is a technique to measure the effect of uncertainties in one or more input parameters on output quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Sensitivity can be regarded, quantitatively, as the derivative of output quantities with respect to input pa- rameters that might have some uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' It has great importance in several Preprint submitted to Computer Methods in Applied Mechanics and Engineering arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='02428v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='NA] 6 Jan 2023 engineering applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Examples of these applications include aerodynamic optimization [1], shape optimization in solid mechanics [2], injection molding [3] and biomedical applications [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' This is just to name few applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Several methods exist to calculate the sensitivities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' the simplest one is the finite difference approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' It includes solving the system repeatedly, using any numerical technique, while varying the parameter of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Afterwards, the derivatives can be approximated by calculating the finite differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' This method becomes impractical when the number of the parameters of interest increases because the number of times the full system needs to be solved will grow exponentially, thus getting the sensitivities will become computationally intractable [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The adjoint method is becoming the state of the art for performing sensi- tivity analysis specially in CFD applications [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The method attempts to solve an adjoint system of equations that is usually derived from the primal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' By solving the adjoint system, one obtains the gradient values with respect to parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Despite the success and increasing widespread of the method, some issues still exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' One of these issues is related to the differentia- bility of the solution with discontinuities appearing in shock wave problems or two-phase flow problems that leads to instabilities [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Machine learning based techniques have been growing rapidly to solve prob- lems governed by partial differential equations (PDEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Physics-informed neu- ral networks (PINN) is one of the fast growing fields that attempts to solve forward and inverse problems governed by PDEs [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' PINN gained wide in- terest due to the lack of need to a big data set since the physics governed by PDEs are used to regularize the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' PINN is powerful due to the use of feed forward neural networks, that are universal approximators [10], as the approx- imation space and the recent advances in automatic differentiation capabilities [11] that facilitated the derivative calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The method has been applied to several fields including solid mechanics [12], fluid mechanics [13, 14], addi- tive manufacturing [15], two-phase flow in porous media [16] and many others [17, 18, 19, 20, 21, 22, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 2 In this article, PINN is used as the base framework to develop the new sensi- tivity analysis technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The main objectives of the article can be summarized as follows: Introducing a new technique to perform sensitivity analysis based on the framework of PINN and we call it SA-PINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Showing the ease and effectiveness of getting the sensitivity with respect to multiple parameters of interest at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Displaying the ability of SA-PINN to get the sensitivities for problems with discontinuities such as moving boundaries or sharp gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The paper is organized in the following manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' In chapter 2, a quick intro- duction to PINN is given followed by explanation to SA-PINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Chapter 3 intro- duces three problems presented in the paper: 1D advection-diffusion problem, 2D Poisson’s problem with multiple parameters of interest and a 1D unsteady two-phase flow in porous media problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Chapter 4 gives the results to the three problems which includes the calculation of some sensitivities of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Chapter 5 offers a summary to the method and a conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Sensitivity Analysis-PINN (SA-PINN) We consider a general partial differential equation of the form ut + µL(u) = 0, x ∈ Ω, t ∈ [0, T] (1) where ut is the time derivative, L a general differential operator and µ a ma- terial parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Initial and boundary conditions for the problems are defined as u(0, x) = u0 (2) u(t, xD) = uD (3) B(u(t, xN)) = f(xN) (4) 3 where B is a differential operator, xD the boundary where Dirichlet bound- ary condition is enforced and xN the boundary where Neumann boundary con- dition is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' PINN The first step to solve this problem with PINN is to choose the approxima- tion space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The choice is a feed forward neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Through automatic differentiation, a combined loss is formed of the residual of the PDE defined at spatio-temporal points called collocation points and the error in the ini- tial/boundary conditions’ enforcement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' A solution can be obtained through updating the weights and biases of the neural network by minimizing the loss function using algorithms such as: gradient-descent, Adam, BFGS, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The loss function can be written as: Loss = λ0 loss0 + λD lossD + λN lossN + λ1 lossr (5) where λi are the weights for each loss term which play an important role in the optimization process and: loss0 = 1 N0 N0 � i=1 r2 0(ti 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' xi 0) = 1 N0 N0 � i=1 ||u(ti 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' xi 0) − ui 0||2 (6) lossD = 1 ND ND � i=1 r2 D(ti D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' xi D) = 1 ND ND � i=1 ||u(ti D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' xi D) − ui D||2 (7) lossN = 1 NN NN � i=1 r2 N(ti N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' xi N) = 1 NN NN � i=1 ||B(u(ti N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' xi N)) − f i N||2 (8) lossr = 1 Nr Nr � i=1 ||r(ti r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' xi r)||2 = 1 Nr Nr � i=1 ||ut + µL(u)||2 (ti r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='xi r) (9) (10) lossi respectively are the losses representing the initial condition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Dirichlet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Neumann boundary conditions and the PDE residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' SA-PINN The main objective of PINN is to find a solution that minimizes the residual of the PDE within the spatiotemporal domain that is represented by the collo- cation points while respecting the initial and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The result is a solution ˆu(t, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' ˆµ) to the PDE at a specific value ˆµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' To perform sensitivity analysis with µ as the input parameter of interest, we would like to find not only the solution ˆu but also the derivative of the solution with respect to µ (the sensitivity) at a given nominal value ˆµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' One way to obtain the sensitivity in PINN is to build a parametric model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' This is done by changing the structure of the neural network to accommodate for another input which is the parameter of interest µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Then, one adds collocation points in the spatiotemporal-parametric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Then, the residual is minimized in the whole parametric domain while respecting the initial and boundary con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Afterwards, the derivative of the solution with respect to µ can be easily obtained through automatic differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The main issue of building such parametric models is that the number of collocation points grows exponen- tially with the number of parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The problem can, then, easily become computationally intractable if we have several parameters of interest which is common in most engineering applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' To overcome this issue, we thought that instead of only minimizing the residual of the PDE, we can also minimize the derivative of the residual with respect to the parameter of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' First, the structure of the neural network should accommodate for µ as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Then, the loss function is formed as the sum of the residual and the derivative of the residual with respect to µ along with the terms to respect the initial and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' This way we make sure that the solution is accurate within a small neighborhood of ˆµ, thus sensitivity can be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' We call this technique SA-PINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The technique can be summarized in points as follows: Choose the neural network to have inputs related to space, time and pa- rameter of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 5 Sample the collocation points only in space and time, however the points will be living in a higher dimension but without adding more points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Create the loss function having terms related to PDE residual, the residual derivative with respect to the parameter of interest and the terms related to the initial and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The modified loss function will then be Lossm = Loss + λ0µ loss0µ + λDµ lossDµ + λNµ lossNµ + λ1µ lossrµ (11) where loss0µ = 1 N0 N0 � i=1 ���� ∂r0(ti 0, xi 0, µi 0) ∂µ ���� (12) lossDµ = 1 ND ND � i=1 ���� ∂rD(ti D, xi D, µi D) ∂µ ���� (13) lossNµ = 1 NN NN � i=1 ���� ∂rN(ti N, xi N, µi N) ∂µ ���� (14) lossrµ = 1 Nr Nr � i=1 ���� ∂r(ti r, xi r, µi r) ∂µ ���� (15) (16) Figure 1 shows a diagram that summarizes the methodology of SA-PINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The parts in orange are the added parts from classical PINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The u − ˆu term represents the mismatch of the solution from the initial and boundary condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' It must be noted that we sample the collocation points only in space and time, but the points have another coordinate µ and all have a nominal value ˆµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 6 Figure 1: Diagram explaining the methodology of SA-PINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Model problems In this section, we introduce the models of the three examples that are used to show the effectiveness of the technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 1D diffusion-advection equation The first example is a steady one-dimensional diffusion-advection equation where we would like to study the effect of perturbations in the diffusion term ϵ on the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The strong form of the problem can be written as follows: ϵ yxx − yx + 1 = 0, x ∈ [0, 1], y(0) = 1, y(1) = 3 (17) The chosen nominal value for ϵ is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' yxx and yx are respectively the second and first order derivatives of the solution y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The weights for the different terms in the loss function are set to 1 for the original PINN terms and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='1 for the added sensitivity terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 7 X dr ne t NN LosS u u-a a(u-a) ne3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 2D Poisson’s problem The next example is a 2-dimensional Poisson’s problem where we have mul- tiple parameters to study their effect on the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The domain is shown in figure 2 where there exists 9 subdomains each having different diffusivity value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' k1 k3 k2 k4 k9 k6 k5 k8 k7 Figure 2: 2D Poisson’s problem domain The strong form of the problem can be written as: k ∆u = −1, in Ω, u = 0, on ∂Ω (18) where Ω is a square with unit sides and k is the diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The 9 subdomains have equal areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The nominal values for the diffusivity is 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' k1 = k2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' = k9 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The main PINN terms weights are set to 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='1 for the added sensitivity terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 1D two-phase flow in porous media In this section, we introduce a 1D two-phase flow in porous media problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The problem is faced in Liquid Transfer Molding composite manufacturing pro- cesses, where resin is injected in a mold that has prepositioned fibrous matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The problem is shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' At t = 0, the domain is initially saturated with one fluid (fluid 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Another fluid (fluid 2) is being injected from the left end at constant pressure pin, while the pressure at the other end is fixed to pout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 8 pin pout Flow front fluid 2 fluid 1 Figure 3: One-dimensional domain (filling problem) The momentum equation can be approximated with Darcy’s law that can be written in 1D as follow: v = − k φµpx (19) where v is the volume average Darcy’s velocity, µ the viscosity, and px the pressure gradient, and φ the porosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Both fluids are assumed to be incom- pressible, therefore, the mass conservation equation reduces to vx = 0 (20) Pressure boundary conditions can prescribed on the inlet and oulet: p(xinlet, t) = pin, p(xoutlet, t) = pout (21) To track the interface between the two fluids, the Volume Of Fluids (VOF) technique is used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' a fraction function c is introduced which takes a value 1 for the resin and 0 for the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The viscosity µ is redefined as µ = cµ2 + (1 − c)µ1 (22) where µ2 and µ1 are the two fluids’ viscosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' c evolves with time according to the following advection equation ct + vcx = 0 (23) 9 where ct and cx are the time and spatial derivative of the fraction function c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Initial and boundary conditions are defined to solve the advection of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' c(x, t = 0) = c0(x), c(xinlet, t) = 1 (24) To sum up, the strong form of the problem can be written as: ct + v cx = 0, x ∈ [0, l], t ∈ [0, T], v = − k φµpx, x ∈ [0, l], t ∈ [0, T], vx = 0, x ∈ [0, l], t ∈ [0, T], µ = cµ2 + (1 − c)µ1 p(0, t) = pin, p(l, t) = pout c(0, t) = 1, c(x, 0) = 0 (25) The parameters of the problem are shown in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Table 1: Parameters used for the two-phase flow problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Parameter Value k 1 µ1 10−5 µ2 1 pin 1 pout 0 l 1 φ 1 The main PINN terms weights are set to 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='01 for the added sensitivity terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The adaptivity algorithm presented in [16] is used to get a better sharper solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 1D diffusion-advection equation The solution u using PINN and SA-PINN is shown in figure 4 along with the analytical solution for ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Figure 4: Solution u at ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='1 using PINN and SA-PINN along with the analytical solution of the 1D advection-diffusion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' From figure 4, we can see that PINN and SA-PINN acturetly captures the analytical solution to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The derivative of the solution with respect to ϵ at to ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='1 is shown in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='75- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='25 - u 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='75 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='50 Analytical PINN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='25- SA-PINN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='0 XFigure 5: ∂u ∂ϵ at ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='1 using PINN and SA-PINN along with the finite difference solution of the 1D advection-diffusion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The reference finite difference solution in figure 5 is obtained by obtaining different PINN solutions near ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='1 and then calculating the derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' We can see that classical PINN fails to predict the derivative, while, SA-PINN accurately predicts the derivative due to the added regularization term in the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The loss function for different values of ϵ is plotted in figure 6 for PINN and SA-PINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 12 4 2 3e/ne 4 6 Finite difference 8 - PINN SA-PINN 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='0 XFigure 6: Loss function for different ϵ values using PINN and SA-PINN of the 1D advection- diffusion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' As seen in figure 6, SA-PINN has the effect of greatly flattening the loss curve in a neighborhood near the nominal value of ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' This leads to better solutions that PINN in the neighborhood and accurate derivative calculation at ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 2D Poisson’s problem The PINN solution of the boundary value problem is shown in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The solution appears to be accurate and agrees with the analytical solution of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='0 PINN SA-PINN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='0 Loss funcion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='14Figure 7: PINN solution of the 2D Poisson’s boundary value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The sensitivity terms ∂u ∂ki can then be plotted to see the effect of the diffu- sivity on the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='6 0.' metadata={'source': 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parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' It can be seen from the figure that the computational time grows linearly when increasing the number of parameters the sensitivity is calculated with respect to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' This happens because the number of collocation points is the same when adding a new term to the loss function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' the added cost is the same when adding new sensitivity terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 1D transient two-phase flow in porous media First, we plot the front location for three different values of k by taking the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='5 level set of the fraction function c in figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' We compare SA-PINN with classical PINN along with the analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 16 1000 computation time (s) 800 600 400 200 - 0 2 4 6 8 no of parametersFigure 10: Flow front location vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' time for three different values of k (k = 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='5 and 2) of the transient two-phase flow in porous media problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' We can notice that SA-PINN provides good results for values of k away from the nominal value k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Classical PINN accurately predicts the solution only at the nominal values, however, away from that values, random solutions were obtained which is clear from the two red lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' In figure 11, we plot the time the flow front reaches x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='5 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' We compare the solution from SA-PINN with the analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='0 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='2 1/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='2 Analytical PINN SA-PINN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='0 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='0 TimeFigure 11: Time at which the flow front reaches x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='5 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' k for the transient two-phase flow in porous media problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' We can see a good estimation of the filling time at different values of k using SA-PINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' This result can be useful in applications of injection processes to estimate the filling time as a function of a parameter of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' Conclusion In the article, we presented a new method to perform sensitivity analysis based on the paradigm of PINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The method is easy to implement using any of the machine learning libraries as TensorFlow or PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' We show, through the examples, that the technique is easy to use when sensitivity with respect to multiple parameters of interest are studied at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' The computation time grows linearly as the parameters increase which is an advantage to the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' We also show through the last example that the method is working for a problem where a discontinuity exists (flow front) and VOF method is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content=' 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='5 Analytical SA-PINN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='3 time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E0T4oBgHgl3EQfhQEq/content/2301.02428v1.pdf'} +page_content='4 kAcknowledgements This study was 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Bio. & Env. Sci. 2022 + +109 | Javier et al. + + +RE +RE +RE +RESEARCH +SEARCH +SEARCH +SEARCH PAPER +PAPER +PAPER +PAPER + + + + + + + + + OPEN ACCESS +OPEN ACCESS +OPEN ACCESS +OPEN ACCESS + +MangngalApp- An integrated package of technology for +COVID- 19 response and rural development: Acceptability and +usability using TAM + +Billy S. Javier*, Leo P. Paliuanan, James Karl A. Agpalza, Jesty S. Agoto + +College of Information and Computing Sciences, Cagayan State University, Aparri, Philippines + +Article published on October 20, 2022 +Key words: Acceptability, COVID-19, Fishers, POTs, Technology acceptance model, ISO 25010 +Abstract +The COVID19 pandemic has challenged universities and organizations to devise mechanisms to uplift the well- +being and welfare of people and communities. In response, the design and development of an integrated package +of technologies, MangngalApp- A web-based portal and mobile responsive application for rural development +served as an opportunity. It showcases different packets of technologies that were outputs of R&D in the field of +fisheries and aqua-culture, innovations that were IP-protected, and technologies that harness locally available +resources for post-harvest development and aiding in sustaining growth and development in the communities. +This paper focused on the usability and acceptability of the MangngalApp implementing a descriptive research +design using the Technology Acceptance Model or TAM and ISO 25010 software quality standards. Constrained +by government health restrictions due to COVID- 19, a Google form-based questionnaire was forwarded to +consented participants via an email with the attached consent and evaluation form. Results revealed that the +MangngalApp was found to be very acceptable and usable, and compliant to ISO 25010 software quality +characteristics to the higher extent. From the results, it is concluded that the developed MangngalApp will be a +usable and responsive technology that aids to rural development especially among target users- fishers, +gatherers, processors, traders, and farmers. Considering compatibility and usefulness, the MangngalApp is +expected to provide greater social development in the community. +*Corresponding Author: Billy S. Javier  billyjavier@csu.edu.ph + + + + + + + + + + + + + + + + + + +Journal of Biodiversity and Environmental Sciences (JBES) +ISSN: 2220-6663 (Print) 2222-3045 (Online) +Vol. 21, No. 4, p. 109-117, 2022 +http://www.innspub.net + + +J. Bio. & Env. Sci. 2022 + +110 | Javier et al. +Introduction +The COVID 19 pandemic has disrupted many +organizations, +government +and +non-government +institutions, +schools, +companies, +and +various +communities. As a result, more displaced workers and +job losses increased, more families sent home, locked +down due to COVID19 restrictions and uncertain of +how and where to obtain immediate income for the +family. The government may have provided financial +assistance to erring families and fed empty stomachs. +However, resources deplete as no concrete measure to +total stop the threat of the on-going pandemic the +Filipino people is enjoying. The Cagayan State +University is mandated to transforming the lives of +people and communities through high quality +instruction, innovative research, development, and +production. Through the years, CSU has been +working hard on innovating technologies that could +help alleviate poverty, increase productivity and +improve socioeconomic status of the communities, +and +help +in +sustaining +and +protecting +the +environment. However, no matter how promising +these +technologies +are +if +these +packages +of +technologies are not widely accessible to target +communities, to its intended stakeholders: fishers, +farmers, gatherers, and processors. In fact, Sharma A, +and Kiranmayi, D (2019) was unable to find in many +literature and studies pertaining to a package of +technologies as an IEC initiative to adopting and +utilizing research-based fisheries technologies, post- +harvest technologies, and aquaculture techniques. +Most of the 124 applications reported focused on +mobile apps for angling, aquaculture management, +aquarium management, marine fisheries, and fisheries +governance, marketing and biology. + +Research project generating innovative technologies +and products has been funded and curated by experts +in +the +various +fields +leading +to +technology +commercialization. These then has to be extended to +communities via available and relevant technologies +so that as an academic institution, it really radiates its +mantra of improving the lives of people and +communities. The MangngalApp research program +was generally geared at providing a solution for a +well-informed +utilization +of +the +packets +of +technologies (POTs) developed as results of scientific +inquiries and experiments of the University and +collaborating agencies. It has been said that +technologies should be utilized by the communities, +adopted via technology-transfer, generating income +from them. However, access to POTs may have not +deliberately +reaching +the +realms +of +coastal +communities. Lack of or limited access to POTs +among fishers, farmers, gatherers, and processors +may cause inefficiency, increased cost for production, +and lower productivity among fishers, fish processors +and gatherers, as well as farmers in the coastal +communities in northern Philippines. + +With aqua-marine as banner program in the Aparri +Campus, a multi-disciplinary research program was +proposed with the hope of generating a package of +technology showcasing the science-based packages of +technologies of university along fishing activities, +seaweed farming, post-harvest, product development +and more. The research is expected to benefit the +coastal communities through provision of mobile- +ready and friendly application accessible to users +aiding to improve productivity, increased awareness +and protection for the environment, and providing +livelihood for women and differently-able persons. +Packages of technologies developed will be best +adopted or utilized in the community once an +integrated package or technology is made available. +Hence, the potential benefits expand from the fishers +in the conduct of and management of their fisheries +activities to any other intended users. Coastal farmers +will be able to uncover scientific ways to conservation +and management of marine species or seaweeds. Fish +processors will have the potential to improve +productivity, creation of jobs, and increased revenues. + +Adapting the vision of the Food and Agriculture +Organization of the United Nations (FAO) on +enhancing the role of small-scale fisheries in +contributing to poverty alleviation and food security, +the project also focused on understanding the +technology awareness, technology adoption practices, +the information needs and seeking behaviors, media + +J. Bio. & Env. Sci. 2022 + +111 | Javier et al. +literacy and media adoption of various stakeholders +in the fishing communities of Northeastern Cagayan +Philippines. In the academe, students and teachers +may benefit from the having obtained the scientific +packages of technologies for instruction purposes, and +an opportunity for more relevant research formulation. +The results of the study hope to provide and cultivate +new knowledge for students, researchers, and teachers. +In so doing, students and teachers may devise projects, +programs, and studies that could add up to the +packages of technologies. Institutions or organization +may have devise appropriate strategies, programs and +plans from data mining and knowledge data discovery +thru the program. + +The emergence of an information, communication, +and education platform through varied technologies +is a must especially in the dissemination of scientific +results and innovations from rigid experiments and +research. Digital visibility is considered an efficient +and reasonable way to publicize the outputs of +innovative +developments +and +research +results +(Magdalinou, 2019). The Technology Acceptance +Model (TAM) is a theory in information systems that +explain how consumers come to embrace the use of a +technology. When consumers are introduced with +new technology, the model argues that a variety of +factors influence their decision on how and when to +use it. TAM has been critiqued for a variety of +reasons, but it is a useful overall framework that is +compatible with several studies examining the +elements that influence older individuals' willingness +to utilize new technology (Braun, 2013). + +This paper generally aims to describe the usability +and acceptability of the developed mobile responsive +web project known as MangngalApp - an integrated +package of technology using open-source web +development platform. + +The assessment of the usability and acceptability of +the MangngalApp using the Technology Acceptance +Model (TAM) focused on (a) Perceived Ease of Use, +(b) Perceived usefulness, (c)Attitudes towards usage, +(d) Behavioral intention to use, and Relevance to the +present job. In addition, the assessment of the +developed MangngalApp based on ISO 25010 +software quality characteristics has been reported. + +Materials and methods +The descriptive research design was implemented in +this part of the project. The assessment of the +usability and acceptability of the developed Mangngal +App using the Technology Acceptance Model or TAM +was participated by 200 non-technical respondents. +These included fishers, farmers, fish processors, +gatherers, and households involved in post-harvest. A +listing of which was taken from municipal agriculture +office +thru +communications. +Meanwhile, +the +assessment of the 20 technical respondents applying +ISO 25010 software quality standards, provided proof +of the compliance in terms of compatibility, +reliability, user-friendliness, security, portability, and +functional suitability. The profile of the technical +respondents is presented herein in table 1. The +survey-questionnaire included some profile data of +respondents, their assessment of the MangngalApp, +and an optional remark or comment part. A consent +form was part of the questionnaire, while prior +presentation or orientation on its use was provided +via Google Meet. + +The researchers took the assistance of partner- +students and community leaders handled by the team +in the locality to share the MangngalApp project and +guide intended users including those involved in +actual fishing, post-harvest development, processing, +gathering, as well as those who are trading. This is a +COVID-19 initiative of the project team in order to +gather sentiments and assessment of those greater +users. On the other hand, the technical respondents +were +communicated +formally +requesting +their +expertise, and provided the team consent to +participate in the assessment. + +The respondents in the evaluation of the technical +compliance, usability and acceptability standards +using TAM included 10 industry practitioners, 10 ICT +teachers +with +experiences +in +databases, +web +development and design, and programming. + +J. Bio. & Env. Sci. 2022 + +112 | Javier et al. +Table 1. Profile of the Technical Respondents. +Participants +Male +Female +Total +% +Classification + + + + +Industry +Practitioners +6 +4 +10 +50.0 +ICT Teachers +5 +5 +10 +50.0 +Area of Interest + + + + +Web Design +2 +3 +5 +25.0 +Web Programming +3 +2 +5 +25.0 +Databases +2 +2 +4 +20.0 +Programming +2 +2 +4 +20.0 +Networks +2 + +2 +10.0 +Years of Relevant ICT Experience + + +1 to 3 +4 +5 +9 +45.0 +4 to 6 +6 +3 +9 +45.0 +More than 6 +1 +1 +2 +10.0 + + + +The participants were notified via email on their +participation in the assessment. A brief orientation +via Google Meet was conducted to provide them +overview of the project. The project team provided the +link of web based MangngalApp project. They were +given at least 2 to 5 weeks to access the web project +and were requested to fill out the evaluation forms via +Google Forms. Treating the assessment of the +Usability and Acceptability of the MangngalApp using +the Technology Acceptance Model, the 4-point Likert +scale was used: 1 being not acceptable and usable to 4 +being very acceptable and usable. The MangngalApp +web portal was developed applying the Design +Science Research (DSR) for Information Systems. The +Design Science Research creates and evaluates IT +artifacts intended to solve identified organizational +problems, (Peffers, 2007). Accessible thru http://cics- +csuaparri.org.ph/mangngalapp, +the +XAMPP +development framework was mainly used. XAMPP is +a cross-platform development tool involving the use +of the PHP scripting language, My SQL database +engine, and Apache web service. Other tools used +included CSS3, HTML5 and JavaScript. + + +Fig. 1. The MangngalApp Ecosystem. +The MangngalApp Web Project is an ecosystem that +involves people in the research and development, +technologies for development and dissemination of +research outputs, people and communities that are +the main reason for this project towards rural +development. The research outputs of the researchers +and scientific organization that were IP-registered are +highlighted for dissemination towards adoption +strategy. Bridging the gap is maximizing the use of +web tools and technologies that are accessible to the +communities. The package of technology available in +the +current +version +contains +14 +IP-registered +technologies showcasing most of the CSU Aparri- +based research and innovations. + +Permissions were sought through the Knowledge and +Technology Management Office and the Office of the +Research and Development, and Extension. End- +users of the project may click on the view process to +see the detailed descriptions, as well as the steps +involved in making, producing, or utilizing the +technology. The project is scalable, it will still house +other registered post-harvest technologies, fisheries- +based products, and technologies supporting the +different +arrays +of +fisheries +and +aquaculture +development for rural use. + + +Fig. 2. Mobile View of the MangngalApp. + +Results and discussions +Assessment of the Usability and Acceptability of the +MangngalApp using the Technology Acceptance +Model (TAM) +Table 3 presents the results of the assessment made by +the technical respondents along the aspects of perceived +ease of use, perceived usefulness, attitudes towards +usage, behavioral intention to use, and job relevance. + +Fishersfarmers,gatherersandprocessors +Inaddition,currenttechnologyadoption +practices,accesstorelevantdata, +preferencesonfishingandfarming +NOODLESEINRICHEDWITHARAMANG +DeVEIOPCDRLENMFAPMOLNA +technologiesthemobileinternetand +VwProcOSS +medialiteracyandtheneedtosupport +activitiesofthefishers,farmers,and +processorswillbeobtained.TheprojectMangngalApp +MangngalApp +Home +PackagesofTechnology +Packs ofTechnology +AboutMangngalApp +NewsandUpdates +describestheinformation seeking +ARAMANG-ENRICHEDPOLVORON +DeVelODer:DR.LENMFARANOLINA +practices,technologyawareness +WewProc心5出 +RESEARCH +TECHNOLOGY +MangngalAppll +APP +Aeelicotlon +DATABASE +mamgalapn.cics顺 +csuanarrlorg.hJ. Bio. & Env. Sci. 2022 + +113 | Javier et al. +Table 1. Assessment of the Usability and Acceptability using TAM. + VAU AU SAU NAU Weighted DV +Aspects of the Technology Acceptance +f +f +f +f +Mean + +Perceived Ease of Use + + + + +3.14 +AU +I feel that using MangngalApp would be easy for me +I feel that my interaction with MangngalApp would be clear +8 +12 +0 +0 +3.40 + +VAU +and understandable +I feel that it would be easy to become skillful at using +6 +12 +1 +0 +3.25 +VAU +MangngalApp +3 +15 +1 +0 +3.10 +AU +I would find MangngalApp to be flexible to interact with +6 +12 +1 +0 +3.25 +VAU +Learning to operate MangngalApp would be easy for me +It would be easy for me to get MangngalApp to do what I +6 +12 +2 +0 +3.20 +AU +want to do +I feel that my ability to determine MangngalApp ease of use +2 +13 +4 +0 +2.90 +AU +is limited by my lack of experience +4 +11 +3 +2 +2.85 +AU +Perceived Usefulness + + + + +3.32 +VAU +Using MangngalApp in disseminating technologies to intended +users would enable me or users to accomplish + + + + + + +tasks more quickly +Using MangngalApp would improve my skills and is useful +11 +8 +1 +0 +3.50 + +VAU +in the fishers and user's needs. +7 +12 +1 +0 +3.30 +VAU +Using MangngalApp would increase my productivity +Using MangngalApp would enhance other users' capabilities +7 +12 +1 +0 +3.30 +VAU +adopting the technology shared. +Using MangngalApp would make it easier to know new +technological updates in fishing, postharvest and related +7 +12 +1 +0 +3.30 +VAU +activities. +I would find MangngalApp useful in helping the fishers and +6 +12 +2 +0 +3.20 +AU +related sectors towards rural development. +7 +12 +1 +0 +3.30 +VAU +Attitudes towards Usage + + + + +3.43 +VAU +I believe it is a good idea to use the MangngalApp web + + + + + + +project +8 +12 +0 +0 +3.40 +VAU +I like the idea of using the MangngalApp web project +8 +12 +0 +0 +3.40 +VAU +Using the MangngalApp is a positive idea +10 +10 +0 +0 +3.50 +VAU +Behavioural Intention to Use + + + + +3.22 +AU +I tend to use the MangngalApp web project for seeking new +innovations in fisheries post-harvest and technologies. +6 +13 +1 +0 + +3.25 +VAU +I tend to use MangngalApp to enhance my interest in related +fishing, aqua-culture, and post-harvest activities +I tend to use the MangngalApp to provide multi-approaches on +sharing and obtaining technological and innovations in +6 +12 +2 +0 +3.20 +AU +fisheries, aqua-marine and post-harvest activities. +6 +12 +2 +0 +3.20 +AU +Relevance of the MangngalApp to Current Job + + +3.35 VAU +In disseminating new packets of technologies along fisheries and +aqua-marine, the usage of MangngalApp is important +8 +11 +1 +0 +3.35 +VAU +In disseminating new packets of technologies along fisheries and +aqua-marine, the usage of MangngalApp is timely relevant +8 +11 +1 +0 +3.35 +VAU +Overall Weighted Mean + +3.29 +VAU +3.25 – 4.00 >> Very acceptable and usable (VAU) +1.75 – 2.49 >> +Somewhat acceptable and +usable (SAU) +2.50 – 3.24 >> Acceptable and usable (AU) +1.00 – 1.74 >> Not acceptable and usable (NAU) + +With an overall mean of 3.29, the assessment of the +MangngalApp along the usability and acceptability +aspects were found to be “very acceptable and usable” +(table 3). Specifically, the assessment of perceived +usefulness (3.32), their attitude towards usage (3.43), +and relevance (3.45) were rated very acceptable and +usable. The perceived usefulness could be associated +to their perceived attitude towards its usage as well as +how relevant the MangngalApp web project specially +to intended users. For the purpose of clarity and +understanding, the project team intended to have the +MangngalApp project be assessed by the fishers, +processors, farmers, traders, and gathers. However, +the team was constrained to do the actual +demonstration due to restrictions of the COVID-19 +virus and high-risk alert levels of cases in the locality. +The team also tried to meet the all intended +participants via virtual setup in a video conferencing + +J. Bio. & Env. Sci. 2022 + +114 | Javier et al. +tool +as +well +as +used +other +strategies +like +communicating with students and leaders in the area. +Feed backs from the students who were parents of the +fishers and farmers as well as processors; said most of +their parents prefer to have the project demonstrated +in face-to-face setup so they could easily grasp the +technology. The team decided to conduct the actual +dissemination and training in the actual users in the +ground upon notice of approval from relevant office +still confirming to minimum health protocols. It is +one of the key future directions the team is looking +forward. + +As presented, the group of non-technical respondents +generally assessed the usability and acceptability of +the MangngalApp as “very acceptable and usable” +with a mean of 3.39. This rating is associated to the +very acceptable and usable descriptive values for +perceived usefulness, attitude towards usage, and job +relevance. Interestingly, more male respondents +perceived higher valuation of the Mangngal App +compared to their female counterparts. Meanwhile, +the technical respondents rated the aspects of TAM as +“acceptable and usable” with a mean of 3.21. Higher +assessment has been made by female industry +practitioners with a mean of 3.38, especially along +usefulness, attitudes towards usage, behavioral +intention to use and job relevance. There were 40 +percent +of +the +respondents +who +rated +the +MangngalApp as overall very acceptable and usable. + +Table 2. Detailed presentation of the assessment of the usability and acceptability +Aspects of TAM +Technical Respondents +Non-Technical Respondents +Male +Female +Weighted +Mean +Descriptive +Value +Male +Female +Weighted +Mean +Descriptive +Value +1. Perceived ease of use +3.07 +3.14 +3.10 +AU +3.23 +3.11 +3.17 +AU +2. Perceived usefulness +3.00 +3.50 +3.27 +VAU +3.47 +3.27 +3.37 +VAU +3. Attitude towards usage +3.33 +3.50 +3.40 +VAU +3.47 +3.47 +3.47 +VAU +4. Behavioral intention to use +3.00 +3.50 +3.20 +AU +3.40 +3.07 +3.23 +AU +5. Job Relevance +3.17 +3.50 +3.30 +VAU +3.40 +3.40 +3.40 +VAU +Overall +3.12 +3.38 +3.32 +AU +3.37 +3.23 +3.30 +VAU + + +3.21 + +AU + +3.39 +VAU + +Percentage of those who rated the +MangngalApp as overall “very acceptable +and usable” + + +40% + + +40% + +Compliance to ISO 25010 software quality characteristics of the developed MangngalApp + +Table 3. Summary table of the assessment of the developed MangngalApp based on ISO 25010 software quality +characteristics. +Indicator +Technical Evaluators +Non-Technical (Fisher) +Overall +WM +DV +WM +DV +WM +DV +Accuracy +3.47 +VHE +3.87 +VHE +3.67 +VHE +Reliability +3.53 +VHE +3.90 +VHE +3.72 +VHE +Security +3.50 +VHE +4.00 +VHE +3.75 +VHE +Functional Suitability +3.60 +VHE +3.93 +VHE +3.77 +VHE +Portability +3.67 +VHE +3.87 +VHE +3.77 +VHE +Usability +3.60 +VHE +3.9 +VHE +3.75 +VHE +Maintainability +3.57 +VHE +3.87 +VHE +3.72 +VHE +Efficiency +3.60 +VHE +3.90 +VHE +3.75 +VHE +Overall Weighted Mean +3.57 +VHE +3.91 +VHE +3.74 +VHE +Legend: +WM– Weighted Mean; DV– Descriptive Value +3.25-4.00 >> Very High Extent (VHE, 1.75-2.49 >> Fair Extent (FE) +2.50-3.24 >> High Extent (HE), 1.00-1.74>> Poor Extent (PE) + +Presented in table the summary table of the +assessment +of +the +MangngalApp +web +project +following +the +ISO +25010 +software +quality +characteristics. The assessment of the technical and +non-technical respondents revealed an overall remark +of excellent with an overall mean of 3.74. Notably, +both groups made a high remark or excellent +highlighting functionality and portability aspects. + +J. Bio. & Env. Sci. 2022 + +115 | Javier et al. +The functionality can be associated to the fact that the +MangngalApp follows a WYSWYG approach making +ease of access and functional. Meanwhile, the +portability aspect could be associated to the project +being compatible to varied devices making it +convenient to users. + +The participants were asked about their problems and +challenges associated to the use of the MangngalApp. +Although the participants are technical evaluators, it +is believed that common issues will be experienced by +the intended users. This includes but not limited to: +a. Internet connectivity issues +b. Not very good using via tablets PC +c. Limited contents only focused to fisheries and +aquaculture +d. Cannot visualize from just an image + +There were comments and suggestions highlighted by +the respondents. This includes but not limited to: +a. Strengthen internet connection in the area +b. Share more techno guides that are easily +understood by intended users +c. Produce video of the steps which are visibly +understood by intended users +d. Add more contents not only along post-harvest +and processing. +e. Translation of contents to Filipino or vernaculars +if possible + + +Moreover, the overall impressions made by the +participants include: +a. MangngalApp as a good project for rural +development +b. The project is impressive +c. Great project especially if with more contents for +the intended users +d. Very good one-stop IEC mechanism + +Considering the above-mentioned, the project team is +looking way forward to scale up the project, fast-track +the translation to Filipino, as well as integrating other +technologies that would benefit the communities for +rural development. The translation is in coordination +with owners of the technology. +Conclusions +The MangngalApp project was found to be very +acceptable and usable based on the assessment of the +technical respondents. There were uncontrolled +issues or problems in the use of the MangngalApp, +the constructive comments and suggestions, as well as +the overall impressions over the project. Based on the +ISO 25010 software quality characteristics, the +respondents generally remark it as “excellent” with an +overall mean of 3.74. + +From the results, it is concluded that the developed +MangngalApp will be a usable and responsive +technology that aids to rural development especially +among target users- fishers, gatherers, processors, +traders, and farmers. Considering compatibility and +usefulness, the MangngalApp is expected to provide +greater social development in the community. + +Social Implications +The use of the MangngalApp would offer greater +opportunity for local users to livelihood development +adopting the technologies being shared from the +output of scientific undertakings at the University and +with collaborators. Meanwhile, the adoption of the +technologies +may +be +undertaken +providing +opportunities for small to medium organizations +towards livelihood development – forging partnership +with the University and other stakeholders and +private institutions. + +Project Limitations +The researchers acknowledge the technical challenge +that may have encountered by the participants as +there were very limited face-to-face presentations +made with intended users, thus may affect the results +in the study. There is a need to perform actual +demonstration +with +them +upon +approval +of +authorities and observing minimum health protocols. + +Recommendations +From the results of the study, it is recommended to +integrate the fully translated content and additional +technologies geared towards full utilization of the +MangngalApp especially creating opportunities for + +J. Bio. & Env. Sci. 2022 + +116 | Javier et al. +livelihood development. Further, the conduct of +extension activities to adopt and utilize the project +accessible in the web is highly encouraged thru +demonstration activities forging collaboration with +fishers and women organizations. In addition, there is +a need to constantly update and make the project +scalable providing other opportunities for rural +development +in +general +especially +when +new +innovations are IP-registered from the research +innovations in fisheries and aqua-marine. The +development of a video production is suggested for +actual demonstration of the processes involved +especially in post-harvest or product development. + +Acknowledgement +The research project would not be a success without +the support of the administration of the Cagayan +State University headed by Dr. Urdujah G. Alvarado, +the kind assistance and support of the RDE for the +funding thru VP for RDE Dr. Junel Guzman, as well +as the commitment and leadership of the Campus +Executive Officer Dr. Simeon R. Rabanal, Jr. The +project team is ever grateful for the usual and +unparalleled support and drive of the Coordinator for +Research and Development Dr. Lenimfa Molina for +sharing the technologies and helping us in the project +contents. Special mention to Ms. Eunice Daluddung +for her patience and assistance to the project team. +Kind appreciation is extended to Dr. Corazon T. +Talamayan for supporting us in the project. Morever, +the assessment of the project as well as how could we +better improve the MangngalApp is greatly attributed +to the self-less sharing of time, effort and expertise of +the industry practitioners and ICT teachers despite +being very busy also. To all the fishers, farmers, +processors, gatherers, and small-scale merchants – we +owe this project to you, as our inspiration of doing the +project towards rural development. Special mention +goes to the member of the review committee in the 2 +in-house reviews conducted – Engr. Gil Mark Hizon of +DOST RO2 and Dr. Emma Ballad of BFAR RO2 for +their constructive comments, guidance and inspiration: +GAD-Focal Person Prof Kristine Lara, Extension +coordinator Josie Bas-ong and KTM Coordinator Dr. +Gilbert Magulod Jr for the inputs and support. +References +Abdelaziz T, Elammari M, Bani W. 2015. +Applying the ISO Standard in Assessing the Quality of +Software Systems. 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Retrieved from globalvincitore.com: +https://www.globalvincitore.com/rise-of-on-demand + +Patel +R. +2019. +On-demand +App +Benefits, +Applications +and +Future. +Retrieved +from +yourstory.com: +https://yourstory.com/mystory/on- +demand-app/amp + +Robbert-Jan +van +der +Burg +KA. +2019. +Investigating the on-demand service characteristics: +an empirical study. Journal of Service Management. +DOI: 10.1108/JOSM-01-2019-0025 +The Strait Times Asia. 2020. Philippines Suffers +worst job losses in 15 years due to Covid-19 and +lockdown. Retrieved June 2021, from The Strait +Times Asia: + +Truong T, Rothschild BJ, Azadivar F. 2005. +Decision Support System for Fisheries Management. +DOI: 10.1145/1162708.1163075 + + + + + + + diff --git a/FNE1T4oBgHgl3EQfEwMv/content/tmp_files/load_file.txt b/FNE1T4oBgHgl3EQfEwMv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e6f3ca4b876fd73e4bfeaa8c1eb066f81920ac39 --- /dev/null +++ b/FNE1T4oBgHgl3EQfEwMv/content/tmp_files/load_file.txt @@ -0,0 +1,435 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf,len=434 +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' & Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' 2022 109 | Javier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' RE RE RE RESEARCH SEARCH SEARCH SEARCH PAPER PAPER PAPER PAPER OPEN ACCESS OPEN ACCESS OPEN ACCESS OPEN ACCESS MangngalApp- An integrated package of technology for COVID- 19 response and rural development: Acceptability and usability using TAM Billy S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Javier*, Leo P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Paliuanan, James Karl A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Agpalza, Jesty S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Agoto College of Information and Computing Sciences, Cagayan State University, Aparri, Philippines Article published on October 20, 2022 Key words: Acceptability, COVID-19, Fishers, POTs, Technology acceptance model, ISO 25010 Abstract The COVID19 pandemic has challenged universities and organizations to devise mechanisms to uplift the well- being and welfare of people and communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' In response, the design and development of an integrated package of technologies, MangngalApp- A web-based portal and mobile responsive application for rural development served as an opportunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' It showcases different packets of technologies that were outputs of R&D in the field of fisheries and aqua-culture, innovations that were IP-protected, and technologies that harness locally available resources for post-harvest development and aiding in sustaining growth and development in the communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' This paper focused on the usability and acceptability of the MangngalApp implementing a descriptive research design using the Technology Acceptance Model or TAM and ISO 25010 software quality standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Constrained by government health restrictions due to COVID- 19, a Google form-based questionnaire was forwarded to consented participants via an email with the attached consent and evaluation form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Results revealed that the MangngalApp was found to be very acceptable and usable, and compliant to ISO 25010 software quality characteristics to the higher extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' From the results, it is concluded that the developed MangngalApp will be a usable and responsive technology that aids to rural development especially among target users- fishers, gatherers, processors, traders, and farmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Considering compatibility and usefulness, the MangngalApp is expected to provide greater social development in the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' *Corresponding Author: Billy S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Javier \uf02a billyjavier@csu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='ph Journal of Biodiversity and Environmental Sciences (JBES) ISSN: 2220-6663 (Print) 2222-3045 (Online) Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' 21, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' 4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' 109-117, 2022 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='innspub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='net J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' & Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' 2022 110 | Javier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Introduction The COVID 19 pandemic has disrupted many organizations, government and non-government institutions, schools, companies, and various communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' As a result, more displaced workers and job losses increased, more families sent home, locked down due to COVID19 restrictions and uncertain of how and where to obtain immediate income for the family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The government may have provided financial assistance to erring families and fed empty stomachs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' However, resources deplete as no concrete measure to total stop the threat of the on-going pandemic the Filipino people is enjoying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The Cagayan State University is mandated to transforming the lives of people and communities through high quality instruction, innovative research, development, and production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Through the years, CSU has been working hard on innovating technologies that could help alleviate poverty, increase productivity and improve socioeconomic status of the communities, and help in sustaining and protecting the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' However, no matter how promising these technologies are if these packages of technologies are not widely accessible to target communities, to its intended stakeholders: fishers, farmers, gatherers, and processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' In fact, Sharma A, and Kiranmayi, D (2019) was unable to find in many literature and studies pertaining to a package of technologies as an IEC initiative to adopting and utilizing research-based fisheries technologies, post- harvest technologies, and aquaculture techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Most of the 124 applications reported focused on mobile apps for angling, aquaculture management, aquarium management, marine fisheries, and fisheries governance, marketing and biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Research project generating innovative technologies and products has been funded and curated by experts in the various fields leading to technology commercialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' These then has to be extended to communities via available and relevant technologies so that as an academic institution, it really radiates its mantra of improving the lives of people and communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The MangngalApp research program was generally geared at providing a solution for a well-informed utilization of the packets of technologies (POTs) developed as results of scientific inquiries and experiments of the University and collaborating agencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' It has been said that technologies should be utilized by the communities, adopted via technology-transfer, generating income from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' However, access to POTs may have not deliberately reaching the realms of coastal communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Lack of or limited access to POTs among fishers, farmers, gatherers, and processors may cause inefficiency, increased cost for production, and lower productivity among fishers, fish processors and gatherers, as well as farmers in the coastal communities in northern Philippines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' With aqua-marine as banner program in the Aparri Campus, a multi-disciplinary research program was proposed with the hope of generating a package of technology showcasing the science-based packages of technologies of university along fishing activities, seaweed farming, post-harvest, product development and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The research is expected to benefit the coastal communities through provision of mobile- ready and friendly application accessible to users aiding to improve productivity, increased awareness and protection for the environment, and providing livelihood for women and differently-able persons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Packages of technologies developed will be best adopted or utilized in the community once an integrated package or technology is made available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Hence, the potential benefits expand from the fishers in the conduct of and management of their fisheries activities to any other intended users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Coastal farmers will be able to uncover scientific ways to conservation and management of marine species or seaweeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Fish processors will have the potential to improve productivity, creation of jobs, and increased revenues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Adapting the vision of the Food and Agriculture Organization of the United Nations (FAO) on enhancing the role of small-scale fisheries in contributing to poverty alleviation and food security, the project also focused on understanding the technology awareness, technology adoption practices, the information needs and seeking behaviors, media J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' & Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' 2022 111 | Javier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' literacy and media adoption of various stakeholders in the fishing communities of Northeastern Cagayan Philippines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' In the academe, students and teachers may benefit from the having obtained the scientific packages of technologies for instruction purposes, and an opportunity for more relevant research formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The results of the study hope to provide and cultivate new knowledge for students, researchers, and teachers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' In so doing, students and teachers may devise projects, programs, and studies that could add up to the packages of technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Institutions or organization may have devise appropriate strategies, programs and plans from data mining and knowledge data discovery thru the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The emergence of an information, communication, and education platform through varied technologies is a must especially in the dissemination of scientific results and innovations from rigid experiments and research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Digital visibility is considered an efficient and reasonable way to publicize the outputs of innovative developments and research results (Magdalinou, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The Technology Acceptance Model (TAM) is a theory in information systems that explain how consumers come to embrace the use of a technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' When consumers are introduced with new technology, the model argues that a variety of factors influence their decision on how and when to use it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=" TAM has been critiqued for a variety of reasons, but it is a useful overall framework that is compatible with several studies examining the elements that influence older individuals' willingness to utilize new technology (Braun, 2013)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' This paper generally aims to describe the usability and acceptability of the developed mobile responsive web project known as MangngalApp - an integrated package of technology using open-source web development platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The assessment of the usability and acceptability of the MangngalApp using the Technology Acceptance Model (TAM) focused on (a) Perceived Ease of Use, (b) Perceived usefulness, (c)Attitudes towards usage, (d) Behavioral intention to use, and Relevance to the present job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' In addition, the assessment of the developed MangngalApp based on ISO 25010 software quality characteristics has been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Materials and methods The descriptive research design was implemented in this part of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The assessment of the usability and acceptability of the developed Mangngal App using the Technology Acceptance Model or TAM was participated by 200 non-technical respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' These included fishers, farmers, fish processors, gatherers, and households involved in post-harvest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' A listing of which was taken from municipal agriculture office thru communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Meanwhile, the assessment of the 20 technical respondents applying ISO 25010 software quality standards, provided proof of the compliance in terms of compatibility, reliability, user-friendliness, security, portability, and functional suitability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The profile of the technical respondents is presented herein in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The survey-questionnaire included some profile data of respondents, their assessment of the MangngalApp, and an optional remark or comment part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' A consent form was part of the questionnaire, while prior presentation or orientation on its use was provided via Google Meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The researchers took the assistance of partner- students and community leaders handled by the team in the locality to share the MangngalApp project and guide intended users including those involved in actual fishing, post-harvest development, processing, gathering, as well as those who are trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' This is a COVID-19 initiative of the project team in order to gather sentiments and assessment of those greater users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' On the other hand, the technical respondents were communicated formally requesting their expertise, and provided the team consent to participate in the assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The respondents in the evaluation of the technical compliance, usability and acceptability standards using TAM included 10 industry practitioners, 10 ICT teachers with experiences in databases, web development and design, and programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' & Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' 2022 112 | Javier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Profile of the Technical Respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Participants Male Female Total % Classification Industry Practitioners 6 4 10 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='0 ICT Teachers 5 5 10 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='0 Area of Interest Web Design 2 3 5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='0 Web Programming 3 2 5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='0 Databases 2 2 4 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='0 Programming 2 2 4 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='0 Networks 2 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='0 Years of Relevant ICT Experience 1 to 3 4 5 9 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='0 4 to 6 6 3 9 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='0 More than 6 1 1 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='0 The participants were notified via email on their participation in the assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' A brief orientation via Google Meet was conducted to provide them overview of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The project team provided the link of web based MangngalApp project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' They were given at least 2 to 5 weeks to access the web project and were requested to fill out the evaluation forms via Google Forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Treating the assessment of the Usability and Acceptability of the MangngalApp using the Technology Acceptance Model, the 4-point Likert scale was used: 1 being not acceptable and usable to 4 being very acceptable and usable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The MangngalApp web portal was developed applying the Design Science Research (DSR) for Information Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The Design Science Research creates and evaluates IT artifacts intended to solve identified organizational problems, (Peffers, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Accessible thru http://cics- csuaparri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='ph/mangngalapp, the XAMPP development framework was mainly used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' XAMPP is a cross-platform development tool involving the use of the PHP scripting language, My SQL database engine, and Apache web service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Other tools used included CSS3, HTML5 and JavaScript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The MangngalApp Ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The MangngalApp Web Project is an ecosystem that involves people in the research and development, technologies for development and dissemination of research outputs, people and communities that are the main reason for this project towards rural development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The research outputs of the researchers and scientific organization that were IP-registered are highlighted for dissemination towards adoption strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Bridging the gap is maximizing the use of web tools and technologies that are accessible to the communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The package of technology available in the current version contains 14 IP-registered technologies showcasing most of the CSU Aparri- based research and innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Permissions were sought through the Knowledge and Technology Management Office and the Office of the Research and Development, and Extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' End- users of the project may click on the view process to see the detailed descriptions, as well as the steps involved in making, producing, or utilizing the technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The project is scalable, it will still house other registered post-harvest technologies, fisheries- based products, and technologies supporting the different arrays of fisheries and aquaculture development for rural use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Mobile View of the MangngalApp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Results and discussions Assessment of the Usability and Acceptability of the MangngalApp using the Technology Acceptance Model (TAM) Table 3 presents the results of the assessment made by the technical respondents along the aspects of perceived ease of use, perceived usefulness, attitudes towards usage, behavioral intention to use, and job relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Fishersfarmers,gatherersandprocessors Inaddition,currenttechnologyadoption practices,accesstorelevantdata, preferencesonfishingandfarming NOODLESEINRICHEDWITHARAMANG DeVEIOPCDRLENMFAPMOLNA technologiesthemobileinternetand VwProcOSS medialiteracyandtheneedtosupport activitiesofthefishers,farmers,and processorswillbeobtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='TheprojectMangngalApp MangngalApp Home PackagesofTechnology Packs ofTechnology AboutMangngalApp NewsandUpdates describestheinformation seeking ARAMANG-ENRICHEDPOLVORON DeVelODer:DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='LENMFARANOLINA practices,technologyawareness WewProc心5出 RESEARCH TECHNOLOGY MangngalAppll APP Aeelicotlon DATABASE mamgalapn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='cics顺 csuanarrlorg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='hJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' & Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' 2022 113 | Javier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Assessment of the Usability and Acceptability using TAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' VAU AU SAU NAU Weighted DV Aspects of the Technology Acceptance f f f f Mean Perceived Ease of Use 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='14 AU I feel that using MangngalApp would be easy for me I feel that my interaction with MangngalApp would be clear 8 12 0 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='40 VAU and understandable I feel that it would be easy to become skillful at using 6 12 1 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='25 VAU MangngalApp 3 15 1 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='10 AU I would find MangngalApp to be flexible to interact with 6 12 1 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='25 VAU Learning to operate MangngalApp would be easy for me It would be easy for me to get MangngalApp to do what I 6 12 2 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='20 AU want to do I feel that my ability to determine MangngalApp ease of use 2 13 4 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='90 AU is limited by my lack of experience 4 11 3 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='85 AU Perceived Usefulness 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='32 VAU Using MangngalApp in disseminating technologies to intended users would enable me or users to accomplish tasks more quickly Using MangngalApp would improve my skills and is useful 11 8 1 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content="50 VAU in the fishers and user's needs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' 7 12 1 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content="30 VAU Using MangngalApp would increase my productivity Using MangngalApp would enhance other users' capabilities 7 12 1 0 3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='30 VAU adopting the technology shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Using MangngalApp would make it easier to know new technological updates in fishing, postharvest and related 7 12 1 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='30 VAU activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' I would find MangngalApp useful in helping the fishers and 6 12 2 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='20 AU related sectors towards rural development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' 7 12 1 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='30 VAU Attitudes towards Usage 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='43 VAU I believe it is a good idea to use the MangngalApp web project 8 12 0 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='40 VAU I like the idea of using the MangngalApp web project 8 12 0 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='40 VAU Using the MangngalApp is a positive idea 10 10 0 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='50 VAU Behavioural Intention to Use 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='22 AU I tend to use the MangngalApp web project for seeking new innovations in fisheries post-harvest and technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' 6 13 1 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='25 VAU I tend to use MangngalApp to enhance my interest in related fishing, aqua-culture, and post-harvest activities I tend to use the MangngalApp to provide multi-approaches on sharing and obtaining technological and innovations in 6 12 2 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='20 AU fisheries, aqua-marine and post-harvest activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' 6 12 2 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='20 AU Relevance of the MangngalApp to Current Job 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='35 VAU In disseminating new packets of technologies along fisheries and aqua-marine, the usage of MangngalApp is important 8 11 1 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='35 VAU In disseminating new packets of technologies along fisheries and aqua-marine, the usage of MangngalApp is timely relevant 8 11 1 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='35 VAU Overall Weighted Mean 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='29 VAU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='25 – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='00 >> Very acceptable and usable (VAU) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='75 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='49 >> Somewhat acceptable and usable (SAU) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='50 – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='24 >> Acceptable and usable (AU) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='00 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='74 >> Not acceptable and usable (NAU) With an overall mean of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='29, the assessment of the MangngalApp along the usability and acceptability aspects were found to be “very acceptable and usable” (table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Specifically, the assessment of perceived usefulness (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='32), their attitude towards usage (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='43), and relevance (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='45) were rated very acceptable and usable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The perceived usefulness could be associated to their perceived attitude towards its usage as well as how relevant the MangngalApp web project specially to intended users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' For the purpose of clarity and understanding, the project team intended to have the MangngalApp project be assessed by the fishers, processors, farmers, traders, and gathers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' However, the team was constrained to do the actual demonstration due to restrictions of the COVID-19 virus and high-risk alert levels of cases in the locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The team also tried to meet the all intended participants via virtual setup in a video conferencing J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' & Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' 2022 114 | Javier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' tool as well as used other strategies like communicating with students and leaders in the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Feed backs from the students who were parents of the fishers and farmers as well as processors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' said most of their parents prefer to have the project demonstrated in face-to-face setup so they could easily grasp the technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The team decided to conduct the actual dissemination and training in the actual users in the ground upon notice of approval from relevant office still confirming to minimum health protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' It is one of the key future directions the team is looking forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' As presented, the group of non-technical respondents generally assessed the usability and acceptability of the MangngalApp as “very acceptable and usable” with a mean of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' This rating is associated to the very acceptable and usable descriptive values for perceived usefulness, attitude towards usage, and job relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Interestingly, more male respondents perceived higher valuation of the Mangngal App compared to their female counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Meanwhile, the technical respondents rated the aspects of TAM as “acceptable and usable” with a mean of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Higher assessment has been made by female industry practitioners with a mean of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='38, especially along usefulness, attitudes towards usage, behavioral intention to use and job relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' There were 40 percent of the respondents who rated the MangngalApp as overall very acceptable and usable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Detailed presentation of the assessment of the usability and acceptability Aspects of TAM Technical Respondents Non-Technical Respondents Male Female Weighted Mean Descriptive Value Male Female Weighted Mean Descriptive Value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Perceived ease of use 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='10 AU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='17 AU 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Perceived usefulness 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='27 VAU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='37 VAU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Attitude towards usage 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='33 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='40 VAU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='47 VAU 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Behavioral intention to use 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='20 AU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='23 AU 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Job Relevance 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='30 VAU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='40 VAU Overall 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='32 AU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='37 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='30 VAU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='21 AU 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='39 VAU Percentage of those who rated the MangngalApp as overall “very acceptable and usable” 40% 40% Compliance to ISO 25010 software quality characteristics of the developed MangngalApp Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Summary table of the assessment of the developed MangngalApp based on ISO 25010 software quality characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Indicator Technical Evaluators Non-Technical (Fisher) Overall WM DV WM DV WM DV Accuracy 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='47 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='87 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='67 VHE Reliability 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='53 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='90 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='72 VHE Security 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='50 VHE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='00 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='75 VHE Functional Suitability 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='60 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='93 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='77 VHE Portability 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='67 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='87 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='77 VHE Usability 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='60 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='9 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='75 VHE Maintainability 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='57 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='87 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='72 VHE Efficiency 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='60 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='90 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='75 VHE Overall Weighted Mean 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='57 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='91 VHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='74 VHE Legend: WM– Weighted Mean;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' DV– Descriptive Value 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='25-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='00 >> Very High Extent (VHE, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='75-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='49 >> Fair Extent (FE) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='50-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='24 >> High Extent (HE), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='00-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='74>> Poor Extent (PE) Presented in table the summary table of the assessment of the MangngalApp web project following the ISO 25010 software quality characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The assessment of the technical and non-technical respondents revealed an overall remark of excellent with an overall mean of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Notably, both groups made a high remark or excellent highlighting functionality and portability aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' & Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' 2022 115 | Javier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The functionality can be associated to the fact that the MangngalApp follows a WYSWYG approach making ease of access and functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Meanwhile, the portability aspect could be associated to the project being compatible to varied devices making it convenient to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The participants were asked about their problems and challenges associated to the use of the MangngalApp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Although the participants are technical evaluators, it is believed that common issues will be experienced by the intended users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' This includes but not limited to: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Internet connectivity issues b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Not very good using via tablets PC c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Limited contents only focused to fisheries and aquaculture d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Cannot visualize from just an image There were comments and suggestions highlighted by the respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' This includes but not limited to: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Strengthen internet connection in the area b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Share more techno guides that are easily understood by intended users c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Produce video of the steps which are visibly understood by intended users d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Add more contents not only along post-harvest and processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Translation of contents to Filipino or vernaculars if possible Moreover, the overall impressions made by the participants include: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' MangngalApp as a good project for rural development b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The project is impressive c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Great project especially if with more contents for the intended users d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Very good one-stop IEC mechanism Considering the above-mentioned, the project team is looking way forward to scale up the project, fast-track the translation to Filipino, as well as integrating other technologies that would benefit the communities for rural development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The translation is in coordination with owners of the technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Conclusions The MangngalApp project was found to be very acceptable and usable based on the assessment of the technical respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' There were uncontrolled issues or problems in the use of the MangngalApp, the constructive comments and suggestions, as well as the overall impressions over the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Based on the ISO 25010 software quality characteristics, the respondents generally remark it as “excellent” with an overall mean of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content='74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' From the results, it is concluded that the developed MangngalApp will be a usable and responsive technology that aids to rural development especially among target users- fishers, gatherers, processors, traders, and farmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Considering compatibility and usefulness, the MangngalApp is expected to provide greater social development in the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Social Implications The use of the MangngalApp would offer greater opportunity for local users to livelihood development adopting the technologies being shared from the output of scientific undertakings at the University and with collaborators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Meanwhile, the adoption of the technologies may be undertaken providing opportunities for small to medium organizations towards livelihood development – forging partnership with the University and other stakeholders and private institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Project Limitations The researchers acknowledge the technical challenge that may have encountered by the participants as there were very limited face-to-face presentations made with intended users, thus may affect the results in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' There is a need to perform actual demonstration with them upon approval of authorities and observing minimum health protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Recommendations From the results of the study, it is recommended to integrate the fully translated content and additional technologies geared towards full utilization of the MangngalApp especially creating opportunities for J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' & Env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' 2022 116 | Javier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' livelihood development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Further, the conduct of extension activities to adopt and utilize the project accessible in the web is highly encouraged thru demonstration activities forging collaboration with fishers and women organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' In addition, there is a need to constantly update and make the project scalable providing other opportunities for rural development in general especially when new innovations are IP-registered from the research innovations in fisheries and aqua-marine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The development of a video production is suggested for actual demonstration of the processes involved especially in post-harvest or product development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Acknowledgement The research project would not be a success without the support of the administration of the Cagayan State University headed by Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Urdujah G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Alvarado, the kind assistance and support of the RDE for the funding thru VP for RDE Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Junel Guzman, as well as the commitment and leadership of the Campus Executive Officer Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Simeon R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Rabanal, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' The project team is ever grateful for the usual and unparalleled support and drive of the Coordinator for Research and Development Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Lenimfa Molina for sharing the technologies and helping us in the project contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Special mention to Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Eunice Daluddung for her patience and assistance to the project team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Kind appreciation is extended to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Corazon T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Talamayan for supporting us in the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Morever, the assessment of the project as well as how could we better improve the MangngalApp is greatly attributed to the self-less sharing of time, effort and expertise of the industry practitioners and ICT teachers despite being very busy also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' To all the fishers, farmers, processors, gatherers, and small-scale merchants – we owe this project to you, as our inspiration of doing the project towards rural development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Special mention goes to the member of the review committee in the 2 in-house reviews conducted – Engr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Gil Mark Hizon of DOST RO2 and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Emma Ballad of BFAR RO2 for their constructive comments, guidance and inspiration: GAD-Focal Person Prof Kristine Lara, Extension coordinator Josie Bas-ong and KTM Coordinator Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Gilbert Magulod Jr for the inputs and support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' References Abdelaziz T, Elammari M, Bani W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' Applying the ISO Standard in Assessing the Quality of Software Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfEwMv/content/2301.02893v1.pdf'} +page_content=' American Journal of Computer Science and Information Engineering 2(3), 28-32.' metadata={'source': 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32, HR-21000 Split, Croatia +cJ. Stefan Institute, Jamova 39, P. O. Box 3000, SI-1001 Ljubljana, Slovenia +E-mail: stefan.antusch@unibas.ch, dorsner@fesb.hr, +kevin.hinze@unibas.ch, shaikh.saad@unibas.ch +Abstract: +We present a minimal Grand Unified Theory model, based on SU(5) +gauge symmetry and a global U(1) Peccei-Quinn symmetry, that predicts the ex- +istence of an ultralight axion dark matter within a narrow mass range of ma ∈ +[0.1, 4.7] neV. This mass window is determined through an interplay between gauge +coupling unification constraints, partial proton decay lifetime limits, and the need to +reproduce the experimentally observed fermion mass spectrum. The entire parame- +ter space of the proposed model will be probed through a synergy between several +low-energy experiments that look for proton decay (Hyper-Kamiokande), axion dark +matter through axion-photon coupling (ABRACADABRA and DMRadio-GUT) and +nucleon electric dipole moments (CASPEr Electric). +arXiv:2301.00809v1 [hep-ph] 2 Jan 2023 + +Contents +1 +Introduction +1 +2 +The model +3 +2.1 +Scalar sector +4 +2.2 +Fermion sector +6 +3 +Peccei-Quinn symmetry and axion dark matter +9 +4 +Unification, axion mass and proton decay +14 +4.1 +Unification +15 +4.2 +Proton decay +15 +4.3 +Numerical procedure +16 +4.4 +Results +18 +5 +Conclusions +24 +A Renormalization group running of the gauge couplings +24 +1 +Introduction +The Standard Model (SM) of elementary particle physics has performed exquisitely +in explaining a multitude of experimental observations. There are, however, several +important questions that evidently require physics beyond the SM in order to be fully +addressed. For example, one of the most important discoveries in particle physics is +the observation of nonzero neutrino masses, whereas neutrinos are strictly massless +within the SM framework. Furthermore, it is well established that approximately +26% of the total energy density of the universe is in the form of the so-called dark +matter that cannot be of the SM origin. This is especially puzzling as the stable +SM matter only represents about 5% of the energy density of the universe. Also, the +strong CP problem — why the QCD θ parameter takes the value 10−10 or less — is +still an open issue within the SM. +It might be that all these issues are related. In fact, the unified gauge theory [1– +6] formulation of the elementary particle interactions is a very popular and successful +tool for tackling the aforementioned shortcomings of the SM. The simplest possible +scenario, among various possible choices of the Grand Unified Theory (GUT) groups, +is the Georgi-Glashow model [3] that embeds the entire SM gauge group within an +– 1 – + +SU(5). In that construction, one 5-dimensional and one 10-dimensional representa- +tion of SU(5) comprise all the fermions of a single SM family. The SU(5) symmetry +is broken down to the SM gauge group when a real Higgs in the adjoint representa- +tion acquires a vacuum expectation value (VEV). The SM symmetry is subsequently +broken to SU(3) × U(1)em by the VEV of the SM Higgs doublet that resides within +a fundamental representation. The Georgi-Glashow model, however, is incomplete +since (i) it fails to achieve gauge coupling unification, (ii) it predicts wrong mass +relations between down-type quarks and charged leptons, and (iii) neutrinos remain +massless as in the SM. On top of that, the Georgi-Glashow model does not address +the strong CP problem, nor does it include a dark matter candidate. +The most compelling new physics resolution of the strong CP problem is given +in terms of the Peccei-Quinn (PQ) symmetry [7, 8]. In the PQ framework, a global +U(1)PQ symmetry is spontaneously broken by a complex scalar leading to a nearly +massless pseudoscalar particle [9–14], namely the “axion”, which can, in turn, serve +as a cold dark matter candidate [15–17]. Intriguingly, as first shown in Ref. [18], the +axion can be embedded within the scalar representation that breaks the GUT sym- +metry. The model presented in Ref. [18] did not, however, address several important +GUT issues, such as neutrino mass generation and gauge coupling unification. For +a sample of models that pursue this particular approach, but with a more realistic +agenda, see Refs. [19–22]. +Our primary interest in this manuscript is to combine the PQ symmetry with +a simple, yet realistic, SU(5) GUT scenario [23, 24] and to investigate the main +predictions of such a setup. The SU(5) proposal [23, 24] in question extends the +particle content of the Georgi-Glashow model by a 35-dimensional Higgs represen- +tation and a 15-dimensional vectorlike fermion representation. Remarkably, within +that scenario, the observed mismatch between the down-type quarks and charged +leptons is intrinsically connected to the neutrino mass generation. More specifically, +the difference between the down-type quark and charged lepton mass matrices is +given by a rank-one matrix. This stipulates that the down-type quarks and charged +leptons have similar, yet, different masses, in accordance with experimental observa- +tions. The neutrino mass matrix, on the other hand, is made out of a sum of two +rank-one matrices. This, in turn, dictates that one of the neutrinos is a strictly mass- +less particle. Moreover, since the model relates these three rank-one matrices, the +neutrino masses consequentially mirror the mismatch between the down-type quark +and charged lepton masses and are thus of the normal hierarchy. +We extend the minimal realistic SU(5) proposal [23, 24] with a PQ symmetry +to address the strong CP problem as well as the origin of dark matter and show +that such a simple extension still preserves the most prominent features of the orig- +inal model. Our detailed study reveals that the proposed setup is highly predictive, +and that the entire parameter space of the theory will be fully tested in the near +future through a combination of several experiments that include the proton de- +– 2 – + +cay experiment Hyper-Kamiokande as well as the axion dark matter experiments +ABRACADABRA, DMRadio-GUT, and CASPEr Electric. +The manuscript is organized as follows. In Sec. 2 we introduce the particle con- +tent and symmetries of the model. The details of the PQ symmetry implementation +and the nature of the axion dark matter are discussed in detail in Sec. 3. A numerical +study of the model is performed in Sec. 4, where we also present the most relevant +experimental predictions. We briefly conclude in Sec. 5. +2 +The model +The model in question comprises 5F i ≡ Fα i, 10F j ≡ T αβ +j += −T βα +j , 15F ≡ Σαβ = +Σβα, 15F ≡ Σαβ, 5H ≡ Λδ, 5′ +H ≡ Λ′ δ, a complex 24H ≡ φα +β, 35H ≡ Φαβγ, and +24V ≡ Γα +β, where Hs, Fs, and V denote whether a given irreducible representation, +i.e., irrep, contains scalars, fermions, or gauge bosons, respectively, i, j (= 1, 2, 3) +represent the generation indices, and α, β, γ, δ (= 1, . . . , 5) are the SU(5) indices. +The decomposition of the SU(5) scalar and fermion irreps under the Standard Model +(SM) gauge group SU(3)×SU(2)×U(1) is presented in Table I. We will sometimes, +SU(5) +SU(3) × SU(2) × U(1) +SU(5) +SU(3) × SU(2) × U(1) +Λ(′) +1 +� +1, 2, + 1 +2 +� +Li +� +1, 2, − 1 +2 +� +5(′) +H ≡ Λ(′) α +Λ(′) +3 +� +3, 1, − 1 +3 +� +5F i ≡ Fα i +dc +i +� +3, 1, + 1 +3 +� +φ0 (1, 1, 0) +Qi +� +3, 2, + 1 +6 +� +φ1 (1, 3, 0) +10F i ≡ T αβ +i +uc +i +� +3, 1, − 2 +3 +� +24H ≡ φα +β +φ3 +� +3, 2, − 5 +6 +� +ec +i (1, 1, +1) +φ3 +� +3, 2, + 5 +6 +� +Σ1(1, 3, −1) +φ8 (8, 1, 0) +15F ≡ Σαβ +Σ3 +� +3, 2, − 1 +6 +� +Φ1 +� +1, 4, − 3 +2 +� +Σ6 +� +6, 1, + 2 +3 +� +Φ3 +� +3, 3, − 2 +3 +� +Σ1 (1, 3, +1) +35H ≡ Φαβγ +Φ6 +� +6, 2, + 1 +6 +� +15F ≡ Σαβ +Σ3 +� +3, 2, + 1 +6 +� +Φ10 +� +10, 1, +1 +� +Σ6 +� +6, 1, − 2 +3 +� +Table I: Content and nomenclature of the scalar and fermion irreps of the proposal +at both the SU(5) and SM levels. α, β, γ (= 1, . . . , 5) are the SU(5) indices while +i(= 1, 2, 3) is a generation index. +for convenience, refer to a given irrep/multiplet by using either its dimensionality +under the gauge group or the associated symbol. +Beside the non-trivial assignment under the Lorentz symmetry, the aforemen- +tioned SU(5) irreps carry the PQ U(1)PQ charges that are presented in Table II. +– 3 – + +SU(5) irrep +5F i +10F i +15F +15F +5H +5′ +H +24H +35H +24V +U(1)PQ charge +− 1 +2 +− 1 +2 +− 1 +2 +− 1 +2 +−1 ++1 ++1 +−1 +0 +Table II: U(1)PQ charge assignment of the model. H, F, and V subscripts denote +scalar, fermion, or gauge boson SU(5) irreps, respectively, while i = 1, 2, 3. +Before we write down and discuss relevant parts of the model Lagrangian, we +briefly justify the proposed particle content. +• 24H breaks the SU(5)×U(1)PQ symmetry. It furthermore provides axion dark +matter (DM), helps to generate unification of the SM gauge coupling constants, +and facilitates a process of creation of the experimentally observed mismatch +between the down-type quark and charged lepton masses. +• 5H and 5′ +H jointly break the SM gauge symmetry down to SU(3)×U(1)em. 5′ +H +also provides the up-type quark masses through its vacuum expectation value +(VEV), whereas 5H and 5′ +H together play an indispensable role in three different +mechanisms that create phenomenologically viable masses for the down-type +quarks, charged leptons, and neutrinos. +• 35H is essential for neutrino mass generation. It also helps to provide the gauge +coupling unification at scales compatible with the existing limits on partial +proton decay lifetimes. +• 15F and 15F participate in the neutrino mass generation mechanism. In addi- +tion to that, these SU(5) irreps are, together with 24H and 5H, instrumental in +addressing the observed mismatch between the down-type quark and charged +lepton masses. +2.1 +Scalar sector +There are several parts of the scalar sector of the model that need to be discussed in +detail. The SU(5) × U(1)PQ symmetry breaking is due to +L ⊃ −µ2φ∗β +α φα +β + ξ1(φ∗β +α φα +β)2 + ξ2φ∗β +α φα +γφ∗γ +δ φδ +β + ξ3φ∗β +α φδ +γφ∗α +β φγ +δ + ξ4φ∗β +α φδ +γφ∗α +δ φγ +β. +(2.1) +The VEV of φα +β that does the SU(5) symmetry breaking reads +⟨φ⟩ = +vφ +√ +15diag(−1, −1, −1, 3/2, 3/2), +(2.2) +where we assume that the VEV of the electrically neutral component of the SU(2) +triplet φ1(∈ 24H) is negligible. The masses of the SM multiplets in 24H that are +generated via Eq. (2.1) are given in Table III. We emphasize that 24H also breaks +– 4 – + +the PQ symmetry while we currently discuss solely the SU(5) symmetry breaking. +(Hence the omission of an overall phase in Eq. (2.2). The exact role of that phase +will be discussed in Sec. 3.) +multiplet +real part mass-squared +imaginary part mass-squared +φ0 (1, 1, 0) +m2 +1 +0 +φ1 (1, 3, 0) +m2 +3 +1 +4m2 +3 + m2 +8 +φ8 (8, 1, 0) +1 +4m2 +3 +m2 +8 +φ3 +� +3, 2, − 5 +6 +� +0 +m2 +5/6 +φ3 +� +3, 2, + 5 +6 +� +0 +m2 +5/6 +Table III: Mass-squared spectrum of a complex irrep 24H. +The potential given by Eq. (2.1) dictates that the imaginary part of φ0(∈ 24H) +is massless. In fact, the axion is mostly composed of that particular state. The real +components of φ3(∈ 24H) and φ3(∈ 24H), on the other hand, provide the necessary +degrees of freedom for the proton decay mediating gauge bosons in 24V to obtain a +mass MGUT, where +M 2 +GUT = 5π +6 αGUTv2 +φ. +(2.3) +Here, MGUT is also the scale of gauge coupling unification, and αGUT is the corre- +sponding SU(5) gauge coupling. +The scalar fields in the fundamental irreps of SU(5) couple via +L ⊃ −1 +2µ2 +Λ(′)Λ(′)†Λ(′) + γΛ(′) +� +Λ(′)†Λ(′)�2 + ζ1 +� +Λ†Λ +� � +Λ′†Λ′� ++ ζ2 +� +Λ†Λ′� � +Λ′†Λ +� +, +(2.4) +where we suppress SU(5) indices. The doublet-triplet spitting, i.e., breaking of the +mass degeneracy between Λ(′) +1 and Λ(′) +3 multiplets, is accomplished via the following +additional terms in the scalar potential: +L ⊃ λΛ(′)Λ(′)†Λ(′)φ†φ + Λ(′)† � +αΛ(′)φ†φ + βΛ(′)φφ†� +Λ(′) ++ +� +κ1Λ′†φ2Λ + κ2 +� +Λ′†Λ +� +φ2 + h.c. +� +. +(2.5) +The VEVs of 5H and 5′ +H that break SU(3) × SU(2) × U(1) down to SU(3) × U(1)em +read ⟨Λ(′)⟩ = (0 +0 +0 +0 +vΛ(′))T. +The lepton number conservation is violated through a single term in the La- +grangian that reads +L ⊃ λΛαΛ′βΛ′γΦαβγ + h.c.. +(2.6) +The neutrino masses will thus be directly proportional to the dimensionless parameter +λ of Eq. (2.6). +– 5 – + +The masses of the SM gauge group multiplets in 35H are determined by the +following SU(5) contractions +L ⊃ µ2 +35ΦΦ∗ + λ0 (ΦΦ∗) φ∗φ + λ1Φαβγ(Φ∗)αδϵ(φ∗)β +δ φγ +ϵ + λ2Φαβϵ(Φ∗)αβδ(φ∗)ϵ +γφγ +δ . +(2.7) +The contractions of Eq. (2.7) yield a single mass-squared relation that reads +M 2 +Φ10 = M 2 +Φ1 − 3M 2 +Φ3 + 3M 2 +Φ6. +(2.8) +The mass spectrum given in Table III and the mass relation presented in Eq. (2.8) +are necessary inputs for the gauge coupling unification analysis. +2.2 +Fermion sector +The Yukawa sector of the model is +L ⊃ Y u +ij 10F i10F j5′ +H + Y d +ij 10F i5F j5∗ +H + Y a +i 15F5F i5∗ +H ++ Y b +i 15F5F i35∗ +H + Y c +i 10F i15F24H + y 15F15F24H + h.c., +(2.9) +where the PQ charge assignment of Table II and the SU(5) indices are all implicitly +understood. The Yukawa matrix elements of the model are Y u +ij ≡ Y u +ji, Y d +ij = Y d∗ +ij ≡ +δijY d +i , Y a +i , Y b +i , Y c +i , and y = y∗, where we have used the freedom to rotate irreps in +the SU(5) group space to reach this particular Yukawa coupling basis. The model +accordingly has nineteen real parameters and fourteen phases in the Yukawa sector +to accommodate all of the masses and mixing parameters of the SM fermions as well +as the masses of fermions in the 15F-15F pair. +The PQ charge assignment forbids a bare-mass term for the 15F-15F pair. The +masses of the associated SM gauge group multiplets are thus generated solely through +the last term of Eq. (2.9), which reads +L ⊃ yvφ +√ +15 +�3 +2Σ1Σ1 + 1 +4Σ3Σ3 − Σ6Σ6 +� ++ h.c., +(2.10) +where the overall phase of 24H, once again, is not shown for simplicity. We subse- +quently define +MΣ1 = y +2 +� +3 +5vφ , +(2.11) +MΣ3 = +y +4 +√ +15vφ , +(2.12) +MΣ6 = − y +√ +15vφ. +(2.13) +It is important to point out that apart from different Clebsch coefficients, all submul- +tiplets within 15F have a common mass scale. (Even though Σ1 and Σ3 mix with the +– 6 – + +fermions in 5F i and 10F i, this does not affect equalities in Eqs. (2.11) and (2.12).) +We will show, later on, that the product yvφ is rather constrained by a requirement +for the model to simultaneously generate large enough unification and neutrino mass +scales. +The masses of the SM fermions are obtained after the breaking of the SM gauge +group down to SU(3) × U(1)em as follows. The down-type quark sector 4 × 4 mass +matrix can be written as +MD = +�vΛY d v′ +φY c +vΛY a MΣ3 +� +, +(2.14) +where we introduce v′ +φ = − 1 +4 +� +5 +3vφ. This matrix can be transformed into a block- +diagonal form comprising a 3 × 3 part denoted Md and a mass parameter MH as +follows +XMDY † = +�Md +0 +0 MH +� +, +(2.15) +where unitary matrices X and Y take the form +X ∼ +� +� +� +1 + +v′2 +φ +M2 +Σ3 Y cY c†�−1/2 +− +� +1 + +v′2 +φ +M2 +Σ3 Y cY c†�−1/2 +v′ +φ +MΣ3 Y c +v′ +φY c† +MH +MΣ3 +MH +� +� , +(2.16) +Y ∼ +� +� +1 +− +vΛv′ +φ +M2 +H (Y d†Y c + +MΣ3 +v′ +φ Y a†) +vΛv′ +φ +M2 +H (Y c†Y d + +MΣ3 +v′ +φ Y a) +1 +� +� , +(2.17) +with +Md ∼ +� +1 + v′2 +φ +M 2 +Σ3 +Y cY c† +�−1/2 � +vΛY d − vΛv′ +φ +MΣ3 +Y cY a +� +, +(2.18) +MH = +� +M 2 +Σ3 + v′ +φ +2Y c†Y c ≈ MΣ3. +(2.19) +Here, 1 = diag(1, 1, 1) while Y c, Y a, and Y d are Yukawa matrices that are featured +in Eq. (2.9). It is clear from Eq. (2.18) that the down-type quark mass matrix Md is +generated through the VEV of 5′ +H and the mixing between fields in 5F i, 10F i, 15F, +and 15F. This is possible due to the fact that Σ3 ∈ 15F and Qi ∈ 10F i transform in +the exact same way under the SM gauge group [25]. +The charged fermion mass matrices of the model can be succinctly written as +Mu = +� +1 + δ2 Y cY c†�− 1 +2 8vΛ′Y u, +(2.20) +Md = +� +1 + δ2 Y cY c†�− 1 +2 vΛ +� +Y d + δ Y cY a� +, +(2.21) +Me = vΛY d, +(2.22) +– 7 – + +where δ = −v′ +φ/MΣ3 and v2 +Λ + v2 +Λ′ = v2 with v = 174 GeV. We note the two most +prominent features of the charged fermion sector. First, Mu can be treated as a +symmetric matrix in the flavor space. Second, a mismatch between the charged lepton +and down-type quark mass matrices is proportional to a rank-one matrix Y cY a. We +again note that we work in the basis where Y u +ij ≡ Y u +ji and Y d +ij = Y d∗ +ij ≡ δijY d +i . This +simply means that vΛY di, i = 1, 2, 3, are the masses of the SM charged leptons. +The neutrino mass matrix elements (Mν)ij, at the leading order, read +(Mν)ij ≈ λv2 +Λ′ +8π2 (Y a +i Y b +j + Y b +i Y a +j ) +MΣ1 +M 2 +Σ1 − M 2 +Φ1 +ln +�M 2 +Σ1 +M 2 +Φ1 +� +≡ m0(Y a +i Y b +j + Y b +i Y a +j ) = (N diag(0, m2, m3) N T)ij , +(2.23) +where m2 and m3 are neutrino mass eigenstates and N is a unitary matrix. Note +that one of the neutrinos is a strictly massless particle due to the fact that Mν is +constructed out of two rank-one matrices with elements Y a +i Y b +j and Y b +i Y a +j . This is +accordingly encoded in the right-hand side of Eq. (2.23). +Since the charged lepton mass matrix in Eq. (2.22) is already in a diagonal form, +we can write that +N = diag(eiην +1, eiην +2, eiην +3)V ∗ +PMNS, +(2.24) +where VPMNS is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) unitary mixing ma- +trix, that is defined as VPMNS = R23U13R12Q, with Q = diag(1, eiβν, 1). Here we use +the PDG parametrization [26] for the R23, U13, and R12 matrices. Note that there is +only one Majorana phase βν appearing in Q due to the fact that one of the neutrinos +is massless. +One especially convenient feature of the neutrino sector is that the matrices Y a +and Y b can be expressed in terms of the PMNS matrix parameters and phases ην +i , +i = 1, 2, 3. Using the parametrization mentioned in Refs. [27, 28] we can write the +two Yukawa coupling vectors Y a and Y b as +Y a T = +ξ +√ +2 +� +� +i r2 N12 + r3 N13 +i r2 N22 + r3 N23 +i r2 N32 + r3 N33 +� +� , Y b T = +1 +√ +2ξ +� +� +−i r2 N12 + r3 N13 +−i r2 N22 + r3 N23 +−i r2 N32 + r3 N33 +� +� , +(2.25) +where r2 = +� +m2/m0 and r3 = +� +m3/m0, and where Nij denotes the ij-th element +of the unitary matrix N. +Moreover, ξ is a dimensionless scaling parameter that +needs to be introduced if one is to accurately scan over all possible phenomenologi- +cally viable entries in Y a and Y b that accommodate experimental observables in the +neutrino sector with utmost certainty. Eq. (2.25) is applicable solely to the normal +neutrino mass hierarchy scenario since that is one of the model predictions, as we will +discuss later. For alternative ways of generating neutrino masses within the SU(5) +framework, see, for example, Refs. [29–38]. +– 8 – + +3 +Peccei-Quinn symmetry and axion dark matter +We discuss the implementation of the PQ symmetry within our setup and elabo- +rate on the model’s main ingredients and experimental detection prospects in the +following. +In the “invisible axion” models [11–14] the PQ symmetry is broken by a scalar +field that carries a non-trivial PQ charge, where the scalar is a singlet under the SM. +We embed this scalar within the 24-dimensional Higgs irrep that is charged under the +U(1)PQ symmetry, as shown in Table II. Consequently, our setup unifies the GUT +and PQ breaking scales. The VEV of 24H ≡ φα +β can be written as +⟨φ⟩ = ˆvφ +√ +2diag +� −1 +√ +15, −1 +√ +15, −1 +√ +15, +3 +2 +√ +15, +3 +2 +√ +15 +� +eiaφ(x)/ˆvφ, +ˆvφ ≡ +√ +2vφ, +(3.1) +where the pseudoscalar part, i.e., field aφ(x), essentially remains massless, whereas +the radial mode acquires a mass of the order of the GUT scale while the global +U(1)PQ symmetry is spontaneously broken with order parameter vφ. To correctly +identify the massless axion, one also needs to include all other Higgses that carry PQ +charges and participate in symmetry breaking. +The non-Hermitian operators that are responsible for the breaking of the re- +phasing symmetry of the three scalar fields are given by the terms in the second +line of Eq. (2.5). The VEVs of neutral components of the SU(2) doublets can be +re-written as +⟨Λ′⟩ = ˆvΛ′ +√ +2e +i +aΛ′ +ˆvΛ′ , +⟨Λ∗⟩ = ˆvΛ +√ +2e +i aΛ +ˆvΛ , +ˆvΛ(′) ≡ +√ +2vΛ(′), +(3.2) +where we take all VEVs to be real, and, as mentioned before, we neglect the VEV +of the SU(2) triplet in 24H. With these assumptions, the axion field is identified +as [39], +a = xΛ′ˆvΛ′aΛ′ + x∗ +ΛˆvΛaΛ + xφˆvφaφ +va +, +v2 +a = x2 +Λ′ˆv2 +Λ′ + x2 +Λˆv2 +Λ + x2 +φˆv2 +φ, +(3.3) +where xi denotes the PQ charge of the corresponding i-th scalar (and x∗ +i = −xi). +Since vφ ∼ 1016 GeV and vΛ(′) ∼ 102 GeV, the axion mostly resides in 24H with +a ≈ aφ. +The axion field must also be orthogonal to the Goldstone field eaten up by the +Z-boson. This translates into the following condition +tan2 β = v2 +Λ′ +v2 +Λ += x∗ +Λ +xΛ′ , +(3.4) +which, in our benchmark charge assignments, fixes tan β = 1. +Here, we do not +present the expression of the SM Higgs mass eigenstate, which can be obtained via +– 9 – + +the diagonalization of the 4 × 4 mass matrix of the CP-even states. The heaviest +one is expected to reside at the GUT scale, and the lightest one is the SM Higgs +boson. Depending on the chosen hierarchy, the remaining two eigenstates — one +coming from the triplet and the other from the pair of doublets — can live anywhere +in between the electroweak and GUT scales. +Now, performing a field-dependent axial transformation that is anomalous under +QCD, the axion can be disentangled from the Yukawa interactions. This transfor- +mation generates the effective anomalous interactions of the following types: +δLeff = αs +8π +a +fa +G �G + +� αem +2πfa +E +N +� a +4F �F . +(3.5) +Here, G (F) is the gluon (photon) field strength tensor, �G ( �F) is its dual, and fa +is the axion decay constant. The effective operator of the form aG �G is the key to +the PQ solution to the strong CP problem. Since these sub-multiplets carry color +and electromagnetic charges, the PQ current has both QCD and electromagnetic +anomalies, with the corresponding anomaly coefficients [40], +N = +� +ψ +Nψ, +E = +� +ψ +Eψ , +(3.6) +where sums are taken over all fermions, which we generically denote by ψ. Using +well-known formulas, +Nψ = xψd(Iψ)T(Cψ), +(3.7) +Eψ = xψd(Cψ)d(Iψ) +� 1 +12(d(Iψ)2 − 1) + Y 2 +ψ +� +, +(3.8) +we obtain |N| = 13/2 and |E| = 52/3, and the domain-wall number is NDW = 2N = +13, which is relevant for cosmology. Subsequently, we find the axion decay constant +to be +fa = va +2N ≈ ˆvφ +2N = +� +3 +10παGUT +MGUT +N +. +(3.9) +Since the decay constant is of the order of the GUT scale, i.e., fa ∼ MGUT, we refer +to the axion as the “GUT axion”. Once strong interactions confine, non-perturbative +QCD effects generate a potential that gives rise to a tiny axion mass [41, 42] +ma = 5.7 neV +�1015 GeV +fa +� += ma = 5.7 neV +�1015 GeV +MGUT +� +N +� +10παGUT +3 +. +(3.10) +This shows that the axion mass is predicted if the grand unification scale MGUT is +known. We accordingly compute the predicted range of the GUT scale within our +model in Sec. 4 by taking into account all relevant constraints. +– 10 – + +10-12 +10-11 +10-10 +10-9 +10-8 +10-20 +10-19 +10-18 +10-17 +10-16 +ma [eV] +gaγγ [GeV-1] +DMRadio-GUT +Res I: ABD +Res II: ABD +Res III: ABD +Broad I: ABD +Broad II: ABD +Broad III: ABD +GUT axion +optomechanical cavity +Figure 1: Expected reach in the ma vs. gaγγ plane for the broadband (Broad) and +resonant (Res) strategies of the ABRACADABRA (ABD) experiment [43]. The blue +line corresponds to the prediction of our proposed model. The projected 3 σ sensitiv- +ity of DMRadio-GUT [44, 45] is also presented in the green shaded region. Further- +more, the expected theoretical reach using the optomechanical cavity method [46] is +shown with solid black lines. See text for more details. +Since the non-observation of proton decay requires the GUT scale to be large, +the axion mass is expected to be around the neV scale within our setup. An axion +in this mass range is extremely weakly coupled to the SM particles due to an ex- +tremely large decay constant. Remarkably, an axion with neV mass can serve as an +excellent dark matter candidate and can be searched for efficiently in direct detection +experiments [47] hunting for ultra-light axions. +Next, we consider the most important axion couplings relevant for experimental +sensitives. +In the low-energy effective Lagrangian for the axion, it is sometimes +convenient to eliminate the axion coupling to the gluons via a field-dependent axial +transformation of the SM quarks. After making such a rotation, the axion coupling +to the photons is given by [42], +L ⊃ αem +2πfa +� E +N − 1.92 +� +� +�� +� +≡gaγγ +a +4F �F, +(3.11) +where the model-dependent quantity, apart from fa (see Eq. (3.9)), in our case, is +given by E/N = 8/3. In fact, the dark matter experiment ABRACADABRA [43] +has great potential to look for an axion dark matter in the mass range of interest. +As shown in Fig. 1, a major part of the parameter space of our theory will be probed +by this dark matter direct detection experiment. Fig. 1 is obtained by varying model +– 11 – + +10-12 +10-11 +10-10 +10-9 +10-8 +10-7 +10-23 +10-21 +10-19 +10-17 +10-15 +10-13 +10-11 +ma [eV] +|gaD| [GeV-2] +CASPEr Electric +phase I +phase II +phase III +spin noise +GUT axion +Figure 2: Axion coupling to the nucleon EDM operator as a function of the axion +mass. The blue band corresponds to the prediction of our model; see text for details. +The shaded regions show the sensitivity projections of CASPEr Electric [48, 49] in +its various phases. Moreover, the ultimate sensitivity limit is given by the nuclear +spin noise. +the parameters while imposing all relevant constraints. The details of our numerical +procedure are relegated to Sec. 4. +Another axion dark matter experiment, the DMRadio-GUT [44, 45], will also +be sensitive in detecting axions with GUT scale decay constant fa ∼ 1016 GeV. +DMRadio-GUT will be far more sensitive compared to its previous two phases, +DMRadio-50L and DMRadio-m3, since it will have a factor of three enhancement +in the field and a factor of ten enhancement in volume relative to DMRadio-m3. The +projected 3 σ sensitivity of DMRadio-GUT is also presented in Fig. 1 by a green +shaded region, which will probe a significant portion of the parameter space. Yet +another proposal utilizing an optomechanical cavity [46] filled with superfluid he- +lium is shown to be highly promising in detecting ultra-light axion dark matter. +This proposed experimental method, with a cavity size of order O(10 m) is expected +to be sensitive to axion-photon couplings for axions with the GUT scale size decay +constant. In Fig. 1, the corresponding theoretical reach is shown with solid black +lines. The ABRACADABRA experiment will be sensitive to axion masses as low +as ma ∼ 2 neV, whereas it is about ma ∼ 0.4 neV and ma ∼ 0.1 neV for DMRadio- +GUT and optomechanical cavity, respectively. A combination of all these axion dark +matter experiments will eventually probe the entire parameter space of the proposed +model. +Intriguingly, ultra-light axion dark matter can also be efficiently searched via +oscillating nucleon electric dipole moments (EDM). As aforementioned, the QCD +– 12 – + +axion solves the strong CP problem by promoting the θ parameter into the dynamical +axion field. Consequently, the effective θ angle gives rise to an EDM for nucleons +sourced by the axion. Owing to the dynamical nature of the axion, this EDM will +change in time, giving rise to unique signals. In the effective Lagrangian, the coupling +of the axion to nucleon n takes the following form, +L ⊃ − i +2gaD aψnσµνγ5ψnF µν . +(3.12) +The nucleon electric dipole moment generated through the above operator is given +by dn = gaDa. In terms of our model parameters, it can be re-written in the following +form [50]: +dn ≈ a 2.4 × 10−16 +fa +e · cm +� +�� +� +gaD +, +(3.13) +with roughly a 40% uncertainty [51], where the decay constant is given in Eq. (3.9). +(See also Refs. [52–54].) +The corresponding coupling as a function of the axion +mass is shown in Fig. 2. As can be seen from this figure, excitingly, the CASPEr +Electric [48, 49] experiment alone will probe almost the entire parameter space of +our model. The width of the band corresponds to the uncertainty in the calculation +as aforementioned. As in Fig. 1, Fig. 2 is also obtained by varying model parameters +by imposing all relevant constraints, which will be discussed later in the text. +Since the axion is ultra-light in our setup, it can constitute the entirety of the +dark matter. It is important to point out that the breaking of the GUT symmetry +to that of the SM gauge group SU(5) × U(1)PQ → SU(3) × SU(2) × U(1) leads +to an overproduction of super-heavy monopoles that must be inflated away. +As +discussed above, spontaneous breaking of the PQ symmetry leads to NDW distinct +degenerate vacua, giving rise to a domain-wall problem, which also requires dilution +to be consistent with cosmology. Both of these problems, along with the horizon +and flatness problems, can be elegantly solved via inflation taking place after the +GUT symmetry breaking. We, however, do not specify the details of the inflationary +dynamics, which is beyond the scope of this work. The amount of axion dark matter +produced then depends on whether the PQ symmetry is restored or not after inflation. +We assume that the U(1)PQ remains broken during inflation and is never restored +afterwards. In such a scenario, the relic abundance of the axion dark matter is given +by [55], +Ωh2 ∼ 0.12 +�5 neV +ma +�1.17 � +θi +1.53 × 10−2 +�2 +, +(3.14) +which shows that the initial value of θi = ai/fa, where ai is the initial value of +the axion field, needs to be somewhat smaller than unity to be consistent with the +– 13 – + +102 +104 +106 +108 +1010 +1012 +1014 +1016 +1018 +10 +20 +30 +40 +50 +60 +μ [GeV] +α-1(μ) +Mϕ1 +Re +Mϕ8 +Re +Mϕ1 +Im +Mϕ3 +Im +Mϕ8 +Im +MΦ3 +MΦ6 +MH2 +MΣ1 +MΣ3 +MΣ6 +MΦ1 +MΦ10 +MT1 +MT2 +α1 +-1 +α2 +-1 +α3 +-1 +α-1 +Figure 3: Example for the choice of the intermediate-scale particle masses giving +gauge coupling unification. +observed dark matter relic abundance Ωh2 ∼ 0.12 ± 0.001 [56]. Thus, for θi ∼ 10−2, +the axion consists of 100% of the dark matter within our setup. +Before we conclude this section, we note that since the PQ symmetry is assumed +not to get restored after inflation, our scenario cannot be tested at gravitational wave +observatories. +4 +Unification, axion mass and proton decay +In our model, the axion decay width fa is connected to the GUT scale MGUT due +to the fact that the adjoint GUT Higgs field 24H simultaneously breaks the SU(5) +and U(1)PQ symmetries. This, in particular, directly relates the axion mass ma to +the GUT scale MGUT via Eq. (3.10). Moreover, since the partial proton lifetimes are +proportional to the fourth power of the GUT scale, our model can be simultaneously +probed with axion dark matter and proton decay experiments. +– 14 – + +4.1 +Unification +The renormalization group equations (RGEs) for the gauge couplings can, at the +2-loop level, be written as [57] +µdα−1 +dµ += − 1 +2π +� +bSM +i ++ +� +J +bJ +i H(µ − MJ) +� +− +1 +8π2 +�� +J +� +bSM +ij + bJ +ijH(µ − MJ) +� +α−1 +j ++ βY +i +� +. +(4.1) +Here, bSM +i +(bSM +ij ) are the SM 1-loop (2-loop) gauge coefficients, while bJ +i (bJ +ij) are the +1-loop (2-loop) gauge coefficients of the multiplets J with intermediate-scale masses +MJ, i.e., MZ < MJ < MGUT. These coefficients are listed in Appendix A. Moreover, +βY +i are the Yukawa contributions and H is the Heaviside step function defined as +H(m) = +� 1, m > 0 +0, +m ≤ 0 . +(4.2) +Note that we neglect the effect of the Yukawa couplings Y a, Y b, and Y c on the running +of the gauge couplings. In order to investigate the viable part of the parameter space +giving gauge coupling unification, we freely vary the masses of the fields φRe +1 , φIm +1 , +φIm +3 , φRe +8 , φIm +8 , Σ1, Σ3, Σ6, Φ1, Φ3, Φ6, Φ10, T1, T2, and H2 respecting the mass +relations presented in Sec. 2. Here, T1,2 and H1,2 refer to the mass eigenstates of the +scalar color triplets and weak doublets, where H1 is identified with the SM Higgs +with 125 GeV mass. We ensure that the scalar leptoquark mediated proton decay +is sufficiently suppressed by taking a lower bound of 3 × 1011 GeV for the masses +of T1 and T2. +The masses of the remaining multiplets are freely varied between +the TeV and the GUT scale. The numerical fit is performed by running the gauge +couplings at the 2-loop level from the GUT scale to the Z mass scale at which a +χ2-function that we define later in detail is minimized. We use the low-scale values +g1 = 0.461425+0.000044 +−0.000043, g2 = 0.65184+0.00018 +−0.00017, g3 = 1.2143+0.0035 +−0.0036 [58] as our input, +where gi = √4παi. +To demonstrate that within our setup, the gauge couplings +can indeed unify, Fig. 3 shows one possible choice of the particle mass spectrum +giving exact gauge coupling unification in agreement with the current proton decay +constraints and also for a choice of the masses of Σ1 and Φ1 giving the correct neutrino +mass scale via Eq. (2.23). +4.2 +Proton decay +The formulae for the proton decay widths of various decay channels can be found in +[59, 60]. For example, the decay width for the proton decay channel having a pion +– 15 – + +and a charged lepton in the final state is given by1 +Γ(p → π0e+ +α) = mpπ +2 +� +1 − m2 +π +m2 +p +�2 +A2 +L +α2 +GUT +M 4 +GUT +(4.3) +× +� +A2 +SL|c(ec +α, d)⟨π0|(ud)LuL|p⟩|2 + A2 +SR|c(eα, dc)⟨π0|(ud)RuL|p⟩|2� +. +Here, mp = 0.9393 GeV and mπ = 0.134 GeV denote the proton and pion masses, +respectively, while AL = 1.2 [62] and ASL(R) encode the leading log renormalization +of the dimension six operators, where2 +ASL(R) = +� +i=1,2,3 +MZ≤MI≤MGUT +� +I +�αi(MI+1) +αi(MI) +� +γL(R)i +bSM +i ++�MZ ≤MJ ≤MI +J +bJ +i , +(4.4) +with γL(R)i = (23(11)/20, 9/4, 2) [63–65]. Moreover, we take the hadron matrix ele- +ments, such as, for example, ⟨π0|(ud)LuL|p⟩ = 0.134(5)(16) GeV2 and ⟨π0|(ud)RuL|p⟩ = +−0.131(4)(13) GeV2, from Refs. [66, 67]. +Finally, the c-coefficients of Eq. (4.3) +read [68–70] +c(ec +α, dβ) = (U † +RU ∗ +L)11(E† +RD∗ +L)αβ + (E† +RU ∗ +L)α1(U † +RD∗ +L)1β , +(4.5) +c(eα, dc +β) = (U † +RU ∗ +L)11(E† +LD∗ +R)αβ , +(4.6) +c(νl, dα, dc +β) = (U † +RD∗ +L)1α(D† +RN)βl , +(4.7) +where the unitary matrices UL/R, EL/R, DL/R, and N diagonalize the SM fermion +mass matrices through the following transformations +Mu = ULM diag +u +U † +R, +Md = DLM diag +d +D† +R, +Me = ELM diag +e +E† +R, +Mν = NM diag +ν +N T. +(4.8) +The current experimental constraints and future sensitivities for the various par- +tial lifetimes that we use in our numerical analysis are presented in Table IV. For a +recent review on the subject, see Ref. [71]. +4.3 +Numerical procedure +We start our numerical analysis by constructing matrices Mu, Me, Y a, Y b, and Y c +at the GUT scale, as described in the next few paragraphs. +Since the up-type quark mass matrix Mu is approximately symmetric, we have +that UR = U ∗ +L. This allows us to construct Mu as +Mu = ULdiag(mu, mc, mt)U T +L , +(4.9) +1The Mathematica package ProtonDecay [61] can be used to compute the decay widths of various +nucleon decay channels. +2If the denominator of the exponent vanishes for some factor, i.e., the 1-loop running of a specific +gauge coupling is constant within a certain interval, the respective factor in Eq. (4.4) is replaced +with exp[γL(R)iα(MI+1)]/(2π). +– 16 – + +decay channel +current bound τp [yrs] +future sensitivity τp [yrs] +p → π0 e+ +2.4 · 1034 [72] +7.8 · 1034 [73] +p → π0 µ+ +1.6 · 1034 [72] +7.7 · 1034 [73] +p → η0 e+ +1.0 · 1034 [74] +4.3 · 1034 [73] +p → η0 µ+ +4.7 · 1033 [74] +4.9 · 1034 [73] +p → K0 e+ +1.1 · 1033 [75] +- +p → K0 µ+ +3.6 · 1033 [76] +- +p → π+ ν +3.9 · 1032 [77] +- +p → K+ ν +6.6 · 1033 [78] +3.2 · 1034 [73] +Table IV: Present experimental bounds on the partial lifetimes τp as well as future +sensitivities for 10 years of runtime, both at 90% confidence level. +where we parametrize the up-type quark mixing matrix UL in terms of the down-type +quark mixing matrix DL, the Cabibbo-Kobayashi-Maskawa (CKM) matrix VCKM, +and five GUT phases βu +1 , βu +2 , ηu +1, ηu +2, and ηu +3, as +UL = DLdiag(eiβu +1 , eiβu +2 , 1)V T +CKMdiag(eiηu +1 , eiηu +2 , eiηu +3 ) . +(4.10) +In our analysis, we set ηu +1 = ηu +2 = ηu +3 = 0 since these three phases do not affect the +proton decay predictions at all. +We set EL = ER = 1 since Me is diagonal and real. This also means that we +can simply construct Me via an equality that reads +Me = diag(me, mµ, mτ). +(4.11) +Y a and Y b are constructed via Eq. (2.25) using the neutrino mixing matrix +N = (eiην +1, eiην +2, eiην +3)V ∗ +PMNS as an input. Note that VPMNS contains the CP violating +phase δν as well as the Majorana phase βν. We furthermore construct Y c to be a +general complex 1 × 3 matrix through +Y c = (yc +1eiηc +1, yc +2eiηc +2, yc +3eiηc +3). +(4.12) +Once the parameter dependence of Mu, Me, Y a, Y b, and Y c is properly accounted +for, as described above, we can also construct Md and Mν that are given by Eqs. (2.21) +and (2.23), respectively. We treat λ in Mν as a free parameter while the two Higgs +VEVs that enter Md and Mν are given by vΛ = vΛ′ = 174/ +√ +2 GeV due to the +constraint that tan β of Eq. (3.4) is equal to one. +In summary, the free parameters for our numerical analysis are the unification +scale MGUT and the corresponding gauge coupling αGUT, the masses of the fields3 +3Note that the masses of the fields φRe +1 , φIm +1 , Σ3, Σ6, Φ10 are obtained via the mass relations +discussed in Sec. 2. +– 17 – + +φIm +3 , φRe +8 , φIm +8 , Σ1, Φ1, Φ3, Φ6, T1, T2, and H2, the phases βu +1 , βu +2 , δν, βν, ην +1, ην +2, +ην +3, the Yukawa parameters yc +1, yc +2, yc +3, the quartic Higgs coupling λ, and the scaling +parameter ξ. These 24 parameters are fitted to the experimental observables that +are the SM gauge couplings g1, g2, and g3, and the down-type quark masses md, +ms, and mb, while requiring that the current proton decay constraints, as given in +Table IV, are satisfied. Note that the charged lepton masses, the up-type quark +masses, the neutrino mass squared differences, the CKM mixing parameters, and the +known PMNS mixing parameters are all automatically accounted for. +Since there are more parameters than observables, proton decay cannot be pre- +dicted sharply in all decay channels as we will discuss in the next section. But, due +to the fact that the neutrino mass matrix is connected to the mismatch between the +charged lepton and down-type quark mass matrices, our model predicts the PMNS +parameters δν and βν to be in relatively narrow intervals. +The gauge couplings are fitted to their low-energy scale values [58] after the 2- +loop level running from the high scale to the low scale is performed. To simplify +the analysis, the down-type quark and neutrino masses are directly fitted at the +high scale, using the high scale values provided in Ref. [79]. +The χ2-function is +obtained comparing the theoretical prediction pi with the experimental central value +ei, normalized with the corresponding experimental standard deviation σi of the i-th +observable via +χ2 = +� +i +�pi − ei +σi +�2 +. +(4.13) +To minimize the χ2-function we apply a differential evolution algorithm. This mini- +mization yields a viable benchmark point and thus proves the viability of our model. +Then, starting from this benchmark point with a flat prior distribution a Markov- +chain-Monte-Carlo (MCMC) analysis involving a Metropolis-Hasting algorithm is +performed, giving us a total of 6 × 106 datapoints. Finally, we use these points to +calculate the highest posterior density (HPD) regions of various quantities. +For the numerical analysis, all parameters are freely varied in such a way that +the perturbativity of all Yukawa and Higgs couplings is satisfied. In particular, the +absolute values of all entries in Y a, Y b, and Y c, as well as the absolute value of λ are +required to be less than or equal to 1. To this end, the scaling parameter ξ ensures +that the full parameter space is covered with the chosen parametrization of the +matrices Y a and Y b. Furthermore, although we fix some model parameters during the +fitting/minimization procedure by directly plugging in experimental central values of +some observables, we still vary these parameters in the subsequent MCMC analysis. +4.4 +Results +In this section, we present the outcome of our numerical study. We are interested in +the full axion mass range, the predictions for partial proton decay lifetimes, and the +– 18 – + +viable range of the Dirac CP and Majorana phases of the PMNS matrix. +The axion mass ma is connected to the GUT scale MGUT and gauge coupling +αGUT via Eq. (3.10). We can therefore obtain the predicted range of the axion mass by +maximizing and minimizing Eq. (3.10). We demand viable gauge coupling unification +and correct neutrino mass scale while making sure that none of the current proton +decay constraints is violated. We find ma ∈ [0.1, 4.7] neV which we present in Figs. 1 +and 2. As discussed in more detail in Sec. 3, this already demonstrates that the full +parameter space will be probed by two kinds of future axion DM experiments that +are sensitive to either the axion to photon coupling or to the nucleon EDM. +To start our numerical analysis, we find a viable benchmark point from a full χ2 +fit. In particular, for the case of normal neutrino mass ordering, we obtain that +Y a = +� +−0.120 + i 0.00943, 0.513 + i 0.200, 0.898 +� +, +(4.14) +Y b = +� +0.109 + i 0.150, 0.348 + i 0.334, 0.195 − i 0.0211 +� +, +(4.15) +Y c = +� +0.00115 + i 0.00198, −0.0532 + i 0.0852, −2.781 − i 0.743 +� +× 10−6, +(4.16) +for MGUT = 1016.2 GeV, mH2 = 103.77 GeV, MT1 = MT2 = 1014.55 GeV, MφRe +1 += +104.39 GeV, MφIm +1 += 104.12 GeV, MφIm +3 += 104.40 GeV, MφRe +8 += 104.09 GeV, MφIm +8 += +103.71 GeV, MΣ1 = 1013.41 GeV, MΣ3 = 1012.63 GeV, MΣ3 = 1013.24 GeV, MΦ1 = +1011.63 GeV, MΦ3 = 105.28 GeV, MΦ6 = 104.18 GeV, MΦ10 = 1011.63 GeV, α−1 +GUT = 15.62, +and λ = 1.00. If the proton decay pull is neglected, this choice of the input param- +eters gives a total χ2 below 0.01. This is thus a perfect fit for the gauge couplings +as well as for the fermion masses and mixings. Moreover, for this benchmark point, +the PMNS Dirac CP phase is given by δν = −48.5◦, whereas the PMNS Majorana +phase is βν = −71.3◦. We note that for the case of inverted neutrino mass ordering, +no good fit-point can be obtained. This is due to the fact that the Yukawa matrix +Y a is needed to generate both the viable neutrino masses and the correct mismatch +between the charged lepton and down-type quark masses. In the case of inverted +ordering, the first two entries in Y a would need to be somewhat larger than the third +entry. This is, however, in conflict with the down-type quark mass fit that requires +the first entry of Y a to be smaller than the second and third entries. Therefore, a +strong prediction of our model is that the neutrinos have normal mass ordering. +From the aforementioned benchmark point, we start an MCMC analysis with a +flat prior. All obtained points are presented in Fig. 4 in a plane of axion mass vs. par- +tial proton decay lifetime in the dominant decay channel p → π0e+. We also present +the future sensitivities of the DM experiments ABRACADABRA, DMRadio-GUT, +and Casper Electric, as discussed in Sec. 3, as well as the future sensitivity of the +proton decay experiment Hyper-Kamiokande, as discussed in Sec. 4.2. Fig. 4 nicely +visualizes how a part of our parameter space can be probed through the synergy be- +tween three different kinds of experiments testing (i) the axion to photon coupling, +(ii) the nucleon EDM, and (iii) proton decay. For example, if the axion mass is +– 19 – + +0 +1 +2 +3 +4 +5 +1034 +1035 +1036 +1037 +1038 +ma [neV] +τ(p → π 0e+) [yrs] +Hyper-K +Super-K +ABD +DMRadio-GUT +CASPEr +Figure 4: The generated points from the MCMC analysis presented in the ma − +τ(p → π0e+) plane. The current Super-Kamiokande bound is represented by a gray +box, while the future Hyper-Kamiokande sensitivity is indicated by a blue dotted +line. Moreover, the projected sensitivity of various axion DM experiments is also +shown: ABRACADABRA (ABD) with a red dotted line, DMRadio-GUT with a +green dotted line, CASPEr Electric with a brown dotted line. For details, see the +main text. +observed to be above 3 neV, proton decay in the decay channel p → π0e+ necessarily +has to be seen by Hyper-Kamiokande if our model is realized in nature. Moreover, +regardless of whether proton decay will be observed by Hyper-Kamiokande, the for- +mer two kinds of experiments will be able to cover the entire parameter space of our +model. +We are also interested in the proton decay predictions of all the different decay +channels within our model. First, we want to obtain the full allowed range for all +partial proton lifetimes, which is for the decay channel p → π0e+ already hinted +in Fig. 4. To this end, we vary all the parameters, including the intermediate-scale +particle masses in the MCMC analysis. The 1 σ (dark) and 2 σ (light) HPD results +of this analysis are shown in Fig. 5. The blue line segments indicate the current +experimental bounds, while the red line segments represent the future sensitivities. +(See, for example, Table IV.) Fig. 5 shows that a part of the predicted 1 σ HPD +interval for the two decay channels p → π0e+ and p → η0e+ will be tested by Hyper- +Kamiokande. +The large uncertainty in these partial lifetime predictions that are +coming from the dependence on the fourth power of the GUT scale can be erased +by considering ratios of specific decay channels.4 Fig. 6 shows the prediction of such +4For recent works analyzing ratios of partial proton decay lifetimes in models with predicted +GUT scale quark-lepton Yukawa ratios, see Refs. [37, 61]. +– 20 – + +p → π0e+ +p → π0 μ+ +p → K+ν +p → π+ν +p → K0e+ p → K0 μ+ +p → η e+ +p → η μ+ +1033 +1034 +1035 +1036 +1037 +1038 +1039 +1040 +1033 +1034 +1035 +1036 +1037 +1038 +1039 +1040 +τp [yrs] +Figure 5: The predicted 1 σ (dark) and 2 σ (light) HPD intervals of the proton +lifetime for various decay channels. The blue (red) line segments indicate the current +(future) experimental bounds (sensitivities) at 90% confidence level. Interestingly, a +part of the predicted 1 σ region for both decay channels p → π0e+ and p → η0e+ lies +within the reach of Hyper-Kamiokande. +p → π0 μ+ +p → π0 e+ +p → K+ ν +_ +p → π0 e+ +p → π+ ν +_ +p → π0 e+ +p → K0 e+ +p → π0 e+ +p → K0 μ+ +p → π0 e+ +p → η e+ +p → π0 e+ +p → η μ+ +p → π0 e+ +1 +3 +10 +30 +100 +300 +1000 +3000 +1 +3 +10 +30 +100 +300 +1000 +3000 +τ +Figure 6: The 1 σ (dark) and 2 σ (light) HPD intervals of ratios of the proton +lifetime of various decay channels. Interestingly, the ratio τ(p → η0e+)/τ(p → π0e+) +(which will partly be tested by Hyper-Kamiokande) is predicted very sharply. +– 21 – + +ratios with the dominant decay channel p → π0e+ in the denominator. Especially +interesting is the prediction for the ratio τ(p → η0e+)/τ(p → π0e+), since both of the +featured decay channels will be partly tested by Hyper-Kamiokande. This ratio is +predicted very sharply. However, such a sharp prediction for this particular ratio is +not only specific to our model but a more common feature of models in which gauge +boson mediated proton decay is dominant and in which the contribution involving +the c-coefficient c(ec, d) dominates the contribution with the c-coefficient c(e, dc). +Nevertheless, these ratios of partial proton lifetimes provide an interesting additional +opportunity to probe our model. +In order to understand the dependence of the different decay channels on the +flavor structure in the fermionic mass matrices, we fix the gauge coupling unification +to the same values as listed below Eq. (4.16) and only vary the parameters in the +mass matrices in the MCMC analysis, computing for each point the proton decay +prediction for each decay channel. We visualize the 1 σ (dark) and 2 σ (light) HPD +results of this analysis in Fig. 7, where the blue line segments indicate the current +experimental bounds at 90% confidence level presented in Table IV. Interestingly, the +partial lifetime is for some channels much more sharply predicted than for others. +The sharp prediction for the decay channels with an antineutrino in the final state is +a general feature for models with a (nearly) symmetric up-type quark mass matrix. +On the other hand, the fact that the partial lifetime of the decay channel p → π0e+ +has such a sharp prediction is uncommon and a nice feature of our model, which also +implies that this decay channel is predicted to be the dominant one.5 +The interesting result that some decay channels have much sharper predictions +than others can be understood by investigating the freedom in the mixing matrices +that are defined in Eq. (4.8). This is demonstrated in the following example, where +we compare the predictions for the two decay channels p → π0e+ and p → π0µ+. In +the chosen basis, the respective c-coefficients of the two decay channels in question +read +c(ec +α, d) = (D∗ +L)α1 + (U ∗ +L)α1(U T +L D∗ +L)11, +(4.17) +c(eα, dc) = (D∗ +R)α1. +(4.18) +As it can be seen from Eq. (2.21), the left mixing of the down-type quark mass +matrix DL strongly depends on the Yukawa matrix Y c, while the right mixing DR +dominantly depends on the Yukawa matrix Y a. Since Y a has to be chosen in such +a way that the correct PMNS parameters and neutrino masses are obtained, there +cannot be a strong hierarchy within Y a entries. On the other hand, a strong hierarchy +of the entries in Y c is required in order to produce the correct mismatch between +the down-type quark and charged lepton masses. Therefore, DR appears to have +5Note, however, that proton decay mediated by the two scalar triplets T1 and T2 could enhance +the decay channel p → K+ν. +– 22 – + +p → π0e+ +p → π0 μ+ +p → K+ν +p → π+ν +p → K0e+ p → K0 μ+ +p → η e+ +p → ημ+ +1033 +1034 +1035 +1036 +1037 +1038 +1039 +1040 +1033 +1034 +1035 +1036 +1037 +1038 +1039 +1040 +τp [yrs] +Figure 7: The 1 σ (dark) and 2 σ (light) HPD intervals of the proton lifetime for +various decay channels for a benchmark scenario with MGUT = 1016.2 GeV. The blue +line segments represent the current experimental bounds at 90% confidence level. +a large mixing, whereas DL is for all points in the MCMC almost equal to the +identity matrix. This, in particular, also implies that the CKM mixing is mostly +coming from UL. Hence, in the case of a positron in the final state, i.e., for α = 1 +in Eqs. (4.17) and (4.18), the contribution coming from c(ec, d) dominates over the +contribution coming from c(e, dc) in the decay width formula (see Eq. (4.3)), since +|(UL)11|, |(DL)11| > |(DR)11|. Contrarily, if an antimuon is in the final state (α = 2), +the contribution involving c(µ, dc) is dominant over the contribution from c(µc, d), +since |(DR)21| > |(UL)21|, |(DL)21|. Now, varying over the full flavor freedom, since we +always roughly have |(UL)11| ≈ |(VCKM)11|, |(DL)11| ≈ 1, the dominating c-coefficient +c(ec, d) only varies by an order 1 factor. This results in a very sharp prediction for +the partial lifetime of the decay channel p → π0e+. On the other hand, |(DR)21| +roughly varies within the interval [0.1, 1], resulting in a much less sharp prediction +for the partial lifetime of the decay width p → π0µ+. +Finally, from our MCMC results, we deduce the HPD intervals of the Dirac CP +and Majorana phase of the PMNS matrix. At 1 σ we obtain δν ∈ [−22.6◦, 34.4◦] +and βν ∈ [−124.1◦, −71.4◦], while our 2 σ HPD results are δν ∈ [−50.7◦, 55.6◦] and +βν ∈ [−132.2◦, −54.1◦]. Future experiments involving these two observables also have +the potential to probe our model and to possibly further reduce the allowed parameter +space. For instance, our 2 σ HPD results for mββ, the effective mass parameter for +the neutrinoless double beta decay, is predicted to be mββ ∈ [1.46, 2.24] meV, well +below the current experimental bound mββ < 61 meV provided by Ref. [80]. +– 23 – + +5 +Conclusions +We present a minimal model of unification based on an SU(5) gauge group augmented +with a Peccei-Quinn symmetry that predicts the existence of ultralight axion dark +matter within a narrow mass range of ma ∈ [0.1, 4.7] neV. This mass window is deter- +mined through an interplay between gauge coupling unification constraints, partial +proton decay lifetime limits, and the need to reproduce the experimentally observed +fermion mass spectrum. The model also predicts that neutrinos are purely of Majo- +rana nature, possessing a normal mass hierarchy spectrum, where one of the neutrinos +is a massless particle. We discuss the gauge boson mediated proton decay signatures +of the model and specify expected partial lifetime ranges for two-body nucleon decays. +Our analysis yields viable 2 σ ranges for the Dirac CP phase δν ∈ [−50.7◦, 55.6◦] and +for the neutrinoless double beta decay mββ ∈ [1.46, 2.24] meV, respectively, through +which the model may be tested in the neutrino experiments. Finally, we demonstrate +that the entire parameter space of the model will be tested through a synergy between +several low-energy experiments that look for proton decay (Hyper-Kamiokande) and +axion dark matter (ABRACADABRA and DMRadio-GUT by measuring the axion- +photon coupling, and CASPEr Electric by measuring the nucleon electric dipole +moments). +A +Renormalization group running of the gauge couplings +The 2-loop renormalization group equations of the SM gauge couplings are given +in Eq. 4.1. Here, we present the 1-loop and 2-loop gauge coefficients of the various +intermediate-scale multiplets listed in Table I. The 1-loop gauge coefficients +� +b1 b2 b3 +� +are +b +φRe +1 +i += +� +0 1 +3 0 +� +, +b +φIm +1 +i += +� +0 1 +3 0 +� +, +b +φIm +3 +i += +� 5 +12 +1 +4 +1 +6 +� +, +b +φIm +3 +i += +� 5 +12 +1 +4 +1 +6 +� +, +b +φRe +8 +i += +� +0 0 1 +2 +� +, +b +φIm +8 +i += +� +0 0 1 +2 +� +, +bT1 +i += +� 1 +15 0 1 +6 +� +, +bT2 +i += +� 1 +15 0 1 +6 +� +, +bΦ1 +i += +� 9 +5 +5 +3 0 +� +, +bΦ3 +i += +� 4 +5 2 1 +2 +� +, +bΦ6 +i += +� 1 +15 1 5 +3 +� +, +bΦ10 +i += +� +2 0 5 +2 +� +, +bΣ1 +i += +� 6 +5 +4 +3 0 +� +, +bΣ1 +i += +� 6 +5 +4 +3 0 +� +, +bΣ3 +i += +� 1 +15 1 2 +3 +� +, +bΣ3 +i += +� 1 +15 1 2 +3 +� +, +bΣ6 +i += +� 16 +15 0 5 +3 +� +, +bΣ6 +i += +� 16 +15 0 5 +3 +� +, +bH2 +i += +� 1 +10 +1 +6 0 +� +, +(A.1) +whereas the 2-loop gauge coefficients read +b +φRe +1 +ij += +� +� +0 0 0 +0 28 +3 0 +0 0 0 +� +� , +b +φIm +1 +ij += +� +� +0 0 0 +0 28 +3 0 +0 0 0 +� +� , +b +φIm +3 +ij += +� +� +25 +12 +15 +4 +20 +3 +5 +4 +13 +4 +4 +5 +6 +3 +2 +11 +3 +� +� , b +φIm +3 +ij += +� +� +25 +12 +15 +4 +20 +3 +5 +4 +13 +4 +4 +5 +6 +3 +2 +11 +3 +� +� , +b +φRe +8 +ij += +� +� +0 0 0 +0 0 0 +0 0 21 +� +� , +b +φIm +8 +ij += +� +� +0 0 0 +0 0 0 +0 0 21 +� +� , +bT1 +ij = +� +� +4 +75 0 16 +15 +0 0 0 +2 +15 0 11 +3 +� +� , +bT2 +ij = +� +� +4 +75 0 16 +15 +0 0 0 +2 +15 0 11 +3 +� +� , +– 24 – + +bΦ1 +ij = +� +� +729 +25 81 0 +27 +245 +3 0 +0 +0 0 +� +� , bΦ3 +ij = +� +� +64 +25 +96 +5 +64 +5 +32 +5 56 32 +8 +5 12 11 +� +� , bΦ6 +ij = +� +� +1 +75 +3 +5 +8 +3 +1 +5 13 40 +1 +3 15 230 +3 +� +� , bΦ10 +ij += +� +� +72 +5 0 144 +0 0 +0 +18 0 195 +� +� , +bΣ1 +ij = +� +� +54 +25 +36 +5 0 +12 +5 +64 +3 0 +0 0 0 +� +� , +bΣ1 +ij = +� +� +54 +25 +36 +5 0 +12 +5 +64 +3 0 +0 0 0 +� +� , +bΣ3 +ij = +� +� +1 +300 +3 +20 +4 +15 +1 +20 +49 +4 +4 +1 +30 +3 +2 +38 +3 +� +� , bΣ3 +ij = +� +� +1 +300 +3 +20 +4 +15 +1 +20 +49 +4 +4 +1 +30 +3 +2 +38 +3 +� +� , +bΣ6 +ij = +� +� +64 +75 0 +32 +3 +0 0 0 +4 +3 0 125 +3 +� +� , bΣ6 +ij = +� +� +64 +75 0 +32 +3 +0 0 0 +4 +3 0 125 +3 +� +� , bH2 +ij = +� +� +9 +50 +9 +10 0 +3 +10 +13 +6 0 +0 0 0 +� +� . +(A.2) +References +[1] J. 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[Addendum: +Phys.Rev.Lett. 117, 109903 (2016)]. +– 30 – + diff --git a/H9AyT4oBgHgl3EQf5vo1/content/tmp_files/load_file.txt b/H9AyT4oBgHgl3EQf5vo1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fd3d1ef05fdd49d8abe43ba59c8ae9f041282b8b --- /dev/null +++ b/H9AyT4oBgHgl3EQf5vo1/content/tmp_files/load_file.txt @@ -0,0 +1,1281 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf,len=1280 +page_content='Fully Testable Axion Dark Matter within a Minimal SU(5) GUT Stefan Antusch,a Ilja Doršner,b,c Kevin Hinze,a and Shaikh Saada aDepartment of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland bUniversity of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture in Split, Ruđera Boškovića 32, HR-21000 Split, Croatia cJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Stefan Institute, Jamova 39, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Box 3000, SI-1001 Ljubljana, Slovenia E-mail: stefan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='antusch@unibas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='ch, dorsner@fesb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='hr, kevin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='hinze@unibas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='ch, shaikh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='saad@unibas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='ch Abstract: We present a minimal Grand Unified Theory model, based on SU(5) gauge symmetry and a global U(1) Peccei-Quinn symmetry, that predicts the ex- istence of an ultralight axion dark matter within a narrow mass range of ma ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='7] neV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This mass window is determined through an interplay between gauge coupling unification constraints, partial proton decay lifetime limits, and the need to reproduce the experimentally observed fermion mass spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The entire parame- ter space of the proposed model will be probed through a synergy between several low-energy experiments that look for proton decay (Hyper-Kamiokande), axion dark matter through axion-photon coupling (ABRACADABRA and DMRadio-GUT) and nucleon electric dipole moments (CASPEr Electric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='00809v1 [hep-ph] 2 Jan 2023 Contents 1 Introduction 1 2 The model 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='1 Scalar sector 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='2 Fermion sector 6 3 Peccei-Quinn symmetry and axion dark matter 9 4 Unification, axion mass and proton decay 14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='1 Unification 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='2 Proton decay 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='3 Numerical procedure 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='4 Results 18 5 Conclusions 24 A Renormalization group running of the gauge couplings 24 1 Introduction The Standard Model (SM) of elementary particle physics has performed exquisitely in explaining a multitude of experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' There are, however, several important questions that evidently require physics beyond the SM in order to be fully addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' For example, one of the most important discoveries in particle physics is the observation of nonzero neutrino masses, whereas neutrinos are strictly massless within the SM framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Furthermore, it is well established that approximately 26% of the total energy density of the universe is in the form of the so-called dark matter that cannot be of the SM origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This is especially puzzling as the stable SM matter only represents about 5% of the energy density of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Also, the strong CP problem — why the QCD θ parameter takes the value 10−10 or less — is still an open issue within the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' It might be that all these issues are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' In fact, the unified gauge theory [1– 6] formulation of the elementary particle interactions is a very popular and successful tool for tackling the aforementioned shortcomings of the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The simplest possible scenario, among various possible choices of the Grand Unified Theory (GUT) groups, is the Georgi-Glashow model [3] that embeds the entire SM gauge group within an – 1 – SU(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' In that construction, one 5-dimensional and one 10-dimensional representa- tion of SU(5) comprise all the fermions of a single SM family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The SU(5) symmetry is broken down to the SM gauge group when a real Higgs in the adjoint representa- tion acquires a vacuum expectation value (VEV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The SM symmetry is subsequently broken to SU(3) × U(1)em by the VEV of the SM Higgs doublet that resides within a fundamental representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The Georgi-Glashow model, however, is incomplete since (i) it fails to achieve gauge coupling unification, (ii) it predicts wrong mass relations between down-type quarks and charged leptons, and (iii) neutrinos remain massless as in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' On top of that, the Georgi-Glashow model does not address the strong CP problem, nor does it include a dark matter candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The most compelling new physics resolution of the strong CP problem is given in terms of the Peccei-Quinn (PQ) symmetry [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' In the PQ framework, a global U(1)PQ symmetry is spontaneously broken by a complex scalar leading to a nearly massless pseudoscalar particle [9–14], namely the “axion”, which can, in turn, serve as a cold dark matter candidate [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Intriguingly, as first shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' [18], the axion can be embedded within the scalar representation that breaks the GUT sym- metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The model presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' [18] did not, however, address several important GUT issues, such as neutrino mass generation and gauge coupling unification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' For a sample of models that pursue this particular approach, but with a more realistic agenda, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' [19–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Our primary interest in this manuscript is to combine the PQ symmetry with a simple, yet realistic, SU(5) GUT scenario [23, 24] and to investigate the main predictions of such a setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The SU(5) proposal [23, 24] in question extends the particle content of the Georgi-Glashow model by a 35-dimensional Higgs represen- tation and a 15-dimensional vectorlike fermion representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Remarkably, within that scenario, the observed mismatch between the down-type quarks and charged leptons is intrinsically connected to the neutrino mass generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' More specifically, the difference between the down-type quark and charged lepton mass matrices is given by a rank-one matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This stipulates that the down-type quarks and charged leptons have similar, yet, different masses, in accordance with experimental observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The neutrino mass matrix, on the other hand, is made out of a sum of two rank-one matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This, in turn, dictates that one of the neutrinos is a strictly mass- less particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Moreover, since the model relates these three rank-one matrices, the neutrino masses consequentially mirror the mismatch between the down-type quark and charged lepton masses and are thus of the normal hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We extend the minimal realistic SU(5) proposal [23, 24] with a PQ symmetry to address the strong CP problem as well as the origin of dark matter and show that such a simple extension still preserves the most prominent features of the orig- inal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Our detailed study reveals that the proposed setup is highly predictive, and that the entire parameter space of the theory will be fully tested in the near future through a combination of several experiments that include the proton de- – 2 – cay experiment Hyper-Kamiokande as well as the axion dark matter experiments ABRACADABRA, DMRadio-GUT, and CASPEr Electric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The manuscript is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 2 we introduce the particle con- tent and symmetries of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The details of the PQ symmetry implementation and the nature of the axion dark matter are discussed in detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' A numerical study of the model is performed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 4, where we also present the most relevant experimental predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We briefly conclude in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 2 The model The model in question comprises 5F i ≡ Fα i, 10F j ≡ T αβ j = −T βα j , 15F ≡ Σαβ = Σβα, 15F ≡ Σαβ, 5H ≡ Λδ, 5′ H ≡ Λ′ δ, a complex 24H ≡ φα β, 35H ≡ Φαβγ, and 24V ≡ Γα β, where Hs, Fs, and V denote whether a given irreducible representation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=', irrep, contains scalars, fermions, or gauge bosons, respectively, i, j (= 1, 2, 3) represent the generation indices, and α, β, γ, δ (= 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' , 5) are the SU(5) indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The decomposition of the SU(5) scalar and fermion irreps under the Standard Model (SM) gauge group SU(3)×SU(2)×U(1) is presented in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We will sometimes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' SU(5) SU(3) × SU(2) × U(1) SU(5) SU(3) × SU(2) × U(1) Λ(′) 1 � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' + 1 2 � Li � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' − 1 2 � 5(′) H ≡ Λ(′) α Λ(′) 3 � 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' − 1 3 � 5F i ≡ Fα i dc i � 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' + 1 3 � φ0 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 0) Qi � 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' + 1 6 � φ1 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 0) 10F i ≡ T αβ i uc i � 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' − 2 3 � 24H ≡ φα β φ3 � 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' − 5 6 � ec i (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' +1) φ3 � 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' + 5 6 � Σ1(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' −1) φ8 (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 0) 15F ≡ Σαβ Σ3 � 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' − 1 6 � Φ1 � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' − 3 2 � Σ6 � 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' + 2 3 � Φ3 � 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' − 2 3 � Σ1 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' +1) 35H ≡ Φαβγ Φ6 � 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' + 1 6 � 15F ≡ Σαβ Σ3 � 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' + 1 6 � Φ10 � 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' +1 � Σ6 � 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' − 2 3 � Table I: Content and nomenclature of the scalar and fermion irreps of the proposal at both the SU(5) and SM levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' α, β, γ (= 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' , 5) are the SU(5) indices while i(= 1, 2, 3) is a generation index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' for convenience, refer to a given irrep/multiplet by using either its dimensionality under the gauge group or the associated symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Beside the non-trivial assignment under the Lorentz symmetry, the aforemen- tioned SU(5) irreps carry the PQ U(1)PQ charges that are presented in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' – 3 – SU(5) irrep 5F i 10F i 15F 15F 5H 5′ H 24H 35H 24V U(1)PQ charge − 1 2 − 1 2 − 1 2 − 1 2 −1 +1 +1 −1 0 Table II: U(1)PQ charge assignment of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' H, F, and V subscripts denote scalar, fermion, or gauge boson SU(5) irreps, respectively, while i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Before we write down and discuss relevant parts of the model Lagrangian, we briefly justify the proposed particle content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 24H breaks the SU(5)×U(1)PQ symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' It furthermore provides axion dark matter (DM), helps to generate unification of the SM gauge coupling constants, and facilitates a process of creation of the experimentally observed mismatch between the down-type quark and charged lepton masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 5H and 5′ H jointly break the SM gauge symmetry down to SU(3)×U(1)em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 5′ H also provides the up-type quark masses through its vacuum expectation value (VEV), whereas 5H and 5′ H together play an indispensable role in three different mechanisms that create phenomenologically viable masses for the down-type quarks, charged leptons, and neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 35H is essential for neutrino mass generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' It also helps to provide the gauge coupling unification at scales compatible with the existing limits on partial proton decay lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 15F and 15F participate in the neutrino mass generation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' In addi- tion to that, these SU(5) irreps are, together with 24H and 5H, instrumental in addressing the observed mismatch between the down-type quark and charged lepton masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='1 Scalar sector There are several parts of the scalar sector of the model that need to be discussed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The SU(5) × U(1)PQ symmetry breaking is due to L ⊃ −µ2φ∗β α φα β + ξ1(φ∗β α φα β)2 + ξ2φ∗β α φα γφ∗γ δ φδ β + ξ3φ∗β α φδ γφ∗α β φγ δ + ξ4φ∗β α φδ γφ∗α δ φγ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='1) The VEV of φα β that does the SU(5) symmetry breaking reads ⟨φ⟩ = vφ √ 15diag(−1, −1, −1, 3/2, 3/2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='2) where we assume that the VEV of the electrically neutral component of the SU(2) triplet φ1(∈ 24H) is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The masses of the SM multiplets in 24H that are generated via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='1) are given in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We emphasize that 24H also breaks – 4 – the PQ symmetry while we currently discuss solely the SU(5) symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (Hence the omission of an overall phase in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The exact role of that phase will be discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=') multiplet real part mass-squared imaginary part mass-squared φ0 (1, 1, 0) m2 1 0 φ1 (1, 3, 0) m2 3 1 4m2 3 + m2 8 φ8 (8, 1, 0) 1 4m2 3 m2 8 φ3 � 3, 2, − 5 6 � 0 m2 5/6 φ3 � 3, 2, + 5 6 � 0 m2 5/6 Table III: Mass-squared spectrum of a complex irrep 24H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The potential given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='1) dictates that the imaginary part of φ0(∈ 24H) is massless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' In fact, the axion is mostly composed of that particular state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The real components of φ3(∈ 24H) and φ3(∈ 24H), on the other hand, provide the necessary degrees of freedom for the proton decay mediating gauge bosons in 24V to obtain a mass MGUT, where M 2 GUT = 5π 6 αGUTv2 φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='3) Here, MGUT is also the scale of gauge coupling unification, and αGUT is the corre- sponding SU(5) gauge coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The scalar fields in the fundamental irreps of SU(5) couple via L ⊃ −1 2µ2 Λ(′)Λ(′)†Λ(′) + γΛ(′) � Λ(′)†Λ(′)�2 + ζ1 � Λ†Λ � � Λ′†Λ′� + ζ2 � Λ†Λ′� � Λ′†Λ � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='4) where we suppress SU(5) indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The doublet-triplet spitting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=', breaking of the mass degeneracy between Λ(′) 1 and Λ(′) 3 multiplets, is accomplished via the following additional terms in the scalar potential: L ⊃ λΛ(′)Λ(′)†Λ(′)φ†φ + Λ(′)† � αΛ(′)φ†φ + βΛ(′)φφ†� Λ(′) + � κ1Λ′†φ2Λ + κ2 � Λ′†Λ � φ2 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='5) The VEVs of 5H and 5′ H that break SU(3) × SU(2) × U(1) down to SU(3) × U(1)em read ⟨Λ(′)⟩ = (0 0 0 0 vΛ(′))T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The lepton number conservation is violated through a single term in the La- grangian that reads L ⊃ λΛαΛ′βΛ′γΦαβγ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='. (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='6) The neutrino masses will thus be directly proportional to the dimensionless parameter λ of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' – 5 – The masses of the SM gauge group multiplets in 35H are determined by the following SU(5) contractions L ⊃ µ2 35ΦΦ∗ + λ0 (ΦΦ∗) φ∗φ + λ1Φαβγ(Φ∗)αδϵ(φ∗)β δ φγ ϵ + λ2Φαβϵ(Φ∗)αβδ(φ∗)ϵ γφγ δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='7) The contractions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='7) yield a single mass-squared relation that reads M 2 Φ10 = M 2 Φ1 − 3M 2 Φ3 + 3M 2 Φ6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='8) The mass spectrum given in Table III and the mass relation presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='8) are necessary inputs for the gauge coupling unification analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='2 Fermion sector The Yukawa sector of the model is L ⊃ Y u ij 10F i10F j5′ H + Y d ij 10F i5F j5∗ H + Y a i 15F5F i5∗ H + Y b i 15F5F i35∗ H + Y c i 10F i15F24H + y 15F15F24H + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=', (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='9) where the PQ charge assignment of Table II and the SU(5) indices are all implicitly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The Yukawa matrix elements of the model are Y u ij ≡ Y u ji, Y d ij = Y d∗ ij ≡ δijY d i , Y a i , Y b i , Y c i , and y = y∗, where we have used the freedom to rotate irreps in the SU(5) group space to reach this particular Yukawa coupling basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The model accordingly has nineteen real parameters and fourteen phases in the Yukawa sector to accommodate all of the masses and mixing parameters of the SM fermions as well as the masses of fermions in the 15F-15F pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The PQ charge assignment forbids a bare-mass term for the 15F-15F pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The masses of the associated SM gauge group multiplets are thus generated solely through the last term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='9), which reads L ⊃ yvφ √ 15 �3 2Σ1Σ1 + 1 4Σ3Σ3 − Σ6Σ6 � + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=', (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='10) where the overall phase of 24H, once again, is not shown for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We subse- quently define MΣ1 = y 2 � 3 5vφ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='11) MΣ3 = y 4 √ 15vφ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='12) MΣ6 = − y √ 15vφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='13) It is important to point out that apart from different Clebsch coefficients, all submul- tiplets within 15F have a common mass scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (Even though Σ1 and Σ3 mix with the – 6 – fermions in 5F i and 10F i, this does not affect equalities in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=') We will show, later on, that the product yvφ is rather constrained by a requirement for the model to simultaneously generate large enough unification and neutrino mass scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The masses of the SM fermions are obtained after the breaking of the SM gauge group down to SU(3) × U(1)em as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The down-type quark sector 4 × 4 mass matrix can be written as MD = �vΛY d v′ φY c vΛY a MΣ3 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='14) where we introduce v′ φ = − 1 4 � 5 3vφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This matrix can be transformed into a block- diagonal form comprising a 3 × 3 part denoted Md and a mass parameter MH as follows XMDY † = �Md 0 0 MH � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='15) where unitary matrices X and Y take the form X ∼ � � � 1 + v′2 φ M2 Σ3 Y cY c†�−1/2 − � 1 + v′2 φ M2 Σ3 Y cY c†�−1/2 v′ φ MΣ3 Y c v′ φY c† MH MΣ3 MH � � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='16) Y ∼ � � 1 − vΛv′ φ M2 H (Y d†Y c + MΣ3 v′ φ Y a†) vΛv′ φ M2 H (Y c†Y d + MΣ3 v′ φ Y a) 1 � � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='17) with Md ∼ � 1 + v′2 φ M 2 Σ3 Y cY c† �−1/2 � vΛY d − vΛv′ φ MΣ3 Y cY a � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='18) MH = � M 2 Σ3 + v′ φ 2Y c†Y c ≈ MΣ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='19) Here, 1 = diag(1, 1, 1) while Y c, Y a, and Y d are Yukawa matrices that are featured in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' It is clear from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='18) that the down-type quark mass matrix Md is generated through the VEV of 5′ H and the mixing between fields in 5F i, 10F i, 15F, and 15F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This is possible due to the fact that Σ3 ∈ 15F and Qi ∈ 10F i transform in the exact same way under the SM gauge group [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The charged fermion mass matrices of the model can be succinctly written as Mu = � 1 + δ2 Y cY c†�− 1 2 8vΛ′Y u, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='20) Md = � 1 + δ2 Y cY c†�− 1 2 vΛ � Y d + δ Y cY a� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='21) Me = vΛY d, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='22) – 7 – where δ = −v′ φ/MΣ3 and v2 Λ + v2 Λ′ = v2 with v = 174 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We note the two most prominent features of the charged fermion sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' First, Mu can be treated as a symmetric matrix in the flavor space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Second, a mismatch between the charged lepton and down-type quark mass matrices is proportional to a rank-one matrix Y cY a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We again note that we work in the basis where Y u ij ≡ Y u ji and Y d ij = Y d∗ ij ≡ δijY d i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This simply means that vΛY di, i = 1, 2, 3, are the masses of the SM charged leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The neutrino mass matrix elements (Mν)ij, at the leading order, read (Mν)ij ≈ λv2 Λ′ 8π2 (Y a i Y b j + Y b i Y a j ) MΣ1 M 2 Σ1 − M 2 Φ1 ln �M 2 Σ1 M 2 Φ1 � ≡ m0(Y a i Y b j + Y b i Y a j ) = (N diag(0, m2, m3) N T)ij , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='23) where m2 and m3 are neutrino mass eigenstates and N is a unitary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Note that one of the neutrinos is a strictly massless particle due to the fact that Mν is constructed out of two rank-one matrices with elements Y a i Y b j and Y b i Y a j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This is accordingly encoded in the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Since the charged lepton mass matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='22) is already in a diagonal form, we can write that N = diag(eiην 1, eiην 2, eiην 3)V ∗ PMNS, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='24) where VPMNS is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) unitary mixing ma- trix, that is defined as VPMNS = R23U13R12Q, with Q = diag(1, eiβν, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Here we use the PDG parametrization [26] for the R23, U13, and R12 matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Note that there is only one Majorana phase βν appearing in Q due to the fact that one of the neutrinos is massless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' One especially convenient feature of the neutrino sector is that the matrices Y a and Y b can be expressed in terms of the PMNS matrix parameters and phases ην i , i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Using the parametrization mentioned in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' [27, 28] we can write the two Yukawa coupling vectors Y a and Y b as Y a T = ξ √ 2 � � i r2 N12 + r3 N13 i r2 N22 + r3 N23 i r2 N32 + r3 N33 � � , Y b T = 1 √ 2ξ � � −i r2 N12 + r3 N13 −i r2 N22 + r3 N23 −i r2 N32 + r3 N33 � � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='25) where r2 = � m2/m0 and r3 = � m3/m0, and where Nij denotes the ij-th element of the unitary matrix N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Moreover, ξ is a dimensionless scaling parameter that needs to be introduced if one is to accurately scan over all possible phenomenologi- cally viable entries in Y a and Y b that accommodate experimental observables in the neutrino sector with utmost certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='25) is applicable solely to the normal neutrino mass hierarchy scenario since that is one of the model predictions, as we will discuss later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' For alternative ways of generating neutrino masses within the SU(5) framework, see, for example, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' [29–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' – 8 – 3 Peccei-Quinn symmetry and axion dark matter We discuss the implementation of the PQ symmetry within our setup and elabo- rate on the model’s main ingredients and experimental detection prospects in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' In the “invisible axion” models [11–14] the PQ symmetry is broken by a scalar field that carries a non-trivial PQ charge, where the scalar is a singlet under the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We embed this scalar within the 24-dimensional Higgs irrep that is charged under the U(1)PQ symmetry, as shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Consequently, our setup unifies the GUT and PQ breaking scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The VEV of 24H ≡ φα β can be written as ⟨φ⟩ = ˆvφ √ 2diag � −1 √ 15, −1 √ 15, −1 √ 15, 3 2 √ 15, 3 2 √ 15 � eiaφ(x)/ˆvφ, ˆvφ ≡ √ 2vφ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='1) where the pseudoscalar part, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=', field aφ(x), essentially remains massless, whereas the radial mode acquires a mass of the order of the GUT scale while the global U(1)PQ symmetry is spontaneously broken with order parameter vφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' To correctly identify the massless axion, one also needs to include all other Higgses that carry PQ charges and participate in symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The non-Hermitian operators that are responsible for the breaking of the re- phasing symmetry of the three scalar fields are given by the terms in the second line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The VEVs of neutral components of the SU(2) doublets can be re-written as ⟨Λ′⟩ = ˆvΛ′ √ 2e i aΛ′ ˆvΛ′ , ⟨Λ∗⟩ = ˆvΛ √ 2e i aΛ ˆvΛ , ˆvΛ(′) ≡ √ 2vΛ(′), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='2) where we take all VEVs to be real, and, as mentioned before, we neglect the VEV of the SU(2) triplet in 24H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' With these assumptions, the axion field is identified as [39], a = xΛ′ˆvΛ′aΛ′ + x∗ ΛˆvΛaΛ + xφˆvφaφ va , v2 a = x2 Λ′ˆv2 Λ′ + x2 Λˆv2 Λ + x2 φˆv2 φ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='3) where xi denotes the PQ charge of the corresponding i-th scalar (and x∗ i = −xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Since vφ ∼ 1016 GeV and vΛ(′) ∼ 102 GeV, the axion mostly resides in 24H with a ≈ aφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The axion field must also be orthogonal to the Goldstone field eaten up by the Z-boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This translates into the following condition tan2 β = v2 Λ′ v2 Λ = x∗ Λ xΛ′ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='4) which, in our benchmark charge assignments, fixes tan β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Here, we do not present the expression of the SM Higgs mass eigenstate, which can be obtained via – 9 – the diagonalization of the 4 × 4 mass matrix of the CP-even states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The heaviest one is expected to reside at the GUT scale, and the lightest one is the SM Higgs boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Depending on the chosen hierarchy, the remaining two eigenstates — one coming from the triplet and the other from the pair of doublets — can live anywhere in between the electroweak and GUT scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Now, performing a field-dependent axial transformation that is anomalous under QCD, the axion can be disentangled from the Yukawa interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This transfor- mation generates the effective anomalous interactions of the following types: δLeff = αs 8π a fa G �G + � αem 2πfa E N � a 4F �F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='5) Here, G (F) is the gluon (photon) field strength tensor, �G ( �F) is its dual, and fa is the axion decay constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The effective operator of the form aG �G is the key to the PQ solution to the strong CP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Since these sub-multiplets carry color and electromagnetic charges, the PQ current has both QCD and electromagnetic anomalies, with the corresponding anomaly coefficients [40], N = � ψ Nψ, E = � ψ Eψ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='6) where sums are taken over all fermions, which we generically denote by ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Using well-known formulas, Nψ = xψd(Iψ)T(Cψ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='7) Eψ = xψd(Cψ)d(Iψ) � 1 12(d(Iψ)2 − 1) + Y 2 ψ � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='8) we obtain |N| = 13/2 and |E| = 52/3, and the domain-wall number is NDW = 2N = 13, which is relevant for cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Subsequently, we find the axion decay constant to be fa = va 2N ≈ ˆvφ 2N = � 3 10παGUT MGUT N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='9) Since the decay constant is of the order of the GUT scale, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=', fa ∼ MGUT, we refer to the axion as the “GUT axion”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Once strong interactions confine, non-perturbative QCD effects generate a potential that gives rise to a tiny axion mass [41, 42] ma = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='7 neV �1015 GeV fa � = ma = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='7 neV �1015 GeV MGUT � N � 10παGUT 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='10) This shows that the axion mass is predicted if the grand unification scale MGUT is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We accordingly compute the predicted range of the GUT scale within our model in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 4 by taking into account all relevant constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' – 10 – 10-12 10-11 10-10 10-9 10-8 10-20 10-19 10-18 10-17 10-16 ma [eV] gaγγ [GeV-1] DMRadio-GUT Res I: ABD Res II: ABD Res III: ABD Broad I: ABD Broad II: ABD Broad III: ABD GUT axion optomechanical cavity Figure 1: Expected reach in the ma vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' gaγγ plane for the broadband (Broad) and resonant (Res) strategies of the ABRACADABRA (ABD) experiment [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The blue line corresponds to the prediction of our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The projected 3 σ sensitiv- ity of DMRadio-GUT [44, 45] is also presented in the green shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Further- more, the expected theoretical reach using the optomechanical cavity method [46] is shown with solid black lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' See text for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Since the non-observation of proton decay requires the GUT scale to be large, the axion mass is expected to be around the neV scale within our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' An axion in this mass range is extremely weakly coupled to the SM particles due to an ex- tremely large decay constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Remarkably, an axion with neV mass can serve as an excellent dark matter candidate and can be searched for efficiently in direct detection experiments [47] hunting for ultra-light axions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Next, we consider the most important axion couplings relevant for experimental sensitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' In the low-energy effective Lagrangian for the axion, it is sometimes convenient to eliminate the axion coupling to the gluons via a field-dependent axial transformation of the SM quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' After making such a rotation, the axion coupling to the photons is given by [42], L ⊃ αem 2πfa � E N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='92 � � �� � ≡gaγγ a 4F �F, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='11) where the model-dependent quantity, apart from fa (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='9)), in our case, is given by E/N = 8/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' In fact, the dark matter experiment ABRACADABRA [43] has great potential to look for an axion dark matter in the mass range of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 1, a major part of the parameter space of our theory will be probed by this dark matter direct detection experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 1 is obtained by varying model – 11 – 10-12 10-11 10-10 10-9 10-8 10-7 10-23 10-21 10-19 10-17 10-15 10-13 10-11 ma [eV] |gaD| [GeV-2] CASPEr Electric phase I phase II phase III spin noise GUT axion Figure 2: Axion coupling to the nucleon EDM operator as a function of the axion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The blue band corresponds to the prediction of our model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' see text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The shaded regions show the sensitivity projections of CASPEr Electric [48, 49] in its various phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Moreover, the ultimate sensitivity limit is given by the nuclear spin noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' the parameters while imposing all relevant constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The details of our numerical procedure are relegated to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Another axion dark matter experiment, the DMRadio-GUT [44, 45], will also be sensitive in detecting axions with GUT scale decay constant fa ∼ 1016 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' DMRadio-GUT will be far more sensitive compared to its previous two phases, DMRadio-50L and DMRadio-m3, since it will have a factor of three enhancement in the field and a factor of ten enhancement in volume relative to DMRadio-m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The projected 3 σ sensitivity of DMRadio-GUT is also presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 1 by a green shaded region, which will probe a significant portion of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Yet another proposal utilizing an optomechanical cavity [46] filled with superfluid he- lium is shown to be highly promising in detecting ultra-light axion dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This proposed experimental method, with a cavity size of order O(10 m) is expected to be sensitive to axion-photon couplings for axions with the GUT scale size decay constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 1, the corresponding theoretical reach is shown with solid black lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The ABRACADABRA experiment will be sensitive to axion masses as low as ma ∼ 2 neV, whereas it is about ma ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='4 neV and ma ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='1 neV for DMRadio- GUT and optomechanical cavity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' A combination of all these axion dark matter experiments will eventually probe the entire parameter space of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Intriguingly, ultra-light axion dark matter can also be efficiently searched via oscillating nucleon electric dipole moments (EDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' As aforementioned, the QCD – 12 – axion solves the strong CP problem by promoting the θ parameter into the dynamical axion field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Consequently, the effective θ angle gives rise to an EDM for nucleons sourced by the axion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Owing to the dynamical nature of the axion, this EDM will change in time, giving rise to unique signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' In the effective Lagrangian, the coupling of the axion to nucleon n takes the following form, L ⊃ − i 2gaD aψnσµνγ5ψnF µν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='12) The nucleon electric dipole moment generated through the above operator is given by dn = gaDa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' In terms of our model parameters, it can be re-written in the following form [50]: dn ≈ a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='4 × 10−16 fa e · cm � �� � gaD , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='13) with roughly a 40% uncertainty [51], where the decay constant is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (See also Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' [52–54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=') The corresponding coupling as a function of the axion mass is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' As can be seen from this figure, excitingly, the CASPEr Electric [48, 49] experiment alone will probe almost the entire parameter space of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The width of the band corresponds to the uncertainty in the calculation as aforementioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 1, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 2 is also obtained by varying model parameters by imposing all relevant constraints, which will be discussed later in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Since the axion is ultra-light in our setup, it can constitute the entirety of the dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' It is important to point out that the breaking of the GUT symmetry to that of the SM gauge group SU(5) × U(1)PQ → SU(3) × SU(2) × U(1) leads to an overproduction of super-heavy monopoles that must be inflated away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' As discussed above, spontaneous breaking of the PQ symmetry leads to NDW distinct degenerate vacua, giving rise to a domain-wall problem, which also requires dilution to be consistent with cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Both of these problems, along with the horizon and flatness problems, can be elegantly solved via inflation taking place after the GUT symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We, however, do not specify the details of the inflationary dynamics, which is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The amount of axion dark matter produced then depends on whether the PQ symmetry is restored or not after inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We assume that the U(1)PQ remains broken during inflation and is never restored afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' In such a scenario, the relic abundance of the axion dark matter is given by [55], Ωh2 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='12 �5 neV ma �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='17 � θi 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='53 × 10−2 �2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='14) which shows that the initial value of θi = ai/fa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' where ai is the initial value of the axion field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' needs to be somewhat smaller than unity to be consistent with the – 13 – 102 104 106 108 1010 1012 1014 1016 1018 10 20 30 40 50 60 μ [GeV] α-1(μ) Mϕ1 Re Mϕ8 Re Mϕ1 Im Mϕ3 Im Mϕ8 Im MΦ3 MΦ6 MH2 MΣ1 MΣ3 MΣ6 MΦ1 MΦ10 MT1 MT2 α1 1 α2 1 α3 1 α-1 Figure 3: Example for the choice of the intermediate-scale particle masses giving gauge coupling unification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' observed dark matter relic abundance Ωh2 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='001 [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Thus, for θi ∼ 10−2, the axion consists of 100% of the dark matter within our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Before we conclude this section, we note that since the PQ symmetry is assumed not to get restored after inflation, our scenario cannot be tested at gravitational wave observatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 4 Unification, axion mass and proton decay In our model, the axion decay width fa is connected to the GUT scale MGUT due to the fact that the adjoint GUT Higgs field 24H simultaneously breaks the SU(5) and U(1)PQ symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This, in particular, directly relates the axion mass ma to the GUT scale MGUT via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Moreover, since the partial proton lifetimes are proportional to the fourth power of the GUT scale, our model can be simultaneously probed with axion dark matter and proton decay experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' – 14 – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='1 Unification The renormalization group equations (RGEs) for the gauge couplings can, at the 2-loop level, be written as [57] µdα−1 dµ = − 1 2π � bSM i + � J bJ i H(µ − MJ) � − 1 8π2 �� J � bSM ij + bJ ijH(µ − MJ) � α−1 j + βY i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='1) Here, bSM i (bSM ij ) are the SM 1-loop (2-loop) gauge coefficients, while bJ i (bJ ij) are the 1-loop (2-loop) gauge coefficients of the multiplets J with intermediate-scale masses MJ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=', MZ < MJ < MGUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' These coefficients are listed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Moreover, βY i are the Yukawa contributions and H is the Heaviside step function defined as H(m) = � 1, m > 0 0, m ≤ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='2) Note that we neglect the effect of the Yukawa couplings Y a, Y b, and Y c on the running of the gauge couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' In order to investigate the viable part of the parameter space giving gauge coupling unification, we freely vary the masses of the fields φRe 1 , φIm 1 , φIm 3 , φRe 8 , φIm 8 , Σ1, Σ3, Σ6, Φ1, Φ3, Φ6, Φ10, T1, T2, and H2 respecting the mass relations presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Here, T1,2 and H1,2 refer to the mass eigenstates of the scalar color triplets and weak doublets, where H1 is identified with the SM Higgs with 125 GeV mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We ensure that the scalar leptoquark mediated proton decay is sufficiently suppressed by taking a lower bound of 3 × 1011 GeV for the masses of T1 and T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The masses of the remaining multiplets are freely varied between the TeV and the GUT scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The numerical fit is performed by running the gauge couplings at the 2-loop level from the GUT scale to the Z mass scale at which a χ2-function that we define later in detail is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We use the low-scale values g1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='461425+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='000044 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='000043, g2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='65184+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='00018 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='00017, g3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='2143+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='0035 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='0036 [58] as our input, where gi = √4παi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' To demonstrate that within our setup, the gauge couplings can indeed unify, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 3 shows one possible choice of the particle mass spectrum giving exact gauge coupling unification in agreement with the current proton decay constraints and also for a choice of the masses of Σ1 and Φ1 giving the correct neutrino mass scale via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='2 Proton decay The formulae for the proton decay widths of various decay channels can be found in [59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' For example, the decay width for the proton decay channel having a pion – 15 – and a charged lepton in the final state is given by1 Γ(p → π0e+ α) = mpπ 2 � 1 − m2 π m2 p �2 A2 L α2 GUT M 4 GUT (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='3) × � A2 SL|c(ec α, d)⟨π0|(ud)LuL|p⟩|2 + A2 SR|c(eα, dc)⟨π0|(ud)RuL|p⟩|2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Here, mp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='9393 GeV and mπ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='134 GeV denote the proton and pion masses, respectively, while AL = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='2 [62] and ASL(R) encode the leading log renormalization of the dimension six operators, where2 ASL(R) = � i=1,2,3 MZ≤MI≤MGUT � I �αi(MI+1) αi(MI) � γL(R)i bSM i +�MZ ≤MJ ≤MI J bJ i , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='4) with γL(R)i = (23(11)/20, 9/4, 2) [63–65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Moreover, we take the hadron matrix ele- ments, such as, for example, ⟨π0|(ud)LuL|p⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='134(5)(16) GeV2 and ⟨π0|(ud)RuL|p⟩ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='131(4)(13) GeV2, from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' [66, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Finally, the c-coefficients of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='3) read [68–70] c(ec α, dβ) = (U † RU ∗ L)11(E† RD∗ L)αβ + (E† RU ∗ L)α1(U † RD∗ L)1β , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='5) c(eα, dc β) = (U † RU ∗ L)11(E† LD∗ R)αβ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='6) c(νl, dα, dc β) = (U † RD∗ L)1α(D† RN)βl , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='7) where the unitary matrices UL/R, EL/R, DL/R, and N diagonalize the SM fermion mass matrices through the following transformations Mu = ULM diag u U † R, Md = DLM diag d D† R, Me = ELM diag e E† R, Mν = NM diag ν N T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='8) The current experimental constraints and future sensitivities for the various par- tial lifetimes that we use in our numerical analysis are presented in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' For a recent review on the subject, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='3 Numerical procedure We start our numerical analysis by constructing matrices Mu, Me, Y a, Y b, and Y c at the GUT scale, as described in the next few paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Since the up-type quark mass matrix Mu is approximately symmetric, we have that UR = U ∗ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This allows us to construct Mu as Mu = ULdiag(mu, mc, mt)U T L , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='9) 1The Mathematica package ProtonDecay [61] can be used to compute the decay widths of various nucleon decay channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 2If the denominator of the exponent vanishes for some factor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=', the 1-loop running of a specific gauge coupling is constant within a certain interval, the respective factor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='4) is replaced with exp[γL(R)iα(MI+1)]/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' – 16 – decay channel current bound τp [yrs] future sensitivity τp [yrs] p → π0 e+ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='4 · 1034 [72] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='8 · 1034 [73] p → π0 µ+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='6 · 1034 [72] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='7 · 1034 [73] p → η0 e+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='0 · 1034 [74] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='3 · 1034 [73] p → η0 µ+ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='7 · 1033 [74] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='9 · 1034 [73] p → K0 e+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='1 · 1033 [75] p → K0 µ+ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='6 · 1033 [76] p → π+ ν 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='9 · 1032 [77] p → K+ ν 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='6 · 1033 [78] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='2 · 1034 [73] Table IV: Present experimental bounds on the partial lifetimes τp as well as future sensitivities for 10 years of runtime, both at 90% confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' where we parametrize the up-type quark mixing matrix UL in terms of the down-type quark mixing matrix DL, the Cabibbo-Kobayashi-Maskawa (CKM) matrix VCKM, and five GUT phases βu 1 , βu 2 , ηu 1, ηu 2, and ηu 3, as UL = DLdiag(eiβu 1 , eiβu 2 , 1)V T CKMdiag(eiηu 1 , eiηu 2 , eiηu 3 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='10) In our analysis, we set ηu 1 = ηu 2 = ηu 3 = 0 since these three phases do not affect the proton decay predictions at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We set EL = ER = 1 since Me is diagonal and real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This also means that we can simply construct Me via an equality that reads Me = diag(me, mµ, mτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='11) Y a and Y b are constructed via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='25) using the neutrino mixing matrix N = (eiην 1, eiην 2, eiην 3)V ∗ PMNS as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Note that VPMNS contains the CP violating phase δν as well as the Majorana phase βν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We furthermore construct Y c to be a general complex 1 × 3 matrix through Y c = (yc 1eiηc 1, yc 2eiηc 2, yc 3eiηc 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='12) Once the parameter dependence of Mu, Me, Y a, Y b, and Y c is properly accounted for, as described above, we can also construct Md and Mν that are given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='21) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='23), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We treat λ in Mν as a free parameter while the two Higgs VEVs that enter Md and Mν are given by vΛ = vΛ′ = 174/ √ 2 GeV due to the constraint that tan β of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='4) is equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' In summary, the free parameters for our numerical analysis are the unification scale MGUT and the corresponding gauge coupling αGUT, the masses of the fields3 3Note that the masses of the fields φRe 1 , φIm 1 , Σ3, Σ6, Φ10 are obtained via the mass relations discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' – 17 – φIm 3 , φRe 8 , φIm 8 , Σ1, Φ1, Φ3, Φ6, T1, T2, and H2, the phases βu 1 , βu 2 , δν, βν, ην 1, ην 2, ην 3, the Yukawa parameters yc 1, yc 2, yc 3, the quartic Higgs coupling λ, and the scaling parameter ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' These 24 parameters are fitted to the experimental observables that are the SM gauge couplings g1, g2, and g3, and the down-type quark masses md, ms, and mb, while requiring that the current proton decay constraints, as given in Table IV, are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Note that the charged lepton masses, the up-type quark masses, the neutrino mass squared differences, the CKM mixing parameters, and the known PMNS mixing parameters are all automatically accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Since there are more parameters than observables, proton decay cannot be pre- dicted sharply in all decay channels as we will discuss in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' But, due to the fact that the neutrino mass matrix is connected to the mismatch between the charged lepton and down-type quark mass matrices, our model predicts the PMNS parameters δν and βν to be in relatively narrow intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The gauge couplings are fitted to their low-energy scale values [58] after the 2- loop level running from the high scale to the low scale is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' To simplify the analysis, the down-type quark and neutrino masses are directly fitted at the high scale, using the high scale values provided in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The χ2-function is obtained comparing the theoretical prediction pi with the experimental central value ei, normalized with the corresponding experimental standard deviation σi of the i-th observable via χ2 = � i �pi − ei σi �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='13) To minimize the χ2-function we apply a differential evolution algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This mini- mization yields a viable benchmark point and thus proves the viability of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Then, starting from this benchmark point with a flat prior distribution a Markov- chain-Monte-Carlo (MCMC) analysis involving a Metropolis-Hasting algorithm is performed, giving us a total of 6 × 106 datapoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Finally, we use these points to calculate the highest posterior density (HPD) regions of various quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' For the numerical analysis, all parameters are freely varied in such a way that the perturbativity of all Yukawa and Higgs couplings is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' In particular, the absolute values of all entries in Y a, Y b, and Y c, as well as the absolute value of λ are required to be less than or equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' To this end, the scaling parameter ξ ensures that the full parameter space is covered with the chosen parametrization of the matrices Y a and Y b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Furthermore, although we fix some model parameters during the fitting/minimization procedure by directly plugging in experimental central values of some observables, we still vary these parameters in the subsequent MCMC analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='4 Results In this section, we present the outcome of our numerical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We are interested in the full axion mass range, the predictions for partial proton decay lifetimes, and the – 18 – viable range of the Dirac CP and Majorana phases of the PMNS matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The axion mass ma is connected to the GUT scale MGUT and gauge coupling αGUT via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We can therefore obtain the predicted range of the axion mass by maximizing and minimizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We demand viable gauge coupling unification and correct neutrino mass scale while making sure that none of the current proton decay constraints is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We find ma ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='7] neV which we present in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' As discussed in more detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 3, this already demonstrates that the full parameter space will be probed by two kinds of future axion DM experiments that are sensitive to either the axion to photon coupling or to the nucleon EDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' To start our numerical analysis, we find a viable benchmark point from a full χ2 fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' In particular, for the case of normal neutrino mass ordering, we obtain that Y a = � −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='120 + i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='00943, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='513 + i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='200, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='898 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='14) Y b = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='109 + i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='150, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='348 + i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='334, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='195 − i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='0211 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='15) Y c = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='00115 + i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='00198, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='0532 + i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='0852, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='781 − i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='743 � × 10−6, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='16) for MGUT = 1016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='2 GeV, mH2 = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='77 GeV, MT1 = MT2 = 1014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='55 GeV, MφRe 1 = 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='39 GeV, MφIm 1 = 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='12 GeV, MφIm 3 = 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='40 GeV, MφRe 8 = 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='09 GeV, MφIm 8 = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='71 GeV, MΣ1 = 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='41 GeV, MΣ3 = 1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='63 GeV, MΣ3 = 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='24 GeV, MΦ1 = 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='63 GeV, MΦ3 = 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='28 GeV, MΦ6 = 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='18 GeV, MΦ10 = 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='63 GeV, α−1 GUT = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='62, and λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' If the proton decay pull is neglected, this choice of the input param- eters gives a total χ2 below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This is thus a perfect fit for the gauge couplings as well as for the fermion masses and mixings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Moreover, for this benchmark point, the PMNS Dirac CP phase is given by δν = −48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='5◦, whereas the PMNS Majorana phase is βν = −71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='3◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We note that for the case of inverted neutrino mass ordering, no good fit-point can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This is due to the fact that the Yukawa matrix Y a is needed to generate both the viable neutrino masses and the correct mismatch between the charged lepton and down-type quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' In the case of inverted ordering, the first two entries in Y a would need to be somewhat larger than the third entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This is, however, in conflict with the down-type quark mass fit that requires the first entry of Y a to be smaller than the second and third entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Therefore, a strong prediction of our model is that the neutrinos have normal mass ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' From the aforementioned benchmark point, we start an MCMC analysis with a flat prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' All obtained points are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 4 in a plane of axion mass vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' par- tial proton decay lifetime in the dominant decay channel p → π0e+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We also present the future sensitivities of the DM experiments ABRACADABRA, DMRadio-GUT, and Casper Electric, as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 3, as well as the future sensitivity of the proton decay experiment Hyper-Kamiokande, as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 4 nicely visualizes how a part of our parameter space can be probed through the synergy be- tween three different kinds of experiments testing (i) the axion to photon coupling, (ii) the nucleon EDM, and (iii) proton decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' For example, if the axion mass is – 19 – 0 1 2 3 4 5 1034 1035 1036 1037 1038 ma [neV] τ(p → π 0e+) [yrs] Hyper-K Super-K ABD DMRadio-GUT CASPEr Figure 4: The generated points from the MCMC analysis presented in the ma − τ(p → π0e+) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The current Super-Kamiokande bound is represented by a gray box, while the future Hyper-Kamiokande sensitivity is indicated by a blue dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Moreover, the projected sensitivity of various axion DM experiments is also shown: ABRACADABRA (ABD) with a red dotted line, DMRadio-GUT with a green dotted line, CASPEr Electric with a brown dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' For details, see the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' observed to be above 3 neV, proton decay in the decay channel p → π0e+ necessarily has to be seen by Hyper-Kamiokande if our model is realized in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Moreover, regardless of whether proton decay will be observed by Hyper-Kamiokande, the for- mer two kinds of experiments will be able to cover the entire parameter space of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We are also interested in the proton decay predictions of all the different decay channels within our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' First, we want to obtain the full allowed range for all partial proton lifetimes, which is for the decay channel p → π0e+ already hinted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' To this end, we vary all the parameters, including the intermediate-scale particle masses in the MCMC analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The 1 σ (dark) and 2 σ (light) HPD results of this analysis are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The blue line segments indicate the current experimental bounds, while the red line segments represent the future sensitivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (See, for example, Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=') Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 5 shows that a part of the predicted 1 σ HPD interval for the two decay channels p → π0e+ and p → η0e+ will be tested by Hyper- Kamiokande.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The large uncertainty in these partial lifetime predictions that are coming from the dependence on the fourth power of the GUT scale can be erased by considering ratios of specific decay channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 6 shows the prediction of such 4For recent works analyzing ratios of partial proton decay lifetimes in models with predicted GUT scale quark-lepton Yukawa ratios, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' [37, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' – 20 – p → π0e+ p → π0 μ+ p → K+ν p → π+ν p → K0e+ p → K0 μ+ p → η e+ p → η μ+ 1033 1034 1035 1036 1037 1038 1039 1040 1033 1034 1035 1036 1037 1038 1039 1040 τp [yrs] Figure 5: The predicted 1 σ (dark) and 2 σ (light) HPD intervals of the proton lifetime for various decay channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The blue (red) line segments indicate the current (future) experimental bounds (sensitivities) at 90% confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Interestingly, a part of the predicted 1 σ region for both decay channels p → π0e+ and p → η0e+ lies within the reach of Hyper-Kamiokande.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' p → π0 μ+ p → π0 e+ p → K+ ν _ p → π0 e+ p → π+ ν _ p → π0 e+ p → K0 e+ p → π0 e+ p → K0 μ+ p → π0 e+ p → η e+ p → π0 e+ p → η μ+ p → π0 e+ 1 3 10 30 100 300 1000 3000 1 3 10 30 100 300 1000 3000 τ Figure 6: The 1 σ (dark) and 2 σ (light) HPD intervals of ratios of the proton lifetime of various decay channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Interestingly, the ratio τ(p → η0e+)/τ(p → π0e+) (which will partly be tested by Hyper-Kamiokande) is predicted very sharply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' – 21 – ratios with the dominant decay channel p → π0e+ in the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Especially interesting is the prediction for the ratio τ(p → η0e+)/τ(p → π0e+), since both of the featured decay channels will be partly tested by Hyper-Kamiokande.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This ratio is predicted very sharply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' However, such a sharp prediction for this particular ratio is not only specific to our model but a more common feature of models in which gauge boson mediated proton decay is dominant and in which the contribution involving the c-coefficient c(ec, d) dominates the contribution with the c-coefficient c(e, dc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Nevertheless, these ratios of partial proton lifetimes provide an interesting additional opportunity to probe our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' In order to understand the dependence of the different decay channels on the flavor structure in the fermionic mass matrices, we fix the gauge coupling unification to the same values as listed below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='16) and only vary the parameters in the mass matrices in the MCMC analysis, computing for each point the proton decay prediction for each decay channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We visualize the 1 σ (dark) and 2 σ (light) HPD results of this analysis in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 7, where the blue line segments indicate the current experimental bounds at 90% confidence level presented in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Interestingly, the partial lifetime is for some channels much more sharply predicted than for others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The sharp prediction for the decay channels with an antineutrino in the final state is a general feature for models with a (nearly) symmetric up-type quark mass matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' On the other hand, the fact that the partial lifetime of the decay channel p → π0e+ has such a sharp prediction is uncommon and a nice feature of our model, which also implies that this decay channel is predicted to be the dominant one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='5 The interesting result that some decay channels have much sharper predictions than others can be understood by investigating the freedom in the mixing matrices that are defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This is demonstrated in the following example, where we compare the predictions for the two decay channels p → π0e+ and p → π0µ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' In the chosen basis, the respective c-coefficients of the two decay channels in question read c(ec α, d) = (D∗ L)α1 + (U ∗ L)α1(U T L D∗ L)11, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='17) c(eα, dc) = (D∗ R)α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='18) As it can be seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='21), the left mixing of the down-type quark mass matrix DL strongly depends on the Yukawa matrix Y c, while the right mixing DR dominantly depends on the Yukawa matrix Y a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Since Y a has to be chosen in such a way that the correct PMNS parameters and neutrino masses are obtained, there cannot be a strong hierarchy within Y a entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' On the other hand, a strong hierarchy of the entries in Y c is required in order to produce the correct mismatch between the down-type quark and charged lepton masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Therefore, DR appears to have 5Note, however, that proton decay mediated by the two scalar triplets T1 and T2 could enhance the decay channel p → K+ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' – 22 – p → π0e+ p → π0 μ+ p → K+ν p → π+ν p → K0e+ p → K0 μ+ p → η e+ p → ημ+ 1033 1034 1035 1036 1037 1038 1039 1040 1033 1034 1035 1036 1037 1038 1039 1040 τp [yrs] Figure 7: The 1 σ (dark) and 2 σ (light) HPD intervals of the proton lifetime for various decay channels for a benchmark scenario with MGUT = 1016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='2 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The blue line segments represent the current experimental bounds at 90% confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' a large mixing, whereas DL is for all points in the MCMC almost equal to the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This, in particular, also implies that the CKM mixing is mostly coming from UL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Hence, in the case of a positron in the final state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=', for α = 1 in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='17) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='18), the contribution coming from c(ec, d) dominates over the contribution coming from c(e, dc) in the decay width formula (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='3)), since |(UL)11|, |(DL)11| > |(DR)11|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Contrarily, if an antimuon is in the final state (α = 2), the contribution involving c(µ, dc) is dominant over the contribution from c(µc, d), since |(DR)21| > |(UL)21|, |(DL)21|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Now, varying over the full flavor freedom, since we always roughly have |(UL)11| ≈ |(VCKM)11|, |(DL)11| ≈ 1, the dominating c-coefficient c(ec, d) only varies by an order 1 factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This results in a very sharp prediction for the partial lifetime of the decay channel p → π0e+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' On the other hand, |(DR)21| roughly varies within the interval [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='1, 1], resulting in a much less sharp prediction for the partial lifetime of the decay width p → π0µ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Finally, from our MCMC results, we deduce the HPD intervals of the Dirac CP and Majorana phase of the PMNS matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' At 1 σ we obtain δν ∈ [−22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='6◦, 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='4◦] and βν ∈ [−124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='1◦, −71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='4◦], while our 2 σ HPD results are δν ∈ [−50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='7◦, 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='6◦] and βν ∈ [−132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='2◦, −54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='1◦].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Future experiments involving these two observables also have the potential to probe our model and to possibly further reduce the allowed parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' For instance, our 2 σ HPD results for mββ, the effective mass parameter for the neutrinoless double beta decay, is predicted to be mββ ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='46, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='24] meV, well below the current experimental bound mββ < 61 meV provided by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' – 23 – 5 Conclusions We present a minimal model of unification based on an SU(5) gauge group augmented with a Peccei-Quinn symmetry that predicts the existence of ultralight axion dark matter within a narrow mass range of ma ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='7] neV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' This mass window is deter- mined through an interplay between gauge coupling unification constraints, partial proton decay lifetime limits, and the need to reproduce the experimentally observed fermion mass spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The model also predicts that neutrinos are purely of Majo- rana nature, possessing a normal mass hierarchy spectrum, where one of the neutrinos is a massless particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' We discuss the gauge boson mediated proton decay signatures of the model and specify expected partial lifetime ranges for two-body nucleon decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Our analysis yields viable 2 σ ranges for the Dirac CP phase δν ∈ [−50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='7◦, 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='6◦] and for the neutrinoless double beta decay mββ ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='46, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='24] meV, respectively, through which the model may be tested in the neutrino experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Finally, we demonstrate that the entire parameter space of the model will be tested through a synergy between several low-energy experiments that look for proton decay (Hyper-Kamiokande) and axion dark matter (ABRACADABRA and DMRadio-GUT by measuring the axion- photon coupling, and CASPEr Electric by measuring the nucleon electric dipole moments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' A Renormalization group running of the gauge couplings The 2-loop renormalization group equations of the SM gauge couplings are given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Here, we present the 1-loop and 2-loop gauge coefficients of the various intermediate-scale multiplets listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' The 1-loop gauge coefficients � b1 b2 b3 � are b φRe 1 i = � 0 1 3 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' b φIm 1 i = � 0 1 3 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' b φIm 3 i = � 5 12 1 4 1 6 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' b φIm 3 i = � 5 12 1 4 1 6 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' b φRe 8 i = � 0 0 1 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' b φIm 8 i = � 0 0 1 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bT1 i = � 1 15 0 1 6 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bT2 i = � 1 15 0 1 6 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bΦ1 i = � 9 5 5 3 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bΦ3 i = � 4 5 2 1 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bΦ6 i = � 1 15 1 5 3 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bΦ10 i = � 2 0 5 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bΣ1 i = � 6 5 4 3 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bΣ1 i = � 6 5 4 3 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bΣ3 i = � 1 15 1 2 3 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bΣ3 i = � 1 15 1 2 3 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bΣ6 i = � 16 15 0 5 3 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bΣ6 i = � 16 15 0 5 3 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bH2 i = � 1 10 1 6 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='1) whereas the 2-loop gauge coefficients read b φRe 1 ij = � � 0 0 0 0 28 3 0 0 0 0 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' b φIm 1 ij = � � 0 0 0 0 28 3 0 0 0 0 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' b φIm 3 ij = � � 25 12 15 4 20 3 5 4 13 4 4 5 6 3 2 11 3 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' b φIm 3 ij = � � 25 12 15 4 20 3 5 4 13 4 4 5 6 3 2 11 3 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' b φRe 8 ij = � � 0 0 0 0 0 0 0 0 21 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' b φIm 8 ij = � � 0 0 0 0 0 0 0 0 21 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bT1 ij = � � 4 75 0 16 15 0 0 0 2 15 0 11 3 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bT2 ij = � � 4 75 0 16 15 0 0 0 2 15 0 11 3 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' – 24 – bΦ1 ij = � � 729 25 81 0 27 245 3 0 0 0 0 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bΦ3 ij = � � 64 25 96 5 64 5 32 5 56 32 8 5 12 11 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bΦ6 ij = � � 1 75 3 5 8 3 1 5 13 40 1 3 15 230 3 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bΦ10 ij = � � 72 5 0 144 0 0 0 18 0 195 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bΣ1 ij = � � 54 25 36 5 0 12 5 64 3 0 0 0 0 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bΣ1 ij = � � 54 25 36 5 0 12 5 64 3 0 0 0 0 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bΣ3 ij = � � 1 300 3 20 4 15 1 20 49 4 4 1 30 3 2 38 3 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bΣ3 ij = � � 1 300 3 20 4 15 1 20 49 4 4 1 30 3 2 38 3 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bΣ6 ij = � � 64 75 0 32 3 0 0 0 4 3 0 125 3 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bΣ6 ij = � � 64 75 0 32 3 0 0 0 4 3 0 125 3 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' bH2 ij = � � 9 50 9 10 0 3 10 13 6 0 0 0 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='2) References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Pati and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Salam, “Is Baryon Number Conserved?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=',” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' 31 (1973) 661–664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Pati and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Salam, “Lepton Number as the Fourth Color,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' D 10 (1974) 275–289.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' [Erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content='D 11, 703–703 (1975)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' [3] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} +page_content=' Georgi 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+page_content=' – 30 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AyT4oBgHgl3EQf5vo1/content/2301.00809v1.pdf'} diff --git a/HtAyT4oBgHgl3EQfS_fv/content/tmp_files/2301.00099v1.pdf.txt b/HtAyT4oBgHgl3EQfS_fv/content/tmp_files/2301.00099v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..740955835e725b9a4264ec7917a50fbf01ba50a7 --- /dev/null +++ b/HtAyT4oBgHgl3EQfS_fv/content/tmp_files/2301.00099v1.pdf.txt @@ -0,0 +1,979 @@ +Quantum phases of lattice dipolar bosons coupled to a high-finesse cavity +Yaghmorassene Hebib,1, ∗ Chao Zhang,2, † Jin Yang,3, ‡ and Barbara Capogrosso-Sansone1 +1Department of Physics, Clark University, Worcester, Massachusetts 01610, USA +2State Key Laboratory of Precision Spectroscopy, +East China Normal University, Shanghai 200062, China +3Department of Physics, Research Laboratory of Electronics, MIT-Harvard Center for Ultracold Atoms, +Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA +Two types of long range interactions, dipolar interaction and cavity-mediated interaction lead +to exotic quantum phases. +Both interactions have been realized and observed in optical lattice +setups. Here, we study quantum phases of dipolar bosons trapped in optical lattices and coupled +to a high-finesse cavity where both dipolar interaction and cavity-mediated interaction coexist. +We perform quantum Monte Carlo simulations, and find that the checkerboard solid is enhanced +and the checkerboard supersolid phase can exist in a wide range of densities (e.g. +0.27 ≲ n ≲ +0.73). Our unbiased numerical results suggest that both solid and supersolid phases can be achieved +experimentally with magnetic atoms coupled to a cavity. +I. +INTRODUCTION +Competition between different interactions or energy +scales is key to understand the physics world. +A sec- +ondary interaction is treated as a perturbation when it is +small compared with the primary interaction. But when +it changes to be dominant, the original eigenenergy and +eigenwavefunction of the system need to reform corre- +spondingly [1]. +Quantum phase transitions also result +from competition. In optical lattice experiments, at low +temperatures, a transition from superfluid to Mott in- +sulator can be realized when on-site interaction starts +dominating over tunneling [2–5]. +The competition be- +tween short-ranged, on-site interaction and long-range +interaction gives rise to novel exotic quantum phases, like +supersolid states [6]. +In optical lattices, two types of long-range interactions +have been realized and observed - dipolar interaction and +cavity-mediated long-range interaction [6–11]. Theoret- +ically, both types of long range interactions have been +studied comprehensively, and numerous results on exotic +quantum phases realized by these interactions have been +reported [12–36]. So far, these two types of long-range +interactions have been studied singly so that a study +on quantum phases in a system with both dipolar and +cavity-mediated long range interactions is still absent to- +day. Experimentally, creating optical lattices and con- +ducting experiments within cavities is not challenging +and has recently attracted lot of attention [37–39]. +In this manuscript, we are interested in a gas of dipo- +lar bosons trapped in an optical lattice and coupled to +a high-finesse cavity. +The particles in the system in- +teract via dipolar interaction and infinite-range, cavity- +mediated interaction. Dipolar interactions are known to +∗ These authors contributed equally: Y. Hebib, C. Zhang +† These authors contributed equally: +C. Zhang, +Y. Hebib; +czhang@lps.ecnu.edu.cn +‡ dypole_jin@mit.edu +stabilize a plethora of charge density waves and super- +solid phases, and cavity-mediated interactions have been +shown to stabilize charge density waves between odd and +even sites. Here, we study the ground-state phase dia- +gram of lattice dipolar bosons in the presence of global- +range interactions by means of quantum Monte Carlo +simulations. On the one hand, we are interested in un- +derstanding how the presence of photon-mediated inter- +actions affects the quantum phases stabilized in dipolar +bosons trapped in optical lattices; on the other hand, +we want to quantify the changes in the phase diagram +of lattice bosons with cavity-mediated interactions when +dipolar interaction is switched on. For the range of pa- +rameter studied, we find that the presence of cavity- +mediated infinite-range interactions enhances robustness +of the checkerboard solid and supersolid phases. +In- +terestingly, the checkerboard supersolid can survive to +filling factors as low as ∼ 0.27, which is comparable +to what currently achievable experimentally with po- +lar molecules [40]. +Moreover, cavity-mediated interac- +tions significantly lower the dipolar interaction strength +needed to observe checkerboard supersolid, facilitating +the observation of such phase with magnetic atoms. +This paper is organized as follows: In Sec. II we in- +troduce the Hamiltonian of the system. In Sec. III we +discuss various phases and the corresponding order pa- +rameters. In Sec. IV and V we present the phase dia- +grams of the above system in the soft core and hard core +case separately. In Sec. VI we discuss the experimental +realization. We conclude the paper in Sec. VII. +II. +HAMILTONIAN +We consider a gas of dipolar bosons trapped in a two- +dimensional square optical lattice inside a high-finesse +optical cavity (see Fig. 1). Dipoles are aligned perpen- +dicular to the optical lattice plane so that the dipolar +interaction is purely repulsive and isotropic. In the sin- +gle band approximation, the system can be described by +the extended Bose-Hubbard model [6, 29]: +arXiv:2301.00099v1 [cond-mat.quant-gas] 31 Dec 2022 + +2 +FIG. 1. Schematic representation of the system. Dipoles are +trapped in a two-dimensional optical lattice and are aligned +parallel to each other along the direction of polarization, de- +termined by an electric/magnetic field. The polarization is +along the z-axis. The lattice is coupled to a high-finesse op- +tical cavity which is represented by two mirrors. +H = −J +� +⟨i,j⟩ +a† +iaj + Us +2 +� +i +ni(ni − 1) + Vdip +2 +� +i,j +ninj +r3 +ij +− Vca +L2 +� � +i∈e +ni − +� +j∈o +nj +�2 +− µ +� +i +ni , +(1) +where the first term is the kinetic energy characterized +by the hopping amplitude J. Here ⟨·⟩ denotes nearest +neighboring sites, a†(a) bosonic creation (annihilation) +operators satisfying the bosonic commutation relations. +The second term is the short-range on-site repulsive in- +teraction with interaction strength Us. Here, ni = a† +iai +is the particle number operator. The third term is the +dipolar interaction term and rij = |ri − rj| is the relative +distance between site i and site j. The fourth term is the +cavity-mediated long-range interaction with interaction +strength Vca, the summations i ∈ e and j ∈ o denote +summing over even and odd lattice sites respectively. In +the last term, µ is the chemical potential. +In the following, we present unbiased results of phase +diagrams of Hamiltonian (1) based on path-integral +quantum Monte Carlo using the worm algorithm [41]. +We have performed the simulations on an L × L = Ns +square lattice system with L = 16, 20, 24, 30 (we choose +the lattice constant a to be our unit of length). We have +imposed periodic boundary conditions in both spatial di- +mensions. We use Ewald summation to account for the +long-range dipolar interaction. The inverse temperature +β is set to β = L. +III. +ORDER PARAMETERS +In this section, we describe the order parameters +used to characterize superfluid (SF) phase, checker- +board solid (CB) phase, checkerboard supersolid (CBSS) +phase. +Specifically, we calculate superfluid density ρs +and structure factor S(π, π). The superfluid density is +calculated in terms of the winding number [42]: ρs = +⟨W2⟩/DLD−2β, where ⟨W2⟩ = �D +i=1⟨W 2 +i ⟩ is the expec- +tation value of winding number square, D is the dimen- +sion of the system and here D = 2, L is the linear system +size, and β is the inverse temperature. +The structure +factor characterizes diagonal long-range order and is de- +fined as: S(k) = � +r,r′ exp [ik · (r − r′)]⟨nrnr′⟩/N, where +N is the particle number. k is the reciprocal lattice vec- +tor. We use k = (π, π) to identify a checkerboard density +pattern. Notice that, in the CBSS, ρs and S(π, π) are +finite simultaneously. +Another quantity we monitor is compressibility defined +as β∆N 2 +L2 +, where ∆N 2 = ⟨(N − ⟨N⟩)2⟩. The compress- +ibility is finite for compressible phases and zero (in the +thermodynamic limit) for incompressible phases. As we +shall discuss below, for large enough interactions, we find +a variety of incompressible phases. +IV. +SOFTCORE CASE +We present our results for the soft core case with fixed +Us/J = 20 and filling n ≤ 1. In Fig. 2, we study how +the presence of cavity-mediated interactions affects the +phase diagram of a soft core dipolar system. In the ab- +sence of cavity interactions, the phase diagram in the +Vdip vs n plane features three solid phases corresponding +to filling factors n = 1/2, 3/4, 1 separated by a super- +solid phase, a succession of incompressible states in the +lower density regime stabilized at rational filling factors, +a Mott insulator phase at unit filling and low dipolar in- +teraction, and a SF phase [25]. In Fig. 2, we show the +phase diagram in the Vdip vs n plane for cavity interaction +Vca/J = 2 (a) and Vca/J = 1 (b). As expected, the CB +solid (vertical purple dashed line) stabilized at half filling +appears for lower values of the dipolar interaction as a +finite cavity-mediated interaction favors the stabilization +of a density-wave between even and odd sites. Specifi- +cally, CB is stabilized for Vdip ∼ 1.13J at Vca = 2J, and +Vdip ∼ 2.3J at Vca = J compared to Vdip ∼ 4.75J at +Vca = 0. +Upon doping the CB solid with particles or holes, the +system enters a CBSS phase. For Vca = 0, on the parti- +cle side, a supersolid exists in the full range 1/2 < n < 1 +(with the exception of n = 3/4) with the density ordering +which differs depending on how close the value of density +is to the solids at n = 1/2, 3/4, or 1. Here, only CBSS is +observed which disappears to a SF phase for large enough +particle doping. +Similarly, the CBSS also appears for +n < 1/2 and is destroyed for large enough doping. We no- + +3 +FIG. 2. +Softcore case: Phase diagram of Hamiltonian (1) +as a function of Vdip/J and particle density n, computed +via quantum Monte Carlo simulations at Us/J = 20. +(a): +Vca = 2J; (b): Vca = J. We observe a checkerboard (CB) +solid at n = 0.5 (dashed purple line), a checkerboard super- +solid (CBSS), a superflluid (SF) phase, and an incompressible +phase (IP, purple region). The green solid region at the tip +of the lobe in (a) marks the first-order phase transition. Blue +and red open circles are second-order transition points cal- +culated using finite size scaling (see text for details). Filled +circles represent first-order phase transition points. +tice that for lower values of dipolar interaction, the CBSS +replaces the succession of solids stabilized at rational fill- +ing factors observed in [25]. In other words, the CBSS on +the hole side is more robust against doping compared to +what observed in [25] and it can survive to filling factor as +low as n ∼ 0.31. The translational order of the CBSS is +destroyed via a second-order phase transition belonging +to the (2 + 1) Ising universality class, leaving the system +in an SF phase. The boundary between the SS and SF +phases (red and blue open circles in Fig. 2(a) and (b)) +is found using standard finite-size scaling. Specifically, +we determine critical points using finite-size scaling for +the static structure factor by plotting S(π, π)L2β/ν ver- +sus density, with scaling coefficient 2β/ν = 1.0366 [43]. +Critical points are determined from the intersection of +S(π, π)L2β/ν curves for different L’s (Fig. 3(a)). Overall, +we notice that the CBSS is very robust against doping as +it can exist for a wide range of densities already at rela- +FIG. 3. Softcore case: (a) Finite size scaling of the structure +factor S(π, π)L1.0366 as a function of filling factor n for system +sizes L = 16 (red circles), 20 (blue up triangles), 24 (green +down triangles), and 30 (orange squares) at Vdip = 2J and +Vca = 2J. +The CBSS to SF phase transition happens at +n = 0.397 ± 0.03. (b) The hysteresis curve of filling factor n +vs chemical potential µ at Vdip = 1.3J and L = 20. +tively small values of Vdip. For example, excluding half- +filling, CBSS exists within the ranges 0.375 < n < 0.62 at +Vdip = 2.5J and Vca = 2J, 0.31 < n < 0.69 at Vdip = 8J +and Vca = 2J, or 0.39 < n < 0.61 Vdip = 4J and Vca = J. +We are not able to resolve the transition between CB +and SF at fixed half filling. At Vca = 2J, we do observe a +direct CB-SF first-order transition in the vicinity of the +tip as confirmed by a discontinuity in density, structure +factor and superfluid stiffness when crossing the CB-SF +boundary at fixed Vdip, and by the hysteretic behavior +of the same quantities ((Fig. 3(b))). +The solid green +area around the tip of lobe in Fig. 2(a) correspond to +the densities for which one would observe phase coexis- +tence. It is well-known that first-order phase transitions +are forbidden in dipolar systems, based on surface ten- +sion arguments. Rather, two distinct phases are sepa- +rated by a macroemulsion phase [44]. Nonetheless, due +to logarithmic size dependence in the surface tension, it +is possible that, for all practical purposes, in a finite- +size system the transition would effectively be of first +order (see e.g. in [21, 26]). For lattice bosons with on- +site and infinite-range interactions and no dipolar inter- + +(a) 14 +12 +10 +8 +6 +CBSS +4 +2 +SF +2 +0 +0.3 +0.4 +0.5 +0.6 +0.7 +(b) +10 +8 +CB +CBSS +6 +4 +SF +2 +J=1 +0.35 0.40 0.45 0.50 0.55 0.60 0.65 +n(a) +1.4 +L=16 +1.2 +- L=20 +1.0 +L=24 +L=30 +0.8 +0.6 +0.4 +0.385 +0.390 +0.395 +0.400 +0.405 +n +(b) 0.50 +0.49 +0.48 +n +0.47 +0.46 +μ4 +FIG. 4. Density maps for different fillings. Each circle corre- +sponds to a single lattice site, and its radius is proportional +to the local density. (a) n = 0.25 (softcore and hardcore), (b) +n = 0.3775 (softcore), (c) n = 0.3375 (hardcore), (d) n = 0.3 +(hardcore). +action, first-order phase transitions have been observed +before (see e.g [32, 36]). Here, we also observe it in the +presence of dipolar interaction. In Fig. 3(b), we show the +hysteresis curve of n vs µ at Vdip = 1.3J and L = 20. We +notice that hysteretic behavior is observed in a narrow +range of µ of ∼ 2%. +For Vca = 2J, we checked under which conditions other +incompressible phases appear in the phase diagram. Con- +sidering the enhanced robustness of CBSS due to non- +zero Vca, in order to start observing solid phases at ra- +tional fillings other than n = 1/2 (purple solid region +in Fig. 2(a)), we need Vdip ∼ 10.75J which is almost ten +times larger than Vdip ∼ 1.13J corresponding to the onset +of CB order. In comparison, for Vca = 0, solids at other +rational filling factors start appearing (on the hole side) +at Vdip ∼ 8J [25], approximately two times Vdip ∼ 4.75J +corresponding to the onset of CB order. As Vdip is in- +creased, this succession of solids that we call incompress- +ible phase (IP) tends to become dense in the filling factor +for a wide range of densities. In Fig. 4, we show exam- +ples of density maps within the purple region. Here, each +circle corresponds to a single lattice site, and its radius is +proportional to the local density. In Fig. 4(a), we show +the star solid at n = 1/4. For other densities, (see e.g. +Fig. 4(b)), the density pattern possesses defects which +appear in order to accommodate certain rational fillings. +All these findings are very similar for Vca = J (not shown +here). It is interesting to notice that for dipolar strength +corresponding to formation of solids at fillings other than +0.5, the CBSS-SF transition appears to become of first +FIG. 5. Softcore case: Vdip = 1.8J, Vca = 2J, and n = 0.5605. +Upon increasing the temperature, thermal fluctuations de- +stroy the checkerboard supersolid phase in favor of a normal +fluid in two steps. First, superfluidity is destroyed and the +checkerboard supersolid becomes a checkerboard solid via a +Kosterlitz–Thouless phase transition. Then, the checkerboard +solid phase melts into a normal fluid via a two-dimensional +Ising transition. +In (a) we show ρs as a function of T/J +for L = 16 (red circles), 20 (blue up triangles), 24 (green +down triangles), and 30 (orange squares). The dashed line +is T/Jπ. +Inset: intersection points between the T/Jπ line +and the ρs versus T/J curves for each L are used to ex- +tract Tc/J ∼ 0.35 ± 0.01. +(b) scaled structure factor with +2β/ν = 0.25 for L = 16 (red circles), 20 (blue up triangles), 24 +(green down triangles), and 30 (orange squares). The crossing +determines the critical temperature Tc/J = 1.47 ± 0.03. +order (filled circles in Fig. 2(a)). Finally, we notice that, +at unit filling, the system is in a Mott insulator state for +the range of dipolar interaction explored. +Let us now turn to a brief discussion on the robustness +of the CBSS against thermal fluctuations. Superfluidity +disappears via a Kosterlitz-Thouless transition [45] while +the solid order melts via a two-dimensional Ising transi- +tion. In Fig. 5, we show the superfluid density ρs as a +function of T/J for L = 16, 20, 24, and 30 (circles, up +triangles, down triangles, squares respectively) at fixed +n = 0.5605, Vdip = 1.8J, and Vca = 2J. In the thermo- +dynamic limit, a universal jump is observed at the critical +temperature given by ρs(Tc) = 2mkBTc/πℏ2. Here, m is +the effective mass in the lattice, m = ℏ2/2Ja2 = 1/2J +in our units (ℏ = 1, kB = 1, lattice step a = 1). In a +finite size system, this jump is smeared out as one can +see in Fig. 5(a). To extract the critical temperature in +the thermodynamic limit, we apply finite-size scaling to +Tc(L). +From renormalization-group analysis one finds +Tc(L) = Tc(∞) + +c +ln2(L), where c is a constant and Tc(L) +is determined from ρs(Tc, L) = 2mkBTc/πℏ2 [42, 46]. +The dashed line in Fig. 5(a) corresponds to ρs = T/Jπ +and its intersection points with each ρs vs. T/J curve +are used to find Tc as shown in the inset. +We find +Tc/J = 0.35 ± 0.01, a slightly higher temperature than +Tc/J ∼ 0.25 reported in [26]. Above this temperature +the system is in a CB. The solid order melts in favor +of a normal fluid via a two-dimensional Ising transition. + +(a) +n= +0.25 +(b) +n = 0.3775 +(c) +n = 0.3375 +(d) +n = 0.3(a) + 0.30 +(b) +>0.55 +L=16 +0.35 + 0.45 +0.25 +-^-- L=20 +0.35 +0 +0.05 +0.1 +0.1$ +0.30 +1/(lnL)2 +L=24 +0.20 +.25 +0.25 +L=30 +s +p +0.15 +F0.20 +S 0.15 +0.10 +L=16 +-^-- L=20 +0.10 +0.05 +-V-- L=24 +0.05 +L=30 +0.00 +0.40 0.45 0.50 0.55 0.60 0.65 0.70 +1.3 +1.4 +1.5 +1.6 +1.7 +T/J +T/J5 +We use standard finite size scaling as shown in Fig. 5(b), +where we plot the scaled structure factor S(π, π)L2β/ν, +with 2β/ν = 0.25 as a function of T/J for L = 16, 20, +24, and 30 (circles, up triangles, down triangles, squares +respectively). The crossing indicates a critical tempera- +ture Tc/J = 1.47 ± 0.03 making the CB solid consider- +ably more robust against thermal fluctuations than what +found in [26] where Tc/J ∼ 0.68 has been reported. +V. +HARDCORE CASE +In Fig. 6, we study how the presence of cavity-mediated +interactions affects the phase diagram of a hardcore dipo- +lar system. +In the absence of cavity interactions, the +phase diagram in the µ/Vdip vs J/Vdip features three +main lobes (we discuss densities n ≤ 0.5, considering the +particle-hole symmetry of the system) corresponding to +solids at filling factors n = 1/2, 1/3, 1/4, a SF phase, +and a supersolid phase stabilized upon doping the solids +with particles or holes [13]. Moreover, for Vdip ≳ 50J a +succession of incompressible states at rational filling fac- +tors is stabilized. In Fig. 6, we show the phase diagram +in the µ/Vdip vs J/Vdip at Vca = 2J (a) and Vca = 5J +(b). +Let us start our discussion with Fig. 6(a), correspond- +ing to Vca = 2J. +The infinite-ranged cavity-mediated +interaction favors density-waves between even and odd +sites. As a result, the lobe corresponding to the stripe +solid at filling 1/3 observed in [13] has disappeared. The +main lobe in Fig. 6(a) corresponds to the CB solid. +The CB order appears at Vdip ∼ 1.15J compared to +Vdip ∼ 3.6J at Vca = 0. This is expected, considering +that cavity-mediated interactions favor the stabilization +of a CB solid. For Vdip ≥ 1.6J, the CB lobe is surrounded +by a CBSS. This supersolid phase survives for densities +0.34 < n < 0.66 (n = 0.5 excluded) at Vdip = 4J, and +0.4 < n < 0.6 (n = 0.5 excluded) at Vdip = 1.8J making +the CBSS more robust against doping than in the case +Vca = 0. A major difference with the phase diagram at +Vca = 0 is that there exists a range of dipolar interac- +tion for which a CBSS does not intervene between the +solid and the SF phase upon doping the solid with parti- +cles or holes. Instead, we found evidence of a first-order +phase transition (solid thick green line) as confirmed by a +discontinuity in density, structure factor, and superfluid +stiffness when crossing the CB-SF boundary at fixed Vdip, +and hysteretic behavior of the same quantities. We notice +that when crossing the CB-SF boundary at fixed filling +n = 1/2, we were not able to resolve neither a first-order +transition nor a microemulsion phase. +For large enough Vdip (Vdip ≳ 15J) a variety of incom- +pressible phases appears, the solid at n = 1/4, 3/4 being +the first ones appearing. In Fig. 4(a) we show the density +map at n = 1/4 which corresponds to a star solid, like- +wise what is observed in the absence of cavity-mediated +interactions. Other incompressible phases are stabilized +at other rational filling factors. We find that in many +cases (see e.g. Fig. 4(c)) the density pattern possesses +defects (which appear in order to accommodate certain +rational fillings). In other cases, we observe some of the +particles being delocalized within a pair of sites but with +no global phase coherence (Fig. 4(d)). +As the cavity interaction is increased to Vca/J = 5 +(Fig. 6 (b)), only quantitative changes are observed: The +CB solid at half filling is always stabilized for any Vdip; +the CBSS disappears in favor of the IP at lower values +of Vdip; the CBSS is more robust against doping (e.g., +at Vdip = 5.0J, CBSS survives for densities within the +range 0.27 ≲ n ≲ 0.73). +In Fig. 7, we study how the presence of dipolar in- +teractions affects the phase diagram of hardcore bosons +in a cavity. In the absence of dipolar interactions, the +phase diagram in the µ/Vca vs J/Vca [36] features a CB +solid, a SF phase, and a coexistence region separating the +CB from the fully filled lattice and the SF phase. The +coexistence region results from robust first order phase +transition between CB and SF or CB and fully filled lat- +tice. In Fig. 7, we show how this phase diagram changes +when dipolar interaction is switched on. The main effect +of dipolar interaction is to suppress the first order CB to +fully-filled-lattice transition, dramatically shrinking the +coexistence region between CB and SF or suppressing +the CB-SF first-order transition altogether. For Vdip = J +(Fig. 7(a)), we observe that CB is surrounded by the SF +phase. The CB-SF transition is of first order but, unlike +the case of Vdip = 0, the range of µ where we observe +hysteretic behavior is small, corresponding to the thick- +ness of the green solid line in Fig. 7 (a). As a result, the +coexistence region is highly suppressed. We also notice +that, as expected, the CB solid is stabilized for smaller +cavity-mediated interaction (Vca = 2.5J), compared to +Vca = 3.1J in the absence of dipolar interactions. As the +dipolar interaction is increased to Vdip = 2J (Fig. 7(b)), +the CB appears for Vca ∼ 1.25 J. Here, we observe a su- +persolid region separating the CB and SF. The CB-SF +first order transition is therefore replaced by a two-step +second order transition CB-CBSS and CBSS-SF. In both +phase diagrams of Fig. 7, we are unable to resolve the +transition at fixed n = 0.5 within the accuracy of our +simulation results. +VI. +EXPERIMENTAL REALIZATION +Experiments with optical lattices within a high fi- +nesse cavity have been realized and have become popular +in state-of-the-art laboratories. Quantum exotic phases +from short-ranged contact interaction and infinite-range +interaction mediated by the cavity has been observed us- +ing Rb [6]. Such systems should be able to be extended to +explore the quantum phases discussed above using mag- +netic polar atoms such as Cr, Er, and Dy [47], and polar +molecules, Er2, KRb, NaK, NaRb [40, 48], and Rydberg +dressing [49, 50]. +The infinite-long range interactions can be tuned by + +6 +◦ ◦ +◦ ◦ +◦ +◦◦ +◦ +◦ +◦ +◦ +◦ +◦ +◦ +◦ +◦ +◦ +◦◦ +◦ +◦ +◦ +◦ +◦ +◦ ◦ +◦ +◦◦ +◦ +◦ ◦ ◦ +◦ +◦ +◦ ◦ +◦ +◦ +◦ +0.0 +0.2 +0.4 +0.6 +0.8 +0 +2 +4 +6 +8 +J/Vdip +μ/Vdip +◦ +◦ +◦ +◦ +◦ +◦ ◦ +◦ +◦◦ +◦ +◦ +◦ +◦ +◦ +◦ +◦ +◦ +◦◦ +◦◦◦ +◦ ◦ +◦ ◦ +◦ ◦ +◦ +◦ +◦ +◦ +◦ +◦ +◦ +◦ +◦ +◦ +◦ +◦ +◦ +◦ ◦ ◦ ◦ ◦ +◦ ◦ +◦ +◦ +◦ +◦ +◦ +◦ +◦ +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1 +2 +3 +4 +5 +6 +7 +8 +J/Vdip +μ/Vdip +(a) (b) +CBSS +SF +IP +CB +CBSS +IP +CBSS +SF +IP +CB +CBSS +IP +A +AB+nicbVBNS8NAEN3Ur1q/Uj16CRbBU02KqBeh4EU9VbAf0Iaw2W7apZtN2J2oJeanePGgiFd/iTf/jds2B219MPB4b4aZeX7MmQLb/ +jYKS8srq2vF9dLG5tb2jlnebakokYQ2ScQj2fGxopwJ2gQGnHZiSXHoc9r2R5cTv31PpWKRuINxTN0QDwQLGMGgJc8st7y0B/QRUoKz7P +jmouaZFbtqT2EtEicnFZSj4ZlfvX5EkpAKIBwr1XsGNwUS2CE06zUSxSNMRnhAe1qKnBIlZtOT8+sQ630rSCSugRYU/X3RIpDpcahrz +tDEM1703E/7xuAsG5mzIRJ0AFmS0KEm5BZE1ysPpMUgJ8rAkmkulbLTLEhPQaZV0CM78y4ukVas6p1Xn9qRSv87jKJ9dICOkIPOUB1 +doQZqIoIe0DN6RW/Gk/FivBsfs9aCkc/soT8wPn8A+IOT1A=Vca/J = 2 +AB+nicbVDLSsNAFJ3UV62vVJdugkVwVRPxtREKbtRVBfuANoTJd +NIOnUzCzI1aYj7FjQtF3Pol7vwbp20W2nrgwuGce7n3Hj/mTIFtfxuFhcWl5ZXiamltfWNzyxvN1WUSEIbJOKRbPtYUc4EbQADTtuxpDj0OW35w8ux37qnUrFI3MEopm6I+4IFjGDQkmeWm17aBfoIKcFZdnhzceKZFbtqT2DNEycnFZSj7plf3V5EkpAKIBwr1XHsGNwUS2CE06 +zUTRSNMRniPu1oKnBIlZtOTs+sfa30rCSugRYE/X3RIpDpUahrztDAM1643F/7xOAsG5mzIRJ0AFmS4KEm5BZI1zsHpMUgJ8pAkmkulbLTLAEhPQaZV0CM7sy/OkeVR1TqvO7XGldp3HUS7aA8dIAedoRq6QnXUQAQ9oGf0it6MJ+PFeDc+pq0FI5/ZQX9gfP4A/Q+T1w=Vca/J = 5 +FIG. 6. Hardcore case: Ground state phase diagram of the system described by Eq. 1 at fixed cavity interaction Vca/J = 2 (a) +and Vca/J = 5 (b) in the plane µ/Vdip vs J/Vdip. We observe a superfluid phase (SF), checkerboard phase (CB), checkerboard +supersolid phase (CBSS), and incompressible phase (IP). Dashed lines correspond to second-order phase transitions while solid- +lines correspond to first-order phase transitions. The gray shadowed region corresponds to J/Vdip ≤ 0.05 which has not been +explored. +adjusting laser intensity, while the dipolar interactions +can be tuned by using different rotational states of the +polar molecules, different projection state along the mag- +netic field quantization axes, and different Rydberg states +for Rydberg dressing. The filling factor and temperatures +can be controlled by using different evaporation depths. +By changing lattice light intensity, different tunnelings +and chemical potentials can be realized. Time of flight +and quantum gas microscopy can be used to detect the +above discussed exotic quantum phases. +Here, we give a simple sketch of parameters used in +experiments. +For magnetic polar atoms Er and Dy in +optical lattices using ultra-violet light ∼ 400 nm or less, +the dipolar interaction can reach to 200 Hz. When tun- +neling rate is around 100 Hz, Vdip/J is around 2. Thus, +by tuning filling factor, SF, CB and CBSS are all able +to be observed, when Vca/J is 2 and Us/J = 20. When +considering polar molecules, dipole moments can be as +large as several Debye. This gives interactions of several +kilo-Hertz to tens of kilo-Hertz even in lattice of 1064 nm +laser light. However, low filling factors and high temper- +atures are two challenges. In this paper, though, we show +that CBSS can be observed for filling as low as n ∼ 0.27, +which is close to what is currently available. The temper- +atures needed to observe different phases and transitions +are around one to several nano-Kelvins. +VII. +CONCLUSION +In conclusion, we have used quantum Monte Carlo by +the worm algorithm to study the phase diagram of dipo- +lar lattice bosons coupled to a high-finesse cavity in both +hardcore and softcore case. The cavity-mediated infinite- +range interaction enhances robustness of the checker- +board solid and supersolid. +As a result, the checker- +board supersolid can already exist at filling factors as +low as ∼ 0.27. The recent advent of molecular quantum +gas microscope gives unprecedented resolution of the lat- +tices [40] which greatly facilitates the optimization of the +cold polar molecule system. Therefore, we expect such +a filling factor to be achieved in the near future, mak- +ing the realization of the checkerboard supersolid phase +with this setup possible. We also observed that cavity- +mediated interactions facilitate observation of checker- +board solid and supersolid phases with magnetic atoms. +As a consequence, based on the results presented, both +polar molecules and magnetic atoms are good candidates +to observe checkerboard supersolidity in optical lattices +coupled to a high-finesse cavity within currently available +experiments. + +7 +FIG. 7. Hardcore case: Ground state phase diagram of the system described by Eq. 1 at fixed dipolar interaction Vdip/J = 1 +(a) and Vdip/J = 2 (b) in the plane µ/Vdip vs J/Vca. We observe a superfluid phase (SF), checkerboard solid phase (CB) and +fully filled phase in (a); superfluid phase (SF), checkerboard phase (CB), checkerboard supersolid phase (CBSS), and fully filled +phase in (b). Dashed lines correspond to second-order phase transitions while the solid thick line corresponds to first-order +phase transition. 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Zeiher, +Physical Review Letters 128, 113602 (2022). + diff --git a/HtAyT4oBgHgl3EQfS_fv/content/tmp_files/load_file.txt b/HtAyT4oBgHgl3EQfS_fv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d654e47b95626105b7b9ae065eb0d1e3c558f66f --- /dev/null +++ b/HtAyT4oBgHgl3EQfS_fv/content/tmp_files/load_file.txt @@ -0,0 +1,821 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf,len=820 +page_content='Quantum phases of lattice dipolar bosons coupled to a high-finesse cavity Yaghmorassene Hebib,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' ∗ Chao Zhang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' † Jin Yang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' ‡ and Barbara Capogrosso-Sansone1 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Clark University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Worcester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Massachusetts 01610,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' USA 2State Key Laboratory of Precision Spectroscopy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' East China Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Shanghai 200062,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' China 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Research Laboratory of Electronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' MIT-Harvard Center for Ultracold Atoms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Massachusetts 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' USA Two types of long range interactions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' dipolar interaction and cavity-mediated interaction lead to exotic quantum phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Both interactions have been realized and observed in optical lattice setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Here, we study quantum phases of dipolar bosons trapped in optical lattices and coupled to a high-finesse cavity where both dipolar interaction and cavity-mediated interaction coexist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' We perform quantum Monte Carlo simulations, and find that the checkerboard solid is enhanced and the checkerboard supersolid phase can exist in a wide range of densities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='27 ≲ n ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Our unbiased numerical results suggest that both solid and supersolid phases can be achieved experimentally with magnetic atoms coupled to a cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' INTRODUCTION Competition between different interactions or energy scales is key to understand the physics world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' A sec- ondary interaction is treated as a perturbation when it is small compared with the primary interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' But when it changes to be dominant, the original eigenenergy and eigenwavefunction of the system need to reform corre- spondingly [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Quantum phase transitions also result from competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In optical lattice experiments, at low temperatures, a transition from superfluid to Mott in- sulator can be realized when on-site interaction starts dominating over tunneling [2–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The competition be- tween short-ranged, on-site interaction and long-range interaction gives rise to novel exotic quantum phases, like supersolid states [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In optical lattices, two types of long-range interactions have been realized and observed - dipolar interaction and cavity-mediated long-range interaction [6–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Theoret- ically, both types of long range interactions have been studied comprehensively, and numerous results on exotic quantum phases realized by these interactions have been reported [12–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' So far, these two types of long-range interactions have been studied singly so that a study on quantum phases in a system with both dipolar and cavity-mediated long range interactions is still absent to- day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Experimentally, creating optical lattices and con- ducting experiments within cavities is not challenging and has recently attracted lot of attention [37–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In this manuscript, we are interested in a gas of dipo- lar bosons trapped in an optical lattice and coupled to a high-finesse cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The particles in the system in- teract via dipolar interaction and infinite-range, cavity- mediated interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Dipolar interactions are known to ∗ These authors contributed equally: Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Hebib, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Zhang † These authors contributed equally: C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Hebib;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' czhang@lps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='ecnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='cn ‡ dypole_jin@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='edu stabilize a plethora of charge density waves and super- solid phases, and cavity-mediated interactions have been shown to stabilize charge density waves between odd and even sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Here, we study the ground-state phase dia- gram of lattice dipolar bosons in the presence of global- range interactions by means of quantum Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' On the one hand, we are interested in un- derstanding how the presence of photon-mediated inter- actions affects the quantum phases stabilized in dipolar bosons trapped in optical lattices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' on the other hand, we want to quantify the changes in the phase diagram of lattice bosons with cavity-mediated interactions when dipolar interaction is switched on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' For the range of pa- rameter studied, we find that the presence of cavity- mediated infinite-range interactions enhances robustness of the checkerboard solid and supersolid phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In- terestingly, the checkerboard supersolid can survive to filling factors as low as ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='27, which is comparable to what currently achievable experimentally with po- lar molecules [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Moreover, cavity-mediated interac- tions significantly lower the dipolar interaction strength needed to observe checkerboard supersolid, facilitating the observation of such phase with magnetic atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' This paper is organized as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' II we in- troduce the Hamiltonian of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' III we discuss various phases and the corresponding order pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' IV and V we present the phase dia- grams of the above system in the soft core and hard core case separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' VI we discuss the experimental realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' We conclude the paper in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' HAMILTONIAN We consider a gas of dipolar bosons trapped in a two- dimensional square optical lattice inside a high-finesse optical cavity (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Dipoles are aligned perpen- dicular to the optical lattice plane so that the dipolar interaction is purely repulsive and isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In the sin- gle band approximation, the system can be described by the extended Bose-Hubbard model [6, 29]: arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='00099v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='quant-gas] 31 Dec 2022 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Schematic representation of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Dipoles are trapped in a two-dimensional optical lattice and are aligned parallel to each other along the direction of polarization, de- termined by an electric/magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The polarization is along the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The lattice is coupled to a high-finesse op- tical cavity which is represented by two mirrors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' H = −J � ⟨i,j⟩ a† iaj + Us 2 � i ni(ni − 1) + Vdip 2 � i,j ninj r3 ij − Vca L2 � � i∈e ni − � j∈o nj �2 − µ � i ni , (1) where the first term is the kinetic energy characterized by the hopping amplitude J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Here ⟨·⟩ denotes nearest neighboring sites, a†(a) bosonic creation (annihilation) operators satisfying the bosonic commutation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The second term is the short-range on-site repulsive in- teraction with interaction strength Us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Here, ni = a† iai is the particle number operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The third term is the dipolar interaction term and rij = |ri − rj| is the relative distance between site i and site j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The fourth term is the cavity-mediated long-range interaction with interaction strength Vca, the summations i ∈ e and j ∈ o denote summing over even and odd lattice sites respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In the last term, µ is the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In the following, we present unbiased results of phase diagrams of Hamiltonian (1) based on path-integral quantum Monte Carlo using the worm algorithm [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' We have performed the simulations on an L × L = Ns square lattice system with L = 16, 20, 24, 30 (we choose the lattice constant a to be our unit of length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' We have imposed periodic boundary conditions in both spatial di- mensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' We use Ewald summation to account for the long-range dipolar interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The inverse temperature β is set to β = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' ORDER PARAMETERS In this section, we describe the order parameters used to characterize superfluid (SF) phase, checker- board solid (CB) phase, checkerboard supersolid (CBSS) phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Specifically, we calculate superfluid density ρs and structure factor S(π, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The superfluid density is calculated in terms of the winding number [42]: ρs = ⟨W2⟩/DLD−2β, where ⟨W2⟩ = �D i=1⟨W 2 i ⟩ is the expec- tation value of winding number square, D is the dimen- sion of the system and here D = 2, L is the linear system size, and β is the inverse temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The structure factor characterizes diagonal long-range order and is de- fined as: S(k) = � r,r′ exp [ik · (r − r′)]⟨nrnr′⟩/N, where N is the particle number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' k is the reciprocal lattice vec- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' We use k = (π, π) to identify a checkerboard density pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Notice that, in the CBSS, ρs and S(π, π) are finite simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Another quantity we monitor is compressibility defined as β∆N 2 L2 , where ∆N 2 = ⟨(N − ⟨N⟩)2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The compress- ibility is finite for compressible phases and zero (in the thermodynamic limit) for incompressible phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' As we shall discuss below, for large enough interactions, we find a variety of incompressible phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' SOFTCORE CASE We present our results for the soft core case with fixed Us/J = 20 and filling n ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 2, we study how the presence of cavity-mediated interactions affects the phase diagram of a soft core dipolar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In the ab- sence of cavity interactions, the phase diagram in the Vdip vs n plane features three solid phases corresponding to filling factors n = 1/2, 3/4, 1 separated by a super- solid phase, a succession of incompressible states in the lower density regime stabilized at rational filling factors, a Mott insulator phase at unit filling and low dipolar in- teraction, and a SF phase [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 2, we show the phase diagram in the Vdip vs n plane for cavity interaction Vca/J = 2 (a) and Vca/J = 1 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' As expected, the CB solid (vertical purple dashed line) stabilized at half filling appears for lower values of the dipolar interaction as a finite cavity-mediated interaction favors the stabilization of a density-wave between even and odd sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Specifi- cally, CB is stabilized for Vdip ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='13J at Vca = 2J, and Vdip ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='3J at Vca = J compared to Vdip ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='75J at Vca = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Upon doping the CB solid with particles or holes, the system enters a CBSS phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' For Vca = 0, on the parti- cle side, a supersolid exists in the full range 1/2 < n < 1 (with the exception of n = 3/4) with the density ordering which differs depending on how close the value of density is to the solids at n = 1/2, 3/4, or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Here, only CBSS is observed which disappears to a SF phase for large enough particle doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Similarly, the CBSS also appears for n < 1/2 and is destroyed for large enough doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' We no- 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Softcore case: Phase diagram of Hamiltonian (1) as a function of Vdip/J and particle density n, computed via quantum Monte Carlo simulations at Us/J = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' (a): Vca = 2J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' (b): Vca = J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' We observe a checkerboard (CB) solid at n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='5 (dashed purple line), a checkerboard super- solid (CBSS), a superflluid (SF) phase, and an incompressible phase (IP, purple region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The green solid region at the tip of the lobe in (a) marks the first-order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Blue and red open circles are second-order transition points cal- culated using finite size scaling (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Filled circles represent first-order phase transition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' tice that for lower values of dipolar interaction, the CBSS replaces the succession of solids stabilized at rational fill- ing factors observed in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In other words, the CBSS on the hole side is more robust against doping compared to what observed in [25] and it can survive to filling factor as low as n ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The translational order of the CBSS is destroyed via a second-order phase transition belonging to the (2 + 1) Ising universality class, leaving the system in an SF phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The boundary between the SS and SF phases (red and blue open circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 2(a) and (b)) is found using standard finite-size scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Specifically, we determine critical points using finite-size scaling for the static structure factor by plotting S(π, π)L2β/ν ver- sus density, with scaling coefficient 2β/ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='0366 [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Critical points are determined from the intersection of S(π, π)L2β/ν curves for different L’s (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 3(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Overall, we notice that the CBSS is very robust against doping as it can exist for a wide range of densities already at rela- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Softcore case: (a) Finite size scaling of the structure factor S(π, π)L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='0366 as a function of filling factor n for system sizes L = 16 (red circles), 20 (blue up triangles), 24 (green down triangles), and 30 (orange squares) at Vdip = 2J and Vca = 2J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The CBSS to SF phase transition happens at n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='397 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' (b) The hysteresis curve of filling factor n vs chemical potential µ at Vdip = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='3J and L = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' tively small values of Vdip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' For example, excluding half- filling, CBSS exists within the ranges 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='375 < n < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='62 at Vdip = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='5J and Vca = 2J, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='31 < n < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='69 at Vdip = 8J and Vca = 2J, or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='39 < n < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='61 Vdip = 4J and Vca = J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' We are not able to resolve the transition between CB and SF at fixed half filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' At Vca = 2J, we do observe a direct CB-SF first-order transition in the vicinity of the tip as confirmed by a discontinuity in density, structure factor and superfluid stiffness when crossing the CB-SF boundary at fixed Vdip, and by the hysteretic behavior of the same quantities ((Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 3(b))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The solid green area around the tip of lobe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 2(a) correspond to the densities for which one would observe phase coexis- tence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' It is well-known that first-order phase transitions are forbidden in dipolar systems, based on surface ten- sion arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Rather, two distinct phases are sepa- rated by a macroemulsion phase [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Nonetheless, due to logarithmic size dependence in the surface tension, it is possible that, for all practical purposes, in a finite- size system the transition would effectively be of first order (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' in [21, 26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' For lattice bosons with on- site and infinite-range interactions and no dipolar inter- (a) 14 12 10 8 6 CBSS 4 2 SF 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='7 (b) 10 8 CB CBSS 6 4 SF 2 J=1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='65 n(a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='4 L=16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='2 L=20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='0 L=24 L=30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='385 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='390 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='395 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='405 n (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='48 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='46 μ4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Density maps for different fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Each circle corre- sponds to a single lattice site, and its radius is proportional to the local density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' (a) n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='25 (softcore and hardcore), (b) n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='3775 (softcore), (c) n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='3375 (hardcore), (d) n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='3 (hardcore).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' action, first-order phase transitions have been observed before (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='g [32, 36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Here, we also observe it in the presence of dipolar interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 3(b), we show the hysteresis curve of n vs µ at Vdip = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='3J and L = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' We notice that hysteretic behavior is observed in a narrow range of µ of ∼ 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' For Vca = 2J, we checked under which conditions other incompressible phases appear in the phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Con- sidering the enhanced robustness of CBSS due to non- zero Vca, in order to start observing solid phases at ra- tional fillings other than n = 1/2 (purple solid region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 2(a)), we need Vdip ∼ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='75J which is almost ten times larger than Vdip ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='13J corresponding to the onset of CB order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In comparison, for Vca = 0, solids at other rational filling factors start appearing (on the hole side) at Vdip ∼ 8J [25], approximately two times Vdip ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='75J corresponding to the onset of CB order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' As Vdip is in- creased, this succession of solids that we call incompress- ible phase (IP) tends to become dense in the filling factor for a wide range of densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 4, we show exam- ples of density maps within the purple region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Here, each circle corresponds to a single lattice site, and its radius is proportional to the local density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 4(a), we show the star solid at n = 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' For other densities, (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 4(b)), the density pattern possesses defects which appear in order to accommodate certain rational fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' All these findings are very similar for Vca = J (not shown here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' It is interesting to notice that for dipolar strength corresponding to formation of solids at fillings other than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='5, the CBSS-SF transition appears to become of first FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Softcore case: Vdip = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='8J, Vca = 2J, and n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='5605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Upon increasing the temperature, thermal fluctuations de- stroy the checkerboard supersolid phase in favor of a normal fluid in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' First, superfluidity is destroyed and the checkerboard supersolid becomes a checkerboard solid via a Kosterlitz–Thouless phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Then, the checkerboard solid phase melts into a normal fluid via a two-dimensional Ising transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In (a) we show ρs as a function of T/J for L = 16 (red circles), 20 (blue up triangles), 24 (green down triangles), and 30 (orange squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The dashed line is T/Jπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Inset: intersection points between the T/Jπ line and the ρs versus T/J curves for each L are used to ex- tract Tc/J ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' (b) scaled structure factor with 2β/ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='25 for L = 16 (red circles), 20 (blue up triangles), 24 (green down triangles), and 30 (orange squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The crossing determines the critical temperature Tc/J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' order (filled circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Finally, we notice that, at unit filling, the system is in a Mott insulator state for the range of dipolar interaction explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Let us now turn to a brief discussion on the robustness of the CBSS against thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Superfluidity disappears via a Kosterlitz-Thouless transition [45] while the solid order melts via a two-dimensional Ising transi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 5, we show the superfluid density ρs as a function of T/J for L = 16, 20, 24, and 30 (circles, up triangles, down triangles, squares respectively) at fixed n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='5605, Vdip = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='8J, and Vca = 2J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In the thermo- dynamic limit, a universal jump is observed at the critical temperature given by ρs(Tc) = 2mkBTc/πℏ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Here, m is the effective mass in the lattice, m = ℏ2/2Ja2 = 1/2J in our units (ℏ = 1, kB = 1, lattice step a = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In a finite size system, this jump is smeared out as one can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' To extract the critical temperature in the thermodynamic limit, we apply finite-size scaling to Tc(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' From renormalization-group analysis one finds Tc(L) = Tc(∞) + c ln2(L), where c is a constant and Tc(L) is determined from ρs(Tc, L) = 2mkBTc/πℏ2 [42, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 5(a) corresponds to ρs = T/Jπ and its intersection points with each ρs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' T/J curve are used to find Tc as shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' We find Tc/J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='01, a slightly higher temperature than Tc/J ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='25 reported in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Above this temperature the system is in a CB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The solid order melts in favor of a normal fluid via a two-dimensional Ising transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' (a) n= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='25 (b) n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='3775 (c) n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='3375 (d) n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='3(a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='30 (b) >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='55 L=16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='25 ^-- L=20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='35 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='1$ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='30 1/(lnL)2 L=24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='25 L=30 s p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='15 F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='20 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='10 L=16 ^-- L=20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='05 V-- L=24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='05 L=30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='7 T/J T/J5 We use standard finite size scaling as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 5(b), where we plot the scaled structure factor S(π, π)L2β/ν, with 2β/ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='25 as a function of T/J for L = 16, 20, 24, and 30 (circles, up triangles, down triangles, squares respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The crossing indicates a critical tempera- ture Tc/J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='03 making the CB solid consider- ably more robust against thermal fluctuations than what found in [26] where Tc/J ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='68 has been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' HARDCORE CASE In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 6, we study how the presence of cavity-mediated interactions affects the phase diagram of a hardcore dipo- lar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In the absence of cavity interactions, the phase diagram in the µ/Vdip vs J/Vdip features three main lobes (we discuss densities n ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='5, considering the particle-hole symmetry of the system) corresponding to solids at filling factors n = 1/2, 1/3, 1/4, a SF phase, and a supersolid phase stabilized upon doping the solids with particles or holes [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Moreover, for Vdip ≳ 50J a succession of incompressible states at rational filling fac- tors is stabilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 6, we show the phase diagram in the µ/Vdip vs J/Vdip at Vca = 2J (a) and Vca = 5J (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Let us start our discussion with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 6(a), correspond- ing to Vca = 2J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The infinite-ranged cavity-mediated interaction favors density-waves between even and odd sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' As a result, the lobe corresponding to the stripe solid at filling 1/3 observed in [13] has disappeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The main lobe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 6(a) corresponds to the CB solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The CB order appears at Vdip ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='15J compared to Vdip ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='6J at Vca = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' This is expected, considering that cavity-mediated interactions favor the stabilization of a CB solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' For Vdip ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='6J, the CB lobe is surrounded by a CBSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' This supersolid phase survives for densities 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='34 < n < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='66 (n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='5 excluded) at Vdip = 4J, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='4 < n < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='6 (n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='5 excluded) at Vdip = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='8J making the CBSS more robust against doping than in the case Vca = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' A major difference with the phase diagram at Vca = 0 is that there exists a range of dipolar interac- tion for which a CBSS does not intervene between the solid and the SF phase upon doping the solid with parti- cles or holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Instead, we found evidence of a first-order phase transition (solid thick green line) as confirmed by a discontinuity in density, structure factor, and superfluid stiffness when crossing the CB-SF boundary at fixed Vdip, and hysteretic behavior of the same quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' We notice that when crossing the CB-SF boundary at fixed filling n = 1/2, we were not able to resolve neither a first-order transition nor a microemulsion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' For large enough Vdip (Vdip ≳ 15J) a variety of incom- pressible phases appears, the solid at n = 1/4, 3/4 being the first ones appearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 4(a) we show the density map at n = 1/4 which corresponds to a star solid, like- wise what is observed in the absence of cavity-mediated interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Other incompressible phases are stabilized at other rational filling factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' We find that in many cases (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 4(c)) the density pattern possesses defects (which appear in order to accommodate certain rational fillings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In other cases, we observe some of the particles being delocalized within a pair of sites but with no global phase coherence (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 4(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' As the cavity interaction is increased to Vca/J = 5 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 6 (b)), only quantitative changes are observed: The CB solid at half filling is always stabilized for any Vdip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' the CBSS disappears in favor of the IP at lower values of Vdip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' the CBSS is more robust against doping (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=', at Vdip = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='0J, CBSS survives for densities within the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='27 ≲ n ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 7, we study how the presence of dipolar in- teractions affects the phase diagram of hardcore bosons in a cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In the absence of dipolar interactions, the phase diagram in the µ/Vca vs J/Vca [36] features a CB solid, a SF phase, and a coexistence region separating the CB from the fully filled lattice and the SF phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The coexistence region results from robust first order phase transition between CB and SF or CB and fully filled lat- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 7, we show how this phase diagram changes when dipolar interaction is switched on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The main effect of dipolar interaction is to suppress the first order CB to fully-filled-lattice transition, dramatically shrinking the coexistence region between CB and SF or suppressing the CB-SF first-order transition altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' For Vdip = J (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 7(a)), we observe that CB is surrounded by the SF phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The CB-SF transition is of first order but, unlike the case of Vdip = 0, the range of µ where we observe hysteretic behavior is small, corresponding to the thick- ness of the green solid line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 7 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' As a result, the coexistence region is highly suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' We also notice that, as expected, the CB solid is stabilized for smaller cavity-mediated interaction (Vca = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='5J), compared to Vca = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='1J in the absence of dipolar interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' As the dipolar interaction is increased to Vdip = 2J (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 7(b)), the CB appears for Vca ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='25 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Here, we observe a su- persolid region separating the CB and SF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The CB-SF first order transition is therefore replaced by a two-step second order transition CB-CBSS and CBSS-SF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' In both phase diagrams of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' 7, we are unable to resolve the transition at fixed n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='5 within the accuracy of our simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' EXPERIMENTAL REALIZATION Experiments with optical lattices within a high fi- nesse cavity have been realized and have become popular in state-of-the-art laboratories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Quantum exotic phases from short-ranged contact interaction and infinite-range interaction mediated by the cavity has been observed us- ing Rb [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' Such systems should be able to be extended to explore the quantum phases discussed above using mag- netic polar atoms such as Cr, Er, and Dy [47], and polar molecules, Er2, KRb, NaK, NaRb [40, 48], and Rydberg dressing [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content=' The infinite-long range interactions can be tuned by 6 ◦ ◦ ◦◦ ◦◦ ◦ ◦◦ ◦ ◦ ◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='8 0 2 4 6 8 J/Vdip μ/Vdip ◦ ◦◦ ◦◦ ◦◦◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='1 ' metadata={'source': 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+page_content='NIOnUzCzI1aYj7FjQtF3Pol7vwbp20W2nrgwuGce7n3Hj/mTIFtfxuFhcWl5ZXiamltfWNzyxvN1WUSEIbJOKRbPtYUc4EbQADTtuxpDj0OW35w8ux37qnUrFI3MEopm6I+4IFjGDQkmeWm17aBfoIKcFZdnhzceKZFbtqT2DNEycnFZSj7plf3V5EkpAKIBwr1XHsGNwUS2CE06 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfS_fv/content/2301.00099v1.pdf'} +page_content='zUTRSNMRniPu1oKnBIlZtOTs+sfa30rCSugRYE/X3RIpDpUahrztDAM1643F/7xOAsG5mzIRJ0AFmS4KEm5BZI1zsHpMUgJ8pAkmkulbLTLAEhPQaZV0CM7sy/OkeVR1TqvO7XGldp3HUS7aA8dIAedoRq6QnXUQAQ9oGf0it6MJ+PFeDc+pq0FI5/ZQX9gfP4A/Q+T1w= −M) have an exponentially smaller statistical +weight and have much less impact on the dynamics. To +calculate the fidelity of the NRQM up to the linear order +in ζ, we only include the states with M = −I and M = +−I + 1 and neglect the others. +The fidelity of the memory protocol is defined as the +overlap between the initial electron state and the re- +trieved one, taking the minimum over all possible initial +states. According to Ref. [19], the minimal fidelity can +be found by considering only two types of initial condi- +tions: (1) the electron spin pointing to the z-axis, (2) the +electron spin lying in the xy-plane. For the first case, +we calculate sz = Tr(Szρf), where ρf is the density ma- +trix for the final retrieved electron state. For the second +case, we calculate sT = +� +s2x + s2y and s0 = Tr(Szρf), +where sx,y = Tr(Sx,yρf). The fidelity F is the minimum + +4 +of the following three quantities: f1 = (1 + sz)/2, f2 = +(1+sz −2s0)/2 and f3 = (1+sT −s2 +0/[4(sz −s0−sT )])/2, +When s0/[2(sz−s0−sT )] /∈ [−1, 1], F takes the minimum +of f1 and f2. +By taking into account of two types of initial bath +states, M = −I and M = −I + 1, and averaging with +probability w(I, M), we calculate sz, s0, sT up to the +linear order in ζ : sz ≈ 1−ζ +� +1 − cos4 γ0 − sin4 γ0 +� +, s0 ≈ +−(ζ/2) +� +1 − cos4 γ0 − sin4 γ0 +� +, sT ≈ 1 − ζ (1 + cos γ0) , +where γ0 = π/ +√ +2.We then obtain the fidelity +F = 1 + sz +2 +≈ 1 − 0.232 ζ . +(7) +Substituting ζ = ∆P/(2 − ∆P), the fidelity is further +simplified as +F ≈ 1 − 0.116∆P . +(8) +The results from Eqs. (7) and (8) are depicted in +Fig. 2(c) as dash-dotted and dotted lines. For compar- +ison, the analytical estimate of fidelity for the original +resonant QM protocol is F ≈ 1 − 1.38∆P [19], which is +also depicted in Fig. 2(c) as the dashed line. Starting +with full polarization, the fidelity of the NRQM proto- +col drops more than ten times slower than that of the +resonant one as the nuclear polarization lowers, demon- +strating the advantages of the NRQM protocol. +V. +NUMERICAL RESULTS +While we have derived the fidelity Eq. (7) at high bath +polarizations and proved the advantage of the NRQM, +the performance of the protocol at low polarizations or +inhomogeneous hyperfine coupling is still in need. This +task is completed by numerical simulations. +First, we consider the case where inhomogeneity is neg- +ligible (Aj = A). We numerically simulate the dynamics +of the electron spin and N = 104 nuclear spins. +The +system is prepared as a tensor product of the electron +state |φ⟩ and the nuclear state |I, M⟩ with a statistical +weight w(I, M). The system evolves under the Hamilto- +nian Eq. (1) and the pulse sequence [XXYY]n for a time +t1 and the electron state is mapped onto the collective +nuclear state. After that, the electron is ejected and a +nuclear mixed state is reduced by tracing out the elec- +tron’s degree of freedom. For the retrieval, another fully +polarized electron is injected in the QD. The system’s +state becomes a tensor product of the electron state and +the reduced nuclear bath state. Then the system evolves +under the same pulse sequence for another time t2. Af- +ter tracing out the nuclear bath’s degree of freedom, we +obtain the final density matrix ρf of the electron. We +plot how the electron observable ⟨Sz⟩ changes over time +during this process in Fig. 2(a) and (b). +For a given +|I, M⟩, the trajectory of ⟨Sz⟩ is a translucent curve with +opacity based on the statistical weight of the bath state +w(I, M). In total, we get blurred lines in the image that +0.2 +0.4 +0.6 +0.8 +1.0 +P +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +F +(a) +0.2 +0.4 +0.6 +0.8 +1.0 +P +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 (b) +Norm. (NRQM) +Narr. (NRQM) +Norm. (RQM) +Narr. (RQM) +FIG. 3. Performance of quantum memory protocols in a bath +of N=20 nuclear spins. (a) QM fidelity for homogeneous bath +polarizations with different widths of Aj for the resonant QM +protocol (RQM) and the NRQM protocol. (b) QM fidelity for +inhomogeneous bath polarizations. +diffuse over time. +The thickest blurred line, which in +fact includes many lines, shows the electron spin’s evolu- +tion with bath states |I, M⟩ where M = −I, I ∈ [0, N/2]. +The rest blurred lines come from bath states |I, M⟩ where +M = −I + 1, M = −I + 2, etc. By averaging over dif- +ferent bath states with statistical weight w(I, M) we cal- +culate the weighted observables for the electron (sz,0,T ) +in a realistic thermal bath ensemble. This ensemble av- +eraged dynamics of the electron are drawn as the yellow +and green solid lines in Fig. 2(a) and (b). +The mini- +mal fidelity is calculated accordingly using (sz,0,T ) [19]. +The times t1 and t2 are tuned to maximize the minimal +fidelity. The NRQM results at different nuclear polariza- +tion P are shown in Fig. 2(c) as a solid line with triangles. +The simulated fidelities for the resonant QM protocol are +also plotted as a solid line with circles for comparison. +As shown in the figure, the NRQM illustrates a signif- +icant improvement in the fidelity over the resonant QM. +The NRQM protocol has a fidelity over 90% at P = 0.5 +and the fidelity is still over 80% at P = 0.3. In stark con- +trast, the resonant QM protocol requires a polarization +greater than 0.8 in order to achieve 80% fidelity [18, 19]. +Clearly, the need for strong nuclear polarization is dra- +matically mitigated for the NRQM protocol. In addition, +the numerics agrees well with our analytical estimate of +the fidelity at high polarizations, implying the validity of +previous analytics. +Second, we consider the case of inhomogeneous hyper- +fine coupling (Aj ̸= A). We numerically simulate the dy- +namics of the electron spin interacting with N = 4×5 nu- +clear spins arrayed in a rectangular lattice, using the ef- +ficient Chebyshev-expansion-based algorithm [45]. More +computer resources would be required to include more +nuclear spins, but N = 20 is adequate to make our +simulations represent bigger systems with a precision of +1/N [19]. In order to compare with previous works, we +adopt the same coupling strengths Aj as in Refs. [19, 20]. + +5 +The values of Aj spread from 0.96 to 0.31, referred to as +the normal distribution. We also consider the case with +decreased QD widths by a factor of 1/ +√ +2 to represent a +narrow distribution of Aj, spreading from 0.92 to 0.09. +Numerical results are presented in Fig. 3(a). As shown +in the figure, the NRQM outperforms significantly the +resonant QM. The fidelity jumps from around 60% up to +80% at P = 0.5 if one switches from the resonant QM to +the NRQM. In addition, normal and narrow distribution +of Aj have little impact on the fidelity. Thus the results +for Aj = A are expected similar to that for Aj ̸= A. Such +an independence of the distribution of Aj indicates that +the results shown in Fig. 2(c) may also be applicable to +the case Aj ̸= A even for N = 104. Therefore, we expect +that the NRQM performs better than the resonant QM +in a realistic QD. +Finally, we look at the case of inhomogeneous nuclear +polarization, which can be produced by dynamic nuclear +polarization (DNP) [27, 29]. In DNP, the speed of an +individual nuclear spin’s polarization is roughly propor- +tional to the square of its hyperfine coupling strength, +resulting in a spatially nonuniform distribution of nu- +clear polarization in the QD. High degree of polariza- +tion occurs at these strongly coupled nuclei. Polariza- +tion of the j-th nuclear spin after DNP is approximately +pj = tanh(βA2 +j), where β is a parameter related to the +number and duration of DNP cycles [31, 46]. It was re- +ported that inhomogeneous nuclear polarization signifi- +cantly improves the performance of a QD-based quantum +memory [20]. The performance improves even more when +NRQM is used to suppress nuclear spin noise. Numerical +simulations are carried out in the same model with the +coupling strength Aj unchanged. As shown in Fig. 3(b), +the NRQM protocol again outperforms the resonant one +for inhomogeneous nuclear polarization, similar to the +homogeneous case. +However, the performance of the +NRQM for the inhomogeneous polarization depends on +the distribution of Aj, which is quite different from that +for the homogeneous case. In circumstances of the “nar- +row” distribution of Aj, the combined scheme shows the +best performance: a fidelity over 95% at a bath polariza- +tion P = 0.5, and a fidelity over 80% at P = 0.2. +VI. +CONCLUSION +We proposed a noise-resistant pulsed quantum mem- +ory protocol that performs coherent state transfer be- +tween the electronic and nuclear spins using Hamiltonian +engineering of the hyperfine interaction. Because of its +strong suppression of nuclear spin noise, the NRQM pro- +tocol reduces the requirement for high nuclear polariza- +tion, making experimental realization of QD-based quan- +tum memory more feasible. In addition, this Hamiltonian +engineering approach may be helpful for further investi- +gations in quantum memory and DNP in other systems +such as NV color centers, doped-ion crystals and atomic +ensembles. +ACKNOWLEDGMENTS +This work is supported by the NSAF under Grant No. +U1930201, National Natural Science Foundation of China +(NSFC) under Grants No. 12274331 and No. 91836101, +and Innovation Program for Quantum Science and Tech- +nology under Grant No. 2021ZD0302100. The numerical +calculations in this paper have been partially done on the +supercomputing system in the Supercomputing Center of +Wuhan University. +[1] H.-J. Briegel, W. D¨ur, J. I. Cirac, and P. Zoller, Phys. +Rev. Lett. 81, 5932 (1998). +[2] S. Wehner, D. Elkouss, and R. Hanson, Science 362, +eaam9288 (2018). +[3] N. Sangouard, +C. Simon, +H. De Riedmatten, +and +N. Gisin, Rev. Mod. Phys. 83, 33 (2011). +[4] S. Ritter, C. N¨olleke, C. Hahn, A. Reiserer, A. Neuzner, +M. Uphoff, M. M¨ucke, E. Figueroa, J. 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B 82, 045314 (2010). + diff --git a/J9AyT4oBgHgl3EQfsPnV/content/tmp_files/load_file.txt b/J9AyT4oBgHgl3EQfsPnV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e98101142909eb2cab77e6ad375766d42c6fcd4a --- /dev/null +++ b/J9AyT4oBgHgl3EQfsPnV/content/tmp_files/load_file.txt @@ -0,0 +1,702 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf,len=701 +page_content='Noise-resistant quantum memory enabled by Hamiltonian engineering Lei Jing,1 Peng Du,1 Hui Tang,1 and Wenxian Zhang1, 2, ∗ 1Key Laboratory of Artificial Micro- and Nano-structures of Ministry of Education, School of Physics and Technology, Wuhan University, Wuhan, Hubei 430072, China 2Wuhan Institute of Quantum Technology, Wuhan, Hubei 430206, China (Dated: January 3, 2023) Nuclear spins in quantum dots are promising candidates for fast and scalable quantum memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' By utilizing the hyperfine interaction between the central electron and its surrounding nuclei, quan- tum information can be transferred to the collective state of the nuclei and be stored for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' However, nuclear spin fluctuations in a partially polarized nuclear bath deteriorate the quan- tum memory fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Here we introduce a noise-resistant protocol to realize fast and high-fidelity quantum memory through Hamiltonian engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' With analytics and numerics, we show that high-fidelity quantum state transfer between the electron and the nuclear spins is achievable at rela- tively low nuclear polarizations, due to the strong suppression of nuclear spin noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' For a realistic quantum dot with 104 nuclear spins, a fidelity surpassing 80% is possible at a polarization as low as 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Our approach reduces the demand for high nuclear polarization, making experimentally realizing quantum memory in quantum dots more feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' INTRODUCTION Quantum memory (QM) is a fundamental building block in quantum computation and quantum commu- nication [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Although efficient and an-hour-long- storage-time QM has been realized in trapped ions and atomic ensembles [6–9], fast and scalable solid state can- didates for a practical QM are still in demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Those solid-state physical systems include Nitrogen-vacancy centers in diamonds, doped ions in crystals, and semicon- ductor quantum dots (QDs) [10–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Among them, the QD-based QM which takes the nuclear spin ensemble as its memory medium, is known for its long storage time, excellent optical and electronic properties, and large-area manufacture potentials, making it an appealing option for quantum information processing [18–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The original QD-based QM protocol is proposed by Taylor, Marcus and Lukin (hereafter referred to as the resonant QM) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' It utilizes the hyperfine interaction to write in and read out the quantum information from the electron spin to nuclear spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Due to the intrin- sic long coherence time of nuclear spins, the quantum information can be stored for up to milliseconds [22, 24– 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' In addition, the writing and reading process could be as fast as nanoseconds because of the strong hyper- fine coupling between the electron and nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' However, the performance of QD-based QM depends sensitively on the nuclear spin polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' To write an arbitrary qubit into the nuclei with 100% fidelity, full polarization is required, which is impossible to achieve in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Recent advances in optical pumping of nuclear spins in GaAs/AlGaAs QDs have illustrated about 80% nuclear polarization [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Even with such a record-high degree of polarization, the fidelity of the resonant QM protocol is still below 80% [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Alternative approaches to en- sure high QM fidelity but at low nuclear polarizations are ∗ Corresponding email: wxzhang@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='cn constantly in great demand, such as nuclear state prepa- ration [21, 28–30], inhomogeneous polarization [20, 31], and the use of noncollinear hyperfine interaction [23, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The major obstacle in QD-based QM protocols stems from the nuclear spin fluctuations, which become promi- nent at low polarizations and degrade significantly the QM fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' To suppress the nuclear spin noises, we pro- pose in this paper a noise-resistant QM (NRQM) proto- col through Hamiltonian engineering of electron-nuclear hyperfine interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' By applying periodically fast π- pulses along x- and y-axis, the hyperfine interaction is ef- fectively transformed into a flip-flop Hamiltonian, which simultaneously flips the electron spin and flops a nuclear spin thus realizes an efficient quantum state transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' More importantly, the effects of nuclear spin fluctuations are strongly suppressed by these pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' With this idea, we realize high-fidelity QM but at relatively low polar- izations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Our scheme is compatible with inhomogeneous polarization and nuclear state preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Better QM performance is achieved by combining them together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' ELECTRON-NUCLEAR SPIN DYNAMICS IN A QD-BASED QM For QDs, the coupling of the s-state conduction elec- tron to a mesoscopic bath of nuclear spins is governed by hyperfine contact interaction [33–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' In a magnetic field B0 along the z-axis, the Hamiltonian is H = g∗ eµBB0Sz + � j AjIj · S, (1) where the first term corresponds the electron Zeeman en- ergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Spin operators S and Ij are for the electron and the j-th nucleus, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' We assume all spins are spin-1/2 for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The coupling strength Aj is given by Aj = A0v0 |ψ(rj)|2 with A0 being the hyper- fine contact interaction constant, v0 the volume of a unit arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='00575v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='mes-hall] 2 Jan 2023 2 cell and |ψ(rj)|2 the probability density of the electron at site rj of the j-th nucleus [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The Aj is a func- tion of position varying in a Gaussian form in a typical QD [16, 17, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The hyperfine interaction term can be rewritten as HD + HΩ, where HD = Sz � j AjIz j and HΩ = 1 2 � j Aj � S+I− j + S−I+ j � with S± = Sx ±iSy and I± j = Ix j ±iIy j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The diagonal term HD produces an effec- tive magnetic field on the electron called the Overhauser field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' By tuning the magnetic field B0 to be equal in mag- nitude and opposite to the Overhauser field, HD may be cancelled and only the flip-flop term HΩ is left, which introduces spin exchange between the electron and the nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Utilizing the flip-flop term HΩ and zeroing the sum of the Zeeman term and HD, Taylor, Marcus and Lukin proposed a resonant QM protocol in a QD [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Starting with a fully polarized nuclear bath |0⟩n = |I0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' , I0⟩n and a spin-down electron |↓⟩e, this flip-flop Hamiltonian HΩ induces a flipped electron spin with a collective nuclear spin excitation |↑⟩e � |1⟩n where |1⟩n = �� j |Aj|2�−1/2 � j Aj |I0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' , (I0 − 1)(j), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' , I0⟩n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Since a spin-up electron and a fully polarized bath stay still due to the conservation of angular momentum, an arbitrary initial electron spin evolves like (α |↑⟩e + β |↓⟩e) ⊗ |0⟩n → |↑⟩e ⊗ (α |0⟩n + iβ |1⟩n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (2) In this way the quantum state of the electron spin is co- herently mapped into the collective mode of the nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The quantum state transition can be turned off by remov- ing the electron from the QD, tuning the magnetic field away from the resonant condition or dynamical decou- pling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Due to nuclear spins’ long coherence time, infor- mation encoded in the nuclear spins can be preserved for a long time [22, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Retrieval of the stored information is simply reversing the process: let the system oscillate for another half cycle under the flip-flop Hamiltonian and the quantum information returns to the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' In practice, full nuclear polarization is difficult to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Incomplete polarization may degrade sig- nificantly the QM performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' For instance, a partly polarized thermal nuclear bath is composed of many different pure nuclear spin states |I, M⟩, ρ = � w(I, M) |I, M⟩ ⟨I, M|, where I is the total angular mo- mentum for N nuclear spins and M = −I, −I + 1, · · · , I is its projection into the z-axis [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' To transfer the qubit back and forth fully between the electron and the nuclear state, all pure states have to be simultaneously in resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' This is roughly the case at high polar- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' However, at low polarization P, the proba- bility of I follows approximately a Gaussian distribu- tion with a width σ = � (1 − P)(1 + P)N/4 increasing as P decreases, indicating that a large number of bath states are off-resonant [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' These off-resonant oscilla- tions damp the Rabi oscillation and deteriorate the QM performance [18, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (a) (b) x y z x y z x y z π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' π" π" x y z x y z (c) 𝐹!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=',# 𝐹$,# 𝐹%,# +1 +1 +1 1 +1 1 +1 1 "𝑆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (𝑡) +1 1 +1 +1 "𝑆$(𝑡) "𝑆%(𝑡) Pulse sequence FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (a) Procedure of NRQM, where the electron spin state is coherently transferred into the collective modes of nu- clei through the flip-flop Hamiltonian HΩ enabled by pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (b) A pulse cycle [XXYY] for Hamiltonian engineering to gen- erate effectively HΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The spin frames (spin operators instead of states) shown as spheres are periodically rotated by the pulses in the interaction picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (c) Time-domain transfor- mations of spin operators ˜Sx,y,z(t) in the interaction frame driven by the periodic pulse sequence, depicted by the matrix- based representation [Fµ,k], where the row is µ = (x, y, z) and the column is k = (1, 2, 3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' NOISE-RESISTANT QUANTUM STATE TRANSFER VIA HAMILTONIAN ENGINEERING To implement the QM in a QD even at low nuclear po- larizations, we need to design a pulse sequence that keeps intact the desired flip-flop Hamiltonian but significantly suppresses the fluctuation of the Overhauser field, which causes the off-resonant oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The developed pulse sequence is [XXYY]n, n cycles of [XXYY] as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 1, where X and Y represent a π pulse that rotates the electron spin 180 degrees around the x- and y-axis, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The pulse interval is τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' It is straightforward to illustrate the effect of the pulse sequence according to the average Hamiltonian theory and the matrix representation [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' In the toggling frame the electron spin operators are periodically rotated by the pulses, leading to the time-dependent Hamiltonian ˜H(t) = � j � AjIx j ˜Sx(t)+AjIy j ˜Sy(t) � +g∗ eµBBeff ˜Sz(t), (3) where Beff = B0 + � j AjIz j /g∗ eµB is the effective mag- netic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Taking the spin operator Sz as an example, the pulse sequence transforms it into ±Sz operators pe- riodically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' For each spin operator Si, we can identify its 3 transformation trajectory as ˜Si(t) = � µ Fµ,kSµ, for tk−1 < t < tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (4) where the F = [Fµ,k] = [Fx,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Fy,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Fz,k] is a 3 × n ma- trix containing only 0 and ±1, tk is the time point at which the pulses are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 1(c), the transformation matrix representation for ˜Si (i = x, y, z) reveals how the pulses alter the system’s spin dynamics in an intuitive way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' By averaging ˜Si(t), we find that the Sz operator is effectively cancelled but a fraction of Sx and Sy remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Thus the pulse sequence zeros the effective magnetic field while maintaining the flip-flop interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' In this way, one easily obtains the zeroth-order average Hamiltonian H (0) = 1 4 � j Aj � S+I− j + S−I+ j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (5) Higher-order terms are neglected since they diminish as τ approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The above analysis indicates that the pulse sequence [XXYY]n indeed generates the desired flip-flop Hamilto- nian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' In addition, this protocol is expected to be robust against magnetic noise in the z-direction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' fluctua- tions of the Overhauser field) because terms containing Sz in the Hamiltonian average to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' In this sense, we re- fer to the protocol as the NRQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Compared to the reso- nant QM protocol, NRQM is independent of the external magnetic field B0 and may outperform the resonant one, particularly at lower nuclear polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' ANALYTICAL RESULTS FOR Aj = A The nuclear bath is composed of 104 to 106 nuclear spins, each having different coupling strength Aj with the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Analytics for the dynamics of the system is challenging [40–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' In the following we consider the case where inhomogeneity is negligible (Aj = A), such that the dynamics of the system under the average Hamil- tonian H (0) is analytically solvable, and we analyze the performance of the NRQM protocol following the deriva- tion in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' For the initial electron state |↑⟩ and the collective nuclear state |I, M⟩, the system’s wave function after time t is |ψ1(t)⟩ = cos(ω1t) |↑⟩ ⊗ |I, M⟩ − i sin(ω1t) |↓⟩ ⊗ |I, M + 1⟩ where ω1 = A � (I − M)(I + M + 1)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' For the initial electron state |↓⟩ the system evolves as |ψ2(t)⟩ = cos(ω2t) |↓⟩⊗|I, M⟩−i sin(ω2t) |↑⟩⊗|I, M − 1⟩, where ω2 = A � (I + M)(I − M + 1)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Obviously, the oscillation frequencies of the dynamics depends on I and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' For a partially polarized nuclear bath in thermal equi- librium, the statistical weight w(M) = CN k θk(1 − θ)k, where CN k is the binomial coefficient, k = N/2 − M and θ = eγ/(1 + eγ) [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The corresponding nuclear polar- ization is P = tanh(γ/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' This distribution is roughly 0 t1 t1+t2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='5 ⟨Sz⟩ (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='0 P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='0 F (c) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (8) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (7) 0 t1 t1+t2 Time −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='5 ⟨Sz⟩ (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Typical dynamics of the electron spin in a partially polarized nuclear spin bath (P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='5) (a) in the resonant QM protocol and (b) in the NRQM protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Because the bath is a mixture of numerous potential states, the system’s evolution can take many different paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' When we depict every path, we get blurred lines in the image that diffuse over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The ensemble averaged dynamics of the electron spin shown as the yellow and green solid lines are damped Rabi oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (c) Fidelity as a function of nuclear bath polarization P with 104 nuclear spins for the resonant QM protocol (solid line with circles) and NRQM protocol (solid line with triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Dash- dotted and dotted lines: analytical estimates from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (7) and (8), respectively, for the NRQM protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Dashed line: analytical estimate for the resonant QM protocol [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The NRQM protocol significantly outperforms the resonant QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Gaussian centered at M = −NP/2 with the variance σ2 = (N/4)(1 − P)(1 + P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The statistical weight of the state |I, M⟩ is w(I, M) = w(M)(CN m − CN m−1)/CN k , ≈ w(M)ζI−|M|, (6) where m = N/2−I, ζ = ∆P/(2−∆P) and ∆P = 1−P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The approximation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (6) holds for small I − |M|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' One immediately finds that for a large value of P (small ∆P and ζ), the states |I, M⟩ with M = −I account for the majority with a proportion of 1−ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Other states with higher I(> −M) have an exponentially smaller statistical weight and have much less impact on the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' To calculate the fidelity of the NRQM up to the linear order in ζ, we only include the states with M = −I and M = −I + 1 and neglect the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The fidelity of the memory protocol is defined as the overlap between the initial electron state and the re- trieved one, taking the minimum over all possible initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' According to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' [19], the minimal fidelity can be found by considering only two types of initial condi- tions: (1) the electron spin pointing to the z-axis, (2) the electron spin lying in the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' For the first case, we calculate sz = Tr(Szρf), where ρf is the density ma- trix for the final retrieved electron state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' For the second case, we calculate sT = � s2x + s2y and s0 = Tr(Szρf), where sx,y = Tr(Sx,yρf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The fidelity F is the minimum 4 of the following three quantities: f1 = (1 + sz)/2, f2 = (1+sz −2s0)/2 and f3 = (1+sT −s2 0/[4(sz −s0−sT )])/2, When s0/[2(sz−s0−sT )] /∈ [−1, 1], F takes the minimum of f1 and f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' By taking into account of two types of initial bath states, M = −I and M = −I + 1, and averaging with probability w(I, M), we calculate sz, s0, sT up to the linear order in ζ : sz ≈ 1−ζ � 1 − cos4 γ0 − sin4 γ0 � , s0 ≈ −(ζ/2) � 1 − cos4 γ0 − sin4 γ0 � , sT ≈ 1 − ζ (1 + cos γ0) , where γ0 = π/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='We then obtain the fidelity F = 1 + sz 2 ≈ 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='232 ζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (7) Substituting ζ = ∆P/(2 − ∆P), the fidelity is further simplified as F ≈ 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='116∆P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (8) The results from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (7) and (8) are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 2(c) as dash-dotted and dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' For compar- ison, the analytical estimate of fidelity for the original resonant QM protocol is F ≈ 1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='38∆P [19], which is also depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 2(c) as the dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Starting with full polarization, the fidelity of the NRQM proto- col drops more than ten times slower than that of the resonant one as the nuclear polarization lowers, demon- strating the advantages of the NRQM protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' NUMERICAL RESULTS While we have derived the fidelity Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (7) at high bath polarizations and proved the advantage of the NRQM, the performance of the protocol at low polarizations or inhomogeneous hyperfine coupling is still in need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' This task is completed by numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' First, we consider the case where inhomogeneity is neg- ligible (Aj = A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' We numerically simulate the dynamics of the electron spin and N = 104 nuclear spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The system is prepared as a tensor product of the electron state |φ⟩ and the nuclear state |I, M⟩ with a statistical weight w(I, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The system evolves under the Hamilto- nian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (1) and the pulse sequence [XXYY]n for a time t1 and the electron state is mapped onto the collective nuclear state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' After that, the electron is ejected and a nuclear mixed state is reduced by tracing out the elec- tron’s degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' For the retrieval, another fully polarized electron is injected in the QD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The system’s state becomes a tensor product of the electron state and the reduced nuclear bath state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Then the system evolves under the same pulse sequence for another time t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Af- ter tracing out the nuclear bath’s degree of freedom, we obtain the final density matrix ρf of the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' We plot how the electron observable ⟨Sz⟩ changes over time during this process in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 2(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' For a given |I, M⟩, the trajectory of ⟨Sz⟩ is a translucent curve with opacity based on the statistical weight of the bath state w(I, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' In total, we get blurred lines in the image that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='0 P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='0 F (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='0 P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='0 (b) Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (NRQM) Narr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (NRQM) Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (RQM) Narr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (RQM) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Performance of quantum memory protocols in a bath of N=20 nuclear spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (a) QM fidelity for homogeneous bath polarizations with different widths of Aj for the resonant QM protocol (RQM) and the NRQM protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' (b) QM fidelity for inhomogeneous bath polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' diffuse over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The thickest blurred line, which in fact includes many lines, shows the electron spin’s evolu- tion with bath states |I, M⟩ where M = −I, I ∈ [0, N/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The rest blurred lines come from bath states |I, M⟩ where M = −I + 1, M = −I + 2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' By averaging over dif- ferent bath states with statistical weight w(I, M) we cal- culate the weighted observables for the electron (sz,0,T ) in a realistic thermal bath ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' This ensemble av- eraged dynamics of the electron are drawn as the yellow and green solid lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 2(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The mini- mal fidelity is calculated accordingly using (sz,0,T ) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The times t1 and t2 are tuned to maximize the minimal fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The NRQM results at different nuclear polariza- tion P are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 2(c) as a solid line with triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The simulated fidelities for the resonant QM protocol are also plotted as a solid line with circles for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' As shown in the figure, the NRQM illustrates a signif- icant improvement in the fidelity over the resonant QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The NRQM protocol has a fidelity over 90% at P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='5 and the fidelity is still over 80% at P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' In stark con- trast, the resonant QM protocol requires a polarization greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='8 in order to achieve 80% fidelity [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Clearly, the need for strong nuclear polarization is dra- matically mitigated for the NRQM protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' In addition, the numerics agrees well with our analytical estimate of the fidelity at high polarizations, implying the validity of previous analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Second, we consider the case of inhomogeneous hyper- fine coupling (Aj ̸= A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' We numerically simulate the dy- namics of the electron spin interacting with N = 4×5 nu- clear spins arrayed in a rectangular lattice, using the ef- ficient Chebyshev-expansion-based algorithm [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' More computer resources would be required to include more nuclear spins, but N = 20 is adequate to make our simulations represent bigger systems with a precision of 1/N [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' In order to compare with previous works, we adopt the same coupling strengths Aj as in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 5 The values of Aj spread from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='96 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='31, referred to as the normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' We also consider the case with decreased QD widths by a factor of 1/ √ 2 to represent a narrow distribution of Aj, spreading from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='92 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Numerical results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' As shown in the figure, the NRQM outperforms significantly the resonant QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The fidelity jumps from around 60% up to 80% at P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='5 if one switches from the resonant QM to the NRQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' In addition, normal and narrow distribution of Aj have little impact on the fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Thus the results for Aj = A are expected similar to that for Aj ̸= A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Such an independence of the distribution of Aj indicates that the results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 2(c) may also be applicable to the case Aj ̸= A even for N = 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Therefore, we expect that the NRQM performs better than the resonant QM in a realistic QD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Finally, we look at the case of inhomogeneous nuclear polarization, which can be produced by dynamic nuclear polarization (DNP) [27, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' In DNP, the speed of an individual nuclear spin’s polarization is roughly propor- tional to the square of its hyperfine coupling strength, resulting in a spatially nonuniform distribution of nu- clear polarization in the QD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' High degree of polariza- tion occurs at these strongly coupled nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Polariza- tion of the j-th nuclear spin after DNP is approximately pj = tanh(βA2 j), where β is a parameter related to the number and duration of DNP cycles [31, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' It was re- ported that inhomogeneous nuclear polarization signifi- cantly improves the performance of a QD-based quantum memory [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' The performance improves even more when NRQM is used to suppress nuclear spin noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Numerical simulations are carried out in the same model with the coupling strength Aj unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 3(b), the NRQM protocol again outperforms the resonant one for inhomogeneous nuclear polarization, similar to the homogeneous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' However, the performance of the NRQM for the inhomogeneous polarization depends on the distribution of Aj, which is quite different from that for the homogeneous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' In circumstances of the “nar- row” distribution of Aj, the combined scheme shows the best performance: a fidelity over 95% at a bath polariza- tion P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='5, and a fidelity over 80% at P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' CONCLUSION We proposed a noise-resistant pulsed quantum mem- ory protocol that performs coherent state transfer be- tween the electronic and nuclear spins using Hamiltonian engineering of the hyperfine interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Because of its strong suppression of nuclear spin noise, the NRQM pro- tocol reduces the requirement for high nuclear polariza- tion, making experimental realization of QD-based quan- tum memory more feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' In addition, this Hamiltonian engineering approach may be helpful for further investi- gations in quantum memory and DNP in other systems such as NV color centers, doped-ion crystals and atomic ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work is supported by the NSAF under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' U1930201, National Natural Science Foundation of China (NSFC) under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 12274331 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 91836101, and Innovation Program for Quantum Science and Tech- nology under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 2021ZD0302100.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Hugues, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Le Gall, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Atat¨ure, Science 364, 62 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' [22] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Chekhovich, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Bachorz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Clarke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Hugues, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Le Gall, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Atat¨ure, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 17, 585 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' [24] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Gillard, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Clarke, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Chekhovich, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Com- mun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Ludwig, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Wieck, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Loss, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Poggio, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Warburton, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 11, 885 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' [26] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Chekhovich, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Hopkinson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Skolnick, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Tar- takovskii, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 6, 6348 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' [27] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Chekhovich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Ulhaq, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Zallo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Ding, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Schmidt, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Skolnick, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 16, 982 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' [28] G.' metadata={'source': 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Atat¨ure, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' 119, 130503 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' [29] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Reilly, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} +page_content=' Taylor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQfsPnV/content/2301.00575v1.pdf'} 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100644 index 0000000000000000000000000000000000000000..fc6d00427215001f9fc28baafd2088522fef99b2 --- /dev/null +++ b/LdE3T4oBgHgl3EQfYQps/content/tmp_files/2301.04486v1.pdf.txt @@ -0,0 +1,1900 @@ +arXiv:2301.04486v1 [math.DS] 11 Jan 2023 +ON PSEUDO-ROTATIONS OF THE ANNULUS +WITH GENERIC ROTATION NUMBER +BARNEY BRAMHAM AND ZHIYUAN ZHANG +ABSTRACT. We show that for a Baire generic rotation number α ∈ R/Z, +the set of area preserving C∞-pseudo-rotations of the annulus A with +rotation number α equals the closure of the set of area preserving C∞- +pseudo-rotations which are smoothly conjugate to the rotation Rα. As a +corollary, a C∞-generic area preserving pseudo-rotation of the annulus +with a Baire generic rotation number α is weakly mixing. +1. INTRODUCTION +In this paper we denote the 2-dimensional annulus by A = R/Z × [0, 1] +equipped with the standard area-form ω. We let F∞ +A denote the set of ω- +preserving C∞ pseudo-rotations of A (the precise definition will appear in +Definition 2.9). Namely, we set +F∞ +A := {f ∈ Diff∞(A, ω) | f is isotopic to Id and has no periodic points}. +The study of pseudo-rotations can be essentially traced back to the ques- +tion of Birkhoff [B41] (see also [H98]) as to whether there are non-trivial +analytic diffeomorphisms of the 2-sphere with 2 fixed points (the existence +of such diffeomorphisms was recently announced by Berger [B22]). The +name “pseudo-rotation”was introduced by B´eguin, Crovisier, Le Roux and +Patou in [BCLP04]. By a result of Franks [F88], for ω-preserving home- +omorphisms of A, the notion of irrational pseudo-rotations in [BCLP04] +coincides with ours. In particular, each f ∈ F∞ +A admits a rotation number +ρ( f) ∈ (0, 1)/Q (see Definition 2.10 for the details). For each α ∈ (0, 1) \ Q +we set +F∞ +A(α) +:= +{f ∈ F∞ +A | ρ( f) = α}, +O∞ +A(α) +:= +{hRαh−1 | h ∈ Diff∞(A, ω)} +where Rα denotes the rotation (x, y) �→ (x + α, y). It is clear that O∞ +A(α) ⊂ +F∞ +A(α) for any α ∈ (0, 1) \ Q. It is not hard to see that F∞ +A(α) is closed in the +C∞-topology for each irrational α. In particular, O∞ +A(α) ⊂ F∞ +A(α) for any +α ∈ (0, 1) \ Q, where the closure is taken in the C∞-topology. Our main +result is the following. +Date: January 12, 2023. +1 + +2 +BARNEY BRAMHAM AND ZHIYUAN ZHANG +THEOREM 1. For a Baire generic α ∈ (0, 1) \ Q, we have +F∞ +A(α) = O∞ +A(α) +where the closure is taken in Diff∞(A). +In other words, for a Baire generic α, any pseudo-rotation f with rotation +number α is the C∞-limit of a sequence fk of area preserving diffeomor- +phisms, which up to a smooth area preserving change of coordinates, is the +standard rotation Rα. In particular f is approximable by integrable systems. +We can see Theorem 1 as a natural analogue of a well-known theorem +of Herman in [H79]. Recall that one of the most prominent results in the +study of circle diffeomorphisms is the following. +THEOREM 2 (Herman-Yoccoz). For any irrational α ∈ R/Z, we denote by +F∞(α) the set of C∞ circle diffeomorphisms with rotation number α, and denote +by O∞(α) the set of C∞ circle diffeomorphisms which are C∞-conjugate to the +standard rotation Rα. Then we have +F∞(α) = +� +O∞(α) = O∞(α) +if α is Diophantine, +O∞(α) ̸= O∞(α) +if α is Liouville. +Here in the above the closures are taken under the C∞-topology. +The above result for Diophantine α was conjectured by Arnold, who +showed in [A78] that any Cω circle diffeomorphism with a Diophantine +rotation number α which is sufficiently close to Rα in the Cω-topology, is +infact Cω-conjugate to Rα. Arnold’s result was then generalised to the C∞- +category by Moser in [M66]. This is the beginning of what is now known as +the Kolmogorov-Arnold-Moser theory. The global picture was for the first +time established by Herman in the seminal paper [H79]. In [H79], Theorem +2 was proved for a subset of α with full Lebesgue measure. Khanin and +Sinai gave in [KS89] a different proof of the main result in [H79] building +on a renormalization theory for circle diffeomorphisms. The Diophantine +part of Theorem 2 was completed by Yoccoz in [Y84]. We also mention +Katznelson-Ornstein’s papers [KO89a, KO89b] on circle diffeomorphisms +with low regularity, and Yoccoz’s paper in [EKMY02] on Cω-linearization +under the sharp arithmetic condition, i.e., H-condition. For a recent survey +of this development and beyond, we refer the reader to [EFK18]. +The Liouville part of Theorem 2 was conjectured by Herman in [H79, +Conjecture 7.1]. In fact Herman already showed that Theorem 2 holds for +a Baire generic set of α, see [H79, Theorem 7.3]. However, his proof was +based on the Diophantine part of Theorem 2 (at least for a full measure +set of α), and used certain properties of the function t �→ ρ(Rt f) of a cir- +cle diffeomorphism f. It is still an open question of Herman whether the + +ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER +3 +Diophantine rigidity holds within pseudo-rotations. Moreover, it is un- +clear how to deform a general pseudo-rotation within the set of pseudo- +rotations, and change the rotation number. This blocks a direct generalisa- +tion of Herman’s approach for pseudo-rotations. The full answer to Her- +man’s conjecture was provided by Yoccoz in [Y95b]. Yoccoz showed that +any C∞ circle diffeomorphsim with a Liouville rotation number can be C∞- +approximated by a quasi-rotation: this is a class of circle diffeomorphisms +which, among other things, admits a renormalization that is a standard ro- +tation. Our proof of Theorem 1 is somewhat similar to the proof of Yoccoz: +we also consider certain renormalizations of a pseudo-rotation. However, +the type of estimates are very different. We are unable to transfer the strong +estimates for circle diffeomorphisms, such as Denjoy’s inequality in [Y84], +to general pseudo-rotations, due to the possible occurrence of complicated +geometry which does not appear in dimension 1. On the other hand, the +area-preserving hypothesis provides us with certain strong C0-estimates +established in [AFLXZ20]. Combining such estimates with a suitable arith- +metic condition, we are able to extract some useful information from a se- +quence of suitably renormalized pseudo-rotations. +We can also compare Theorem 1 with the main result in [B15b], which +says that any smooth area preserving pseudo-rotation f on the closed 2- +disc, meaning that f fixes the origin and has no periodic points on the +annulus complementary to the origin, is the C0-limit of smooth periodic +disc maps fk, that each fix the origin. In the latter there are no restrictions +on the (irrational) rotation number of f, but in this current paper our inte- +grable approximations are in every way stronger: 1) In [B15b] the sequence +of approximations fk, while C∞-smooth, only converge in the C0 topology +to f. 2) In [B15b] the fk’s are not necessarily area preserving. 3) In [B15b] +the approximation maps fk have rational rotation numbers pk/qk ∈ Q con- +verging to the rotation number α of f as k → ∞. One cannot perturb the fk +in [B15b] in an obvious way to make the rotation number equal to α while +retaining closeness to f. On the other hand, it is easy to modify the fk’s in +the current article, if one so wishes, to make the rotation number rational +and keep the closeness to f. In short, the fact that in this paper we are able +to find integrable approximations without altering the rotation number is +also a stronger conclusion than in [B15b]. +Another motivation behind our result is the important work of Anosov +and Katok [AK70] and the extensions in Fayad-Saprykina [FS05], see also +Fayad-Katok [FK04], in which, for generic rotation numbers, more pre- +cisely all Liouville rotation numbers, examples of pseudo-rotations are con- +structed which are dynamically interesting, that is, not conjugate to a rota- +tion. These “exotic”pseudo-rotations of Anosov-Katok lie, by construction, +in the C∞-closure of ∪t∈QO∞ +A(t) rather than the closure of O∞ +A(α). It how- +ever follows from our main result that, for a possibly smaller Baire-generic +set of rotation numbers than the Liouville numbers, that the Anosov-Katok + +4 +BARNEY BRAMHAM AND ZHIYUAN ZHANG +constructions do indeed lie in the closure of O∞ +A(α) with fixed rotation num- +ber α. Moreover, we have the following interesting corollary. +COROLLARY A. For a Baire generic α ∈ (0, 1) \ Q, the set of weakly mixing +pseudo-rotations in F∞ +A(α) forms a Baire set with empty interior, with respect to +the C∞ topology. +Proof. On the one hand, Anosov and Katok, see [AK70], show that for a +Baire generic α, weak mixing is a C∞-generic property in O∞ +A(α). Thus by +Theorem 1 weak mixing is a C∞-generic property in F∞ +A(α). +On the other hand, the second statement follows since elements of O∞ +A(α) +are never weak mixing and by Theorem 1 the complement F∞ +A(α)\O∞ +A(α) +has empty interior. +□ +REMARK 1. Corollary A is seen to be rather sharp in the following sense: +(1) the genericity of α cannot be improved into any subset of (0, 1) with pos- +itive Lebesgue measure. +This follows from the KAM result of Fayad- +Krikorian [FK09] (attributed by the authors to Herman), that a neigh- +borhood of Rα in F∞ +A(α) lies in O∞ +A(α), for any Diophantine α. +(2) Weakly mixing cannot be replaced by mixing. In fact, it follows from the +proof of [B15a] and [AFLXZ20] that for a Baire generic α, F∞ +A(α) contains +no topologically mixing maps. See also Theorem 3. +Recently, Avila and Krikorian have announced 1 an improvement of The- +orem 1: for every non-Brjuno α, one has F∞ +A(α) = O∞ +A(α). Moreover, they +have announced the following result: for every pseudo-rotation f in an +open neighborhood of the rigid rotations on D, there exists a sequence of +area-preserving diffeomorphism hn such that hn f h−1 +n +converges to Rρ( f ) in +the C∞ topology. Their method involves delicate estimates on high iterates +of the maps, while our method for getting this weaker result relies only on +rather soft estimates. +Acknowledgements. Z.Z. would like to thank Artur Avila and Rapha¨el +Krikorian for discussion on one occasion. Z.Z. would also like to acknowl- +edge the online talk by Rapha¨el Krikorian during the Workshop “Between +Dynamics and Spectral Theory ”at the Simons Center for Geometry and +Physics back in 2016, which inspired this article. +This work was initi- +ated in 2019 while the authors were at the Institute for Advanced Study +both supported by the National Science Foundation under Grant No. DMS- +1638352. We thank them for their hospitality and excellent working envi- +ronment. B.B. was also partially supported by the SFB/TRR 191 ‘Symplec- +tic Structures in Geometry, Algebra and Dynamics’, funded by the DFG (B1 +281071066 – TRR 191) +1See the minicourse of Krikorian in the program “Renormalization and universality in +Conformal Geometry, Dynamics, Random Processes, and Field Theory ”in 2020 at Simons +Center for Geometry and Physics. + +ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER +5 +NOTATION +In the rest of this paper, we use the following notation. For a subset +A ⊂ R2 we denote by Int(A) the interior of A. For a Cr diffeomorphism +f : U → V between open subsets U, V ⊂ R2 and r ∈ N, we set ∥Dr f∥ = +maxx∈U,|α|=r{∥∂α f(x)∥}, ∥f∥Cr = sup1≤ℓ≤r ∥Dℓ f∥ and ∥f∥Diffr(U) = ∥f∥Cr(U) + +∥f −1∥Cr(V). For Cr diffeomorphisms f, g : U → V, we denote dDiffr(U)( f, g) = +max(∥f −1g∥Diffr(U), ∥g−1 f∥Diffr(U)). We abbreviate ∥f∥Diffr(U), resp. dDiffr(U), +as ∥f∥Diffr, resp. dDiffr, when there is no confusion. For two homeomor- +phisms f, g : U → V, we denote ∥f − g∥ = supx∈U d( f(x), g(x)), where +d(·, ·) is the euclidean metric. +2. PRELIMINARIES +We denote the boundary components of A by +Bi := R/Z × {i} +i = 0, 1. +The universal covering of A is ˜A = R × [0, 1]. We write ˜Bi = R × {i} for +i = 0, 1. We let +π : ˜A → A +be the natural projection, and let T : ˜A → ˜A be the translation in the first +coordinate, i.e., T(x, y) = (x + 1, y). Given a homeomorphism f : A → +A there is a lift F : ˜A → ˜A of f, also a homeomorphism, unique up to +composition by a power of T, such that πF = f π. Moreover, any such lift +commutes with T. +We have the following lemma, whose proof is elementary and left to the +readers. +LEMMA 2.1. Suppose f : A → A is a homeomorphism with ∥f − Id∥ < 1/2. +Then there is a unique lift F : ˜A → ˜A satisfying ∥F − Id∥ < 1/2. This lift +satisfies +d +� +F( ˜x), ˜x +� = d +� +f(x), x +� +(2.1) +for all x ∈ A and all ˜x ∈ ˜A with π( ˜x) = x. +DEFINITION 2.1. We say that γ ⊂ A is a simple regular curve connecting B0 +and B1 if γ = φ([0, 1]) where φ : [0, 1] → A is an injective continuous map +mapping 0, resp. 1, into B0, resp. B1, that maps (0, 1) into A\∂A. We define +simple regular curves in ˜A connecting ˜B0 and ˜B1 in a similar way. +DEFINITION 2.2. For a pair of simple regular curves γ1, γ2 in ˜A we will say +that γ1 is to the left of γ2, or γ2 is to the right of γ1, and write +γ1 < γ2 +if γ1 ∩ γ2 = ∅ and γ1 lies in the component of ˜A\γ2 containing points with +arbitrarily negative R-coordinate. This defines a partial ordering on simple +regular curves in ˜A. + +6 +BARNEY BRAMHAM AND ZHIYUAN ZHANG +DEFINITION 2.3. Let f be a homeomorphism of A. A simple regular curve +γ in A connecting B0 and B1, satisfying +γ ∩ f(γ) = ∅ +is called a Brouwer curve for f. A Brouwer curve γ is called smooth if γ is +a C∞-curve. Brouwer curves for homeomorphisms of ˜A are defined in a +similar way. +Note that if F is a lift of a homeomorphism f : A → A and ˜γ is a lift of a +Brouwer curve γ ⊂ A for f, then +TkF( ˜γ) ∩ ˜γ = ∅ +∀k ∈ Z. +DEFINITION 2.4. Let f be a homeomorphism of A and Q ≥ 2 an integer. +We say that γ is a Q-good smooth curve if γ is a smooth Brouwer curve for +each of the maps f, f 2, · · · , f Q−1. +DEFINITION 2.5. Let γ1, γ2 be two disjoint simple regular curves in A con- +necting B0 and B1. There is a unique closed region R in A with left bound- +ary γ1 and right boundary γ2. More precisely, if each γi is oriented from +B0 to B1, then ∂R ∩ (A\∂A) has orientation agreeing with γ2 − γ1. We say +that R is the region bounded by (γ1, γ2). +DEFINITION 2.6. Let {Kr}r≥1 be an increasing sequence of positive real +numbers. Let U, V be smooth surfaces, possibly with boundary, and let +φ : U → V be a C∞-diffeomorphism. We say that φ is {Kr}r≥1-smooth if +∥φ∥Diffr(U) < Kr +∀r ∈ Z+. +DEFINITION 2.7. Let γ be a Brouwer curve for f, and let R ⊂ A be the +closed region bounded by (γ, f(γ)). We say that an orientation preserving +C∞ -diffeomorphism +φ : U → R′ +from an open neighborhood U ⊂ +˜A of [0, 1]2 to an open neighborhood +R′ ⊂ A of R is an admissible coordinate for (R, f) if the following hold: +(1) φ has constant Jacobian, +(2) φ satisfies +φ({0} × [0, 1]) += +γ, +(2.2) +φ({1} × [0, 1]) += +f(γ), +(2.3) +φ([0, 1] × {0}) +⊂ +B0, +(2.4) +φ([0, 1] × {1}) +⊂ +B1, +(2.5) +(3) there is a neighborhood UL of {0} × [0, 1] in U, so that +f φ(x) = φT(x) +∀x ∈ UL. +Without loss of generality we may assume that T(UL) ⊂ U by choosing UL +sufficiently small. + +ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER +7 +We make a similar definition for lifts. Namely, let ˜γ be a lift of a Brouwer +curve γ for f, let F be an lift of f such that F( ˜γ) is on the right of ˜γ, and let +˜R be the region bounded by ( ˜γ, F( ˜γ)). We say that a C∞-diffeomorphism +φ : U → R′ +from an open neighborhood U of [0, 1]2 in ˜A to an open neighborhood R′ +of ˜R in ˜A is an admissible coordinate for ( ˜R, F) if φ satisfies the analogous +properties above with (R, A, f) replaced by ( ˜R, ˜A, F). +REMARK 2. We notice that by item (3) in Definition 2.7, an admissible coordinate +φ : U → R′ is determined by its restriction to [0, 1]2 in the following sense: if +φi : Ui → Ri, i = 1, 2 are two admissible coordinates for (R, f) such that +φ1|[0,1]2 = φ2|[0,1]2, then there exists an open neighborhood U3 of [0, 1]2 in ˜A such +that φ1|U3 = φ2|U3. For this reason, we will sometimes identify an admissible +coordinate for (R, f) or (R, F) with a map from [0, 1]2 to R. +DEFINITION 2.8. Let r ∈ N and K ∈ (0, ∞). We say that φ : [0, 1]2 → R +is a (r, K)-admissible coordinate for (R, f) if φ is an admissible coordinate, in +the sense of Definition 2.7, defined on an open neighborhood U of [0, 1]2, +satisfying +∥φ∥Diffr(U) < K. +Analogously for lifts: We say that φ : U → +˜R is a (r, K)-admissible coor- +dinate for ( ˜R, F) if φ is an admissible coordinate for ( ˜R, F), in the sense +of Definition 2.7, defined on an open neighborhood U of [0, 1]2, satisfying +∥φ∥Diffr(U) < K. +DEFINITION 2.9. A pseudo-rotation is a non-wandering homeomorphism f : +A → A that is isotopic to the identity, maps B0 (resp. B1) to itself, and has +no periodic points. +Recall that a homeomorphism f : A → A is said to be non-wandering if +for every open subet U ⊂ A, there exists an integer n > 0 such that f n(U) ∩ +U ̸= ∅. In this paper we will only be considering C∞-smooth pseudo- +rotations that preserve a smooth area form ω. By a slight abuse of notation, +we denote by ω both the area form on ˜A and A. If f is an ω-preserving +diffeomorphism then so is each lift F an ω-preserving diffeomorphism. +DEFINITION 2.10. Let f : A → A be a pseudo-rotation with lift F : ˜A → ˜A. +Denote by p1 : ˜A → R the projection on the first coordinate. We define +ρ(F) ∈ R by +ρ(F) := lim +n→∞ +1 +n(p1(Fn(x, y)) − x) +x ∈ R, y ∈ [0, 1]. +It is known that the limit on the right hand side above exists for all (x, y) +and is independent of (x, y), see [F88, FH12]. Moreover, we always have +ρ(F) /∈ Q, see [F88]. For any pseudo-rotation f, there exists a unique lift F + +8 +BARNEY BRAMHAM AND ZHIYUAN ZHANG +of f such that ρ(F) ∈ (0, 1) \ Q, which we denote by ρ( f). We call ρ( f) the +rotation number of f. +Let us recall a few well known facts about the best rational approxima- +tions to α ∈ (0, 1) \ Q and its continued fraction expansion. Readers can +consult [HW, Chapters X, XI] for more details. +First, for x ∈ R we will write +⌊x⌋ +:= +max{n ∈ Z | n ≤ x} ∈ Z, +q(x) +:= +⌊1/x⌋ ∈ Z +for the integer parts of x and 1/x respectively. Moreover, if x ∈ R \ Q then +∥x∥R/Z := d(x, Z) ∈ (0, 1/2). +The Gauss map G : [0, 1) → [0, 1) is defined by +G(x) = 1 +x − q(x) +on (0, 1) and G(0) := 0. If x is irrational then so is G(x). +DEFINITION 2.11. For any α ∈ (0, 1) \ Q, we define the sequences (αn)n≥0 +and (βn)n≥0 in (0, 1) \ Q by +α0 := α, +αn +:= +Gn(α0) +∀n ≥ 1, +βn +:= +n +∏ +i=0 +αi +∀n ≥ 0. +Furthermore we define sequences of non-negative integers (an)n≥0, (qn)n≥0 +as follows: +a0 := 0, +an +:= +q(αn−1) +∀n ≥ 1, +(2.6) +q0 := 1, +q1 := q(α), +qn+2 +:= +qn + qn+1an+2 +∀n ≥ 1. +(2.7) +We also define (pn)n≥0 by p0 = 0, and for n ≥ 1, define pn to be the closest +integer to qnα, which is unique by irrationality of α. We will use the notation +αn(α), qn(α) and pn(α) when it is necessary to indicate the dependence of +the sequences on α. +Note that +α−1 +n−1 = an + αn +∀n ≥ 1, +(2.8) +since αn = G(αn−1) = 1/αn−1 − q(αn−1) = 1/αn−1 − an. It is also well +known that +pn+1qn − pnqn+1 += +(−1)n +(2.9) +βn = (−1)n(qnα − pn) +> +0 +∀n ≥ 0. +(2.10) + +ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER +9 +In particular for each n ≥ 1, βn = |qnα − pn| = d(qnα, Z), and +α = +1 +a1 + +1 +a2 + +1 +... + +1 +an + αn +pn +qn += +1 +a1 + +1 +a2 + +1 +... + 1 +an +For any n ≥ 1, the integers pn and qn are relatively prime, and pn/qn is +called the n-th best rational approximation of α. The following simple in- +equalities are known, and will be used later: +1 +2qn+1 +< +1 +qn + qn+1 +< βn < +1 +qn+1 +, +(2.11) +αn, q(αn)−1 ∈ +� +qn +2qn+1 +, 2qn +qn+1 +� +. +(2.12) +3. EXISTENCE OF A BROUWER CURVE WITH UNIFORM BOUNDS +In order to avoid lengthy computations, in the rest of the paper we will +introduce various increasing functions to keep track of parameter depen- +dence. We say that a function A : U → R defined on an open subset +U ⊂ Rn is increasing if for any x = (x1, · · · , xn) and y = (y1, · · · , yn) ∈ U +with xi ≥ yi for all 1 ≤ i ≤ n, we have A(x) ≥ A(y). We define decreas- +ing functions in a similar way. Typically, all the variables and values of the +increasing/decreasing functions that we will consider lie in R+. +The following theorem, essentially proven in [AFLXZ20], allows one to +control the Cr-distance of a pseudo-rotation to the identity in terms of its +rotation number. +THEOREM 3. There is a sequence of increasing functions +Ar : (0, 1/2) × R+ → R+, +lim +t→0 Ar(t, · ) ≡ 0 +for each r ∈ N, so that for any {Kr}r≥1-smooth ω-preserving pseudo-rotation +f : A → A there holds +∥f − Id∥ +< +A0(∥ρ( f)∥R/Z, K1), +∥Dr f∥ +< +Ar(∥ρ( f)∥R/Z, Kr+1), +∀r ≥ 1. +Proof. By [AFLXZ20, Corollary A], we know that +∥f − Id∥ < (1 + 2K1)∥ρ( f)∥1/2 +R/Z =: A0(∥ρ( f)∥R/Z, K1). + +10 +BARNEY BRAMHAM AND ZHIYUAN ZHANG +By the Hadamard-Kolmogorov convexity theorem, we get corresponding +bounds on the “inbetween” derivatives: for any r ≥ 1 +∥Dr f∥ +≤ +Cr∥f − Id∥ +1 +r+1∥f∥ +r +r+1 +Diffr+1 +≤ +Cr∥ρ( f)∥ +1 +2(r+1) +R/Z (1 + 2Kr+1) =: Ar(∥ρ( f)∥R/Z, Kr+1). +□ +By Theorem 3, there is a decreasing function +ρ∗ : R+ → (0, 1/2) +(3.1) +such that for any smooth ω-preserving pseudo-rotation f satisfying +∥ρ( f)∥R/Z < ρ∗(∥D f∥), +(3.2) +we have +∥f − Id∥ ≤ A0(∥ρ( f)∥R/Z, ∥D f∥) < A0(ρ∗(∥D f∥), ∥D f∥) < 1/2. +Then by Lemma 2.1, there is a unique lift F of f satisfying +∥F − Id∥ = ∥f − Id∥ < 1/2. +(3.3) +LEMMA 3.1. There are increasing functions +C′, C′′ : R+ × (0, 1) → R+ +with limt→0 C′′(t, ·) ≡ 0 and +C′′(α, K−1) < ρ∗(Kα−1) +∀(α, K) ∈ (0, 1) × (1, ∞) +(3.4) +such that the following holds: if α ∈ (0, 1)\Q satisfies +α +∈ +(0, ρ∗(K)), +(3.5) +G(α) +< +C′′(α, K−1) +(3.6) +for some K > 1, where ρ∗ is given in (3.1), then every C1-smooth ω-preserving +pseudo-rotation with ∥D f∥ < K and ρ( f) = α satisfies +inf +x∈A d(x, f(x)) ≥ C′(α, K−1). +(3.7) +Proof. Fix K > 0 and α ∈ (0, 1)\Q satisfying (3.5) and (3.6). Let f be a +C1-smooth ω-preserving pseudo-rotation with ∥D f∥ < K and ρ( f) = α. +Denote by F the lift of f with ρ(F) = ρ( f). We abbrieviate q := q(α) = +⌊1/α⌋. +Choose the function C′′ : R+ × (0, 1) → R+, with limt→0 C′′(t, ·) ≡ 0, +sufficiently small so that for each t ∈ (0, 1), the condition +s < C′′(t, K−1) +implies +A0(s, Kt−1) < 1/2. +(3.8) + +ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER +11 +Without loss of generality we may arrange that C′′ is increasing and satis- +fies (3.4). By Theorem 3, we have +∥f q − Id∥ ≤ A0(∥ρ( f q)∥R/Z, ∥D f q∥) ≤ A0(∥qρ( f)∥R/Z, Kq). +Since ∥qρ( f)∥R/Z = 1 − qα = αG(α) ≤ G(α) and Kq ≤ Kα−1, the above +gives +∥f q − Id∥ ≤ A0(G(α), Kα−1). +Thus, if α satisfies (3.6), then by (3.8), +∥f q − Id∥ < 1/2. +(3.9) +The unique lift of f q with rotation number in (−1/2, 1/2) is T−1Fq, since +ρ(T−1Fq) = qρ(F) − 1 = qα − 1 ∈ (−1/2, 1/2). So by Lemma 2.1 +∥Fq − T∥ = ∥T−1Fq − Id∥ = ∥f q − Id∥. +Thus for all ˜x ∈ ˜A +d(Fq( ˜x), ˜x) ≥ d(T( ˜x), ˜x) − d(Fq( ˜x), T( ˜x)) > 1 − 1/2 = 1/2. +(3.10) +On the other hand, for each ˜x ∈ ˜A, there holds +d(Fq( ˜x), ˜x) ≤ +q−1 +∑ +j=0 +d +� +Fj+1( ˜x), Fj( ˜x) +� ≤ +q−1 +∑ +j=0 +∥DF∥jd +� +F( ˜x), ˜x +� ≤ Cd(F( ˜x), ˜x) +where C = Kq−1 +K−1 ≤ Kα−1−1 +K−1 . Combining the last line with (3.10) and (3.3), we +obtain +∥F( ˜x) − ˜x∥ ≥ 1 +2C +∀ ˜x ∈ ˜A. +(3.11) +By Lemma 2.1 and (3.5), we conclude that (3.7) holds if we define C′ by +C′(t, K−1) = 1 +2 +K − 1 +Kt−1 − 1. +□ +DEFINITION 3.1. For each K > 1 let L(K) be the set of α ∈ (0, 1/2) \ Q such +that +α +∈ +� +0, ρ∗(K) +� +, +(3.12) +G(α) +< +C′′(α, K−1) +(3.13) +where ρ∗ is from (3.1) and C′′ is from Lemma 3.1. +LEMMA 3.2. There is a sequence of increasing functions +C′ +r : R2 ++ → R+ +∀r ≥ 1, +such that the following is true. If {Kr}r≥1 is any sequence in (1, ∞) and α ∈ +L(K1), then every {Kr}r≥1-smooth ω-preserving pseudo-rotation f with ρ( f) = +α has for each r ∈ N a Brouwer curve γ = γr for which the closed region in A +bounded by (γ, f(γ)) admits a (r, C′ +r(α−1, Kr+1))-admissible coordinate. + +12 +BARNEY BRAMHAM AND ZHIYUAN ZHANG +Proof. Fix a sequence (Kr)r≥1 in (1, ∞). Fix r ≥ 1. For each c > 0 and K ≥ 1 +define the following subset of Diffr(A, ω): +Hr(c, K) := +� +g ∈ Diffr(A, ω) +��� ∥g∥Diffr ≤ K, +inf +x∈A d(x, g(x)) ≥ c +� +. +Denote by +Hr+1(c, K) ⊂ Diffr(A, ω) +the closure of Hr+1(c, K) in the Cr-topology. We observe that Hr+1(c, K) +is compact in Diffr(A) and contains only diffeomorphisms without fixed +points. Moreover, for c′ ≥ c and K′ ≤ K we have Hr+1(c′, K′) ⊂ Hr+1(c, K). +We now prove a version of the lemma for elements of Hr+1(c, K), for +each fixed c > 0, K ≥ 1. Then we argue that the union of Hr+1(c, Kr+1) +over c ∈ (0, 1] contains all pseudo-rotations satisfying the assumptions of +the Lemma. +To this end, fix c > 0, K ≥ 1 and consider g ∈ Hr+1(c, K). Since g has no +fixed points, a strong refinement of Brouwer’s plane translation theorem +due to Guillou [G94, Th´eor`em 5.1] yields a C0 Brouwer curve γ0 for g in +the sense of Definition 2.3. Any sufficiently C0-close smooth approxima- +tion of γ0 that continues to connect the two boundary components yields a +smooth Brouwer curve γ for g. Clearly we can choose such a γ to meet both +boundary components of the annulus orthogonally. We can then apply the +following lemma. +LEMMA 3.3. Let r ≥ 1 and θ ∈ (0, 1). Suppose g ∈ Diffr,θ(A, ω) has a smooth +Brouwer curve γ that meets both boundary components orthogonally. Then there +exist D = D(g, γ, r) > 1, and a neighborhood V of g in Diffr,θ(A, ω) such +that for every g′ ∈ V, the region (γ, g′(γ)) has admissible coordinates φ′ whose +Cr-norm is bounded by D. +Proof. Let R denote the region (γ, g(γ)). We first construct a C∞-diffeomorphism +ψL with constant Jacobian ω(R) from a neighborhood of {0} × [0, 1] in ˜A +onto its image a neighborhood of γ in A. Indeed, without loss of gen- +erality γ : [0, 1] → A meets the boundary of A orthogonally near both +end points and is parametrised by arclength so that ∥ ˙γ∥ = L is constant, +L ≥ 1. Then n := −i ˙γ/L is a normal vector field along γ and the map +˜A → R2, (x, y) �→ γ(y) + xω(R)n(y)/L extends γ to a smooth diffeomor- +phism from a sufficiently small tubular neighborhood of {0} × [0, 1] in ˜A to +a neighborhood of the image of γ in A, with constant Jacobian ω(R) along +{0} × [0, 1] and also near to the boundary of ˜A. The Jacobian of this map +away from {0} × [0, 1] depends only on the y variable and therefore by a +further change of coordinates it is easily modified to have constant Jacobian +on a whole neighborhood of {0} × [0, 1] while still mapping {0} × [0, 1] to +γ. The resulting map, which we denote by ψL, clearly has finite Cr,θ-norm. +Then gψLT−1 is a Cr,θ-diffeomorphism ψR with constant Jacobian ω(R) +from a neighborhood of {1} × [0, 1] in ˜A onto its image, a neighborhood + +ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER +13 +of g(γ) in A. We construct the chart φ : [0, 1]2 → A by first specifying its +restriction to a neighborhood of {0} × [0, 1], resp. {1} × [0, 1], to be ψL, resp. +ψR. Then we extend it by hand to neighborhoods of [0, 1] × {0} and [0, 1] × +{1}, also with constant Jacobian ω(R). Finally, using [A10, Corollary 4] (or, +in our simple application, using directly [A10, Theorem 3] which follows +from [DM90]), we extend the map to all of [0, 1]2 so as to have constant +Jacobian, and the Cr norm of the resulting map is bounded in terms of the +Cr,θ norm of g. By construction the conditions in Definition 2.7 hold. +Each g has a C0-small (and hence also Cr-small) neighborhood in Diffr(A, ω) +for which the same γ can be applied. The uniform bounds for the Cr-norm +of φ is an immediate consequence of the construction. We omit the proof of +this latter fact and refer the readers to [A10, DM90] for details. +□ +By Lemma 3.3 and by compactness of Hr+1(c, K) ⊂ Diffr,1/2(A, ω) ⊂ +Diffr(A, ω), we find a finite collection of neighborhoods (with respect to +the Cr,1/2-topology) as in Lemma 3.3 whose union covers Hr+1(c, K), and +thus obtain a uniform bound Er(c, K) > 0 on the Cr-norm of admissible +coordinates that applies to all elements of Hr+1(c, K). Due to the inclusions +Hr+1(c′, K′) ⊂ Hr+1(c, K) for c′ > c, K′ < K, we can assume that Er(c, K) is +decreasing in c and increasing in K. +Now, suppose f is as in Lemma 3.2. That is, f is a (Kr)r≥1-smooth ω- +preserving pseudo-rotation with rotation number ρ( f) = α ∈ L(K1). By +Lemma 3.1, +f ∈ Hr+1 +� +C′(α, K−1 +1 ), Kr+1 +� +. +Thus there exists a region (γ, f(γ)) having admissible coordinates whose +Cr-norm is bounded by Er(C′(α, K−1 +1 ), Kr+1), for some smooth Brouwer +curve γ. Hence Lemma 3.2 holds with C′ +r(α−1, Kr+1) := Er(C′(α, K−1 +1 ), Kr+1). +Evidently C′ +r is an increasing function, since C′ is increasing and Er(c, K) is +decreasing in c and increasing in K. +□ +It will be useful to fix the following notation: +DEFINITION 3.2. For each integer r ≥ 1 and each K > 1 let Lr(K) be the set +of α ∈ L(K) such that +A0(αG(α), Kq(α)) < K−q(α)C′ +r(α−1, K)−1 +(3.14) +where C′ +r : R2 ++ → R+, r ≥ 1 are the increasing functions produced by +Lemma 3.2. +We will see later that each Lr(K) is non-empty for each integer r ≥ 1 and +each K > 1. Moreover, by (3.13) and (3.14) we have +Lr(K′) ⊂ Lr(K) +∀K′ > K. +(3.15) +Now we can strengthen Lemma 3.2. Indeed, we show that if more restric- +tions are placed on the rotation number of a ω-preserving pseudo-rotation + +14 +BARNEY BRAMHAM AND ZHIYUAN ZHANG +f, then the produced Brouwer curve is actually Q-good, for some large Q +depending on the rotation number of f: +PROPOSITION 3.1. Let f be a {Kr}r≥1-smooth ω-preserving pseudo-rotation. If +ρ( f) ∈ Lr(Kr+1) for some integer r ≥ 1, then the Brouwer curve γ for f produced +by Lemma 3.2 (corresponding to r) is q(ρ( f))-good. +Note that the condition ρ( f) ∈ Lr(Kr+1) in this Proposition includes the +condition on ρ( f) used in Lemma 3.2, since Lr(K) ⊂ L(K) by definition. +Proof. We denote by dH the Hausdorff distance on ˜A, i.e. for any two sub- +sets A, B of ˜A, +dH(A, B) = +sup +x∈A,y∈B +max(d(x, B), d(y, A)). +Set α = ρ( f) ∈ Lr(Kr+1) for some r ≥ 1. Let γ be the Brouwer curve +given by Lemma 3.2 for this value of r. Let R ⊂ A be the region bounded +by (γ, f(γ)) and let ψ : R → [0, 1] × [0, 1] be the inverse of some (r, C′ +r(α−1, Kr+1))- +admissible coordinates produced by Lemma 3.2. Applying the intermedi- +ate value theorem to ψ yields +dH(γ, f(γ)) > C′ +r(α−1, Kr+1)−1. +(3.16) +Fix a lift ˜γ ⊂ ˜A of γ, and let F be the lift of f for which ρ(F) = α. Since +˜γ ∩ F( ˜γ) = ∅ it follows from α > 0 and the order of boundary points that +˜γ < F( ˜γ). By Definition 2.7, (3.16) and Kr+1 ≥ K1, we can see that +dH(Fq(α)−1( ˜γ), Fq(α)( ˜γ)) +> +K−q(α) +1 +dH( ˜γ, F( ˜γ)) +≥ +K−q(α) +r+1 C′ +r(α−1, Kr+1)−1. +(3.17) +Here we implicitely used that the analogue of (3.16) holds for the lifts, since +dH( ˜γ, F( ˜γ)) ≥ dH(γ, f(γ)) holds - infact for all choice of lifts F and ˜γ. +Moreover, since F is injective, for each i ∈ N we have Fi( ˜γ) ∩ Fi+1( ˜γ) = ∅ +and so from the order of boundary points we have for all i ∈ N, +˜γ < F( ˜γ) < F2( ˜γ) < · · · < Fi( ˜γ). +(3.18) +Thus it suffices to show that +Fq(α)−1( ˜γ) < T ˜γ +(3.19) +and it will follow that the iterates F( ˜γ), . . . , Fq(α)−1( ˜γ) all lie strictly in the +region between ˜γ and T( ˜γ) in ˜A and therefore that the iterates f(γ), . . . , +f q(α)−1(γ) are all disjoint from γ as required. +To show (3.19) we argue by contradiction. Assuming (3.19) is not true, +we have +Fq(α)−1( ˜γ) ∩ T ˜γ ̸= ∅ +(3.20) +because of the order of boundary points. We will show that this means that +Fq(α)−1( ˜γ) and Fq(α)( ˜γ) pass somewhere very close to each other, because +Fq(α)( ˜γ) is close to T ˜γ. This will contradict (3.17). + +ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER +15 +First, notice that by α ∈ Lr(Kr+1), (3.4) and that ρ∗ is decreasing, we have +∥ρ( f q(α))∥R/Z += +|q(α)α − 1| = αG(α) < G(α) < C′′(α, K−1 +r+1) +< +ρ∗(Kα−1 +r+1) ≤ ρ∗(∥D f∥q(α)) ≤ ρ∗(∥D f q(α)∥). +Then by (3.3), we have +dH(Fq(α)( ˜γ), T ˜γ) ≤ ∥Fq(α) − T∥ = ∥T−1Fq(α) − Id∥ = ∥f q(α) − Id∥. +The last equality can be justified by Lemma 2.1, just as in the proof of +Lemma 3.1. Thus by Theorem 3 and Kr+1 ≥ K1, we have +dH(Fq(α)( ˜γ), T ˜γ) +≤ +A0(∥ρ( f q(α))∥R/Z, Kq(α) +1 +) +≤ +A0(αG(α), Kq(α) +r+1 ). +(3.21) +Then along with (3.17), (3.14), and the hypothesis α ∈ Lr(Kr+1), we have +dH(Fq(α)−1( ˜γ), T ˜γ) +≥ +dH(Fq(α)( ˜γ), Fq(α)−1( ˜γ)) − dH(Fq(α)( ˜γ), T ˜γ) +≥ +K−q(α) +r+1 C′ +r(α−1, Kr+1)−1 − A0(αG(α), Kq(α) +r+1 ) > 0. +However this contradicts (3.20). Thus we have (3.19). +□ +4. SMOOTH DOMAIN BOUNDED BY GOOD CURVES +4.1. Renormalization of pseudo-rotations. Given a C∞ pseudo-rotation f +on A and an integer n ≥ 1, we denote by Fn the unique lift of f n to ˜A such +that ρ(Fn) ∈ (0, 1). Given a smooth Brouwer curve γn for f n, we let ˜γn be +an arbitrary lift of γn to ˜A. We let Ωn be the unique closed region in ˜A +bounded by ( ˜γn, Fn( ˜γn)). +Given a C∞ admissible coordinate H : [0, 1]2 → Ωn for (Ωn, Fn) (see +Remark 2), we can uniquely extend H to a C∞ diffeomorphism of ˜A with +constant Jacobian, denoted again by H, satisfying +HT = FnH. +(4.1) +We notice that although the Cr norms of Fn and T are uniformly bounded +throughout ˜A, the Cr norm of H need not be uniformly bounded. However +it is clear from (4.1) that for any integer L > 0, the norm ∥DrH∥[−L,L]×[0,1] is +bounded in terms of L and ∥Fn∥Diffr( ˜A). +Denote α = ρ( f) ∈ (0, 1) \ Q. We have ρ(F1) = α. We abbrieviate +Fa,b := TbFa +1 +∀a, b ∈ Z. +By our previous definition, we have Fn = Fn,−⌊nα⌋. Denote by J : ˜A → ˜A +the orientation reversing diffeomorphism +J(x, y) = (−x, y). +Notice that by (4.1) we have +JH−1FnHJ = JTJ = T−1. +(4.2) + +16 +BARNEY BRAMHAM AND ZHIYUAN ZHANG +We set +Fa,b +H := JH−1Fa,bHJ +∀a, b ∈ Z. +(4.3) +Since Fa,b is ω-preserving and commutes with Fn, and since H has con- +stant Jacobian, we deduce from (4.2) that Fa,b +H +is also ω-preserving, and +commutes with T. Consequently Fa,b +H descends to an ω-preserving C∞ dif- +feomorphism f a,b +H : A → A. +We have the following lemma. +LEMMA 4.1. There is a sequence of increasing functions {Er : N2 × R2+ → +R+}r≥1 such that the following is true. Let f be a {Kr}r≥1-smooth ω-preserving +pseudo-rotation with ρ( f) = α; let n ̸= 0 be an integer; let Ωn be the region +bounded by ( ˜γn, Fn( ˜γn)) where ˜γn is some lift of a Brouwer curve for f n. Assume +that (Ωn, Fn) admits a (r, Lr)-admissible coordinate H. Then for any a, b ∈ Z +with a⌊nα⌋ + bn ̸= 0, f a,b +H is a pseudo-rotation in Diff∞(A, ω) with +ρ( f a,b +H ) = +�−aα − b +{nα} +� +. +Moreover, we have +∥f a,b +H ∥Diffr ≤ Er(|a|, |b|, Kr, Lr). +Proof. We first show that f a,b +H has no periodic points. Assume to the con- +trary there are integers p ∈ Z, q > 0 and some z ∈ ˜A such that +(Fa,b +H )q(z) = Tp(z). +Then by (4.2) and (4.3) we have +z = (Fa,b +H )qT−p(z) = JH−1F(qa+pn) +1 +T(qb−p⌊nα⌋)HJ(z). +(4.4) +However, (4.4) and the condition a⌊nα⌋ + bn ̸= 0 implies that HJ(z) de- +scends to a perodic point for f, which contradicts the hypothesis that f is a +pseudo-rotation. We conclude that f a,b +H is a pseudo-rotation in Diff∞(A, ω). +To compute ρ( f a,b +H ), it suffices to study the trajectory of an arbitrary z ∈ +˜A under the iterates of (Fa,b +H )qT−p using the second equality in (4.4); for +example a point on the boundary. This standard argument is left to the +readers. +For the Cr-norm of f a,b +H , it suffices to control the Cr-norm of Fa,b +H restricted +to (−1, 2) × [0, 1]. +By (4.3), ∥DrFa,b +H ∥(−1,2)×[0,1] depends only on the Cr- +norms of Fa,b, and the Cr-norm of J and H on +Fa,b +H ((−1, 2) × [0, 1]) ∪ (−1, 2) × [0, 1]. +This yields a bound depending only on r, |a|, |b|, Kr, Lr as required. +□ + +ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER +17 +4.2. Finding a good curve. In the following statement we consider n ∈ N +even, so that qnα − pn > 0. +PROPOSITION 4.1. There is a sequence of increasing functions {Gr : R2 ++ → +R+}r≥1 such that the following is true. +Let f be a pseudo-rotation in Diff∞(A, ω), let F be a lift with ρ(F) = ρ( f) = +α ∈ (0, 1) \ Q, and let n ≥ 2 be an even integer. Suppose γ ⊂ A is a smooth +Brouwer curve for f qn with a lift γ† ⊂ ˜A such that the closed region Ω in ˜A +bounded by (γ†, Fqn,−pn(γ†)) admits an (r, Kr)-admissible coordinate +H : [0, 1]2 → Ω +for some r ≥ 1. Suppose further that γn ⊂ A is an an+2-good smooth curve for +f qn+1,−pn+1 +H +,2 for which the closed region Ωn ⊂ A bounded by (γn, f qn+1,−pn+1 +H +(γn)) +admits an (r, Mr)-admissible coordinate. Then there exists a qn+2-good smooth +curve ˆγ ⊂ A for f, for which the closed region Ω∗ ⊂ A bounded by ( f qn+1( ˆγ), ˆγ) +admits a (r, Gr(Kr, Mr))-admissible coordinate for f qn+1. +Proof. We set +S0 = Fqn,−pn, +S = Fqn+1,−pn+1. +As explained in Section 4.1, we extend H to a C∞-diffeomorphism of ˜A +with constant Jacobian by the formula +HT = S0H. +By definition, we know that +(1) JH−1S0HJ = JTJ = T−1 on ˜A; +(2) ˜S := JH−1SHJ commutes with T on ˜A, and descends to f qn+1,−pn+1 +H +. +Moreover, we have ρ( ˜S) = ρ( f qn+1,−pn+1 +H +) = αn+1 ∈ (0, 1 +2). +Let ˜γ be an arbitrary lift of γn. As γn is a simple regular curve and is +disjoint from f qn+1,−pn+1 +H +(γn), we know by item (1), (2) above that ˜γ is also a +simple regular curve, and +T−1( ˜γ) ∩ ˜γ = ˜S( ˜γ) ∩ ˜γ = ∅. +(4.5) +We let γ′ = HJ( ˜γ). Then γ′ is also a simple regular curve, and by (4.5) and +item (1), (2) above, we have +S0(γ′) ∩ γ′ = S(γ′) ∩ γ′ = ∅. +(4.6) +Moreover, we can see that +S(γ′) < γ′ < S0(γ′) +(4.7) +by considering their boundary points on B0 and B1. We let ˆγ = π(γ′) (recall +that π is the canonical projection from ˜A to A). +LEMMA 4.2. The curve ˆγ is a simple regular curve connecting B0 and B1. Namely, +it has no self-intersection. +2Recall that by Lemma 4.1 we have ρ( f qn+1,−pn+1 +H +) = αn+1 and an+2 = q(αn+1). + +18 +BARNEY BRAMHAM AND ZHIYUAN ZHANG +Proof. Since ˆγ = π(γ′), and since γ′ is a simple regular curve in ˜A con- +necting B0 to B1, it suffices to show that γ′ is disjoint from all its translates +Tk(γ′) for k ∈ Z\{0}. Since γ′ connects the two boundary components it +is enough to show γ′ is disjoint from T(γ′). +By definition and by (2.9), we have T = Sqn+1 +0 +S−qn. Then by (4.7), we +obtain that γ′ < T(γ′). This completes the proof. +□ +By item (2) above, the region ˜Ωn bounded by ( ˜γ, ˜S( ˜γ)) is a lift of Ωn +Moreover, the (r, Mr)-admissible coordinate for Ωn lifts to a (r, Mr)-admissible +coordinate for ˜Ωn. We notice that the push forward HJ( ˜Ωn) is the region in +˜A between S(γ′) and γ′. Moreover, from the proof of Lemma 4.2, we see +that +T−1(γ′) < S(γ′) < γ′. +Thus the map π induces a diffeomorphism from HJ( ˜Ωn) to the region in +A between f qn+1( ˆγ) and ˆγ, that is, to the region Ω∗. We conclude that the +(r, Mr)-admissible coordinate for ˜Ωn, after composing with HJ and project- +ing, yield (r, Gr(Kr, Mr))-admissible coordinates for Ω∗, for some functions +Gr as in the proposition. +It remains to show that ˆγ is a qn+2-good curve. We divide the proof into +two cases. +Case I: Assume that there are integers p and 0 < k < qn+2 such that +kα + p < 0 and +Fk,p(γ′) ∩ γ′ ̸= ∅. +(4.8) +We assume further that for any integers p′ and 0 < k′ < qn+2 such that +k′α + p′ < 0 and Fk′,p′(γ′) ∩ γ′ ̸= ∅, we have +k′α + p′ ≤ kα + p. +• First, we observe that either qn+2 > k ≥ qn+2 − qn or +−{qnα} = pn − qnα < kα + p < 0. +Indeed, if k < qn+2 − qn and pn − qnα > kα + p, then we have k + qn < qn+2 +and +kα + p < (k + qn)α + p − pn < 0. +In particular, the endpoints of S0Fk,p(γ′) are on the left hand side of those +of γ′. Then by (4.8) and S0Fk,p(γ′) > Fk,p(γ′), we have that +Fk+qn,−pn+p(γ′) ∩ γ′ ̸= ∅. +This contradicts the choice of k. +• If qn+2 > k ≥ qn+2 − qn, then by (4.6), (4.8) and (2.7), we know that +k > qn+1. We notice that +kα + p < (k − qn+1)α + p + pn+1 < 0. + +ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER +19 +In particular, the endpoints of S−1Fk,p(γ′) are on the left hand side of those +of γ′. Then by (4.8) and S−1Fk,p(γ′) > Fk,p(γ′), we have that +Fk−qn+1,p+pn+1(γ′) ∩ γ′ ̸= ∅. +This contradicts the choice of k. +• If −{qnα} < kα + p < 0, then we have +k ∈ {qn+1, · · · , an+2qn+1}. +If k = iqn+1 with i > 1, then we have +−1 < i(qn+1α − pn+1) < (i − 1)(qn+1α − pn+1) = (k − qn+1)α − (i − 1)pn+1 < 0. +Then we must have p = −ipn+1, and the endpoints of S−1Fk,p(γ′) are on +the left hand side of those of γ′. Then by (4.8) and S−1Fk,p(γ′) > Fk,p(γ′), +we have that +Fk−qn+1,−(i−1)pn+1(γ′) ∩ γ′ ̸= ∅. +If k = qn+1, we would have +Fqn+1(γ′) ∩ (γ′ + Z) = ∅. +Both cases contradict the choice of k. +Case II: Assume that there are integers p and 0 < k < qn+2 such that +kα + p > 0 and +Fk,p(γ′) ∩ γ′ ̸= ∅. +(4.9) +We assume further that for any integers p′ and 0 < k′ < qn+2 such that +k′α + p′ > 0 and Fk′,p′(γ′) ∩ γ′ ̸= ∅, we have +k′α + p′ ≥ kα + p. +• First, we observe that either qn ≥ k > 0 or +{qnα} = qnα − pn > kα + p > 0. +Indeed, if k > qn and qnα − pn < kα + p, then we have qn+2 > k − qn > 0 +and +kα + p > (k − qn)α + p + pn > 0. +In particular, the endpoints of S−1 +0 Fk,p(γ′) are on the right hand side of +those of γ′. Then by (4.9) and Fk−qn,pn+p(γ′) = S−1 +0 Fk,p(γ′) < Fk,p(γ′), we +have that +Fk−qn,pn+p(γ′) ∩ γ′ ̸= ∅. +This contradicts the choice of k. +• If qn ≥ k > 0, then by (4.6) and (4.9), we have qn > k and consequently +k + qn+1 < qn+2. Moreover we notice that +kα + p > (k + qn+1)α + p − pn+1 > 0. + +20 +BARNEY BRAMHAM AND ZHIYUAN ZHANG +In particular, the endpoints of SFk,p(γ′) are on the right hand side of those +of γ′. By (4.9) and SFk,p(γ′) < Fk,p(γ′), we have that +Fk+qn,p−pn(γ′) ∩ γ′ ̸= ∅. +This contradicts the choice of k. +• If {qnα} > kα + p > 0 (and 0 < k < qn+2), then it is straightforward to +verify that +(k, p) ∈ {(qn + iqn+1, pn + ipn+1) | 0 < i < an+2}. +By the hypothesis that γn is a an+2-good curve, we have +T−1 ˜Si( ˜γ) ∩ ˜γ = ∅ +∀1 ≤ i < an+2. +By T−1 = JH−1S0HJ, ˜S = JH−1SHJ and γ′ = HJ( ˜γ), we obtain +Fqn+iqn+1,−pn−ipn+1(γ′) ∩ γ′ = S0Si(γ′) ∩ γ′ = ∅ +for all integer 0 < i < an+2. This again gives a contradiction. +In summary, we see that for all integer 0 < i < qn+2, +Fi(γ′) ∩ (γ′ + Z) = ∅. +Hence ˆγ is qn+2-good. This completes the proof of Proposition 4.1. +□ +We can now prove the following: +COROLLARY B. For each integer r ≥ 1, there exist increasing functions Pr : +R2 ++ → R+ and Wr : N × R+ → R+ such that the following is true. Suppose +that there is an odd integer n ≥ 3 such that +(4.10) +qn > Pr(Kr+2, qn−1), +qn+1 > Pr(Kr+2, qn) +and +qn+2 > Pr(Kr+2, qn+1) +where {qk}k≥0 is the sequence of denominators associated to some α ∈ (0, 1)\Q. +Then for any {Kk}k≥1-smooth pseudo-rotation f ∈ Diff∞(A, ω) with ρ( f) = α +has a qn+1-good smooth curve γ such that the closed region in A bounded by +( f qn(γ), γ) admits a (r, Wr(qn+1, Kr+2))-admissible coordinate for f −qn. +Proof. Fix r ≥ 1. Let {Lk : N × R+ → R+}k≥1 be a sequence of increas- +ing functions independent of f such that for each integer m ≥ 0, f m is +{Lk(m, Kk)}k≥1-smooth. Let us abbreviate +L′ +k := Lk(qn−1, Kk) +∀k ≥ 1. +For each n ≥ 2 we have ρ( f qn−1) = βn−1 ∈ (0, 1/2). If (4.10) holds and Pr is +chosen appropriately, then +βn−1 ∈ L(L′ +1). +(4.11) +This allows us to apply Lemma 3.2 to f qn−1. By Lemma 3.2, there is a smooth +Brouwer curve γ for f qn−1 for which the closed region R′ +n−1 ⊂ A bounded + +ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER +21 +by (γ, f qn−1(γ)) admits a (r + 1, Yr+1)-admissible coordinate H′ : [0, 1]2 → +R′ +n−1 where +Yr+1 := C′ +r+1(β−1 +n−1, L′ +r+2). +(4.12) +We lift H′ to an admissible coordinate H : [0, 1]2 → Rn−1 where Rn−1 +is a lift of R′ +n−1 in ˜A. More precisely, Rn−1 is a connected component of +π−1(R′ +n−1). As in Section 4.1, we extend H to a diffeomorphism of ˜A sat- +isfying +HT = Fqn−1,−pn−1H +and set fn := f qn,−pn +H +: A → A. That is, fn is the projection of Fqn,−pn +H +:= +(HJ)−1Fqn,−pn(HJ) : ˜A → ˜A. By Lemma 4.1, fn is a pseudo-rotation in +Diff∞(A, ω) with ρ( fn) = βn/βn−1 = αn such that +∥fn∥Diffr+1 ≤ Er+1(qn, pn, Kr+1, Yr+1). +(4.13) +By (4.12), and using that +Kr+1 ≤ Kr+2, +max(pn, β−1 +n−1) ≤ 2qn +we can rewrite (4.13) as +∥fn∥Diffr+1 < Vr+1(qn, Kr+2) +(4.14) +where Vr+1 : N × R+ → (1, ∞) is an increasing function independent of f. +Now we choose Pr : R2+ → R+ to be any increasing function that is +sufficiently large that the following conditions are fulfilled: +Pr(D, y) +> +2yρ∗(Vr+1(y, D))−1 +(4.15) +2y +Pr(D, y) +< +C′′ +� 1 +2y, Vr+1(y, D)−1 +� +, +(4.16) +for all D, y > 0, where ρ∗ is the decreasing function introduced in (3.1). +We moreover choose Pr sufficiently large that whenever x > Pr(D, y) there +holds +A0 +�2y +x , Vr+1(y, D)2y +� +< +Vr+1(y, D)−2yC′ +r(2y, Vr+1(y, D))−1 +(4.17) +for all D, y > 0. This we can arrange because limt→0 A0(t, ·) = 0. +Assume now that (4.10) holds for a fixed odd integer n ≥ 3. By (4.15) +and then the second inequality in (4.10), we have +2qn +< +ρ∗(Vr+1(qn, Kr+2))Pr(Kr+2, qn) < ρ∗(Vr+1(qn, Kr+2))qn+1 +and therefore by (2.12) +αn < 2qn +qn+1 +< +ρ∗(Vr+1(qn, Kr+2)). +(4.18) + +22 +BARNEY BRAMHAM AND ZHIYUAN ZHANG +Also, by (2.12), the third inequality in (4.10), and (4.16), we obtain +G(αn) = αn+1 +< +2qn+1 +qn+2 +< +2qn+1 +Pr(Kr+2, qn+1) +(4.19) +< +C′′ +� +1 +2qn+1 +, Vr+1(qn+1, Kr+2)−1 +� +< +C′′(αn, Vr+1(qn, Kr+2)−1) +(4.20) +where the last inequality uses the monotonicity of C′′. We claim that +ρ( fn) = αn ∈ Lr(Vr+1(qn, Kr+2)) +(4.21) +where Lr is as in Definition 3.2. Equivalently, we show that +(4.22) +A0 +� +αnG(αn), Vr+1(qn, Kr+2)q(αn)� +< Vr+1(qn, Kr+2)−q(αn)C′ +r +� +α−1 +n , Vr+1(qn, Kr+2) +�−1 +. +Using (2.12) we have αnG(αn) = αnαn+1 < 4qn/qn+2 ≤ 2qn+1/qn+2 and +q(αn) ≤ 2qn+1 and α−1 +n +< 2qn+1/qn ≤ 2qn+1. Therefore by the monotonicity +of A0, and by (4.17) together with the third inequality in (4.10), we obtain +A0 +� +αnG(αn), Vr+1(qn, Kr+2)q(αn)� +< +A0 +�2qn+1 +qn+2 +, Vr+1(qn+1, Kr+2)2qn+1 +� +< +Vr+1(qn+1, Kr+2)−2qn+1C′ +r(2qn+1, Vr+1(qn+1, Kr+2))−1. +Now inequality (4.22) follows from the monotonicity of Vr+1 and C′ +r, and +because α−1 +n +< 2qn+1/qn ≤ 2qn+1 from (2.12). This proves (4.21). +Combining (4.21) with (4.14) we see that fn satisfies the hypotheses of +Proposition 3.1, and therefore fn has a q(ρ( fn))-good Brouwer curve γn say. +By (2.8) q(ρ( fn)) = q(αn) = an+1. Thus γn is an an+1-good Brouwer curve +for fn. Moreover, by Lemma 3.2, the region in A bounded by (γn, fn(γn)) +admits an (r, C′ +r(α−1 +n , ˆKr+1))-admissible coordinate, provided ˆKr+1 ≥ ∥fn∥Diffr+1 +and ρ( fn) = αn ∈ L( ˆK) where ˆK ≥ ∥fn∥Diff1. By (4.21) and (4.14) we may +take ˆKr+1 = ˆK = Vr+1(qn, Kr+2) and conclude that the region in A bounded +by (γn, fn(γn)) admits a (r, Ur)-admissible coordinate where +Ur := C′ +r(α−1 +n , Vr+1(qn, Kr+2)). +We define Wr by +Wr(q, K) := Gr(C′ +r(2q, Lr+1(q, K)), C′ +r(2q, Vr+1(q, K))). +By (2.11) and (2.12), we have β−1 +n−1, α−1 +n +< 2qn+1. Then by Kr+2 ≥ Kr+1, we +have +C′ +r(2qn+1, Lr+1(qn+1, Kr+2)) > Yr +and +C′ +r(2qn+1, Vr+1(qn+1, Kr+2)) > Ur. +Thus we have +Wr(qn+1, Kr+2) ≥ Gr(Yr, Ur). + +ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER +23 +Then by Proposition 4.1, there is a qn+1-good smooth curve of f, denoted by +ˆγ, such that the region in A bounded by ( f qn( ˆγ), ˆγ) admits a (r, Wr(qn+1, Kr+2))- +admissible coordinate for f −qn. +□ +5. CONSTRUCTION OF APPROXIMANTS +This section is mostly occupied by the proof of the following theorem, +from which the main result of this paper, Theorem 1, will then follow easily. +THEOREM 4. For each (r0, M, ǫ) ∈ Z≥2 × N × (0, 1], there is an increasing +function +P = Pr0,ǫ,M : N → R+ +so that for any pseudo-rotation f ∈ Diff∞(A, ω) with +∥f∥Diffr0+2(A) < M, +(5.1) +whose rotation number ρ( f) = α ∈ (0, 1) \ Q satisfies the property that there +exists an odd integer n ≥ 3 for which +qn > P(qn−1), +qn+1 > P(qn) +and +qn+2 > P(qn+1), +(5.2) +then there exists h0 ∈ Diff∞(A, ω) with +dDiffr0−1(A)(h0Rαh−1 +0 , f) < ǫ. +(5.3) +Proof. Fix some (r0, M, ǫ) in Z≥2 × N × (0, 1]. Let f ∈ Diff∞(A, ω) denote +a pseudo-rotation and set α = ρ( f). +It will be convenient to use the following notation: for an increasing +function S : N → N we define +C(S) := +� +θ ∈ (0, 1)\Q | ∃n ∈ N odd, so that +(5.4) +qn(θ) > S(qn−1(θ)), qn+1(θ) > S(qn(θ)), qn+2(θ) > S(qn+1(θ)) +� +where {pn(θ)/qn(θ)}n≥0 is the sequence of continued fractions of θ intro- +duced in Section 2. Our successive restrictions on the rotation number α +will take the form: +α ∈ C(Si) +(5.5) +for a finite collection of functions S1, S2, . . . to be determined. We will set +P := max +i +Si +and then when α ∈ C(P), all conditions in (5.5) will be met. +For our first condition on the rotation number, set +S1 := Pr0(M, ·) +(5.6) +where Pr0 : R2+ → R+ is defined in Corollary B. From now on we assume +α ∈ C(S1). + +24 +BARNEY BRAMHAM AND ZHIYUAN ZHANG +Fix any odd n ≥ 3 for which +qn > S1(qn−1), +qn+1 > S1(qn) +and +qn+2 > S1(qn+1). +(5.7) +Then by Corollary B, f has a qn+1-good Brouwer curve γ ⊂ A for which +the closed region +D0 ⊂ A +bounded by ( f qn(γ), γ) admits a (r0, ˆWr0)-admissible coordinate for f −qn, +where +ˆWr0 := Wr0(qn+1, Kr0+2). +Recall that γ is qn+1-good for f means that the curves f i(γ), 0 ≤ i ≤ qn+1 − +1 are mutually disjoint. Then it is clear that the curves +f qn(γ), · · · , f qn+qn+1−1(γ) +(5.8) +are mutually disjoint as well. In particular f qn+1(γ) and f qn(γ) are disjoint +and so we can consider the closed region +D ⊂ A +bounded by ( f qn(γ), f qn+1(γ)). +CLAIM 1. The annulus A is covered by the regions +D, f(D), · · · , f qn−1(D), f qn(D0), · · · , f qn+1−1(D0) +with mutually disjoint interiors. Moreover, the intersection of any two such neigh- +boring regions equals one of the curves in (5.8). +Proof. The restriction of f to the boundary circle B0 is an orientation pre- +serving homeomorphism with rotation number α. It is then a well-known +fact that for any fixed x0 ∈ B0 the intervals +[ f qn+i(x0), f qn+1+i(x0)], +0 ≤ i ≤ qn − 1, +and +[ f qn+j(x0), f j(x0)], +qn ≤ j ≤ qn+1 − 1 +together form a covering for B0 with mutually disjoint interiors. In partic- +ular we may take x0 := γ ∩ B0. An analogous statement holds for the other +boundary component B1. The claim then follows from the disjointness of +the curves in (5.8) and our definition of simple regular curve. +□ +Fix a lift ˜γ of γ to ˜A. Let F : ˜A → ˜A be the unique lift of f satisfying +ρ(F) = ρ( f) ∈ (0, 1). Recall the notation from section 4.1 that Fa,b = TbFa +for all a, b ∈ Z. We define ˜D0 ⊂ ˜A to be the region bounded by +� +Fqn,−pn( ˜γ), ˜γ +� +and ˜D ⊂ ˜A to be the region bounded by +� +Fqn,−pn( ˜γ), Fqn+1,−pn+1( ˜γ) +� +. +We claim that T−pnFqn( ˜γ) is contained in the fundamental domain of π +bounded by T−1 ˜γ and ˜γ. Indeed, this follows from comparing the order of + +ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER +25 +their endpoints, and by the disjointness of their projections f qn(γ) and γ. +Thus π restricts to a diffeomorphism from ˜D0 to D0. Similarly, π restricts +to a diffeomorphism from ˜D to D as well. +The following qn+1 regions have disjoint interiors: +˜Bi := +� +Fi( ˜D) +0 ≤ i ≤ qn − 1, +Fi( ˜D0) +qn ≤ i ≤ qn+1 − 1. +(5.9) +By Claim 1, ˜A is covered by the union of the following regions with mutu- +ally disjoint interiors: +T−k ˜Bj, +k ∈ Z, 0 ≤ j ≤ qn+1 − 1. +(5.10) +The intersection of any two neighboring regions in (5.10) therefore equals +to one of the following curves +T−kFj( ˜γ), +k ∈ Z, qn ≤ j ≤ qn + qn+1 − 1. +(5.11) +We denote by Γ the union of the curves in (5.11). The region +U0 := +qn+1−1 +� +i=0 +int( ˜Bi) +(5.12) +satisfies that: +(1) Tj(U0) ∩ Tk(U0) = ∅ for any j ̸= k ∈ Z; +(2) ∪k∈ZTk(U0) = ˜A \ Γ. +By Corollary B, there is an admissible coordinate φ (see Definition 2.7) +from a neighborhood of [0, 1]2 in ˜A to a neighborhood of D0, satisfying +(2.2)-(2.5) for ( f −qn, f qn(γ)) in place of ( f, γ), and +∥φ∥Diffr0 +< +ˆWr0, +φ ◦ T(x) += +f −qn ◦ φ(x) +∀x ∈ V +where V is some neighborhood of {0} × [0, 1] in ˜A. As we have seen in +Remark 2, there is no loss of information by regarding φ as a mapping from +[0, 1]2 to D0. Since the restriction π : ˜D0 → D0 is a diffeomorphism we may +set +ψ := π−1 ◦ φ +which therefore gives us a C∞-smooth diffeomorphism that extends to a +C∞-smooth diffeomorphism with constant Jacobian from a neighborhood +of [0, 1]2 in ˜A to a neighborhood of ˜D0 satisfying +∥ψ∥Diffr0 +< +ˆWr0, +(5.13) +ψ ◦ T(x) += +F−qn,pn ◦ ψ(x) +∀x ∈ V. +(5.14) +There is a unique extension to a C∞-diffeomorphism with constant Jacobian +to the whole strip ˜A, which we still denote by +ψ : ˜A → ˜A + +26 +BARNEY BRAMHAM AND ZHIYUAN ZHANG +and which satisfies +ψ ◦ T = F−qn,pn ◦ ψ. +(5.15) +To summarise thus far, we have proven the following: +LEMMA 5.1. For each (r0, M) ∈ Z≥2 × N there is an increasing function +S1 = S1(r0, M) : N → R+ +so that for any pseudo-rotation f ∈ Diff∞(A, ω) satisfying (5.1) whose rotation +number ρ( f) = α satisfies the property that there exists an odd integer n ≥ 3 for +which +qn > S1(qn−1), +qn+1 > S1(qn) +and +qn+2 > S1(qn+1), +there exists a qn+1-good Brouwer curve γ ⊂ A, so that for any lift ˜γ ∈ ˜A, there +exists a C∞-diffeomorphism, +ψ : ˜A → ˜A +with constant Jacobian which satisfies +∥ψ∥Diffr0([0,1]2) < ˆWr0 := Wr0(qn+1, M) +(5.16) +where Wr0 is the function produced by Corollary B, and +ψ ◦ T = F−qn,pn ◦ ψ +(5.17) +on the whole of ˜A, where F is the unique lift of f with rotation number in (0, 1). +Moreover ψ maps [0, 1]2 onto the region ˜D0 bounded by +� ˜Fqn,−pn( ˜γ), ˜γ +� +. +To continue our proof of Theorem 4 we require the following: +LEMMA 5.2. For each (r0, M) ∈ Z≥2 × N, there is an increasing function +Qr0,M : R2 ++ → R+ +with +lim +t→0 Qr0,M(t, ·) ≡ 0, +and there is an increasing function +S2 = S2(r0, M) : N → R+ +with S2 ≥ S1, so that for any pseudo-rotation f ∈ Diff∞(A, ω) satisfying (5.1) +whose rotation number ρ( f) = α satisfies the property that there exists an odd +integer n ≥ 3 for which +qn > S1(qn−1), +qn+1 > S1(qn) +and +qn+2 > S1(qn+1), +then the following holds for the unique lift F : ˜A → ˜A of f with ρ(F) ∈ (0, 1): +let γ be a qn+1-good Brouwer curve for f given by Lemma 5.13, and let ˜γ be any +lift of γ. Let ˜D and ˜D0 be the regions in ˜A bounded by +� +Fqn,−pn( ˜γ), Fqn+1,−pn+1( ˜γ) +� +, +� +Fqn,−pn( ˜γ), ˜γ +� +3Lemma 5.1 is applicable since by hypothesis S2 ≥ S1. + +ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER +27 +respectively. Then there are open neighborhoods ˆD of ˜D and ˆD0 of ˜D0; a C∞- +diffeomorphism +h : ˆD → ˆD0; +and neighborhoods UL of Fqn,−pn( ˜γ) and UR of Fqn+1,−pn+1( ˜γ), so that +h|UL = Id, +h|UR = F−qn+1,pn+1, +(5.18) +∥h − Id∥Diffr0( ˜D, ˜D0) < Qr0,M +� +βn+1, qn+1 +� +. +(5.19) +Proof. By Lemma 5.1 there is a diffeomorphism ψ : ˜A → ˜A satisfying (5.16) +and which maps [0, 1]2 onto ˜D0. Set +γ′ := Fqn+1,−pn+1( ˜γ). +By letting qn+2 be sufficiently large depending on qn+1 and M, we have +A0(βn+1, Mqn+1) < 1/2 +where A0 is produced by Theorem 3. Then +∥f qn+1 − Id∥ ≤ A0(∥ρ( f qn+1)∥R/Z, Mqn+1) = A0(βn+1, Mqn+1) < 1/2. +Using Theorem 3, Lemma 2.1 and (2.11), we obtain +dH( ˜γ, γ′) = dH( f qn+1(γ), γ) +< +A0(βn+1, Mqn+1). +(5.20) +Notice that by (5.16) and (5.17) we also have that ∥ψ∥Diffr0([0,2]×[0,1]) is bounded +in terms of qn+1, M, r0. Then by (5.20) and by letting βn+1 be sufficiently +large depending on qn+1, M, r0, we have +γ′ ⊂ ψ([1/2, 3/2] × [0, 1]). +We define +γ′′ = ψ−1(γ′). +Then +{1/2} × [0, 1] < γ′′ < {3/2} × [0, 1] +if βn+1 is sufficiently large depending on qn+1, M, r0. Denote by U ′ the re- +gion in ˜A bounded by {0} × [0, 1] and γ′′. It remains to construct a C∞- +diffeomorphism +ϕ : U ′ → [0, 1]2 +such that ϕ equals Id near {0} × [0, 1]; equals ψ−1F−qn+1,pn+1ψ near γ′′; and +tends to Id in the Cr0-topology as βn+1 tends to 0 for each fixed qn+1, M, r0. +Indeed, after the above ϕ is constructed, we can define h = ψϕψ−1. +By Theorem 3, for given qn+1, M, r0, we see that F−qn+1,pn+1 tends to Id in +the Cr0-topology as qn+2 tends to infinity. Thus it suffices to construct ϕ by +smooth interpolation. +□ + +28 +BARNEY BRAMHAM AND ZHIYUAN ZHANG +Continuing the proof of Theorem 4, we first construct a periodic approx- +imation of f as follows. We define G on U0 (see (5.12)) by +G|U = + + + + + + + + + + + +F +U = ∪qn−2 +i=0 int( ˜Bi), +FqnhF1−qn +U = int( ˜Bqn−1), +F +U = ∪qn+1−2 +i=qn +int( ˜Bi), +Tpn+1h−1F1−qn+1 +U = int( ˜Bqn+1−1). +(5.21) +By property (1),(2) below (5.12), we may extend G to a self-map of ˜A \ Γ, +still denoted by G, satisfying +GT = TG. +(5.22) +Moreover, by (5.21) the following identity holds on ˜A \ Γ: +Gqn+1 = Tpn+1. +(5.23) +By construction, for any two curves γa, γb in (5.11), there are integers i, j +such that +γb = TiGj(γa). +(5.24) +By (5.21) in the definition of G, we also have that +˜Bi = Gi( ˜B0), +0 ≤ i ≤ qn+1 − 1. +(5.25) +We denote +˜Bqn+1 := Gqn+1( ˜B0) = Tpn+1( ˜B0). +(5.26) +Now we set +φ := ψ−1h +where ψ is in Lemma 5.1 and h is in Lemma 5.2. +LEMMA 5.3. There is a neighborhood V′ ⊂ ˜A of Fqn+1,−pn+1( ˜γ) so that Gqn ex- +tends to a C∞ map on V′, and +Tφ(x) = φT−pnGqn(x), +x ∈ V′. +(5.27) +Proof. To see this, we first notice that +Fqn+1,−pn+1( ˜γ) = ˜B0 ∩ Tpn−pn+1( ˜Bqn+1−qn), +and ˜B0 ∪ Tpn−pn+1( ˜Bqn+1−qn) is a neighborhood of Fqn+1,−pn+1( ˜γ) in ˜A. Let +UL, UR be given by Lemma 5.2. The set V′ defined by +V′ = Fqn+1−qn,−pn+1+pn(UL) ∩ UR ∩ ( ˜B0 ∪ Tpn−pn+1( ˜Bqn+1−qn)) +is a neighborhood of Fqn+1,−pn+1( ˜γ). +By (5.21), (5.25) and (5.22), we have +Gqn|Int( ˜B0) += +Fqnh, +(5.28) +Gqn|Tpn−pn+1(Int( ˜Bqn+1−qn)) += +Tpnh−1Fqn−qn+1Tpn+1−pn. +(5.29) + +ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER +29 +By Lemma 5.2, we see that +RHS of (5.28)|V′ = RHS of (5.29)|V′ = Tpn+1Fqn−qn+1. +Then it is clear that Gqn extends to a C∞ map on V′. Again by Lemma 5.2, +we have for any x ∈ V′ +φT−pnGqn(x) += +ψ−1hTpn+1−pnFqn−qn+1(x) = ψ−1Tpn+1−pnFqn−qn+1(x), +Tφ(x) += +Tψ−1h(x) = Tψ−1T−pn+1Fqn+1(x). +Thus (5.27) follows from (5.17). +□ +We have the following corollary. +COROLLARY C. The map G on ˜A \ Γ extends to an element in Diff∞( ˜A). +Proof. By (5.23), clearly Gqn+1 extends to a smooth map Tpn+1 on ˜A. By +Lemma 5.3, (5.22) and (5.24), we see that Gqn also extends to a map in +Diff∞( ˜A). To conclude the proof it suffices to notice that +G = (Gqn)−pn+1(Gqn+1)pn. +□ +By a slight abuse of notation, we again write G ∈ Diff∞( ˜A) for the ex- +tension provided in Corollary C. Clearly (5.22) continues to hold. Conse- +quently G descends to a periodic diffeomorphism g ∈ Diff∞(A) satisfying +gqn+1 = Id. +We define map ˜H : ˜A → ˜A by +˜H(x) := T−kqn+1+jpn+1φG−jTk(x), +x ∈ T−k ˜Bj +(5.30) +for all k ∈ Z and all 0 ≤ j ≤ qn+1 − 1. Then by (5.23), (5.25), (5.26) and +(5.30) we have +˜HT = Tqn+1 ˜H, +˜HG = Tpn+1 ˜H. +(5.31) +We set H1 := Dqn+1 ˜H, where +Dλ(x, y) = (λ−1x, y), +λ ∈ R \ {0}. +Notice that for every λ ∈ R \ {0} and every p ∈ R we have +DλTpD−1 +λ += T p +λ +where for each c ∈ R, Tc : ˜A → ˜A denotes the map Tc(x, y) = (x + c, y). +Then by (5.31), we see that H1 : ˜A → ˜A satisfies +TH1 = H1T, +H1G = Tpn+1/qn+1H1. +(5.32) +We now show that ˜H ∈ Diff∞( ˜A). By construction, ˜H is C∞-smooth in +the interior of each T−k ˜Bj, k ∈ Z, 0 ≤ j ≤ qn+1 − 1. Thus it remains to show +that ˜H is C∞ in a neighborhood of each of the curves in (5.11). + +30 +BARNEY BRAMHAM AND ZHIYUAN ZHANG +By (5.9), we can see that the set +˜B∗ := ˜B0 ∪ T−pn ˜Bqn +(5.33) +is a neighborhood of T−pnFqn( ˜γ). By (5.24), we deduce that every curve +in (5.11) has a neighborhood of the form TiGj( ˜B∗). By (5.31), it remains to +verify that ˜H is C∞ over ˜B∗. This follows from a similar argument as in +the proof of Lemma 5.3 using (5.27) and (5.30). Consequently ˜H and H1 +are C∞-smooth diffeomorphisms of ˜A. Then by (5.32) H1 descends to a +C∞-diffeomorphism h1 : A → A satisfying +g = h−1 +1 Rpn+1/qn+1h1. +(5.34) +Moreover, we have the claim that: +∥H1∥Diffr0( ˜A), and hence ∥h1∥Diffr0(A) as well, are bounded in terms of qn+1, M, r0. +Indeed, by definition it is clear that ∥φ∥Diffr0 is bounded in terms of +qn+1, M, r0; and for each integer 0 ≤ j ≤ qn+1 − 1, ∥G∥Diffr0( ˜Bj, ˜Bj+1) is bounded +in terms of qn+1, M, r0. Our claim follows immediately from (5.30) and the +fact that H1 commutes with T. +Recall that r0, M and ǫ are fixed. In the following we will show that: +LEMMA 5.4. For sufficiently fast growing P depending on r0, M and ǫ the follow- +ing holds. Assume n satisfies the conditions in Lemma 5.1 and Lemma 5.2; and +moreover (5.2) holds for n. Then we have: +(1) dDiffr0(A)( f, g) < ǫ/4; +(2) dDiffr0(A)(h−1 +1 Rαh1, g) < ǫ/4; +(3) there is h2 ∈ Diff∞(A) such that (h2h1)∗ω = ω and +dDiffr0−1(A)(h−1 +1 h−1 +2 Rαh2h1, h−1 +1 Rαh1) < ǫ/4. +Proof. Proof of (1): By (5.21), there is D > 0 depending only on r0 such that +dDiffr0(A)( f, g) ≤ D max(dDiffr0(A)( f qnhf −qn, Id), dDiffr0(A)(h−1 f −qn+1, Id)). +By Theorem 3, Lemma 5.2, and by letting qn+2 be sufficiently large depend- +ing on ǫ, r0, M, qn+1, we have +dDiffr0(A)( f qnhf −qn, Id), +dDiffr0(A)(h−1 f −qn+1, Id) < (4D)−1ǫ. +Thus we have +dDiffr0(A)( f, g) < ǫ/4. +Proof of (2): Since ∥h1∥Diffr0(A) admits an upper bound depending only on +qn+1, M and r0, by letting qn+2 be sufficiently large depending on ǫ, r0, M, qn+1, +we have +dDiffr0(A)(h−1 +1 Rαh1, g) < ǫ/4. + +ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER +31 +Proof of (3): Set +λ0 = det(H1) = +1 +qn+1 +det( ˜H) ∈ C∞( ˜A). +Recall that both ψ and F have constant Jacobians, hence by (5.30), on each +˜Bj we have +log λ0 = log det h ◦ (G−j) + log det(G−jFj) ◦ Fj + c1 +for some constant c1. On T−k ˜Bj we have a similar formula. Thus +sup +1≤r≤r0 +∥Dr log λ0∥ ≤ (1 + dDiffr0(A)( f, Id))Cqn+1r0(dDiffr0(A)(h, Id) + dDiffr0(A)(g, f)). +By Lemma 5.2 and the argument above for bounding dDiffr0(A)(g, f), we see +that sup1≤r≤r0 ∥Dr log λ0∥ can be made arbitrarily small by making qn+2 +sufficiently large while keeping qn+1, M and r0 fixed. By (5.32), we have +� +[0,1]2 λ0dω = 1. +Then it is direct to see, for some absolute constant C > 0, that +∥λ0 − 1∥ < C∥D log λ0∥ +given that ∥D log λ0∥ < 1. Consequently, for any δ > 0, we have ∥λ0 − +1∥Cr0 < δ if qn+2 is sufficiently large depending on δ, qn+1, M and r0. +By Dacorogna-Moser’s theorem ([DM90, Theorem 1]), there exists h2 ∈ +Diff∞(A) such that +(h2)∗(λ0ω) = ω. +Moreover, by [DM90, Theorem 2 and Lemma 3], we can choose h2 with +dDiffr0−1(A)(h2, Id) arbitrarily small provided ∥λ0 − 1∥Cr0 is sufficiently small. +In summary, if qn+2 is sufficiently large depending on ǫ, qn+1, M and r0, +we can choose h2 ∈ Diff∞(A) sufficiently close to Id in Diffr0(A) so that +dDiffr0−1(A)(h−1 +1 h−1 +2 Rαh2h1, h−1 +1 Rαh1) < ǫ/4. +□ +We set h0 = h2h1. By Lemma 5.4(3) we have h0 ∈ Diff∞(A, ω). Then by +Lemma 5.4, we have +dDiffr0(A)(h−1 +0 Rαh0, f) < ǫ +This completes the proof of Theorem 4. +□ +We can now prove the main result with the aid of the following lemma +whose straightforward proof is omitted. + +32 +BARNEY BRAMHAM AND ZHIYUAN ZHANG +LEMMA 5.5. For any function P : N → N, the set +C := {α ∈ (0, 1) \ Q +| +∃n ≥ 3 odd such that qn(α) > P(qn−1(α)), +qn+1(α) > P(qn(α)), qn+2(α) > P(qn+1(α))} +is open and dense in (0, 1) \ Q. +Proof of Theorem 1. We have seen in the introduction that O∞ +A(α) ⊂ F∞ +A(α) +for any α ∈ (0, 1)/Q. +It remains to show that for a Baire generic α ∈ +(0, 1)/Q +F∞ +A(α) ⊂ O∞ +A(α) +(5.35) +with the closure taken in the C∞-topology. For each tuple (r0, M, k) in Z3 +≥1, +with r0 ≥ 2, let Pr0,M,k−1 be the function produced by Theorem 4. Therefore +by Lemma 5.5 the set +A(r0, M, k) := +� +α ∈ (0, 1) \ Q | there is an odd n ≥ 3 such that +qn(α) > Pr0,M,k−1(qn−1(α)), +qn+1(α) > Pr0,M,k−1(qn(α)) +and +qn+2(α) > Pr0,M,k−1(qn+1(α)) +� +is open and dense in (0, 1) \ Q. Thus the countable intersection +A := ∩r≥2 ∩k≥1 ∩M≥2A(r, M, k) +is a residual subset of (0, 1) \Q. Fix α ∈ A and suppose f ∈ F∞ +A(α). Then for +any ǫ > 0 and r ∈ N with r ≥ 2 choose M ∈ N so that ∥f∥Diffr+2(A) < M, +then by α ∈ A(r, M, ⌈ǫ⌉−1) and by Theorem 4 there exists h ∈ Diff∞(A, ω) +so that +dCr−1( f, hRαh−1) < ǫ. +This gives F∞ +A(α) ⊂ O∞ +A(α) with the closure in the Cr−1-topology. Since r is +arbitrary we easily conclude (5.35) holds with closure in the C∞-topology. +□ +REFERENCES +[AK70] D. ANOSOV AND A. KATOK, New examples in smooth ergodic theory, Trans. +Moscow Math. Soc., 199 (1970), 1-35. +[A78] V. I. ARNOLD, Mathematical methods of classical mechanics (Appendix 9), Berline- +Heidelberg-New York: Springer 1978. +[A10] A. AVILA, On the regularization of conservative maps, Acta Math., 205(1):5-18, 2010. +[AFLXZ20] A. AVILA, B. FAYAD, P. LE CALVEZ, D. XU AND Z. ZHANG, On mixing dif- +feomorphisms of the disk, Invent. Math., 220, 673–714 (2020). +[B22] +P. 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YOCCOZ, Centralisateurs et conjugaison diff´erentiable des diff´emorphismes +du cercle, Ast´erisque, 231 (1995), p. 89-242. +RUHR UNIVERSITY BOCHUM +Email address: barney.bramham@rub.edu +CNRS, INSTITUT GALIL´EE, UNIVERSIT´E PARIS 13 +Email address: zhiyuan.zhang@math.univ-paris13.fr + diff --git a/LdE3T4oBgHgl3EQfYQps/content/tmp_files/load_file.txt b/LdE3T4oBgHgl3EQfYQps/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6b1692a99d6243a4feea12bfcd2bab8e19cea4ec --- /dev/null +++ b/LdE3T4oBgHgl3EQfYQps/content/tmp_files/load_file.txt @@ -0,0 +1,1119 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf,len=1118 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='04486v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='DS] 11 Jan 2023 ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER BARNEY BRAMHAM AND ZHIYUAN ZHANG ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We show that for a Baire generic rotation number α ∈ R/Z, the set of area preserving C∞-pseudo-rotations of the annulus A with rotation number α equals the closure of the set of area preserving C∞- pseudo-rotations which are smoothly conjugate to the rotation Rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' As a corollary, a C∞-generic area preserving pseudo-rotation of the annulus with a Baire generic rotation number α is weakly mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' INTRODUCTION In this paper we denote the 2-dimensional annulus by A = R/Z × [0, 1] equipped with the standard area-form ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We let F∞ A denote the set of ω- preserving C∞ pseudo-rotations of A (the precise definition will appear in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Namely, we set F∞ A := {f ∈ Diff∞(A, ω) | f is isotopic to Id and has no periodic points}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The study of pseudo-rotations can be essentially traced back to the ques- tion of Birkhoff [B41] (see also [H98]) as to whether there are non-trivial analytic diffeomorphisms of the 2-sphere with 2 fixed points (the existence of such diffeomorphisms was recently announced by Berger [B22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The name “pseudo-rotation”was introduced by B´eguin, Crovisier, Le Roux and Patou in [BCLP04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By a result of Franks [F88], for ω-preserving home- omorphisms of A, the notion of irrational pseudo-rotations in [BCLP04] coincides with ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' In particular, each f ∈ F∞ A admits a rotation number ρ( f) ∈ (0, 1)/Q (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='10 for the details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For each α ∈ (0, 1) \\ Q we set F∞ A(α) := {f ∈ F∞ A | ρ( f) = α}, O∞ A(α) := {hRαh−1 | h ∈ Diff∞(A, ω)} where Rα denotes the rotation (x, y) �→ (x + α, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' It is clear that O∞ A(α) ⊂ F∞ A(α) for any α ∈ (0, 1) \\ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' It is not hard to see that F∞ A(α) is closed in the C∞-topology for each irrational α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' In particular, O∞ A(α) ⊂ F∞ A(α) for any α ∈ (0, 1) \\ Q, where the closure is taken in the C∞-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Our main result is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Date: January 12, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' 1 2 BARNEY BRAMHAM AND ZHIYUAN ZHANG THEOREM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For a Baire generic α ∈ (0, 1) \\ Q, we have F∞ A(α) = O∞ A(α) where the closure is taken in Diff∞(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' In other words, for a Baire generic α, any pseudo-rotation f with rotation number α is the C∞-limit of a sequence fk of area preserving diffeomor- phisms, which up to a smooth area preserving change of coordinates, is the standard rotation Rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' In particular f is approximable by integrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We can see Theorem 1 as a natural analogue of a well-known theorem of Herman in [H79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Recall that one of the most prominent results in the study of circle diffeomorphisms is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' THEOREM 2 (Herman-Yoccoz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For any irrational α ∈ R/Z, we denote by F∞(α) the set of C∞ circle diffeomorphisms with rotation number α, and denote by O∞(α) the set of C∞ circle diffeomorphisms which are C∞-conjugate to the standard rotation Rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then we have F∞(α) = � O∞(α) = O∞(α) if α is Diophantine, O∞(α) ̸= O∞(α) if α is Liouville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Here in the above the closures are taken under the C∞-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The above result for Diophantine α was conjectured by Arnold, who showed in [A78] that any Cω circle diffeomorphism with a Diophantine rotation number α which is sufficiently close to Rα in the Cω-topology, is infact Cω-conjugate to Rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Arnold’s result was then generalised to the C∞- category by Moser in [M66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' This is the beginning of what is now known as the Kolmogorov-Arnold-Moser theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The global picture was for the first time established by Herman in the seminal paper [H79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' In [H79], Theorem 2 was proved for a subset of α with full Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Khanin and Sinai gave in [KS89] a different proof of the main result in [H79] building on a renormalization theory for circle diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The Diophantine part of Theorem 2 was completed by Yoccoz in [Y84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We also mention Katznelson-Ornstein’s papers [KO89a, KO89b] on circle diffeomorphisms with low regularity, and Yoccoz’s paper in [EKMY02] on Cω-linearization under the sharp arithmetic condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=', H-condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For a recent survey of this development and beyond, we refer the reader to [EFK18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The Liouville part of Theorem 2 was conjectured by Herman in [H79, Conjecture 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' In fact Herman already showed that Theorem 2 holds for a Baire generic set of α, see [H79, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' However, his proof was based on the Diophantine part of Theorem 2 (at least for a full measure set of α), and used certain properties of the function t �→ ρ(Rt f) of a cir- cle diffeomorphism f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' It is still an open question of Herman whether the ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER 3 Diophantine rigidity holds within pseudo-rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Moreover, it is un- clear how to deform a general pseudo-rotation within the set of pseudo- rotations, and change the rotation number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' This blocks a direct generalisa- tion of Herman’s approach for pseudo-rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The full answer to Her- man’s conjecture was provided by Yoccoz in [Y95b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Yoccoz showed that any C∞ circle diffeomorphsim with a Liouville rotation number can be C∞- approximated by a quasi-rotation: this is a class of circle diffeomorphisms which, among other things, admits a renormalization that is a standard ro- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Our proof of Theorem 1 is somewhat similar to the proof of Yoccoz: we also consider certain renormalizations of a pseudo-rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' However, the type of estimates are very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We are unable to transfer the strong estimates for circle diffeomorphisms, such as Denjoy’s inequality in [Y84], to general pseudo-rotations, due to the possible occurrence of complicated geometry which does not appear in dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' On the other hand, the area-preserving hypothesis provides us with certain strong C0-estimates established in [AFLXZ20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Combining such estimates with a suitable arith- metic condition, we are able to extract some useful information from a se- quence of suitably renormalized pseudo-rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We can also compare Theorem 1 with the main result in [B15b], which says that any smooth area preserving pseudo-rotation f on the closed 2- disc, meaning that f fixes the origin and has no periodic points on the annulus complementary to the origin, is the C0-limit of smooth periodic disc maps fk, that each fix the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' In the latter there are no restrictions on the (irrational) rotation number of f, but in this current paper our inte- grable approximations are in every way stronger: 1) In [B15b] the sequence of approximations fk, while C∞-smooth, only converge in the C0 topology to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' 2) In [B15b] the fk’s are not necessarily area preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' 3) In [B15b] the approximation maps fk have rational rotation numbers pk/qk ∈ Q con- verging to the rotation number α of f as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' One cannot perturb the fk in [B15b] in an obvious way to make the rotation number equal to α while retaining closeness to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' On the other hand, it is easy to modify the fk’s in the current article, if one so wishes, to make the rotation number rational and keep the closeness to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' In short, the fact that in this paper we are able to find integrable approximations without altering the rotation number is also a stronger conclusion than in [B15b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Another motivation behind our result is the important work of Anosov and Katok [AK70] and the extensions in Fayad-Saprykina [FS05], see also Fayad-Katok [FK04], in which, for generic rotation numbers, more pre- cisely all Liouville rotation numbers, examples of pseudo-rotations are con- structed which are dynamically interesting, that is, not conjugate to a rota- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' These “exotic”pseudo-rotations of Anosov-Katok lie, by construction, in the C∞-closure of ∪t∈QO∞ A(t) rather than the closure of O∞ A(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' It how- ever follows from our main result that, for a possibly smaller Baire-generic set of rotation numbers than the Liouville numbers, that the Anosov-Katok 4 BARNEY BRAMHAM AND ZHIYUAN ZHANG constructions do indeed lie in the closure of O∞ A(α) with fixed rotation num- ber α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Moreover, we have the following interesting corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' COROLLARY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For a Baire generic α ∈ (0, 1) \\ Q, the set of weakly mixing pseudo-rotations in F∞ A(α) forms a Baire set with empty interior, with respect to the C∞ topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' On the one hand, Anosov and Katok, see [AK70], show that for a Baire generic α, weak mixing is a C∞-generic property in O∞ A(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Thus by Theorem 1 weak mixing is a C∞-generic property in F∞ A(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' On the other hand, the second statement follows since elements of O∞ A(α) are never weak mixing and by Theorem 1 the complement F∞ A(α)\\O∞ A(α) has empty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' □ REMARK 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Corollary A is seen to be rather sharp in the following sense: (1) the genericity of α cannot be improved into any subset of (0, 1) with pos- itive Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' This follows from the KAM result of Fayad- Krikorian [FK09] (attributed by the authors to Herman), that a neigh- borhood of Rα in F∞ A(α) lies in O∞ A(α), for any Diophantine α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (2) Weakly mixing cannot be replaced by mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' In fact, it follows from the proof of [B15a] and [AFLXZ20] that for a Baire generic α, F∞ A(α) contains no topologically mixing maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' See also Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Recently, Avila and Krikorian have announced 1 an improvement of The- orem 1: for every non-Brjuno α, one has F∞ A(α) = O∞ A(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Moreover, they have announced the following result: for every pseudo-rotation f in an open neighborhood of the rigid rotations on D, there exists a sequence of area-preserving diffeomorphism hn such that hn f h−1 n converges to Rρ( f ) in the C∞ topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Their method involves delicate estimates on high iterates of the maps, while our method for getting this weaker result relies only on rather soft estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' would like to thank Artur Avila and Rapha¨el Krikorian for discussion on one occasion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' would also like to acknowl- edge the online talk by Rapha¨el Krikorian during the Workshop “Between Dynamics and Spectral Theory ”at the Simons Center for Geometry and Physics back in 2016, which inspired this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' This work was initi- ated in 2019 while the authors were at the Institute for Advanced Study both supported by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' DMS- 1638352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We thank them for their hospitality and excellent working envi- ronment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' was also partially supported by the SFB/TRR 191 ‘Symplec- tic Structures in Geometry, Algebra and Dynamics’, funded by the DFG (B1 281071066 – TRR 191) 1See the minicourse of Krikorian in the program “Renormalization and universality in Conformal Geometry, Dynamics, Random Processes, and Field Theory ”in 2020 at Simons Center for Geometry and Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER 5 NOTATION In the rest of this paper, we use the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For a subset A ⊂ R2 we denote by Int(A) the interior of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For a Cr diffeomorphism f : U → V between open subsets U, V ⊂ R2 and r ∈ N, we set ∥Dr f∥ = maxx∈U,|α|=r{∥∂α f(x)∥}, ∥f∥Cr = sup1≤ℓ≤r ∥Dℓ f∥ and ∥f∥Diffr(U) = ∥f∥Cr(U) + ∥f −1∥Cr(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For Cr diffeomorphisms f, g : U → V, we denote dDiffr(U)( f, g) = max(∥f −1g∥Diffr(U), ∥g−1 f∥Diffr(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We abbreviate ∥f∥Diffr(U), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' dDiffr(U), as ∥f∥Diffr, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' dDiffr, when there is no confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For two homeomor- phisms f, g : U → V, we denote ∥f − g∥ = supx∈U d( f(x), g(x)), where d(·, ·) is the euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' PRELIMINARIES We denote the boundary components of A by Bi := R/Z × {i} i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The universal covering of A is ˜A = R × [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We write ˜Bi = R × {i} for i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We let π : ˜A → A be the natural projection, and let T : ˜A → ˜A be the translation in the first coordinate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=', T(x, y) = (x + 1, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Given a homeomorphism f : A → A there is a lift F : ˜A → ˜A of f, also a homeomorphism, unique up to composition by a power of T, such that πF = f π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Moreover, any such lift commutes with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We have the following lemma, whose proof is elementary and left to the readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' LEMMA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Suppose f : A → A is a homeomorphism with ∥f − Id∥ < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then there is a unique lift F : ˜A → ˜A satisfying ∥F − Id∥ < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' This lift satisfies d � F( ˜x), ˜x � = d � f(x), x � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1) for all x ∈ A and all ˜x ∈ ˜A with π( ˜x) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' DEFINITION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We say that γ ⊂ A is a simple regular curve connecting B0 and B1 if γ = φ([0, 1]) where φ : [0, 1] → A is an injective continuous map mapping 0, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' 1, into B0, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' B1, that maps (0, 1) into A\\∂A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We define simple regular curves in ˜A connecting ˜B0 and ˜B1 in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' DEFINITION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For a pair of simple regular curves γ1, γ2 in ˜A we will say that γ1 is to the left of γ2, or γ2 is to the right of γ1, and write γ1 < γ2 if γ1 ∩ γ2 = ∅ and γ1 lies in the component of ˜A\\γ2 containing points with arbitrarily negative R-coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' This defines a partial ordering on simple regular curves in ˜A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' 6 BARNEY BRAMHAM AND ZHIYUAN ZHANG DEFINITION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let f be a homeomorphism of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' A simple regular curve γ in A connecting B0 and B1, satisfying γ ∩ f(γ) = ∅ is called a Brouwer curve for f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' A Brouwer curve γ is called smooth if γ is a C∞-curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Brouwer curves for homeomorphisms of ˜A are defined in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Note that if F is a lift of a homeomorphism f : A → A and ˜γ is a lift of a Brouwer curve γ ⊂ A for f, then TkF( ˜γ) ∩ ˜γ = ∅ ∀k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' DEFINITION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let f be a homeomorphism of A and Q ≥ 2 an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We say that γ is a Q-good smooth curve if γ is a smooth Brouwer curve for each of the maps f, f 2, · · · , f Q−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' DEFINITION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let γ1, γ2 be two disjoint simple regular curves in A con- necting B0 and B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' There is a unique closed region R in A with left bound- ary γ1 and right boundary γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' More precisely, if each γi is oriented from B0 to B1, then ∂R ∩ (A\\∂A) has orientation agreeing with γ2 − γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We say that R is the region bounded by (γ1, γ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' DEFINITION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let {Kr}r≥1 be an increasing sequence of positive real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let U, V be smooth surfaces, possibly with boundary, and let φ : U → V be a C∞-diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We say that φ is {Kr}r≥1-smooth if ∥φ∥Diffr(U) < Kr ∀r ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' DEFINITION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let γ be a Brouwer curve for f, and let R ⊂ A be the closed region bounded by (γ, f(γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We say that an orientation preserving C∞ -diffeomorphism φ : U → R′ from an open neighborhood U ⊂ ˜A of [0, 1]2 to an open neighborhood R′ ⊂ A of R is an admissible coordinate for (R, f) if the following hold: (1) φ has constant Jacobian, (2) φ satisfies φ({0} × [0, 1]) = γ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2) φ({1} × [0, 1]) = f(γ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='3) φ([0, 1] × {0}) ⊂ B0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='4) φ([0, 1] × {1}) ⊂ B1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='5) (3) there is a neighborhood UL of {0} × [0, 1] in U, so that f φ(x) = φT(x) ∀x ∈ UL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Without loss of generality we may assume that T(UL) ⊂ U by choosing UL sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER 7 We make a similar definition for lifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Namely, let ˜γ be a lift of a Brouwer curve γ for f, let F be an lift of f such that F( ˜γ) is on the right of ˜γ, and let ˜R be the region bounded by ( ˜γ, F( ˜γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We say that a C∞-diffeomorphism φ : U → R′ from an open neighborhood U of [0, 1]2 in ˜A to an open neighborhood R′ of ˜R in ˜A is an admissible coordinate for ( ˜R, F) if φ satisfies the analogous properties above with (R, A, f) replaced by ( ˜R, ˜A, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' REMARK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We notice that by item (3) in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='7, an admissible coordinate φ : U → R′ is determined by its restriction to [0, 1]2 in the following sense: if φi : Ui → Ri, i = 1, 2 are two admissible coordinates for (R, f) such that φ1|[0,1]2 = φ2|[0,1]2, then there exists an open neighborhood U3 of [0, 1]2 in ˜A such that φ1|U3 = φ2|U3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For this reason, we will sometimes identify an admissible coordinate for (R, f) or (R, F) with a map from [0, 1]2 to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' DEFINITION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let r ∈ N and K ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We say that φ : [0, 1]2 → R is a (r, K)-admissible coordinate for (R, f) if φ is an admissible coordinate, in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='7, defined on an open neighborhood U of [0, 1]2, satisfying ∥φ∥Diffr(U) < K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Analogously for lifts: We say that φ : U → ˜R is a (r, K)-admissible coor- dinate for ( ˜R, F) if φ is an admissible coordinate for ( ˜R, F), in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='7, defined on an open neighborhood U of [0, 1]2, satisfying ∥φ∥Diffr(U) < K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' DEFINITION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' A pseudo-rotation is a non-wandering homeomorphism f : A → A that is isotopic to the identity, maps B0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' B1) to itself, and has no periodic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Recall that a homeomorphism f : A → A is said to be non-wandering if for every open subet U ⊂ A, there exists an integer n > 0 such that f n(U) ∩ U ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' In this paper we will only be considering C∞-smooth pseudo- rotations that preserve a smooth area form ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By a slight abuse of notation, we denote by ω both the area form on ˜A and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' If f is an ω-preserving diffeomorphism then so is each lift F an ω-preserving diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' DEFINITION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let f : A → A be a pseudo-rotation with lift F : ˜A → ˜A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Denote by p1 : ˜A → R the projection on the first coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We define ρ(F) ∈ R by ρ(F) := lim n→∞ 1 n(p1(Fn(x, y)) − x) x ∈ R, y ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' It is known that the limit on the right hand side above exists for all (x, y) and is independent of (x, y), see [F88, FH12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Moreover, we always have ρ(F) /∈ Q, see [F88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For any pseudo-rotation f, there exists a unique lift F 8 BARNEY BRAMHAM AND ZHIYUAN ZHANG of f such that ρ(F) ∈ (0, 1) \\ Q, which we denote by ρ( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We call ρ( f) the rotation number of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let us recall a few well known facts about the best rational approxima- tions to α ∈ (0, 1) \\ Q and its continued fraction expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Readers can consult [HW, Chapters X, XI] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' First, for x ∈ R we will write ⌊x⌋ := max{n ∈ Z | n ≤ x} ∈ Z, q(x) := ⌊1/x⌋ ∈ Z for the integer parts of x and 1/x respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Moreover, if x ∈ R \\ Q then ∥x∥R/Z := d(x, Z) ∈ (0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The Gauss map G : [0, 1) → [0, 1) is defined by G(x) = 1 x − q(x) on (0, 1) and G(0) := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' If x is irrational then so is G(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' DEFINITION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For any α ∈ (0, 1) \\ Q, we define the sequences (αn)n≥0 and (βn)n≥0 in (0, 1) \\ Q by α0 := α, αn := Gn(α0) ∀n ≥ 1, βn := n ∏ i=0 αi ∀n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Furthermore we define sequences of non-negative integers (an)n≥0, (qn)n≥0 as follows: a0 := 0, an := q(αn−1) ∀n ≥ 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='6) q0 := 1, q1 := q(α), qn+2 := qn + qn+1an+2 ∀n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='7) We also define (pn)n≥0 by p0 = 0, and for n ≥ 1, define pn to be the closest integer to qnα, which is unique by irrationality of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We will use the notation αn(α), qn(α) and pn(α) when it is necessary to indicate the dependence of the sequences on α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Note that α−1 n−1 = an + αn ∀n ≥ 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='8) since αn = G(αn−1) = 1/αn−1 − q(αn−1) = 1/αn−1 − an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' It is also well known that pn+1qn − pnqn+1 = (−1)n (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='9) βn = (−1)n(qnα − pn) > 0 ∀n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='10) ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER 9 In particular for each n ≥ 1, βn = |qnα − pn| = d(qnα, Z), and α = 1 a1 + 1 a2 + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' + 1 an + αn pn qn = 1 a1 + 1 a2 + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' + 1 an For any n ≥ 1, the integers pn and qn are relatively prime, and pn/qn is called the n-th best rational approximation of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The following simple in- equalities are known, and will be used later: 1 2qn+1 < 1 qn + qn+1 < βn < 1 qn+1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='11) αn, q(αn)−1 ∈ � qn 2qn+1 , 2qn qn+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='12) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' EXISTENCE OF A BROUWER CURVE WITH UNIFORM BOUNDS In order to avoid lengthy computations, in the rest of the paper we will introduce various increasing functions to keep track of parameter depen- dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We say that a function A : U → R defined on an open subset U ⊂ Rn is increasing if for any x = (x1, · · · , xn) and y = (y1, · · · , yn) ∈ U with xi ≥ yi for all 1 ≤ i ≤ n, we have A(x) ≥ A(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We define decreas- ing functions in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Typically, all the variables and values of the increasing/decreasing functions that we will consider lie in R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The following theorem, essentially proven in [AFLXZ20], allows one to control the Cr-distance of a pseudo-rotation to the identity in terms of its rotation number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' THEOREM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' There is a sequence of increasing functions Ar : (0, 1/2) × R+ → R+, lim t→0 Ar(t, · ) ≡ 0 for each r ∈ N, so that for any {Kr}r≥1-smooth ω-preserving pseudo-rotation f : A → A there holds ∥f − Id∥ < A0(∥ρ( f)∥R/Z, K1), ∥Dr f∥ < Ar(∥ρ( f)∥R/Z, Kr+1), ∀r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By [AFLXZ20, Corollary A], we know that ∥f − Id∥ < (1 + 2K1)∥ρ( f)∥1/2 R/Z =: A0(∥ρ( f)∥R/Z, K1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' 10 BARNEY BRAMHAM AND ZHIYUAN ZHANG By the Hadamard-Kolmogorov convexity theorem, we get corresponding bounds on the “inbetween” derivatives: for any r ≥ 1 ∥Dr f∥ ≤ Cr∥f − Id∥ 1 r+1∥f∥ r r+1 Diffr+1 ≤ Cr∥ρ( f)∥ 1 2(r+1) R/Z (1 + 2Kr+1) =: Ar(∥ρ( f)∥R/Z, Kr+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' □ By Theorem 3, there is a decreasing function ρ∗ : R+ → (0, 1/2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1) such that for any smooth ω-preserving pseudo-rotation f satisfying ∥ρ( f)∥R/Z < ρ∗(∥D f∥), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2) we have ∥f − Id∥ ≤ A0(∥ρ( f)∥R/Z, ∥D f∥) < A0(ρ∗(∥D f∥), ∥D f∥) < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1, there is a unique lift F of f satisfying ∥F − Id∥ = ∥f − Id∥ < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='3) LEMMA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' There are increasing functions C′, C′′ : R+ × (0, 1) → R+ with limt→0 C′′(t, ·) ≡ 0 and C′′(α, K−1) < ρ∗(Kα−1) ∀(α, K) ∈ (0, 1) × (1, ∞) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='4) such that the following holds: if α ∈ (0, 1)\\Q satisfies α ∈ (0, ρ∗(K)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='5) G(α) < C′′(α, K−1) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='6) for some K > 1, where ρ∗ is given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1), then every C1-smooth ω-preserving pseudo-rotation with ∥D f∥ < K and ρ( f) = α satisfies inf x∈A d(x, f(x)) ≥ C′(α, K−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Fix K > 0 and α ∈ (0, 1)\\Q satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let f be a C1-smooth ω-preserving pseudo-rotation with ∥D f∥ < K and ρ( f) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Denote by F the lift of f with ρ(F) = ρ( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We abbrieviate q := q(α) = ⌊1/α⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Choose the function C′′ : R+ × (0, 1) → R+, with limt→0 C′′(t, ·) ≡ 0, sufficiently small so that for each t ∈ (0, 1), the condition s < C′′(t, K−1) implies A0(s, Kt−1) < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='8) ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER 11 Without loss of generality we may arrange that C′′ is increasing and satis- fies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By Theorem 3, we have ∥f q − Id∥ ≤ A0(∥ρ( f q)∥R/Z, ∥D f q∥) ≤ A0(∥qρ( f)∥R/Z, Kq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Since ∥qρ( f)∥R/Z = 1 − qα = αG(α) ≤ G(α) and Kq ≤ Kα−1, the above gives ∥f q − Id∥ ≤ A0(G(α), Kα−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Thus, if α satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='6), then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='8), ∥f q − Id∥ < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='9) The unique lift of f q with rotation number in (−1/2, 1/2) is T−1Fq, since ρ(T−1Fq) = qρ(F) − 1 = qα − 1 ∈ (−1/2, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' So by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1 ∥Fq − T∥ = ∥T−1Fq − Id∥ = ∥f q − Id∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Thus for all ˜x ∈ ˜A d(Fq( ˜x), ˜x) ≥ d(T( ˜x), ˜x) − d(Fq( ˜x), T( ˜x)) > 1 − 1/2 = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='10) On the other hand, for each ˜x ∈ ˜A, there holds d(Fq( ˜x), ˜x) ≤ q−1 ∑ j=0 d � Fj+1( ˜x), Fj( ˜x) � ≤ q−1 ∑ j=0 ∥DF∥jd � F( ˜x), ˜x � ≤ Cd(F( ˜x), ˜x) where C = Kq−1 K−1 ≤ Kα−1−1 K−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Combining the last line with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='3), we obtain ∥F( ˜x) − ˜x∥ ≥ 1 2C ∀ ˜x ∈ ˜A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='11) By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='5), we conclude that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='7) holds if we define C′ by C′(t, K−1) = 1 2 K − 1 Kt−1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' □ DEFINITION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For each K > 1 let L(K) be the set of α ∈ (0, 1/2) \\ Q such that α ∈ � 0, ρ∗(K) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='12) G(α) < C′′(α, K−1) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='13) where ρ∗ is from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1) and C′′ is from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' LEMMA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' There is a sequence of increasing functions C′ r : R2 + → R+ ∀r ≥ 1, such that the following is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' If {Kr}r≥1 is any sequence in (1, ∞) and α ∈ L(K1), then every {Kr}r≥1-smooth ω-preserving pseudo-rotation f with ρ( f) = α has for each r ∈ N a Brouwer curve γ = γr for which the closed region in A bounded by (γ, f(γ)) admits a (r, C′ r(α−1, Kr+1))-admissible coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' 12 BARNEY BRAMHAM AND ZHIYUAN ZHANG Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Fix a sequence (Kr)r≥1 in (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Fix r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For each c > 0 and K ≥ 1 define the following subset of Diffr(A, ω): Hr(c, K) := � g ∈ Diffr(A, ω) ��� ∥g∥Diffr ≤ K, inf x∈A d(x, g(x)) ≥ c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Denote by Hr+1(c, K) ⊂ Diffr(A, ω) the closure of Hr+1(c, K) in the Cr-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We observe that Hr+1(c, K) is compact in Diffr(A) and contains only diffeomorphisms without fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Moreover, for c′ ≥ c and K′ ≤ K we have Hr+1(c′, K′) ⊂ Hr+1(c, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We now prove a version of the lemma for elements of Hr+1(c, K), for each fixed c > 0, K ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then we argue that the union of Hr+1(c, Kr+1) over c ∈ (0, 1] contains all pseudo-rotations satisfying the assumptions of the Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' To this end, fix c > 0, K ≥ 1 and consider g ∈ Hr+1(c, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Since g has no fixed points, a strong refinement of Brouwer’s plane translation theorem due to Guillou [G94, Th´eor`em 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1] yields a C0 Brouwer curve γ0 for g in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Any sufficiently C0-close smooth approxima- tion of γ0 that continues to connect the two boundary components yields a smooth Brouwer curve γ for g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Clearly we can choose such a γ to meet both boundary components of the annulus orthogonally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We can then apply the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' LEMMA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let r ≥ 1 and θ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Suppose g ∈ Diffr,θ(A, ω) has a smooth Brouwer curve γ that meets both boundary components orthogonally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then there exist D = D(g, γ, r) > 1, and a neighborhood V of g in Diffr,θ(A, ω) such that for every g′ ∈ V, the region (γ, g′(γ)) has admissible coordinates φ′ whose Cr-norm is bounded by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let R denote the region (γ, g(γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We first construct a C∞-diffeomorphism ψL with constant Jacobian ω(R) from a neighborhood of {0} × [0, 1] in ˜A onto its image a neighborhood of γ in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Indeed, without loss of gen- erality γ : [0, 1] → A meets the boundary of A orthogonally near both end points and is parametrised by arclength so that ∥ ˙γ∥ = L is constant, L ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then n := −i ˙γ/L is a normal vector field along γ and the map ˜A → R2, (x, y) �→ γ(y) + xω(R)n(y)/L extends γ to a smooth diffeomor- phism from a sufficiently small tubular neighborhood of {0} × [0, 1] in ˜A to a neighborhood of the image of γ in A, with constant Jacobian ω(R) along {0} × [0, 1] and also near to the boundary of ˜A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The Jacobian of this map away from {0} × [0, 1] depends only on the y variable and therefore by a further change of coordinates it is easily modified to have constant Jacobian on a whole neighborhood of {0} × [0, 1] while still mapping {0} × [0, 1] to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The resulting map, which we denote by ψL, clearly has finite Cr,θ-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then gψLT−1 is a Cr,θ-diffeomorphism ψR with constant Jacobian ω(R) from a neighborhood of {1} × [0, 1] in ˜A onto its image, a neighborhood ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER 13 of g(γ) in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We construct the chart φ : [0, 1]2 → A by first specifying its restriction to a neighborhood of {0} × [0, 1], resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' {1} × [0, 1], to be ψL, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' ψR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then we extend it by hand to neighborhoods of [0, 1] × {0} and [0, 1] × {1}, also with constant Jacobian ω(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Finally, using [A10, Corollary 4] (or, in our simple application, using directly [A10, Theorem 3] which follows from [DM90]), we extend the map to all of [0, 1]2 so as to have constant Jacobian, and the Cr norm of the resulting map is bounded in terms of the Cr,θ norm of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By construction the conditions in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='7 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Each g has a C0-small (and hence also Cr-small) neighborhood in Diffr(A, ω) for which the same γ can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The uniform bounds for the Cr-norm of φ is an immediate consequence of the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We omit the proof of this latter fact and refer the readers to [A10, DM90] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' □ By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='3 and by compactness of Hr+1(c, K) ⊂ Diffr,1/2(A, ω) ⊂ Diffr(A, ω), we find a finite collection of neighborhoods (with respect to the Cr,1/2-topology) as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='3 whose union covers Hr+1(c, K), and thus obtain a uniform bound Er(c, K) > 0 on the Cr-norm of admissible coordinates that applies to all elements of Hr+1(c, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Due to the inclusions Hr+1(c′, K′) ⊂ Hr+1(c, K) for c′ > c, K′ < K, we can assume that Er(c, K) is decreasing in c and increasing in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Now, suppose f is as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' That is, f is a (Kr)r≥1-smooth ω- preserving pseudo-rotation with rotation number ρ( f) = α ∈ L(K1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1, f ∈ Hr+1 � C′(α, K−1 1 ), Kr+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Thus there exists a region (γ, f(γ)) having admissible coordinates whose Cr-norm is bounded by Er(C′(α, K−1 1 ), Kr+1), for some smooth Brouwer curve γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Hence Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2 holds with C′ r(α−1, Kr+1) := Er(C′(α, K−1 1 ), Kr+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Evidently C′ r is an increasing function, since C′ is increasing and Er(c, K) is decreasing in c and increasing in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' □ It will be useful to fix the following notation: DEFINITION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For each integer r ≥ 1 and each K > 1 let Lr(K) be the set of α ∈ L(K) such that A0(αG(α), Kq(α)) < K−q(α)C′ r(α−1, K)−1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='14) where C′ r : R2 + → R+, r ≥ 1 are the increasing functions produced by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We will see later that each Lr(K) is non-empty for each integer r ≥ 1 and each K > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Moreover, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='14) we have Lr(K′) ⊂ Lr(K) ∀K′ > K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='15) Now we can strengthen Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Indeed, we show that if more restric- tions are placed on the rotation number of a ω-preserving pseudo-rotation 14 BARNEY BRAMHAM AND ZHIYUAN ZHANG f, then the produced Brouwer curve is actually Q-good, for some large Q depending on the rotation number of f: PROPOSITION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let f be a {Kr}r≥1-smooth ω-preserving pseudo-rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' If ρ( f) ∈ Lr(Kr+1) for some integer r ≥ 1, then the Brouwer curve γ for f produced by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2 (corresponding to r) is q(ρ( f))-good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Note that the condition ρ( f) ∈ Lr(Kr+1) in this Proposition includes the condition on ρ( f) used in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2, since Lr(K) ⊂ L(K) by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We denote by dH the Hausdorff distance on ˜A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' for any two sub- sets A, B of ˜A, dH(A, B) = sup x∈A,y∈B max(d(x, B), d(y, A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Set α = ρ( f) ∈ Lr(Kr+1) for some r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let γ be the Brouwer curve given by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2 for this value of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let R ⊂ A be the region bounded by (γ, f(γ)) and let ψ : R → [0, 1] × [0, 1] be the inverse of some (r, C′ r(α−1, Kr+1))- admissible coordinates produced by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Applying the intermedi- ate value theorem to ψ yields dH(γ, f(γ)) > C′ r(α−1, Kr+1)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='16) Fix a lift ˜γ ⊂ ˜A of γ, and let F be the lift of f for which ρ(F) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Since ˜γ ∩ F( ˜γ) = ∅ it follows from α > 0 and the order of boundary points that ˜γ < F( ˜γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='7, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='16) and Kr+1 ≥ K1, we can see that dH(Fq(α)−1( ˜γ), Fq(α)( ˜γ)) > K−q(α) 1 dH( ˜γ, F( ˜γ)) ≥ K−q(α) r+1 C′ r(α−1, Kr+1)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='17) Here we implicitely used that the analogue of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='16) holds for the lifts, since dH( ˜γ, F( ˜γ)) ≥ dH(γ, f(γ)) holds - infact for all choice of lifts F and ˜γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Moreover, since F is injective, for each i ∈ N we have Fi( ˜γ) ∩ Fi+1( ˜γ) = ∅ and so from the order of boundary points we have for all i ∈ N, ˜γ < F( ˜γ) < F2( ˜γ) < · · · < Fi( ˜γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='18) Thus it suffices to show that Fq(α)−1( ˜γ) < T ˜γ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='19) and it will follow that the iterates F( ˜γ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' , Fq(α)−1( ˜γ) all lie strictly in the region between ˜γ and T( ˜γ) in ˜A and therefore that the iterates f(γ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' , f q(α)−1(γ) are all disjoint from γ as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' To show (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='19) we argue by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Assuming (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='19) is not true, we have Fq(α)−1( ˜γ) ∩ T ˜γ ̸= ∅ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='20) because of the order of boundary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We will show that this means that Fq(α)−1( ˜γ) and Fq(α)( ˜γ) pass somewhere very close to each other, because Fq(α)( ˜γ) is close to T ˜γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' This will contradict (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER 15 First, notice that by α ∈ Lr(Kr+1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='4) and that ρ∗ is decreasing, we have ∥ρ( f q(α))∥R/Z = |q(α)α − 1| = αG(α) < G(α) < C′′(α, K−1 r+1) < ρ∗(Kα−1 r+1) ≤ ρ∗(∥D f∥q(α)) ≤ ρ∗(∥D f q(α)∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='3), we have dH(Fq(α)( ˜γ), T ˜γ) ≤ ∥Fq(α) − T∥ = ∥T−1Fq(α) − Id∥ = ∥f q(α) − Id∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The last equality can be justified by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1, just as in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Thus by Theorem 3 and Kr+1 ≥ K1, we have dH(Fq(α)( ˜γ), T ˜γ) ≤ A0(∥ρ( f q(α))∥R/Z, Kq(α) 1 ) ≤ A0(αG(α), Kq(α) r+1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='21) Then along with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='17), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='14), and the hypothesis α ∈ Lr(Kr+1), we have dH(Fq(α)−1( ˜γ), T ˜γ) ≥ dH(Fq(α)( ˜γ), Fq(α)−1( ˜γ)) − dH(Fq(α)( ˜γ), T ˜γ) ≥ K−q(α) r+1 C′ r(α−1, Kr+1)−1 − A0(αG(α), Kq(α) r+1 ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' However this contradicts (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Thus we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' SMOOTH DOMAIN BOUNDED BY GOOD CURVES 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Renormalization of pseudo-rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Given a C∞ pseudo-rotation f on A and an integer n ≥ 1, we denote by Fn the unique lift of f n to ˜A such that ρ(Fn) ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Given a smooth Brouwer curve γn for f n, we let ˜γn be an arbitrary lift of γn to ˜A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We let Ωn be the unique closed region in ˜A bounded by ( ˜γn, Fn( ˜γn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Given a C∞ admissible coordinate H : [0, 1]2 → Ωn for (Ωn, Fn) (see Remark 2), we can uniquely extend H to a C∞ diffeomorphism of ˜A with constant Jacobian, denoted again by H, satisfying HT = FnH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1) We notice that although the Cr norms of Fn and T are uniformly bounded throughout ˜A, the Cr norm of H need not be uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' However it is clear from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1) that for any integer L > 0, the norm ∥DrH∥[−L,L]×[0,1] is bounded in terms of L and ∥Fn∥Diffr( ˜A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Denote α = ρ( f) ∈ (0, 1) \\ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We have ρ(F1) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We abbrieviate Fa,b := TbFa 1 ∀a, b ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By our previous definition, we have Fn = Fn,−⌊nα⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Denote by J : ˜A → ˜A the orientation reversing diffeomorphism J(x, y) = (−x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Notice that by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1) we have JH−1FnHJ = JTJ = T−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2) 16 BARNEY BRAMHAM AND ZHIYUAN ZHANG We set Fa,b H := JH−1Fa,bHJ ∀a, b ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='3) Since Fa,b is ω-preserving and commutes with Fn, and since H has con- stant Jacobian, we deduce from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2) that Fa,b H is also ω-preserving, and commutes with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Consequently Fa,b H descends to an ω-preserving C∞ dif- feomorphism f a,b H : A → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' LEMMA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' There is a sequence of increasing functions {Er : N2 × R2+ → R+}r≥1 such that the following is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let f be a {Kr}r≥1-smooth ω-preserving pseudo-rotation with ρ( f) = α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' let n ̸= 0 be an integer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' let Ωn be the region bounded by ( ˜γn, Fn( ˜γn)) where ˜γn is some lift of a Brouwer curve for f n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Assume that (Ωn, Fn) admits a (r, Lr)-admissible coordinate H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then for any a, b ∈ Z with a⌊nα⌋ + bn ̸= 0, f a,b H is a pseudo-rotation in Diff∞(A, ω) with ρ( f a,b H ) = �−aα − b {nα} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Moreover, we have ∥f a,b H ∥Diffr ≤ Er(|a|, |b|, Kr, Lr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We first show that f a,b H has no periodic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Assume to the con- trary there are integers p ∈ Z, q > 0 and some z ∈ ˜A such that (Fa,b H )q(z) = Tp(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='3) we have z = (Fa,b H )qT−p(z) = JH−1F(qa+pn) 1 T(qb−p⌊nα⌋)HJ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='4) However, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='4) and the condition a⌊nα⌋ + bn ̸= 0 implies that HJ(z) de- scends to a perodic point for f, which contradicts the hypothesis that f is a pseudo-rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We conclude that f a,b H is a pseudo-rotation in Diff∞(A, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' To compute ρ( f a,b H ), it suffices to study the trajectory of an arbitrary z ∈ ˜A under the iterates of (Fa,b H )qT−p using the second equality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' for example a point on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' This standard argument is left to the readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For the Cr-norm of f a,b H , it suffices to control the Cr-norm of Fa,b H restricted to (−1, 2) × [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='3), ∥DrFa,b H ∥(−1,2)×[0,1] depends only on the Cr- norms of Fa,b, and the Cr-norm of J and H on Fa,b H ((−1, 2) × [0, 1]) ∪ (−1, 2) × [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' This yields a bound depending only on r, |a|, |b|, Kr, Lr as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' □ ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Finding a good curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' In the following statement we consider n ∈ N even, so that qnα − pn > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' PROPOSITION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' There is a sequence of increasing functions {Gr : R2 + → R+}r≥1 such that the following is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let f be a pseudo-rotation in Diff∞(A, ω), let F be a lift with ρ(F) = ρ( f) = α ∈ (0, 1) \\ Q, and let n ≥ 2 be an even integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Suppose γ ⊂ A is a smooth Brouwer curve for f qn with a lift γ† ⊂ ˜A such that the closed region Ω in ˜A bounded by (γ†, Fqn,−pn(γ†)) admits an (r, Kr)-admissible coordinate H : [0, 1]2 → Ω for some r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Suppose further that γn ⊂ A is an an+2-good smooth curve for f qn+1,−pn+1 H ,2 for which the closed region Ωn ⊂ A bounded by (γn, f qn+1,−pn+1 H (γn)) admits an (r, Mr)-admissible coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then there exists a qn+2-good smooth curve ˆγ ⊂ A for f, for which the closed region Ω∗ ⊂ A bounded by ( f qn+1( ˆγ), ˆγ) admits a (r, Gr(Kr, Mr))-admissible coordinate for f qn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We set S0 = Fqn,−pn, S = Fqn+1,−pn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' As explained in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1, we extend H to a C∞-diffeomorphism of ˜A with constant Jacobian by the formula HT = S0H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By definition, we know that (1) JH−1S0HJ = JTJ = T−1 on ˜A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (2) ˜S := JH−1SHJ commutes with T on ˜A, and descends to f qn+1,−pn+1 H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Moreover, we have ρ( ˜S) = ρ( f qn+1,−pn+1 H ) = αn+1 ∈ (0, 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let ˜γ be an arbitrary lift of γn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' As γn is a simple regular curve and is disjoint from f qn+1,−pn+1 H (γn), we know by item (1), (2) above that ˜γ is also a simple regular curve, and T−1( ˜γ) ∩ ˜γ = ˜S( ˜γ) ∩ ˜γ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='5) We let γ′ = HJ( ˜γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then γ′ is also a simple regular curve, and by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='5) and item (1), (2) above, we have S0(γ′) ∩ γ′ = S(γ′) ∩ γ′ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='6) Moreover, we can see that S(γ′) < γ′ < S0(γ′) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='7) by considering their boundary points on B0 and B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We let ˆγ = π(γ′) (recall that π is the canonical projection from ˜A to A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' LEMMA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The curve ˆγ is a simple regular curve connecting B0 and B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Namely, it has no self-intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' 2Recall that by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1 we have ρ( f qn+1,−pn+1 H ) = αn+1 and an+2 = q(αn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' 18 BARNEY BRAMHAM AND ZHIYUAN ZHANG Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Since ˆγ = π(γ′), and since γ′ is a simple regular curve in ˜A con- necting B0 to B1, it suffices to show that γ′ is disjoint from all its translates Tk(γ′) for k ∈ Z\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Since γ′ connects the two boundary components it is enough to show γ′ is disjoint from T(γ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By definition and by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='9), we have T = Sqn+1 0 S−qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='7), we obtain that γ′ < T(γ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' □ By item (2) above, the region ˜Ωn bounded by ( ˜γ, ˜S( ˜γ)) is a lift of Ωn Moreover, the (r, Mr)-admissible coordinate for Ωn lifts to a (r, Mr)-admissible coordinate for ˜Ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We notice that the push forward HJ( ˜Ωn) is the region in ˜A between S(γ′) and γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Moreover, from the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2, we see that T−1(γ′) < S(γ′) < γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Thus the map π induces a diffeomorphism from HJ( ˜Ωn) to the region in A between f qn+1( ˆγ) and ˆγ, that is, to the region Ω∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We conclude that the (r, Mr)-admissible coordinate for ˜Ωn, after composing with HJ and project- ing, yield (r, Gr(Kr, Mr))-admissible coordinates for Ω∗, for some functions Gr as in the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' It remains to show that ˆγ is a qn+2-good curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We divide the proof into two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Case I: Assume that there are integers p and 0 < k < qn+2 such that kα + p < 0 and Fk,p(γ′) ∩ γ′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='8) We assume further that for any integers p′ and 0 < k′ < qn+2 such that k′α + p′ < 0 and Fk′,p′(γ′) ∩ γ′ ̸= ∅, we have k′α + p′ ≤ kα + p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' First, we observe that either qn+2 > k ≥ qn+2 − qn or −{qnα} = pn − qnα < kα + p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Indeed, if k < qn+2 − qn and pn − qnα > kα + p, then we have k + qn < qn+2 and kα + p < (k + qn)α + p − pn < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' In particular, the endpoints of S0Fk,p(γ′) are on the left hand side of those of γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='8) and S0Fk,p(γ′) > Fk,p(γ′), we have that Fk+qn,−pn+p(γ′) ∩ γ′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' This contradicts the choice of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' If qn+2 > k ≥ qn+2 − qn, then by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='6), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='7), we know that k > qn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We notice that kα + p < (k − qn+1)α + p + pn+1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER 19 In particular, the endpoints of S−1Fk,p(γ′) are on the left hand side of those of γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='8) and S−1Fk,p(γ′) > Fk,p(γ′), we have that Fk−qn+1,p+pn+1(γ′) ∩ γ′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' This contradicts the choice of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' If −{qnα} < kα + p < 0, then we have k ∈ {qn+1, · · · , an+2qn+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' If k = iqn+1 with i > 1, then we have −1 < i(qn+1α − pn+1) < (i − 1)(qn+1α − pn+1) = (k − qn+1)α − (i − 1)pn+1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then we must have p = −ipn+1, and the endpoints of S−1Fk,p(γ′) are on the left hand side of those of γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='8) and S−1Fk,p(γ′) > Fk,p(γ′), we have that Fk−qn+1,−(i−1)pn+1(γ′) ∩ γ′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' If k = qn+1, we would have Fqn+1(γ′) ∩ (γ′ + Z) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Both cases contradict the choice of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Case II: Assume that there are integers p and 0 < k < qn+2 such that kα + p > 0 and Fk,p(γ′) ∩ γ′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='9) We assume further that for any integers p′ and 0 < k′ < qn+2 such that k′α + p′ > 0 and Fk′,p′(γ′) ∩ γ′ ̸= ∅, we have k′α + p′ ≥ kα + p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' First, we observe that either qn ≥ k > 0 or {qnα} = qnα − pn > kα + p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Indeed, if k > qn and qnα − pn < kα + p, then we have qn+2 > k − qn > 0 and kα + p > (k − qn)α + p + pn > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' In particular, the endpoints of S−1 0 Fk,p(γ′) are on the right hand side of those of γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='9) and Fk−qn,pn+p(γ′) = S−1 0 Fk,p(γ′) < Fk,p(γ′), we have that Fk−qn,pn+p(γ′) ∩ γ′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' This contradicts the choice of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' If qn ≥ k > 0, then by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='9), we have qn > k and consequently k + qn+1 < qn+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Moreover we notice that kα + p > (k + qn+1)α + p − pn+1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' 20 BARNEY BRAMHAM AND ZHIYUAN ZHANG In particular, the endpoints of SFk,p(γ′) are on the right hand side of those of γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='9) and SFk,p(γ′) < Fk,p(γ′), we have that Fk+qn,p−pn(γ′) ∩ γ′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' This contradicts the choice of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' If {qnα} > kα + p > 0 (and 0 < k < qn+2), then it is straightforward to verify that (k, p) ∈ {(qn + iqn+1, pn + ipn+1) | 0 < i < an+2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By the hypothesis that γn is a an+2-good curve, we have T−1 ˜Si( ˜γ) ∩ ˜γ = ∅ ∀1 ≤ i < an+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By T−1 = JH−1S0HJ, ˜S = JH−1SHJ and γ′ = HJ( ˜γ), we obtain Fqn+iqn+1,−pn−ipn+1(γ′) ∩ γ′ = S0Si(γ′) ∩ γ′ = ∅ for all integer 0 < i < an+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' This again gives a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' In summary, we see that for all integer 0 < i < qn+2, Fi(γ′) ∩ (γ′ + Z) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Hence ˆγ is qn+2-good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' This completes the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' □ We can now prove the following: COROLLARY B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For each integer r ≥ 1, there exist increasing functions Pr : R2 + → R+ and Wr : N × R+ → R+ such that the following is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Suppose that there is an odd integer n ≥ 3 such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='10) qn > Pr(Kr+2, qn−1), qn+1 > Pr(Kr+2, qn) and qn+2 > Pr(Kr+2, qn+1) where {qk}k≥0 is the sequence of denominators associated to some α ∈ (0, 1)\\Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then for any {Kk}k≥1-smooth pseudo-rotation f ∈ Diff∞(A, ω) with ρ( f) = α has a qn+1-good smooth curve γ such that the closed region in A bounded by ( f qn(γ), γ) admits a (r, Wr(qn+1, Kr+2))-admissible coordinate for f −qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Fix r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let {Lk : N × R+ → R+}k≥1 be a sequence of increas- ing functions independent of f such that for each integer m ≥ 0, f m is {Lk(m, Kk)}k≥1-smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let us abbreviate L′ k := Lk(qn−1, Kk) ∀k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For each n ≥ 2 we have ρ( f qn−1) = βn−1 ∈ (0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' If (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='10) holds and Pr is chosen appropriately, then βn−1 ∈ L(L′ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='11) This allows us to apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2 to f qn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2, there is a smooth Brouwer curve γ for f qn−1 for which the closed region R′ n−1 ⊂ A bounded ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER 21 by (γ, f qn−1(γ)) admits a (r + 1, Yr+1)-admissible coordinate H′ : [0, 1]2 → R′ n−1 where Yr+1 := C′ r+1(β−1 n−1, L′ r+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='12) We lift H′ to an admissible coordinate H : [0, 1]2 → Rn−1 where Rn−1 is a lift of R′ n−1 in ˜A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' More precisely, Rn−1 is a connected component of π−1(R′ n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' As in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1, we extend H to a diffeomorphism of ˜A sat- isfying HT = Fqn−1,−pn−1H and set fn := f qn,−pn H : A → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' That is, fn is the projection of Fqn,−pn H := (HJ)−1Fqn,−pn(HJ) : ˜A → ˜A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1, fn is a pseudo-rotation in Diff∞(A, ω) with ρ( fn) = βn/βn−1 = αn such that ∥fn∥Diffr+1 ≤ Er+1(qn, pn, Kr+1, Yr+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='13) By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='12), and using that Kr+1 ≤ Kr+2, max(pn, β−1 n−1) ≤ 2qn we can rewrite (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='13) as ∥fn∥Diffr+1 < Vr+1(qn, Kr+2) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='14) where Vr+1 : N × R+ → (1, ∞) is an increasing function independent of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Now we choose Pr : R2+ → R+ to be any increasing function that is sufficiently large that the following conditions are fulfilled: Pr(D, y) > 2yρ∗(Vr+1(y, D))−1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='15) 2y Pr(D, y) < C′′ � 1 2y, Vr+1(y, D)−1 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='16) for all D, y > 0, where ρ∗ is the decreasing function introduced in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We moreover choose Pr sufficiently large that whenever x > Pr(D, y) there holds A0 �2y x , Vr+1(y, D)2y � < Vr+1(y, D)−2yC′ r(2y, Vr+1(y, D))−1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='17) for all D, y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' This we can arrange because limt→0 A0(t, ·) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Assume now that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='10) holds for a fixed odd integer n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='15) and then the second inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='10), we have 2qn < ρ∗(Vr+1(qn, Kr+2))Pr(Kr+2, qn) < ρ∗(Vr+1(qn, Kr+2))qn+1 and therefore by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='12) αn < 2qn qn+1 < ρ∗(Vr+1(qn, Kr+2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='18) 22 BARNEY BRAMHAM AND ZHIYUAN ZHANG Also, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='12), the third inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='10), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='16), we obtain G(αn) = αn+1 < 2qn+1 qn+2 < 2qn+1 Pr(Kr+2, qn+1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='19) < C′′ � 1 2qn+1 , Vr+1(qn+1, Kr+2)−1 � < C′′(αn, Vr+1(qn, Kr+2)−1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='20) where the last inequality uses the monotonicity of C′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We claim that ρ( fn) = αn ∈ Lr(Vr+1(qn, Kr+2)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='21) where Lr is as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Equivalently, we show that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='22) A0 � αnG(αn), Vr+1(qn, Kr+2)q(αn)� < Vr+1(qn, Kr+2)−q(αn)C′ r � α−1 n , Vr+1(qn, Kr+2) �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='12) we have αnG(αn) = αnαn+1 < 4qn/qn+2 ≤ 2qn+1/qn+2 and q(αn) ≤ 2qn+1 and α−1 n < 2qn+1/qn ≤ 2qn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Therefore by the monotonicity of A0, and by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='17) together with the third inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='10), we obtain A0 � αnG(αn), Vr+1(qn, Kr+2)q(αn)� < A0 �2qn+1 qn+2 , Vr+1(qn+1, Kr+2)2qn+1 � < Vr+1(qn+1, Kr+2)−2qn+1C′ r(2qn+1, Vr+1(qn+1, Kr+2))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Now inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='22) follows from the monotonicity of Vr+1 and C′ r, and because α−1 n < 2qn+1/qn ≤ 2qn+1 from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' This proves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='21) with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='14) we see that fn satisfies the hypotheses of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1, and therefore fn has a q(ρ( fn))-good Brouwer curve γn say.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='8) q(ρ( fn)) = q(αn) = an+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Thus γn is an an+1-good Brouwer curve for fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Moreover, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2, the region in A bounded by (γn, fn(γn)) admits an (r, C′ r(α−1 n , ˆKr+1))-admissible coordinate, provided ˆKr+1 ≥ ∥fn∥Diffr+1 and ρ( fn) = αn ∈ L( ˆK) where ˆK ≥ ∥fn∥Diff1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='14) we may take ˆKr+1 = ˆK = Vr+1(qn, Kr+2) and conclude that the region in A bounded by (γn, fn(γn)) admits a (r, Ur)-admissible coordinate where Ur := C′ r(α−1 n , Vr+1(qn, Kr+2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We define Wr by Wr(q, K) := Gr(C′ r(2q, Lr+1(q, K)), C′ r(2q, Vr+1(q, K))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='12), we have β−1 n−1, α−1 n < 2qn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then by Kr+2 ≥ Kr+1, we have C′ r(2qn+1, Lr+1(qn+1, Kr+2)) > Yr and C′ r(2qn+1, Vr+1(qn+1, Kr+2)) > Ur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Thus we have Wr(qn+1, Kr+2) ≥ Gr(Yr, Ur).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER 23 Then by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1, there is a qn+1-good smooth curve of f, denoted by ˆγ, such that the region in A bounded by ( f qn( ˆγ), ˆγ) admits a (r, Wr(qn+1, Kr+2))- admissible coordinate for f −qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' CONSTRUCTION OF APPROXIMANTS This section is mostly occupied by the proof of the following theorem, from which the main result of this paper, Theorem 1, will then follow easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' THEOREM 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For each (r0, M, ǫ) ∈ Z≥2 × N × (0, 1], there is an increasing function P = Pr0,ǫ,M : N → R+ so that for any pseudo-rotation f ∈ Diff∞(A, ω) with ∥f∥Diffr0+2(A) < M, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1) whose rotation number ρ( f) = α ∈ (0, 1) \\ Q satisfies the property that there exists an odd integer n ≥ 3 for which qn > P(qn−1), qn+1 > P(qn) and qn+2 > P(qn+1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2) then there exists h0 ∈ Diff∞(A, ω) with dDiffr0−1(A)(h0Rαh−1 0 , f) < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Fix some (r0, M, ǫ) in Z≥2 × N × (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let f ∈ Diff∞(A, ω) denote a pseudo-rotation and set α = ρ( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' It will be convenient to use the following notation: for an increasing function S : N → N we define C(S) := � θ ∈ (0, 1)\\Q | ∃n ∈ N odd, so that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='4) qn(θ) > S(qn−1(θ)), qn+1(θ) > S(qn(θ)), qn+2(θ) > S(qn+1(θ)) � where {pn(θ)/qn(θ)}n≥0 is the sequence of continued fractions of θ intro- duced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Our successive restrictions on the rotation number α will take the form: α ∈ C(Si) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='5) for a finite collection of functions S1, S2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We will set P := max i Si and then when α ∈ C(P), all conditions in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='5) will be met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For our first condition on the rotation number, set S1 := Pr0(M, ·) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='6) where Pr0 : R2+ → R+ is defined in Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' From now on we assume α ∈ C(S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' 24 BARNEY BRAMHAM AND ZHIYUAN ZHANG Fix any odd n ≥ 3 for which qn > S1(qn−1), qn+1 > S1(qn) and qn+2 > S1(qn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='7) Then by Corollary B, f has a qn+1-good Brouwer curve γ ⊂ A for which the closed region D0 ⊂ A bounded by ( f qn(γ), γ) admits a (r0, ˆWr0)-admissible coordinate for f −qn, where ˆWr0 := Wr0(qn+1, Kr0+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Recall that γ is qn+1-good for f means that the curves f i(γ), 0 ≤ i ≤ qn+1 − 1 are mutually disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then it is clear that the curves f qn(γ), · · · , f qn+qn+1−1(γ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='8) are mutually disjoint as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' In particular f qn+1(γ) and f qn(γ) are disjoint and so we can consider the closed region D ⊂ A bounded by ( f qn(γ), f qn+1(γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' CLAIM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The annulus A is covered by the regions D, f(D), · · · , f qn−1(D), f qn(D0), · · · , f qn+1−1(D0) with mutually disjoint interiors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Moreover, the intersection of any two such neigh- boring regions equals one of the curves in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The restriction of f to the boundary circle B0 is an orientation pre- serving homeomorphism with rotation number α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' It is then a well-known fact that for any fixed x0 ∈ B0 the intervals [ f qn+i(x0), f qn+1+i(x0)], 0 ≤ i ≤ qn − 1, and [ f qn+j(x0), f j(x0)], qn ≤ j ≤ qn+1 − 1 together form a covering for B0 with mutually disjoint interiors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' In partic- ular we may take x0 := γ ∩ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' An analogous statement holds for the other boundary component B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The claim then follows from the disjointness of the curves in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='8) and our definition of simple regular curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' □ Fix a lift ˜γ of γ to ˜A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let F : ˜A → ˜A be the unique lift of f satisfying ρ(F) = ρ( f) ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Recall the notation from section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1 that Fa,b = TbFa for all a, b ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We define ˜D0 ⊂ ˜A to be the region bounded by � Fqn,−pn( ˜γ), ˜γ � and ˜D ⊂ ˜A to be the region bounded by � Fqn,−pn( ˜γ), Fqn+1,−pn+1( ˜γ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We claim that T−pnFqn( ˜γ) is contained in the fundamental domain of π bounded by T−1 ˜γ and ˜γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Indeed, this follows from comparing the order of ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER 25 their endpoints, and by the disjointness of their projections f qn(γ) and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Thus π restricts to a diffeomorphism from ˜D0 to D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Similarly, π restricts to a diffeomorphism from ˜D to D as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The following qn+1 regions have disjoint interiors: ˜Bi := � Fi( ˜D) 0 ≤ i ≤ qn − 1, Fi( ˜D0) qn ≤ i ≤ qn+1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='9) By Claim 1, ˜A is covered by the union of the following regions with mutu- ally disjoint interiors: T−k ˜Bj, k ∈ Z, 0 ≤ j ≤ qn+1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='10) The intersection of any two neighboring regions in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='10) therefore equals to one of the following curves T−kFj( ˜γ), k ∈ Z, qn ≤ j ≤ qn + qn+1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='11) We denote by Γ the union of the curves in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The region U0 := qn+1−1 � i=0 int( ˜Bi) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='12) satisfies that: (1) Tj(U0) ∩ Tk(U0) = ∅ for any j ̸= k ∈ Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (2) ∪k∈ZTk(U0) = ˜A \\ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By Corollary B, there is an admissible coordinate φ (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='7) from a neighborhood of [0, 1]2 in ˜A to a neighborhood of D0, satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='5) for ( f −qn, f qn(γ)) in place of ( f, γ), and ∥φ∥Diffr0 < ˆWr0, φ ◦ T(x) = f −qn ◦ φ(x) ∀x ∈ V where V is some neighborhood of {0} × [0, 1] in ˜A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' As we have seen in Remark 2, there is no loss of information by regarding φ as a mapping from [0, 1]2 to D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Since the restriction π : ˜D0 → D0 is a diffeomorphism we may set ψ := π−1 ◦ φ which therefore gives us a C∞-smooth diffeomorphism that extends to a C∞-smooth diffeomorphism with constant Jacobian from a neighborhood of [0, 1]2 in ˜A to a neighborhood of ˜D0 satisfying ∥ψ∥Diffr0 < ˆWr0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='13) ψ ◦ T(x) = F−qn,pn ◦ ψ(x) ∀x ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='14) There is a unique extension to a C∞-diffeomorphism with constant Jacobian to the whole strip ˜A, which we still denote by ψ : ˜A → ˜A 26 BARNEY BRAMHAM AND ZHIYUAN ZHANG and which satisfies ψ ◦ T = F−qn,pn ◦ ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='15) To summarise thus far, we have proven the following: LEMMA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For each (r0, M) ∈ Z≥2 × N there is an increasing function S1 = S1(r0, M) : N → R+ so that for any pseudo-rotation f ∈ Diff∞(A, ω) satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1) whose rotation number ρ( f) = α satisfies the property that there exists an odd integer n ≥ 3 for which qn > S1(qn−1), qn+1 > S1(qn) and qn+2 > S1(qn+1), there exists a qn+1-good Brouwer curve γ ⊂ A, so that for any lift ˜γ ∈ ˜A, there exists a C∞-diffeomorphism, ψ : ˜A → ˜A with constant Jacobian which satisfies ∥ψ∥Diffr0([0,1]2) < ˆWr0 := Wr0(qn+1, M) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='16) where Wr0 is the function produced by Corollary B, and ψ ◦ T = F−qn,pn ◦ ψ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='17) on the whole of ˜A, where F is the unique lift of f with rotation number in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Moreover ψ maps [0, 1]2 onto the region ˜D0 bounded by � ˜Fqn,−pn( ˜γ), ˜γ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' To continue our proof of Theorem 4 we require the following: LEMMA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For each (r0, M) ∈ Z≥2 × N, there is an increasing function Qr0,M : R2 + → R+ with lim t→0 Qr0,M(t, ·) ≡ 0, and there is an increasing function S2 = S2(r0, M) : N → R+ with S2 ≥ S1, so that for any pseudo-rotation f ∈ Diff∞(A, ω) satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1) whose rotation number ρ( f) = α satisfies the property that there exists an odd integer n ≥ 3 for which qn > S1(qn−1), qn+1 > S1(qn) and qn+2 > S1(qn+1), then the following holds for the unique lift F : ˜A → ˜A of f with ρ(F) ∈ (0, 1): let γ be a qn+1-good Brouwer curve for f given by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='13, and let ˜γ be any lift of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let ˜D and ˜D0 be the regions in ˜A bounded by � Fqn,−pn( ˜γ), Fqn+1,−pn+1( ˜γ) � , � Fqn,−pn( ˜γ), ˜γ � 3Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1 is applicable since by hypothesis S2 ≥ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER 27 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then there are open neighborhoods ˆD of ˜D and ˆD0 of ˜D0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' a C∞- diffeomorphism h : ˆD → ˆD0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' and neighborhoods UL of Fqn,−pn( ˜γ) and UR of Fqn+1,−pn+1( ˜γ), so that h|UL = Id, h|UR = F−qn+1,pn+1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='18) ∥h − Id∥Diffr0( ˜D, ˜D0) < Qr0,M � βn+1, qn+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='19) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1 there is a diffeomorphism ψ : ˜A → ˜A satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='16) and which maps [0, 1]2 onto ˜D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Set γ′ := Fqn+1,−pn+1( ˜γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By letting qn+2 be sufficiently large depending on qn+1 and M, we have A0(βn+1, Mqn+1) < 1/2 where A0 is produced by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then ∥f qn+1 − Id∥ ≤ A0(∥ρ( f qn+1)∥R/Z, Mqn+1) = A0(βn+1, Mqn+1) < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Using Theorem 3, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='11), we obtain dH( ˜γ, γ′) = dH( f qn+1(γ), γ) < A0(βn+1, Mqn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='20) Notice that by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='16) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='17) we also have that ∥ψ∥Diffr0([0,2]×[0,1]) is bounded in terms of qn+1, M, r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='20) and by letting βn+1 be sufficiently large depending on qn+1, M, r0, we have γ′ ⊂ ψ([1/2, 3/2] × [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We define γ′′ = ψ−1(γ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then {1/2} × [0, 1] < γ′′ < {3/2} × [0, 1] if βn+1 is sufficiently large depending on qn+1, M, r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Denote by U ′ the re- gion in ˜A bounded by {0} × [0, 1] and γ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' It remains to construct a C∞- diffeomorphism ϕ : U ′ → [0, 1]2 such that ϕ equals Id near {0} × [0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' equals ψ−1F−qn+1,pn+1ψ near γ′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' and tends to Id in the Cr0-topology as βn+1 tends to 0 for each fixed qn+1, M, r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Indeed, after the above ϕ is constructed, we can define h = ψϕψ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By Theorem 3, for given qn+1, M, r0, we see that F−qn+1,pn+1 tends to Id in the Cr0-topology as qn+2 tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Thus it suffices to construct ϕ by smooth interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' □ 28 BARNEY BRAMHAM AND ZHIYUAN ZHANG Continuing the proof of Theorem 4, we first construct a periodic approx- imation of f as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We define G on U0 (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='12)) by G|U = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 F U = ∪qn−2 i=0 int( ˜Bi), FqnhF1−qn U = int( ˜Bqn−1), F U = ∪qn+1−2 i=qn int( ˜Bi), Tpn+1h−1F1−qn+1 U = int( ˜Bqn+1−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='21) By property (1),(2) below (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='12), we may extend G to a self-map of ˜A \\ Γ, still denoted by G, satisfying GT = TG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='22) Moreover, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='21) the following identity holds on ˜A \\ Γ: Gqn+1 = Tpn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='23) By construction, for any two curves γa, γb in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='11), there are integers i, j such that γb = TiGj(γa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='24) By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='21) in the definition of G, we also have that ˜Bi = Gi( ˜B0), 0 ≤ i ≤ qn+1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='25) We denote ˜Bqn+1 := Gqn+1( ˜B0) = Tpn+1( ˜B0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='26) Now we set φ := ψ−1h where ψ is in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1 and h is in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' LEMMA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' There is a neighborhood V′ ⊂ ˜A of Fqn+1,−pn+1( ˜γ) so that Gqn ex- tends to a C∞ map on V′, and Tφ(x) = φT−pnGqn(x), x ∈ V′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='27) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' To see this, we first notice that Fqn+1,−pn+1( ˜γ) = ˜B0 ∩ Tpn−pn+1( ˜Bqn+1−qn), and ˜B0 ∪ Tpn−pn+1( ˜Bqn+1−qn) is a neighborhood of Fqn+1,−pn+1( ˜γ) in ˜A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Let UL, UR be given by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The set V′ defined by V′ = Fqn+1−qn,−pn+1+pn(UL) ∩ UR ∩ ( ˜B0 ∪ Tpn−pn+1( ˜Bqn+1−qn)) is a neighborhood of Fqn+1,−pn+1( ˜γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='21), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='25) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='22), we have Gqn|Int( ˜B0) = Fqnh, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='28) Gqn|Tpn−pn+1(Int( ˜Bqn+1−qn)) = Tpnh−1Fqn−qn+1Tpn+1−pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='29) ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER 29 By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2, we see that RHS of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='28)|V′ = RHS of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='29)|V′ = Tpn+1Fqn−qn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then it is clear that Gqn extends to a C∞ map on V′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Again by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2, we have for any x ∈ V′ φT−pnGqn(x) = ψ−1hTpn+1−pnFqn−qn+1(x) = ψ−1Tpn+1−pnFqn−qn+1(x), Tφ(x) = Tψ−1h(x) = Tψ−1T−pn+1Fqn+1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Thus (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='27) follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' □ We have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' COROLLARY C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' The map G on ˜A \\ Γ extends to an element in Diff∞( ˜A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='23), clearly Gqn+1 extends to a smooth map Tpn+1 on ˜A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='3, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='22) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='24), we see that Gqn also extends to a map in Diff∞( ˜A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' To conclude the proof it suffices to notice that G = (Gqn)−pn+1(Gqn+1)pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' □ By a slight abuse of notation, we again write G ∈ Diff∞( ˜A) for the ex- tension provided in Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Clearly (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='22) continues to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Conse- quently G descends to a periodic diffeomorphism g ∈ Diff∞(A) satisfying gqn+1 = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We define map ˜H : ˜A → ˜A by ˜H(x) := T−kqn+1+jpn+1φG−jTk(x), x ∈ T−k ˜Bj (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='30) for all k ∈ Z and all 0 ≤ j ≤ qn+1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='23), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='25), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='26) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='30) we have ˜HT = Tqn+1 ˜H, ˜HG = Tpn+1 ˜H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='31) We set H1 := Dqn+1 ˜H, where Dλ(x, y) = (λ−1x, y), λ ∈ R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Notice that for every λ ∈ R \\ {0} and every p ∈ R we have DλTpD−1 λ = T p λ where for each c ∈ R, Tc : ˜A → ˜A denotes the map Tc(x, y) = (x + c, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='31), we see that H1 : ˜A → ˜A satisfies TH1 = H1T, H1G = Tpn+1/qn+1H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='32) We now show that ˜H ∈ Diff∞( ˜A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By construction, ˜H is C∞-smooth in the interior of each T−k ˜Bj, k ∈ Z, 0 ≤ j ≤ qn+1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Thus it remains to show that ˜H is C∞ in a neighborhood of each of the curves in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' 30 BARNEY BRAMHAM AND ZHIYUAN ZHANG By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='9), we can see that the set ˜B∗ := ˜B0 ∪ T−pn ˜Bqn (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='33) is a neighborhood of T−pnFqn( ˜γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='24), we deduce that every curve in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='11) has a neighborhood of the form TiGj( ˜B∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='31), it remains to verify that ˜H is C∞ over ˜B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' This follows from a similar argument as in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='3 using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='27) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Consequently ˜H and H1 are C∞-smooth diffeomorphisms of ˜A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='32) H1 descends to a C∞-diffeomorphism h1 : A → A satisfying g = h−1 1 Rpn+1/qn+1h1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='34) Moreover, we have the claim that: ∥H1∥Diffr0( ˜A), and hence ∥h1∥Diffr0(A) as well, are bounded in terms of qn+1, M, r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Indeed, by definition it is clear that ∥φ∥Diffr0 is bounded in terms of qn+1, M, r0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' and for each integer 0 ≤ j ≤ qn+1 − 1, ∥G∥Diffr0( ˜Bj, ˜Bj+1) is bounded in terms of qn+1, M, r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Our claim follows immediately from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='30) and the fact that H1 commutes with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Recall that r0, M and ǫ are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' In the following we will show that: LEMMA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For sufficiently fast growing P depending on r0, M and ǫ the follow- ing holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Assume n satisfies the conditions in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='1 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' and moreover (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2) holds for n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then we have: (1) dDiffr0(A)( f, g) < ǫ/4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (2) dDiffr0(A)(h−1 1 Rαh1, g) < ǫ/4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' (3) there is h2 ∈ Diff∞(A) such that (h2h1)∗ω = ω and dDiffr0−1(A)(h−1 1 h−1 2 Rαh2h1, h−1 1 Rαh1) < ǫ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Proof of (1): By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='21), there is D > 0 depending only on r0 such that dDiffr0(A)( f, g) ≤ D max(dDiffr0(A)( f qnhf −qn, Id), dDiffr0(A)(h−1 f −qn+1, Id)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By Theorem 3, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2, and by letting qn+2 be sufficiently large depend- ing on ǫ, r0, M, qn+1, we have dDiffr0(A)( f qnhf −qn, Id), dDiffr0(A)(h−1 f −qn+1, Id) < (4D)−1ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Thus we have dDiffr0(A)( f, g) < ǫ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Proof of (2): Since ∥h1∥Diffr0(A) admits an upper bound depending only on qn+1, M and r0, by letting qn+2 be sufficiently large depending on ǫ, r0, M, qn+1, we have dDiffr0(A)(h−1 1 Rαh1, g) < ǫ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' ON PSEUDO-ROTATIONS OF THE ANNULUS WITH GENERIC ROTATION NUMBER 31 Proof of (3): Set λ0 = det(H1) = 1 qn+1 det( ˜H) ∈ C∞( ˜A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Recall that both ψ and F have constant Jacobians, hence by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='30), on each ˜Bj we have log λ0 = log det h ◦ (G−j) + log det(G−jFj) ◦ Fj + c1 for some constant c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' On T−k ˜Bj we have a similar formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Thus sup 1≤r≤r0 ∥Dr log λ0∥ ≤ (1 + dDiffr0(A)( f, Id))Cqn+1r0(dDiffr0(A)(h, Id) + dDiffr0(A)(g, f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='2 and the argument above for bounding dDiffr0(A)(g, f), we see that sup1≤r≤r0 ∥Dr log λ0∥ can be made arbitrarily small by making qn+2 sufficiently large while keeping qn+1, M and r0 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='32), we have � [0,1]2 λ0dω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then it is direct to see, for some absolute constant C > 0, that ∥λ0 − 1∥ < C∥D log λ0∥ given that ∥D log λ0∥ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Consequently, for any δ > 0, we have ∥λ0 − 1∥Cr0 < δ if qn+2 is sufficiently large depending on δ, qn+1, M and r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By Dacorogna-Moser’s theorem ([DM90, Theorem 1]), there exists h2 ∈ Diff∞(A) such that (h2)∗(λ0ω) = ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Moreover, by [DM90, Theorem 2 and Lemma 3], we can choose h2 with dDiffr0−1(A)(h2, Id) arbitrarily small provided ∥λ0 − 1∥Cr0 is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' In summary, if qn+2 is sufficiently large depending on ǫ, qn+1, M and r0, we can choose h2 ∈ Diff∞(A) sufficiently close to Id in Diffr0(A) so that dDiffr0−1(A)(h−1 1 h−1 2 Rαh2h1, h−1 1 Rαh1) < ǫ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' □ We set h0 = h2h1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='4(3) we have h0 ∈ Diff∞(A, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='4, we have dDiffr0(A)(h−1 0 Rαh0, f) < ǫ This completes the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' □ We can now prove the main result with the aid of the following lemma whose straightforward proof is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' 32 BARNEY BRAMHAM AND ZHIYUAN ZHANG LEMMA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For any function P : N → N, the set C := {α ∈ (0, 1) \\ Q | ∃n ≥ 3 odd such that qn(α) > P(qn−1(α)), qn+1(α) > P(qn(α)), qn+2(α) > P(qn+1(α))} is open and dense in (0, 1) \\ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' We have seen in the introduction that O∞ A(α) ⊂ F∞ A(α) for any α ∈ (0, 1)/Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' It remains to show that for a Baire generic α ∈ (0, 1)/Q F∞ A(α) ⊂ O∞ A(α) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='35) with the closure taken in the C∞-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' For each tuple (r0, M, k) in Z3 ≥1, with r0 ≥ 2, let Pr0,M,k−1 be the function produced by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Therefore by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='5 the set A(r0, M, k) := � α ∈ (0, 1) \\ Q | there is an odd n ≥ 3 such that qn(α) > Pr0,M,k−1(qn−1(α)), qn+1(α) > Pr0,M,k−1(qn(α)) and qn+2(α) > Pr0,M,k−1(qn+1(α)) � is open and dense in (0, 1) \\ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Thus the countable intersection A := ∩r≥2 ∩k≥1 ∩M≥2A(r, M, k) is a residual subset of (0, 1) \\Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Fix α ∈ A and suppose f ∈ F∞ A(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Then for any ǫ > 0 and r ∈ N with r ≥ 2 choose M ∈ N so that ∥f∥Diffr+2(A) < M, then by α ∈ A(r, M, ⌈ǫ⌉−1) and by Theorem 4 there exists h ∈ Diff∞(A, ω) so that dCr−1( f, hRαh−1) < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' This gives F∞ A(α) ⊂ O∞ A(α) with the closure in the Cr−1-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' Since r is arbitrary we easily conclude (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='35) holds with closure in the C∞-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' □ REFERENCES [AK70] D.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content=' RUHR UNIVERSITY BOCHUM Email address: barney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='bramham@rub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='edu CNRS, INSTITUT GALIL´EE, UNIVERSIT´E PARIS 13 Email address: zhiyuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='zhang@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='univ-paris13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} +page_content='fr' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE3T4oBgHgl3EQfYQps/content/2301.04486v1.pdf'} diff --git a/LdFRT4oBgHgl3EQfEzcl/content/tmp_files/2301.13477v1.pdf.txt b/LdFRT4oBgHgl3EQfEzcl/content/tmp_files/2301.13477v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6aafa1c8bad27b7c716f1d97e9227143ef7edd3e --- /dev/null +++ b/LdFRT4oBgHgl3EQfEzcl/content/tmp_files/2301.13477v1.pdf.txt @@ -0,0 +1,2382 @@ +arXiv:2301.13477v1 [quant-ph] 31 Jan 2023 +Pre-Born–Oppenheimer Dirac–Coulomb–Breit computations for +two-body systems +D´avid Ferenc1 and Edit M´atyus1, ∗ +1ELTE, E¨otv¨os Lor´and University, Institute of Chemistry, +P´azm´any P´eter s´et´any 1/A, Budapest, H-1117, Hungary +(Dated: February 1, 2023) +The sixteen-component, no-pair Dirac–Coulomb–Breit equation, derived from the Bethe–Salpeter +equation, is solved in a variational procedure using Gaussian-type basis functions for the example +of positronium, muonium, hydrogen atom, and muonic hydrogen. The α fine-structure-constant +dependence of the variational energies, through fitting a function of αn and αnlnα terms, shows +excellent agreement with the relevant energy expressions of the (perturbative) non-relativistic QED +framework, and thereby, establishes a solid reference for the development of a computational rela- +tivistic QED approach. +The positronium, Ps = {e−, e+}, muonium, Mu = {e−, µ+}, hydrogen atom, H = {e−, p+}, +and muonic-hydrogen, µH = {µ−, p+}, are the simplest, yet some of the most extensively studied +bound-state systems. Their simplicity allows for the high-precision evaluation of energy corrections +arising from special relativity and interactions from the matter and photon fields [1, 2]. The high- +precision spectroscopy experiments [3–10] together with the theoretical results (see Refs. [2, 11] +and references therein) provide stringent test for validity of quantum electrodynamics (QED) in +the low-energy range and probe physics beyond the Standard Model [12–16]. Ps is a candidate for +precision free-fall experiments to test QED and gravity [17], H and µH are the stars of the famous +proton-size puzzle [18–20], while Mu has attracted interest in relation with the muon’s anomalous +magnetic moment [10, 21]. +For bound-state systems, it is relevant to have a wave equation that can be solved to obtain a +good zeroth-order description. So far, the non-relativistic Schr¨odinger equation has been used as +reference, which has analytic solution for two-body systems. Then, relativistic and QED corrections +have been derived corresponding to increasing orders of the α fine-structure constant. We call these +corrections, for short, non-relativistic QED (nrQED) corrections. A recent review [11] provides +an excellent overview of the extensive literature of higher-order nrQED corrections to positronium +energies. Corrections up to α6 order (in natural units, α4Eh in hartree atomic units) are considered +complete, and ongoing work is about α7 order corrections. Some of the calculations have been +carried out not only for equal but arbitrary spin-1/2 fermion masses. +In the present work, we do not aim to reproduce the formally derived nrQED expressions, but +initiate an alternative approach to the two-particle relativistic QED problem based on a zeroth- +order wave equation in which special relativity is already accounted for. The theoretical framework +for this (computational) relativistic QED program is provided by the Bethe–Salpeter equation [22], +derived from field theory [23], and its Salpeter–Sucher exact equal-time form [24, 25], which provides +us a no-pair, two-particle relativistic wave equation, +(H + H∆)Ψ = EΨ , +(1) +which has the form of a Schr¨odinger-like wave equation, for which high-precision numerical solution +techniques can be adapted. The Ψ wave function in Eq. (1) depends only on the (spatial) Cartesian +coordinates of the particles, H is the positive-energy projected two-electron Hamiltonian with +instantaneous (Coulomb or Coulomb–Breit) interaction (I), +H = h1 + h2 + Λ++IΛ++ , +(2) +hi = cαipi + βimic2 + U1[4] (i = 1, 2) is the one-particle Dirac Hamiltonian in which U can +account for an external static Coulomb field (if there is any), and Λ++ projects to the positive- +energy (electronic) subspace of the h1 + h2 non-interacting two-fermion problem. For short, we +call H the no-pair Dirac–Coulomb (DC) or Dirac–Coulomb–Breit (DCB) Hamiltonian. +Pair corrections, retardation, and radiative corrections are included in the H∆ term, Eq. (1) +[25–27]. Contribution of H∆ to atomic and molecular energies (QED) can be expected to be small, + +2 +and hence, it can be treated as perturbation to the no-pair Hamiltonian. +This framework offers a perturbative approach based on a relativistic reference, alternative to +earlier work using a non-relativistic reference state. Evaluation of the already formulated per- +turbative correction with H∆ is left for future developments, which appears to be possible along +the lines reviewed in Ref. [28]. Although analytic evaluation of the energy and its corrections is +not possible in this framework, the numerical results can be converged to high precision, which is +demonstrated in the present work. +To compute no-pair, two-particle bound states, let us start with defining overall, center-of-mass, +Rµ = (T, R), and relative, rµ = (t, r), covariant space-time coordinates as +Rµ = +m1 +m1 + m2 +rµ +1 + +m2 +m1 + m2 +rµ +2 +(3) +and +rµ = rµ +1 − rµ +2 . +(4) +Then, following Salpeter and Bethe [22], the wave function of an isolated system can be factorized +as +φ(r1, r2) = e−iPνRνΦ(rµ) +(5) +with the total four-momentum, P ν = (E, P ). By choosing the zero-total-momentum frame, P = 0, +we obtain +φ(r1, r2) = e−iET Φ(rµ) , +(6) +where E is the total energy of the system. It is important to note that Φ(rµ), which describes the +internal motion, depends on rµ = (t, r), i.e., not only on the r relative coordinates, but also on the +t relative time of the particles. Fourier transformation with respect to this relative time variable +yields the relative-energy dependent wave function +˜Φ(ε, r) = +� ∞ +−∞ +dt +(2π)1/2 e−iεtΦ(t, r) . +(7) +In the exact equal-time formalism of Salpeter [24] and Sucher [25], the equal-time (t = 0) wave +function appears, which depends only on the spatial coordinates, +Ψ(r) = +� ∞ +−∞ +dε ˜Φ(ε, r) , +(8) +and the relative-energy dependence of the problem is accounted for in H∆ in Eq. (1) [25]. +To obtain the Hamiltonian for the relative motion, the chain rule for the coordinate transforma- +tion, Eqs. (3) and (4), is used, and it is also considered that contribution from terms containing +∇R vanishes due to the Eq. (5) choice of the ansatz for an isolated system and our choice of a +P = 0 zero-momentum-frame description, Eq. (6). Hence, the spatial momentum operators in this +framework can be replaced according to +p1 = −i∇1 → p = −i∇ +and +p2 = −i∇2 → −p = i∇ , +(9) +where ∇(= ∇r) collects the partial derivatives with respect to the r relative displacement vector +components. This simple replacement ‘rule’ can be used to construct expressions for the relative +motion from the two-particle expressions [29–32]. As a result, the no-pair Dirac–Coulomb–Breit +Hamiltonian for the relative motion is obtained as +H(1, 2) = Λ++ + + + +V 1[4] +−cσ[4] +2 +· p +cσ[4] +1 +· p +B[4] +−cσ[4] +2 +· p V 1[4] − 2m2c21[4] +B[4] +cσ[4] +1 +· p +cσ[4] +1 +· p +B[4] +V 1[4] − 2m1c21[4] +−cσ[4] +2 +· p +B[4] +cσ[4] +1 +· p +−cσ[4] +2 +· p +V 1[4] − 2m12c21[4] + + + Λ++ +(10) + +3 +with m12 = m1 + m2, p = −i( ∂ +∂rx , +∂ +∂ry , +∂ +∂rz ), σ[4] +1 += (σx ⊗ 1[2], σy ⊗ 1[2], σz ⊗ 1[2]) and σ[4] +2 += +(1[2] ⊗ σx, 1[2] ⊗ σy, 1[2] ⊗ σz), where σx, σy, and σz are the 2 × 2 Pauli matrices. We note that the +operator in Eq. (10) contains a −2mic2 shift (i = 1, 2) to match the non-relativistic energy scale. +Furthermore, the Coulomb interaction, +V = q1q2 +r +(11) +is along the diagonal, whereas the Breit interaction, +B[4] = −q1q2 +�1 +r σ[4] +1 · σ[4] +2 − 1 +2 +�� +σ[4] +1 · ∇ +� � +σ[4] +2 · ∇ +� +r +�� +(12) +can be found on the anti diagonal of the Hamiltonian. +The Λ++ positive-energy projector in Eq. (10) corresponds to the positive-energy (‘electronic’) +states of the ‘bare’, non-interacting Hamiltonian, i.e., Eq. (10) without Λ++ and without the V 1[4] +and B[4] interaction blocks. Although the Λ++ free-particle projector in momentum space has an +analytic form [33], we constructed it numerically in coordinate space by computing the eigenstates +of the bare, non-interacting Hamiltonian over the space spanned by the basis functions used for the +interacting computation. The positive-energy states were identified with the simple energy cutting +approach (which can be checked by the complex scaling procedure) [30]. +The no-pair Dirac–Coulomb and Dirac–Coulomb–Breit Hamiltonians are bounded from below +(the positive-energy block, which is considered in this work, is decoupled from the rest), hence the +HΨ = EΨ wave equation can be solved using the variational procedure. +For a single particle, the (four-component) wave function is conveniently partitioned to large (l, +first two) and small (s, last two) components. A good basis representation must fulfill a simple +symmetry relation, which is necessary to provide a correct matrix representation (Mx) for the +Mx(p)Mx(p) = Mx(p2) identity [34]. The simplest implementation of this relation is provided by +the (restricted) kinetic balance (KB) condition [35, 36], +ϕs = σ[2] · p +2mc ϕl +(13) +for the basis function of the ϕs small and ϕl large components. Two(many)-particle relativistic +quantities can be constructed with the block-wise (also called Tracy–Singh) direct product [29– +32, 37–40], which allows us to retain the large-small block structure, used already to write Eq. (10). +The corresponding two-particle function, with highlighting the large (l) and small (s) component +blocks, is +ϕ = + + + + +ϕll +ϕls +ϕsl +ϕss + + + + . +(14) +For a variational procedure, we used the simplest two-particle generalization of the one-particle +kinetic balance, Eq. (13), and implemented it in the sense of a transformation or metric [29–32, 35]: +HKB = X†HX , +X = diag + +1[4], − +� +σ[4] +2 · p +� +2m2c +, +� +σ[4] +1 · p +� +2m1c +, − +� +σ[4] +1 · p +� � +σ[4] +2 · p +� +4m1m2c2 + + . +(15) +We also note that the X balance matrix used in this work can be ‘obtained’ from the balance +used for the Born–Oppenheimer systems [29–32] through the p1 → p and p2 → −p replacement, +Eq. (9). The fundamental ‘guiding principle’ for our construction of the two-particle balance has +been solely to have a correct matrix representation of the Mx(p)Mx(p) = Mx(p2) identity, since the +positive-energy projected Hamiltonian is bounded from below. The transformed DCB Hamiltonian + +4 +is +HKB = X†H(1, 2)X = + + + + + + +D[4] +1 +p2 +2m2 1[4] +p2 +2m1 1[4] +B[4] +1 +p2 +2m2 1[4] +D[4] +2 +B[4] +2 +p4 +8c2m1m2 +2 1[4] +p2 +2m1 1[4] +B[4] +3 +D[4] +3 +p4 +8c2m2 +1m2 1[4] +B[4] +4 +p4 +8c2m1m2 +2 1[4] +p4 +8c2m2 +1m2 1[4] +D[4] +4 + + + + + + +(16) +with the diagonal blocks, +D[4] +1 += V 1[4] +(17) +D[4] +2 += (σ2 · p)V 1[4](σ2 · p) +4m2 +2c2 +− p2 +2m2 +1[4] +(18) +D[4] +3 += (σ1 · p)V 1[4](σ1 · p) +4m2 +1c2 +− p2 +2m1 +1[4] +(19) +D[4] +4 += (σ1 · p)(σ2 · p)V 1[4](σ1 · p)(σ2 · p) +16m2 +1m2 +2c4 +− +m12p4 +8m2 +1m2 +2c2 1[4] , +(20) +and the anti-diagonal blocks including the Breit interaction, Eq. (12), +B[4] +1 += −B[4](σ1 · p)(σ2 · p) +4c2m1m2 +(21) +B[4] +2 += −(σ2 · p2)B[4](σ1 · p) +4c2m1m2 +(22) +B[4] +3 += −(σ1 · p)B[4](σ2 · p) +4c2m1m2 +(23) +B[4] +4 += −(σ2 · p)(σ1 · p)B[4] +4c2m1m2 +. +(24) +The identity in the X-KB metric is +IKB = X†X = diag +� +1[4], +p2 +4c2m2 +2 +1[4], +p2 +4c2m2 +1 +1[4], +p4 +16c4m2 +1m2 +2 +1[4] +� +. +(25) +Then, the sixteen-component wave function is written as a linear-combination of spinor functions, +Ψ(r) = +Nb +� +i=1 +16 +� +χ=1 +ciχfi(r)dχ , +(26) +where the dχ spinor basis vectors are sixteen-dimensional unit vectors, (dχ)ρ = δχρ (χ, ρ = +1, . . . , 16). +For the fi spatial functions, we use spherically symmetric Gaussian functions (Se, +L = 0 orbital angular momentum and p = +1 even (e) parity), +fi(r) = e−ζir2 +(27) +with ζi > 0 (to ensure square integrability). +We optimized the ζi Gaussian exponents (i = +1, . . . , Nb) by minimization of the non-relativistic 1Se ground-state energy to a pEh(= 10−12 Eh) +precision range using quadruple precision arithmetic. Convergence of the non-relativistic and rela- +tivistic energies with respect to the basis size is shown in Table S6. For selected systems and basis +sizes, we continued the optimization of the ζi parameters by minimization of the no-pair DC(B) +energy, and the computation remained variationally stable, the energy ‘converged from above’. +(This variationally stable behaviour was absent during minimization of the relevant energy level of +the bare DC Hamiltonian.) We also note that there are no triplet contributions to the ground state +(1 1Se +0) (p. 419 of Ref. [41]), since even-parity 3P e states do not exist for a pseudo-one-particle + +5 +system (in contrast to helium-like systems [42]). +In addition to variational no-pair DC and DCB computations, we computed the first-order +perturbative Breit correction to the nth DC energy by [31, 32] +EDC⟨B⟩,n = EDC,n + ⟨ΨDC,n|X†BXΨDC,n⟩ +(28) +where B is a sixteen-dimensional matrix with the B[4] blocks on its anti diagonal. The second-order +perturbative Breit correction is computed as +EDCB2,n = EDC⟨B⟩,n + +� +i̸=n +��⟨ΨDC,i|X†BXΨDC,n⟩ +��2 +EDC,i − EDC,n +. +(29) +The outlined algorithm has been implemented in the QUANTEN computer program, which is +used as a molecular physics ‘platform’ for pre-Born–Oppenheimer, non-adiabatic, upper- and lower- +bound, perturbative- and variational relativistic developments [29–32, 42–52]. Hartree atomic units +are used, and the speed of light is c = α−1a0Eh/ℏ with α−1 = 137.035 999 084 [53]. +All computed no-pair energies are listed in Table S6, their change with the basis size can be used +to assess their convergence. Further minimization tests for the DC(B) energy did not reveal major +changes. +For direct comparison of the computed no-pair energies with the current state-of-the-art nrQED +values, we have (numerically) determined the α dependence of the no-pair energies. +For this +reason, we repeated the no-pair computations using the {α ∈ α0 ± n | n ∈ {−50, . . ., 51}} series +of the interaction constant, where α0 labels the value taken from Ref. [53]. Then, we fitted the +function +F(α) = ε0 + α2ε2 + α3ε3 + α4 ln(α)ε′ +4 + α4ε4 +(30) +to the series of the no-pair energies. Inclusion of higher-order, e.g., α5 and α5 ln α, terms in Eq. (30) +did not make any visible difference at the current numerical precision). A small fitting error was +obtained, which had orders of magnitude smaller root-mean-squared deviation than the estimated +energy convergence, Table S6), and smooth convergence of the fitted coefficients was observed +with respect to the basis set size (Tables S1–S4). To obtain consistent results, it was essential to +include also the α4 ln α term in Eq. (30), a simple α polynomial was insufficient to represent the +high-precision no-pair energies (Table S6). +Table II shows the comparison of the α-dependence of the no-pair energies (fitted coefficients) and +the nrQED corrections that were readily available to us or we could obtain with short calculation +(Supplementary Material). Excellent agreement is observed. The numerical deviation (uncertainty) +of the coefficients fitted to the variational results is on the order of the convergence error of the +no-pair energies that can be estimated from Table S6. +Regarding the large mass, m2 → ∞, limit and comparison with the one-electron Dirac energy, it +is necessary to consider that the (bare) one-electron Dirac equation is with-pair (and correct for one +electron). At α3Eh order, the one-electron Dirac limit is recovered from our no-pair computations, +by appending the no-pair energy with the (one) pair correction. +For m2 → ∞, the one-pair +Coulomb correction, Eq. (3.9) of Ref. 54, is +E(3) +C1 (m1, ∞) = +lim +m2→∞ E(3) +C1 (m1, m2) += +lim +m2→∞ +2µ3 +3π +� 2 +m2 +1 +− +1 +m1m2 ++ 2 +m2 +2 +� += 4m1 +3π . +(31) +In Table III, we can (numerically) observe that the large m2 limit of the ε3 coefficient, obtained +from fitting F(α) to the no-pair energies, converges to −E(3) +C1 (1, ∞), and hence cancel with the pair +corrections (the two-pair contribution, Eq. (S12), vanishes) for m2 → ∞. Thereby, the one-particle +Dirac limit is recovered at order α3Eh. Furthermore, we also note that the Breit contribution +vanishes as m2 → ∞ (Table S5). + +6 +In this work, a computational relativistic quantum electrodynamics approach was put forward +based on the exact equal-time Bethe–Salpeter equation. +It is demonstrated that a relativistic +reference state can be converged to a sub-parts-per-billion relative precision by variational solution +of the no-pair Dirac–Coulomb(–Breit) wave equation including the dominant, instantaneous part +of the electromagnetic interaction. The α fine-structure dependence of the computed energies are +in excellent agreement with the formal non-relativistic QED results corresponding to polynomial +and logarithmic corrections in α, up to α6 ln α order in natural units (α4 ln αEh). Perturbative +retardation, radiative, and pair corrections to the no-pair relativistic states had been formulated +long ago, and their evaluation with the high-precision relativistic reference states computed in this +work will be carried out in subsequent work. +Financial support of the European Research Council through a Starting Grant (No. 851421) is +gratefully acknowledged. DF thanks a doctoral scholarship from the ´UNKP-22-4 New National +Excellence Program of the Ministry for Innovation and Technology from the source of the National +Research, Development, and Innovation Fund (´UNKP-22-4-I-ELTE-51). + +7 +Table I. Convergence of the no-pair Dirac–Coulomb(–Breit) energies, in Eh, computed in this work. The +spatial basis, Eq. (27), used in the relativistic computation was parameterized by (numerical) minimization +of the non-relativistic energy, Enr. The numerical value for the analytic (∞) non-relativistic energy is shown +for reference. +Nb +Enr +EDC +EDC⟨B⟩ +EDCB2 +EDCB +Ps (m2/m1 = 1): +10 +−0.249 999 665 988 4 −0.249 997 227 989 +−0.250 016 969 603 +−0.250 016 992 755 +−0.250 016 992 809 +20 +−0.249 999 999 919 4 −0.249 997 552 650 +−0.250 017 362 124 +−0.250 017 403 806 +−0.250 017 404 023 +30 +−0.249 999 999 996 8 −0.249 997 552 766 +−0.250 017 362 426 +−0.250 017 404 153 +−0.250 017 404 371 +40 +−0.249 999 999 999 6 −0.249 997 552 778 +−0.250 017 362 470 +−0.250 017 404 205 +−0.250 017 404 425 +50 +−0.249 999 999 999 9 −0.249 997 552 780 +−0.250 017 362 477 +−0.250 017 404 214 +−0.250 017 404 433 +∞ +−0.250 000 000 000 0 +Mu (m2/m1 = 206.7682830): +10 +−0.497 592 269 419 4 −0.497 598 739 220 +−0.497 599 489 904 +−0.497 599 489 917 +−0.497 599 489 918 +20 +−0.497 593 472 285 4 −0.497 600 024 240 +−0.497 600 780 916 +−0.497 600 780 959 +−0.497 600 780 959 +30 +−0.497 593 472 874 8 −0.497 600 025 977 +−0.497 600 782 839 +−0.497 600 782 891 +−0.497 600 782 891 +40 +−0.497 593 472 910 8 −0.497 600 026 241 +−0.497 600 783 176 +−0.497 600 783 235 +−0.497 600 783 235 +50 +−0.497 593 472 915 7 −0.497 600 026 282 +−0.497 600 783 233 +−0.497 600 783 295 +−0.497 600 783 295 +∞ +−0.497 593 472 917 1 +H (m2/m1 = 1836.15267343): +10 +−0.499 727 019 644 9 −0.499 733 723 658 +−0.499 733 809 460 +−0.499 733 809 460 +−0.499 733 809 460 +20 +−0.499 727 839 067 5 −0.499 734 617 695 +−0.499 734 704 007 +−0.499 734 704 008 +−0.499 734 704 008 +30 +−0.499 727 839 669 3 −0.499 734 619 508 +−0.499 734 705 842 +−0.499 734 705 843 +−0.499 734 705 843 +40 +−0.499 727 839 706 0 −0.499 734 619 795 +−0.499 734 706 138 +−0.499 734 706 139 +−0.499 734 706 140 +50 +−0.499 727 839 710 9 −0.499 734 619 840 +−0.499 734 706 186 +−0.499 734 706 187 +−0.499 734 706 187 +∞ +−0.499 727 839 712 4 +µH (m2/m1 = 8.88024337) : +10 −92.920 263 579 73 +−92.920 730 693 26 +−92.923 396 814 39 +−92.923 397 816 36 +−92.923 397 817 07 +20 −92.920 416 825 53 +−92.920 890 799 40 +−92.923 572 907 75 +−92.923 575 558 50 +−92.923 575 566 96 +30 −92.920 417 297 88 +−92.920 891 278 83 +−92.923 573 403 13 +−92.923 576 058 44 +−92.923 576 066 97 +40 −92.920 417 310 07 +−92.920 891 312 69 +−92.923 573 493 64 +−92.923 576 164 19 +−92.923 576 173 06 +50 −92.920 417 311 03 +−92.920 891 313 65 +−92.923 573 494 58 +−92.923 576 165 15 +−92.923 576 174 01 +∞ −92.920 417 311 31 + +8 +Table II. Comparison of variational no-pair results and nrQED corrections. The F(α) = ε0 +α2ε2 +α3ε3 + +α4 ln(α)ε′ +4 + α4ε4 function was fitted to the no-pair energies to obtain the coefficients (var-fit). All values +correspond to hartree atomic units. (All fitted coefficients are listed in Table S5.) +DC +DC⟨B⟩ +DCB +ε2 +ε3 +ε′ +4 +ε2 +ε3 +ε2 +ε3 +Ps = {e−, e+}: +var-fit +0.046 875 −0.128 8 +−0.063 4 +−0.328 125 +0.280 2 +−0.328 125 +0.189 9 +nrQED a +0.046 875 −0.128 8 +−0.062 5 +−0.328 125 +0.280 3 +−0.328 125 +αn(δεn) b +−4.5·10−12 +7.2·10−12 +2.6·10−12 +−2.3·10−11 +5.5·10−11 +2.3·10−11 +Mu = {e−, µ+}: +var-fit +−0.120 227 −0.419 3 +−0.967 2 +−0.134 526 −0.407 1 +−0.134 526 −0.407 2 +nrQED a +−0.120 227 −0.419 3 +−0.134 528 +−0.134 528 +αn(δεn) b +−4.7·10−11 −1.2·10−11 +−1.0·10−10 +−1.1·10−10 +H = {e−, p+}: +var-fit +−0.124 455 −0.423 8 +−0.983 7 +−0.126 086 −0.422 4 +−0.126 086 −0.422 4 +nrQED a +−0.124 456 −0.423 8 +−0.126 087 +−0.126 087 +αn(δεn) b +−6.1·10−11 −1.0·10−11 +−6.8·10−11 +−6.8·10−11 +µH = {µ−, p+}: +var-fit +−8.437 67 −67.886 +−130.550 2 +−59.154 212 −18.860 6 +−59.154 120 −24.865 6 +nrQED a +−8.437 70 −67.899 +−59.154 516 +−59.154 516 +αn(δεn) b +−1.7 · 10−9 +−5.4 · 10−9 +−1.6 · 10−8 +−2.1 · 10−8 +a The nrQED corrections, with the corresponding references, are collected in the Supplementary Material. +b αn(δεn), in Eh, with the δεn = E(n) − εn difference of the nrQED value and the fitted coefficient. +Table III. Large mass, m2 → ∞, limit, of the α3Eh-order fitted coefficient of the no-pair DC energy, +Eq. (30). (m1 = 1 corresponds to the electron mass.) +m2 +ε3 +Ps += {e−, e+} +1 +−0.128 8 +Mu = {e−, µ+} +206.7682830 +−0.419 3 +H += {e−, p+} +1836.15267343 +−0.423 8 +10H = {e−, 10p+} +18361.5267343 +−0.424 3 +−E(3) +C1 (1, m2) Eq. (31) +∞ +−0.424 413... + +9 +Supplementary Material +Pre-Born–Oppenheimer Dirac–Coulomb–Breit computations for +two-body systems +D´avid Ferenc1 and Edit M´atyus1,∗ +1 ELTE, E¨otv¨os Lor´and University, Institute of Chemistry, +P´azm´any P´eter s´et´any 1/A, Budapest, H-1117, Hungary +∗ edit.matyus@ttk.elte.hu +(Dated: January 31, 2022) +Contents: +S1. Non-relativistic QED expressions compiled from the literature +S2. Matrix elements +S3. Expectation values and mass-dependent correction formulae +S4. Convergence tables +S5. Fitted coefficients +References + +10 +NON-RELATIVISTIC QED EXPRESSIONS COMPILED FROM THE LITERATURE +In the non-relativistic QED approach, the energy is obtained by evaluating E(n) terms for in- +creasing powers of α as corrections to the E(0) +nr +non-relativistic energy (which is of α0 order in +hartree atomic units), +E = E(0) +nr + α2E(2) + α3E(3) + . . . +(S1) +The Schr¨odinger equation of two-particle systems has a closed analytic solution, and the ground- +state energy reads as +E(0) +nr = −µ +2 +(S2) +with the reduced mass +µ = +m1m2 +m1 + m2 +. +(S3) +The next, (non-vanishing) α2Eh-order correction is the sum of two terms arising from the Coulomb +and the Breit (non-retarded part of transverse) interactions, +α2E(2) +DCB = α2E(2) +DC + α2E(2) +B +, +(S4) +which is obtained by calculating the expectation value of the following operators [41], +H(2) +DC = −1 +8 +� 1 +m3 +1 ++ 1 +m3 +2 +� +(p2)2 − π +2 +� 1 +m2 +1 ++ 1 +m2 +2 +� +δ(r) +(S5) +H(2) +B += − +1 +2m1m2r +� +p2 + r(rp)p +r2 +� +− +2π +m1m2 +δ(r) +(S6) +with the non-relativistic ground-state wave function. These expectation values can be written in +a closed, analytic form (Secs. and ). For the 1 1Se state, they are +E(2) +DC = EMV + ED +(S7) +with +EMV = −5 +8µ4 +� 1 +m3 +1 ++ 1 +m3 +2 +� +ED = µ3 +2 +� 1 +m2 +1 ++ 1 +m2 +2 +� +, +(S8) +from the first and second terms of Eq. (S5), respectively, and +E(2) +B += Eoo + Ess , +(S9) +where +Eoo = − +µ3 +m1m2 +and +Ess = − 2µ3 +m1m2 +(S10) +from the two terms of Eq. (S6). +Figure S1 shows the m2 dependence of the corrections for the m1 = 1 case. Interestingly, the +relativistic DC correction vanishes for the m2 = 0.209 and m2 = 4.791 values. In contrast to the +non-relativistic energy, the mass-dependence of the corrections is not only through the reduced +mass of the constituent particles. +The correction arising from non-crossed photons at α3Eh order has been reported by Fulton and + +11 +−20 +0 +20 +−20 +0 +20 +0 +5 +10 +E [µEh] +E(2) +DC +ED +EMV +E [µEh] +m2/me +E(2) +DCB +ED +EMV +Eoo +Ess +Figure S1. +Dependence of the α2Eh-order Dirac–Coulomb (top) and Dirac–Coulomb–Breit (bottom) +energy corrections of two-particle systems on the m2 particle mass with m1 = 1(me). For m1 = 1, the +E(2) +DC correction vanishes for the m2 = 0.208 71 and 4.791 29 values, i.e., up to α2Eh the DC relativistic +energy equals the non-relativistic energy. +Martin (Eq. (3.7) of Ref. [54]), +E(3) +C0,2(m1, m2) = −2µ3 +3π +� 2 +m2 +1 ++ +1 +m1m2 ++ 2 +m2 +2 +� +. +(S11) +We note that this expression contains the sum of the no-pair and the two-pair corrections (indicated +by the ‘0,2’ subscript). To the best of our knowledge, the single and two-pair Coulomb corrections +separately do not have any simple form for general m1, m2 masses, but can be calculated from the +integral (two-pair part of Eqs. (3.1a)–(3.6) of Ref. 54): +E(3) +C2 (m1, m2) = −2µ3 +π +� ∞ +0 +dk (E1(k) − m1)(E2(k) − m2) +E1(k) + E2(k) + m1 + m2 +with +Ei(k) = +� +m2 +i + k2 . +(S12) +This integral can be evaluated by using (for example) a symbolic algebra program, and the resulting +(lengthy) expression can be evaluated for the selected m1 and m2 masses. For the special case of +m1 = m2 = 1, the integral simplifies to, Eq. (4.26b) of Ref. [25], +E(3) +C2 (1, 1) = − 1 +8π +�5 +3 − π +2 +� +, +(S13) +which together with using the simple expression for the non-crossed photon correction of Fulton and +Martin, Eq. (S11), can be used to obtain the third-order perturbative no-pair Coulomb correction +for unit masses, m1 = m2 = 1: +E(3) +C0 (1, 1) = E(3) +C0,2(1, 1) − E(3) +C2 (1, 1) = − 1 +8π +�π +2 + 5 +3 +� +≈ −0.128 815 , +(S14) +which is the same as the third-order no-pair Coulomb correction reported by Sucher (for the two +electrons in helium, Eq. (3.99) of Ref. [25]). The zero- plus two-pair contribution, Eq. (S11), for +unit masses is +E(3) +C0,2(1, 1) = − 5 +12π ≈ −0.132 629 . +(S15) +The E(3) +C0 no-pair contribution was not separately reported in the literature for non-unit masses, + +12 +Table S1. Non-relativistic energy and perturbative correction values, in Hartree atomic units, calculated +using the analytic expressions, Eqs. (S1)–(S14), compiled from the literature [25, 41, 54]. The m2/m1 mass +ratios are taken from Ref. 53. +m2/m1 +E(0) +nr +E(2) +DC +E(2) +DCB +E(3) +C0,2 +E(3) +C2 +E(3) +C0 +Ps = {e−, e+} +1 +−0.250 000 000 000 +0.046 875 +−0.328 125 +−0.132 629 +−0.003 815 +−0.128 815a +Mu= {e−, µ+} +206.7682830 +−0.497 593 472 917 +−0.120 227 +−0.134 528 +−0.419 336 +−3.6 · 10−6 +−0.419 332 +H += {e−, p+} 1836.15267343 +−0.499 727 839 712 +−0.124 456 +−0.126 087 +−0.423 836 +−4.7 · 10−8 +−0.423 836 +µH= {µ−, p+} +8.88024337 −92.920 417 311 307 +−8.437 699 −59.154 516 −68.110 857 +−0.210 9268 −67.899 930 +a Eq. (S14) was used for Ps [25]. +and we calculated it for the relevant mass values using Eqs. (S11) and (S12) (Table S1). +The α3Eh-order contribution from a single instantaneous Breit photon exchange including also +the Coulomb ladder is known for unit masses (positronium) [25], +E(3) +B (1, 1) = 1 +2π +� +1 + π +2 +� +≈ 0.409 155 . +(S16) +The expansion coefficients of the bound-state energy also involve terms with logarithm of α. In +the no-pair DC energy the α3 ln α terms are absent, yet there are logarithmic terms in the fourth +order i.e. α4 ln α. For equal masses, a simple expression can be found for the DC correction in +Ref. [55] (and Ref. [27]). +MATRIX ELEMENTS +The spatial basis functions are +fµ(r) = +�2ζµ +π +�3/4 +e−ζµr2 = Nµe−ζµr2 . +(S17) +The following notation is used, i, j, k, l ∈ {x, y, z} denote Cartesian components +ζµν = ζµ + ζν +(S18) +� +d3r = +� ∞ +−∞ +dx dy dz +(S19) +� +d3r e−ζr2 = +�π +ζ +�3/2 +(S20) +r = +� +x2 + y2 + z2 +(S21) +Nµ = +�2ζµ +π +�3/4 +. +(S22) +In what follows, the Einstein summation convention is understood for the i, j, k, l Cartesian indices. +The derivatives of the basis functions are +∂ie−ζµr2 = −2ζµrie−ζµr2 +(S23) +∂j∂ie−ζµr2 = +� +−2ζµδij + 4ζ2 +µrirj +� +e−ζµr2 +(S24) +∂k∂j∂ie−ζµr2 = +� +4ζ2 +µ(δijrk + δikrj + δkjri) − 8ζ3 +µrirjrk +� +e−ζµr2 . +(S25) + +13 +A useful integrals for the calculation of the Coulomb matrix elements include +� ∞ +0 +dt (a + t2)−3/2 = 1 +a +(S26) +� ∞ +0 +dt (a + t2)−5/2 = +1 +3a2 +(S27) +� ∞ +0 +dt (a + t2)−7/2 = +1 +15a3 . +(S28) +⟨fµ|fν⟩ = NµNν +� +d3r e−(ζµ+ζν)r2 = (4ζµζν)3/4 +ζ3/2 +µν +(S29) +⟨fµ|V |fν⟩ =NµNν +� +d3r 1 +r e−ζµνr2 = NµNν +2 +√π +� ∞ +0 +dt +� +d3r 1 +r e−(ζµν+t2)r2 += NµNν +2 +√π +� ∞ +0 +dt +π3/2 +(ζµν + t2)3/2 = NµNν +2π +ζµν += +� +32 +π +(ζµζν)3/4 +ζµν +(S30) +⟨fµ|∇2|fν⟩ =NµNν +� +d3r +� +−2δiiζµ + 4ζ2 +µriri +� +e−ζµνr2 += −6ζµ(4ζµζν)3/4 +ζ3/2 +µν ++ 12 +√ +2ζ11/4 +µ +ζ3/4 +ν +ζ5/2 +µν +(S31) +⟨fµ|∇2∇2|fν⟩ =NµNν +� +d3r +� +−2ζµδii + 4ζ2 +µriri +� � +−2ζνδjj + 4ζ2 +νrjrj +� +e−ζµνr2 += NµNν +� +4ζµζνδiiδjj +� +d3r e−ζµνr2 +− 8ζµζ2 +νδii +� +d3r rjrje−ζµνr2 +− 8ζνζ2 +µδjj +� +d3r ririe−ζµνr2 ++ 16ζ2 +νζ2 +µ +� +d3r rjrjririe−ζµνr2� +(S32) +36NµNνζµζν +� +d3r e−ζµνr2 = 36NµNνζµζνπ3/2ζ−3/2 +µν +(S33) +−24NµNνζµζ2 +ν +� +d3r rjrje−ζµνr2 = −36NµNνζµζ2 +νπ3/2ζ−5/2 +µν +(S34) + +14 +The sum of the first three terms is zero: +36NµNνζµζνπ3/2 +ζ3/2 +µν +− 36NµNνζµζ2 +νπ3/2 +ζ5/2 +µν +− 36NµNνζνζ2 +µπ3/2 +ζ5/2 +µν += 36NµNνζµζνπ3/2ζµν +ζ3/2 +µν ζµν +− 36NµNνζµζ2 +νπ3/2 +ζ5/2 +µν +− 36NµNνζνζ2 +µπ3/2 +ζ5/2 +µν += 36NµNνζµζνπ3/2(ζµ + ζν) +ζ5/2 +µν +− 36NµNνζµζ2 +νπ3/2 +ζ5/2 +µν +− 36NµNνζνζ2 +µπ3/2 +ζ5/2 +µν += 36NµNνζ2 +µζνπ3/2 +ζ5/2 +µν ++ 36NµNνζµζ2 +νπ3/2 +ζ5/2 +µν +− 36NµNνζµζ2 +νπ3/2 +ζ5/2 +µν +− 36NµNνζνζ2 +µπ3/2 +ζ5/2 +µν += 0 +(S35) +16NµNνζ2 +νζ2 +µ +� +d3r r2r2e−ζµνr2 = 60NµNνζ2 +νζ2 +µπ3/2ζ−7/2 +µν += 120 +√ +2ζ11/4 +µ +ζ11/4 +ν +ζ7/2 +µν +(S36) +⟨∂ifµ|V |∂jfν⟩ = NµNν +8π +3 ζµζνζ−2 +µν = +� +2 +π +16 +3 +ζ7/4 +µ +ζ7/4 +ν +ζ2µν +δij +(S37) +⟨∂i∂jfµ|V |∂k∂lfν⟩ =NµNν +� 8πζµζν +ζµν +δijδkl +− 16πζµζ2 +ν +3ζ2µν +δijδkl − 16πζνζ2 +µ +3ζ2µν +δijδkl ++ 64πζ2 +kζ2 +ν +15ζ3µν +(δilδjk + δikδjl + δijδkl) +� +(S38) +It is convenient to introduce the following integrals +F1(i, j) = +� +d3r 1 +r rirjfµfν +(S39) += NµNν +1 +3πδijζ−2 +µν = δij +4 +3 +� +2 +π ζ3/4 +µ +ζ3/4 +ν +ζ−2 +µν +(S40) +F2(i, j, k, l) = +� +d3r 1 +r rirjrkrlfµfν +(S41) += NµNν +2 +15πζ−3 +µν (δilδjk + δikδjl + δijδkl) += 8 +15 +� +2 +π ζ3/4 +µ +ζ3/4 +ν +ζ−3 +µν (δilδjk + δikδjl + δijδkl) . +(S42) +The Breit terms are expressed with the following functions +I1(µ, ν, i, j, k, l) = +� +d3r (∂ifµ) rj +r (∂k∂lfν) += 4ζµζνδklF1(i, j) − 8ζµζ2 +νF2(i, j, k, l) +(S43) +I2(µ, ν, i, j, k, l) = +� +d3r fµ +ri +r (∂j∂k∂lfν) += 4ζ2 +ν [δjkF1(i, l) + δjlF1(i, k) + δklF1(i, j)] − 8ζ3 +νF2(i, j, k, l) . +(S44) + +15 +The matrix elements of the Breit operator are +⟨fµ|B1|fν⟩ = +1 +4c2m1m2 +⟨fµ|{(σ1 · ∇)(σ2 · ∇)r} (σ1 · ∇)(σ2 · ∇)|fν⟩ += σ1iσ2jσ1kσ2l +4c2m1m2 +⟨fµ|{∂i∂jr} ∂k∂l|fν⟩ += σ1iσ2jσ1kσ2l +4c2m1m2 +� +− ⟨∂ifµ|rj +r |∂k∂lfν⟩ − ⟨fµ|rj +r |∂i∂k∂lfν⟩ +� += σ1iσ2jσ1kσ2l +4c2m1m2 +[−I1(µ, ν, i, j, k, l) − I2(µ, ν, j, i, k, l)] +(S45) +⟨fµ|B2|fν⟩ = +1 +4c2m1m2 +⟨fµ|(σ2 · ∇) {(σ1 · ∇)(σ2 · ∇)r} (σ1 · ∇)|fν⟩ += σ2iσ1jσ2kσ1l +4c2m1m2 +⟨fµ|∂i{∂j∂kr}∂l|fν⟩ += −σ2iσ1jσ2kσ1l +4c2m1m2 +⟨∂ifµ| +� +∂j +rk +r +� +|∂lfν⟩ += σ2iσ1jσ2kσ1l +4c2m1m2 +� +⟨∂j∂ifµ|rk +r |∂lfν⟩ + ⟨∂ifµ|rk +r |∂j∂lfν⟩ +� += σ2iσ1jσ2kσ1l +4c2m1m2 +[I1(ν, µ, l, k, i, j) + I1(µ, ν, i, k, j, l)] +(S46) +⟨fµ|B3|fν⟩ = +1 +4c2m1m2 +⟨fµ|(σ1 · ∇) {(σ1 · ∇)(σ2 · ∇)r} (σ2 · ∇)|fν⟩ += σ1iσ1jσ2kσ2l +4c2m1m2 +⟨fµ|∂i{∂j∂kr}∂l|fν⟩ += −σ1iσ1jσ2kσ2l +4c2m1m2 +⟨∂ifµ| +� +∂j +rk +r +� +|∂lfν⟩ += σ1iσ1jσ2kσ2l +4c2m1m2 +� +⟨∂j∂ifµ|rk +r |∂lfν⟩ + ⟨∂ifµ|rk +r |∂j∂lfν⟩ +� += σ1iσ1jσ2kσ2l +4c2m1m2 +[I1(ν, µ, l, k, i, j) + I1(µ, ν, i, k, j, l)] +(S47) +⟨fµ|B4|fν⟩ = +1 +4c2m1m2 +⟨fµ|(σ2 · ∇)(σ1 · ∇) {(σ1 · ∇)(σ2 · ∇)r}|fν⟩ += σ2iσ1jσ1kσ2l +4c2m1m2 +⟨fµ|∂i∂j {∂k∂lr}|fν⟩ += σ2iσ1jσ1kσ2l +4c2m1m2 +⟨∂i∂jfµ| +� +∂k +rl +r +� +|fν⟩ += σ2iσ1jσ1kσ2l +4c2m1m2 +� +− ⟨∂k∂i∂jfµ|rl +r |fν⟩ − ⟨∂i∂jfµ|rl +r |∂kfν⟩ +� += σ2iσ1jσ1kσ2l +4c2m1m2 +[−I2(ν, µ, l, i, j, k) − I1(ν, µ, k, l, i, j)] +(S48) +EXPECTATION VALUES AND MASS-DEPENDENT CORRECTION FORMULAE +The necessary expectation values with arbitrary reduced mass are +En,l = − µ +2n2 +(S49) +⟨n, l|r−1|n, l⟩ = µ +n2 +(S50) +⟨n, l|r−2|n, l⟩ = +2µ2 +n3(2l + 1) +(S51) +⟨n, l|δ(r)|n, l⟩ = |ψn,l(0)|2 = µ3 +πn3 δl0 +(S52) + +16 +The following operators contribute to the α2Eh DC energy +HMV = −1 +8 +� 1 +m3 +1 ++ 1 +m3 +2 +� +(p2)2 +(S53) +HD = −π +2 +� 1 +m2 +1 ++ 1 +m2 +2 +� +δ(r) , +(S54) +and the additional terms from the Breit interaction (for 1S states) are +Hoo = − +1 +2m1m2r +� +p2 + r(rp)p +r2 +� +(S55) +Hss = − +2π +m1m2 +δ(r) . +(S56) +We use (pp. 421–422 of Ref. [41]) +⟨p2⟩ = 2µ +�� +E + 1 +r +�� +(S57) +⟨(p2)2⟩ = 4µ2 +�� +E + 1 +r +�2� ++ 16µ2π|ψ(0)|2 − 16µ2π|ψ(0)|2δl0 +(S58) +�p2 +r + r(rp)p +r3 +� += 4µ +�1 +r +� +E + 1 +r +�� +− 4π|ψ(0)|2 − l(l + 1)⟨r−3⟩ +(S59) +to obtain the required expectation values. For l = 0, we get +−1 +8 +� 1 +m3 +1 ++ 1 +m3 +2 +� +⟨(p2)2⟩ = µ4 +2 +� 1 +m3 +1 ++ 1 +m3 +2 +� � 3 +4n4 − 2 +n3 +� +(S60) +π +2 +� 1 +m2 +1 ++ 1 +m2 +2 +� +|ψ(0)|2 = µ3 +2n3 +� 1 +m2 +1 ++ 1 +m2 +2 +� +(S61) +− +1 +2m1m2 +�1 +r +� +p2 + r(rp)p +r2 +�� += +1 +2m1m2 +�2µ3 +n4 − 8µ3 +n3 + 4µ3 +n3 +� +(S62) +− +2π +m1m2 +⟨δ(r)⟩ = − +2µ3 +m1m2n3 . +(S63) + +17 +CONVERGENCE TABLES +Convergence of the fitted ε0, ε2, ε3, ε4, and ε′ +4 coefficients of Eq. (30) with respect to the Nb number +of basis functions. +Table S2. Ps +Nb +ε0 +ε2 +ε3 +ε′ +4 +ε4 +DC +10 +−0.249 999 665 675 +0.045 946 +0.048 6 +3.734 6 +8.537 5 +20 +−0.249 999 999 911 +0.046 879 +−0.130 3 +−0.135 4 +−0.140 8 +30 +−0.250 000 000 004 +0.046 877 +−0.129 4 +−0.088 9 +0.008 8 +40 +−0.250 000 000 001 +0.046 875 +−0.128 9 +−0.066 7 +0.074 7 +50 +−0.250 000 000 000 +0.046 875 +−0.128 8 +−0.063 4 +0.083 8 +DC⟨B⟩ +10 +−0.249 999 666 654 +−0.325 371 +−0.115 3 +−9.451 5 −22.409 5 +20 +−0.249 999 999 906 +−0.328 113 +0.278 7 +−0.218 4 +−0.401 2 +30 +−0.249 999 999 975 +−0.328 125 +0.280 8 +−0.173 5 +−0.302 2 +40 +−0.249 999 999 993 +−0.328 125 +0.280 3 +−0.196 8 +−0.375 3 +50 +−0.249 999 999 999 +−0.328 125 +0.280 2 +−0.202 6 +−0.392 3 +DCB +10 +−0.249 999 666 272 +−0.325 514 +−0.040 2 +−2.918 2 +−6.200 3 +20 +−0.249 999 999 841 +−0.328 118 +0.189 5 +0.246 4 +−0.590 5 +30 +−0.249 999 999 977 +−0.328 124 +0.189 7 +0.230 8 +−0.654 0 +40 +−0.249 999 999 994 +−0.328 125 +0.189 8 +0.230 3 +−0.657 4 +50 +−0.249 999 999 996 +−0.328 125 +0.189 9 +0.232 9 +−0.649 6 +Table S3. Mu +Nb +ε0 +ε2 +ε3 +ε′ +4 +ε4 +DC +10 +−0.497 592 269 583 +−0.120 332 +0.144 8 +20.427 9 +58.866 4 +20 +−0.497 593 472 091 +−0.120 167 +−0.429 1 +−1.264 1 +−1.406 6 +30 +−0.497 593 472 817 +−0.120 215 +−0.421 2 +−1.023 3 +−0.754 9 +40 +−0.497 593 472 893 +−0.120 225 +−0.419 6 +−0.975 0 +−0.624 3 +50 +−0.497 593 472 904 +−0.120 227 +−0.419 3 +−0.967 2 +−0.603 3 +DC⟨B⟩ +10 +−0.497 592 269 591 +−0.134 452 +0.141 0 +19.985 8 +57.659 7 +20 +−0.497 593 472 089 +−0.134 461 +−0.416 9 +−1.255 4 +−1.446 6 +30 +−0.497 593 472 817 +−0.134 513 +−0.409 0 +−1.015 1 +−0.796 3 +40 +−0.497 593 472 892 +−0.134 524 +−0.407 3 +−0.966 6 +−0.665 2 +50 +−0.497 593 472 904 +−0.134 526 +−0.407 1 +−0.958 9 +−0.644 5 +DCB +10 +−0.497 592 269 590 +−0.134 452 +0.141 0 +19.988 0 +57.663 3 +20 +−0.497 593 472 089 +−0.134 461 +−0.417 0 +−1.253 9 +−1.445 0 +30 +−0.497 593 472 817 +−0.134 513 +−0.409 1 +−1.013 6 +−0.794 5 +40 +−0.497 593 472 893 +−0.134 524 +−0.407 5 +−0.965 0 +−0.662 8 +50 +−0.497 593 472 904 +−0.134 526 +−0.407 2 +−0.957 3 +−0.641 9 + +18 +Table S4. H +Nb +ε0 +ε2 +ε3 +ε′ +4 +ε4 +DC +10 +−0.499 727 021 003 +−0.124 246 +0.049 6 +19.391 8 +58.146 5 +20 +−0.499 727 838 870 +−0.124 393 +−0.433 8 +−1.285 0 +−1.441 8 +30 +−0.499 727 839 611 +−0.124 443 +−0.425 7 +−1.040 9 +−0.781 0 +40 +−0.499 727 839 687 +−0.124 453 +−0.424 1 +−0.991 7 +−0.647 9 +50 +−0.499 727 839 699 +−0.124 455 +−0.423 8 +−0.983 7 +−0.626 5 +DC⟨B⟩ +10 +−0.499 727 021 056 +−0.125 860 +0.049 3 +19.339 4 +57.998 3 +20 +−0.499 727 838 869 +−0.126 024 +−0.432 4 +−1.283 9 +−1.446 3 +30 +−0.499 727 839 611 +−0.126 074 +−0.424 3 +−1.039 8 +−0.785 6 +40 +−0.499 727 839 687 +−0.126 084 +−0.422 7 +−0.990 5 +−0.652 4 +50 +−0.499 727 839 699 +−0.126 086 +−0.422 4 +−0.982 6 +−0.630 9 +DCB +10 +−0.499 727 021 001 +−0.125 860 +0.049 3 +19.339 9 +57.999 3 +20 +−0.499 727 838 869 +−0.126 024 +−0.432 4 +−1.283 9 +−1.446 2 +30 +−0.499 727 839 611 +−0.126 074 +−0.424 3 +−1.039 8 +−0.785 6 +40 +−0.499 727 839 687 +−0.126 084 +−0.422 7 +−0.990 5 +−0.652 4 +50 +−0.499 727 839 699 +−0.126 086 +−0.422 4 +−0.982 6 +−0.630 9 +Table S5. µH +Nb +ε0 +ε2 +ε3 +ε′ +4 +ε4 +DC +10 −92.920 263 622 900 +−8.580 427 +21.237 3 +3 178.582 7 +9 149.317 3 +20 −92.920 416 817 900 +−8.436 173 −68.216 1 +−142.593 7 +−79.362 0 +30 −92.920 417 290 800 +−8.436 688 −68.097 8 +−138.442 5 +−67.746 6 +40 −92.920 417 309 200 +−8.437 667 −67.885 9 +−130.535 5 +−44.976 0 +50 −92.920 417 310 100 +−8.437 667 −67.886 2 +−130.550 2 +−45.021 0 +DC⟨B⟩ +10 −92.920 263 643 100 −58.741 568 +7.760 3 +1 575.605 5 +4 889.224 2 +20 −92.920 416 808 100 −59.150 377 −19.385 6 +−125.924 0 +−171.285 6 +30 −92.920 417 286 500 −59.150 638 −19.459 2 +−131.205 7 +−188.305 9 +40 −92.920 417 307 300 −59.154 213 −18.860 2 +−113.954 0 +−142.929 1 +50 −92.920 417 308 300 −59.154 212 −18.860 6 +−113.969 5 +−142.976 5 +DCB +10 −92.920 263 637 500 −58.744 360 +9.491 5 +1 756.008 4 +5 236.465 9 +20 −92.920 416 792 300 −59.150 698 −25.262 7 +−103.891 6 +−194.837 7 +30 −92.920 417 273 300 −59.151 372 −25.181 3 +−101.424 0 +−187.901 1 +40 −92.920 417 306 600 −59.154 115 −24.867 0 +−94.725 5 +−172.162 0 +50 −92.920 417 307 500 −59.154 119 −24.865 6 +−94.658 1 +−171.955 8 + +19 +FITTED COEFFICIENTS +Table S6. Coefficients of the fitted F(α) = ε0 + α2ε2 + α3ε3 + α4 ln(α)ε′ +4 + α4ε4 polynomial to the no-pair +energies evaluated for a series of α values using the largest basis sets generated in this work. All values +correspond to hartree atomic units. +ε0 +ε2 +ε3 +ε′ +4 +ε4 +Ps (m2/m1 = 1): +EDC +−0.250 000 000 000 +0.046 875 +−0.128 8 +−0.063 4 +0.084 +EDC⟨B⟩ +−0.249 999 999 999 +−0.328 125 +0.280 2 +−0.202 6 +−0.392 +EDCB2 +−0.249 999 999 993 +−0.328 126 +0.190 3 +0.271 1 +−0.425 +EDCB +−0.249 999 999 996 +−0.328 125 +0.189 9 +0.232 9 +−0.650 +Mu (m2/m1 = 206.7682830): +EDC +−0.497 593 472 904 +−0.120 227 +−0.419 3 +−0.967 2 +−0.603 +EDC⟨B⟩ +−0.497 593 472 904 +−0.134 526 +−0.407 1 +−0.958 9 +−0.644 +EDCB2 +−0.497 593 472 904 +−0.134 526 +−0.407 2 +−0.957 3 +−0.642 +EDCB +−0.497 593 472 904 +−0.134 526 +−0.407 2 +−0.957 3 +−0.642 +H (m2/m1 = 1836.15267343): +EDC +−0.499 727 839 699 +−0.124 455 +−0.423 8 +−0.983 7 +−0.626 +EDC⟨B⟩ +−0.499 727 839 699 +−0.126 086 +−0.422 4 +−0.982 6 +−0.631 +EDCB2 +−0.499 727 839 699 +−0.126 086 +−0.422 4 +−0.982 6 +−0.631 +EDCB +−0.499 727 839 699 +−0.126 086 +−0.422 4 +−0.982 6 +−0.631 +µH (m2/m1 = 8.88024337): +EDC +−92.920 417 310 141 +−8.437 667 −67.886 2 −130.550 2 −45.021 +EDC⟨B⟩ +−92.920 417 308 281 −59.154 212 −18.860 6 −113.969 5 −142.977 +EDCB2 +−92.920 417 307 444 −59.154 145 −24.853 3 −93.505 9 −164.387 +EDCB +−92.920 417 307 523 −59.154 119 −24.865 6 −94.658 1 −171.956 +Table S7. Relative importance, in ppm (10−6), of the Dirac–Coulomb and Breit contributions with respect +to the mass ratio of the two fermions. +m2 +m1 +EDC−Enr +|Enr| +EDC⟨B⟩−EDC +|EDC| +EDCB2 −EDC⟨B⟩ +|EDC⟨B⟩| +EDCB−EDCB2 +|EDCB2| +Ps += {e−, e+} +1 +9.788 9 +−79.238 8 +−0.166 9 +−0.000 9 +µH += {µ−, p+} +8.88024337 +−5.101 2 +−28.865 4 +−0.028 7 +−0.000 1 +Mu = {e−, µ+} +206.768283 +−13.170 1 +−1.521 2 +−0.000 1 +0.000 0 +H += {e−, p+} 1836.15267343 −13.567 6a +−0.172 8 +0.000 0 +0.000 0 +µH∞= {µ−, p+ +∞} +∞ +−13.313 2 +0 +0 +0 +H∞ = {e−, p+ +∞} +∞ +−13.313 2 +0 +0 +0 +a By adding the α3Eh one-pair Coulomb correction to the no-pair DC energy, Eq. 31, we obtain +(EDC + E(3) +C1 − Enr)/Enr = −13.237 6. + +20 +∗ edit.matyus@ttk.elte.hu +[1] H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Springer, +Berlin, 1957). +[2] M. +I. +Eides, +H. +Grotch, +and +V. +A. +Shelyuto, +Theory +of +light +hydrogenlike +atoms, +Phys. Rep. 342, 63 (2001). +[3] M. S. Fee, A. P. 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T46, 252 (1993). + diff --git a/LdFRT4oBgHgl3EQfEzcl/content/tmp_files/load_file.txt b/LdFRT4oBgHgl3EQfEzcl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0cb6e4e3f9cb4cf0c4ec5a7b1f9e43d154437322 --- /dev/null +++ b/LdFRT4oBgHgl3EQfEzcl/content/tmp_files/load_file.txt @@ -0,0 +1,1574 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf,len=1573 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='13477v1 [quant-ph] 31 Jan 2023 Pre-Born–Oppenheimer Dirac–Coulomb–Breit computations for two-body systems D´avid Ferenc1 and Edit M´atyus1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' ∗ 1ELTE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' E¨otv¨os Lor´and University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Institute of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' P´azm´any P´eter s´et´any 1/A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Budapest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' H-1117,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Hungary (Dated: February 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 2023) The sixteen-component,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' no-pair Dirac–Coulomb–Breit equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' derived from the Bethe–Salpeter equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' is solved in a variational procedure using Gaussian-type basis functions for the example of positronium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' muonium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' hydrogen atom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' and muonic hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The α fine-structure-constant dependence of the variational energies, through fitting a function of αn and αnlnα terms, shows excellent agreement with the relevant energy expressions of the (perturbative) non-relativistic QED framework, and thereby, establishes a solid reference for the development of a computational rela- tivistic QED approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The positronium, Ps = {e−, e+}, muonium, Mu = {e−, µ+}, hydrogen atom, H = {e−, p+}, and muonic-hydrogen, µH = {µ−, p+}, are the simplest, yet some of the most extensively studied bound-state systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Their simplicity allows for the high-precision evaluation of energy corrections arising from special relativity and interactions from the matter and photon fields [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The high- precision spectroscopy experiments [3–10] together with the theoretical results (see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [2, 11] and references therein) provide stringent test for validity of quantum electrodynamics (QED) in the low-energy range and probe physics beyond the Standard Model [12–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Ps is a candidate for precision free-fall experiments to test QED and gravity [17], H and µH are the stars of the famous proton-size puzzle [18–20], while Mu has attracted interest in relation with the muon’s anomalous magnetic moment [10, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' For bound-state systems, it is relevant to have a wave equation that can be solved to obtain a good zeroth-order description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' So far, the non-relativistic Schr¨odinger equation has been used as reference, which has analytic solution for two-body systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Then, relativistic and QED corrections have been derived corresponding to increasing orders of the α fine-structure constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' We call these corrections, for short, non-relativistic QED (nrQED) corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' A recent review [11] provides an excellent overview of the extensive literature of higher-order nrQED corrections to positronium energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Corrections up to α6 order (in natural units, α4Eh in hartree atomic units) are considered complete, and ongoing work is about α7 order corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Some of the calculations have been carried out not only for equal but arbitrary spin-1/2 fermion masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' In the present work, we do not aim to reproduce the formally derived nrQED expressions, but initiate an alternative approach to the two-particle relativistic QED problem based on a zeroth- order wave equation in which special relativity is already accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The theoretical framework for this (computational) relativistic QED program is provided by the Bethe–Salpeter equation [22], derived from field theory [23], and its Salpeter–Sucher exact equal-time form [24, 25], which provides us a no-pair, two-particle relativistic wave equation, (H + H∆)Ψ = EΨ , (1) which has the form of a Schr¨odinger-like wave equation, for which high-precision numerical solution techniques can be adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The Ψ wave function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (1) depends only on the (spatial) Cartesian coordinates of the particles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' H is the positive-energy projected two-electron Hamiltonian with instantaneous (Coulomb or Coulomb–Breit) interaction (I),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' H = h1 + h2 + Λ++IΛ++ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (2) hi = cαipi + βimic2 + U1[4] (i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 2) is the one-particle Dirac Hamiltonian in which U can account for an external static Coulomb field (if there is any),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' and Λ++ projects to the positive- energy (electronic) subspace of the h1 + h2 non-interacting two-fermion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' For short, we call H the no-pair Dirac–Coulomb (DC) or Dirac–Coulomb–Breit (DCB) Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Pair corrections, retardation, and radiative corrections are included in the H∆ term, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (1) [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Contribution of H∆ to atomic and molecular energies (QED) can be expected to be small, 2 and hence, it can be treated as perturbation to the no-pair Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' This framework offers a perturbative approach based on a relativistic reference, alternative to earlier work using a non-relativistic reference state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Evaluation of the already formulated per- turbative correction with H∆ is left for future developments, which appears to be possible along the lines reviewed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Although analytic evaluation of the energy and its corrections is not possible in this framework, the numerical results can be converged to high precision, which is demonstrated in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' To compute no-pair, two-particle bound states, let us start with defining overall, center-of-mass, Rµ = (T, R), and relative, rµ = (t, r), covariant space-time coordinates as Rµ = m1 m1 + m2 rµ 1 + m2 m1 + m2 rµ 2 (3) and rµ = rµ 1 − rµ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (4) Then, following Salpeter and Bethe [22], the wave function of an isolated system can be factorized as φ(r1, r2) = e−iPνRνΦ(rµ) (5) with the total four-momentum, P ν = (E, P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' By choosing the zero-total-momentum frame, P = 0, we obtain φ(r1, r2) = e−iET Φ(rµ) , (6) where E is the total energy of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' It is important to note that Φ(rµ), which describes the internal motion, depends on rµ = (t, r), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=', not only on the r relative coordinates, but also on the t relative time of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Fourier transformation with respect to this relative time variable yields the relative-energy dependent wave function ˜Φ(ε, r) = � ∞ −∞ dt (2π)1/2 e−iεtΦ(t, r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (7) In the exact equal-time formalism of Salpeter [24] and Sucher [25], the equal-time (t = 0) wave function appears, which depends only on the spatial coordinates, Ψ(r) = � ∞ −∞ dε ˜Φ(ε, r) , (8) and the relative-energy dependence of the problem is accounted for in H∆ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (1) [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' To obtain the Hamiltonian for the relative motion, the chain rule for the coordinate transforma- tion, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (3) and (4), is used, and it is also considered that contribution from terms containing ∇R vanishes due to the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (5) choice of the ansatz for an isolated system and our choice of a P = 0 zero-momentum-frame description, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Hence, the spatial momentum operators in this framework can be replaced according to p1 = −i∇1 → p = −i∇ and p2 = −i∇2 → −p = i∇ , (9) where ∇(= ∇r) collects the partial derivatives with respect to the r relative displacement vector components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' This simple replacement ‘rule’ can be used to construct expressions for the relative motion from the two-particle expressions [29–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' As a result,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' the no-pair Dirac–Coulomb–Breit Hamiltonian for the relative motion is obtained as H(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 2) = Λ++ \uf8eb \uf8ec \uf8ed V 1[4] −cσ[4] 2 p cσ[4] 1 p B[4] −cσ[4] 2 p V 1[4] − 2m2c21[4] B[4] cσ[4] 1 p cσ[4] 1 p B[4] V 1[4] − 2m1c21[4] −cσ[4] 2 p B[4] cσ[4] 1 p −cσ[4] 2 p V 1[4] − 2m12c21[4] \uf8f6 \uf8f7 \uf8f8 Λ++ (10) 3 with m12 = m1 + m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' p = −i( ∂ ∂rx ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' ∂ ∂ry ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' ∂ ∂rz ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' σ[4] 1 = (σx ⊗ 1[2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' σy ⊗ 1[2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' σz ⊗ 1[2]) and σ[4] 2 = (1[2] ⊗ σx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 1[2] ⊗ σy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 1[2] ⊗ σz),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' where σx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' σy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' and σz are the 2 × 2 Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' We note that the operator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (10) contains a −2mic2 shift (i = 1, 2) to match the non-relativistic energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Furthermore, the Coulomb interaction, V = q1q2 r (11) is along the diagonal, whereas the Breit interaction, B[4] = −q1q2 �1 r σ[4] 1 · σ[4] 2 − 1 2 �� σ[4] 1 · ∇ � � σ[4] 2 · ∇ � r �� (12) can be found on the anti diagonal of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The Λ++ positive-energy projector in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (10) corresponds to the positive-energy (‘electronic’) states of the ‘bare’, non-interacting Hamiltonian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (10) without Λ++ and without the V 1[4] and B[4] interaction blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Although the Λ++ free-particle projector in momentum space has an analytic form [33], we constructed it numerically in coordinate space by computing the eigenstates of the bare, non-interacting Hamiltonian over the space spanned by the basis functions used for the interacting computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The positive-energy states were identified with the simple energy cutting approach (which can be checked by the complex scaling procedure) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The no-pair Dirac–Coulomb and Dirac–Coulomb–Breit Hamiltonians are bounded from below (the positive-energy block, which is considered in this work, is decoupled from the rest), hence the HΨ = EΨ wave equation can be solved using the variational procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' For a single particle, the (four-component) wave function is conveniently partitioned to large (l, first two) and small (s, last two) components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' A good basis representation must fulfill a simple symmetry relation, which is necessary to provide a correct matrix representation (Mx) for the Mx(p)Mx(p) = Mx(p2) identity [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The simplest implementation of this relation is provided by the (restricted) kinetic balance (KB) condition [35, 36], ϕs = σ[2] · p 2mc ϕl (13) for the basis function of the ϕs small and ϕl large components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Two(many)-particle relativistic quantities can be constructed with the block-wise (also called Tracy–Singh) direct product [29– 32, 37–40], which allows us to retain the large-small block structure, used already to write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The corresponding two-particle function, with highlighting the large (l) and small (s) component blocks, is ϕ = \uf8eb \uf8ec \uf8ec \uf8ed ϕll ϕls ϕsl ϕss \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (14) For a variational procedure, we used the simplest two-particle generalization of the one-particle kinetic balance, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (13), and implemented it in the sense of a transformation or metric [29–32, 35]: HKB = X†HX , X = diag \uf8eb \uf8ed1[4], − � σ[4] 2 · p � 2m2c , � σ[4] 1 · p � 2m1c , − � σ[4] 1 · p � � σ[4] 2 · p � 4m1m2c2 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (15) We also note that the X balance matrix used in this work can be ‘obtained’ from the balance used for the Born–Oppenheimer systems [29–32] through the p1 → p and p2 → −p replacement, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The fundamental ‘guiding principle’ for our construction of the two-particle balance has been solely to have a correct matrix representation of the Mx(p)Mx(p) = Mx(p2) identity, since the positive-energy projected Hamiltonian is bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The transformed DCB Hamiltonian 4 is HKB = X†H(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 2)X = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed D[4] 1 p2 2m2 1[4] p2 2m1 1[4] B[4] 1 p2 2m2 1[4] D[4] 2 B[4] 2 p4 8c2m1m2 2 1[4] p2 2m1 1[4] B[4] 3 D[4] 3 p4 8c2m2 1m2 1[4] B[4] 4 p4 8c2m1m2 2 1[4] p4 8c2m2 1m2 1[4] D[4] 4 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (16) with the diagonal blocks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' D[4] 1 = V 1[4] (17) D[4] 2 = (σ2 · p)V 1[4](σ2 · p) 4m2 2c2 − p2 2m2 1[4] (18) D[4] 3 = (σ1 · p)V 1[4](σ1 · p) 4m2 1c2 − p2 2m1 1[4] (19) D[4] 4 = (σ1 · p)(σ2 · p)V 1[4](σ1 · p)(σ2 · p) 16m2 1m2 2c4 − m12p4 8m2 1m2 2c2 1[4] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (20) and the anti-diagonal blocks including the Breit interaction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (12), B[4] 1 = −B[4](σ1 · p)(σ2 · p) 4c2m1m2 (21) B[4] 2 = −(σ2 · p2)B[4](σ1 · p) 4c2m1m2 (22) B[4] 3 = −(σ1 · p)B[4](σ2 · p) 4c2m1m2 (23) B[4] 4 = −(σ2 · p)(σ1 · p)B[4] 4c2m1m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (24) The identity in the X-KB metric is IKB = X†X = diag � 1[4], p2 4c2m2 2 1[4], p2 4c2m2 1 1[4], p4 16c4m2 1m2 2 1[4] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (25) Then, the sixteen-component wave function is written as a linear-combination of spinor functions, Ψ(r) = Nb � i=1 16 � χ=1 ciχfi(r)dχ , (26) where the dχ spinor basis vectors are sixteen-dimensional unit vectors, (dχ)ρ = δχρ (χ, ρ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' , 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' For the fi spatial functions, we use spherically symmetric Gaussian functions (Se, L = 0 orbital angular momentum and p = +1 even (e) parity), fi(r) = e−ζir2 (27) with ζi > 0 (to ensure square integrability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' We optimized the ζi Gaussian exponents (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' , Nb) by minimization of the non-relativistic 1Se ground-state energy to a pEh(= 10−12 Eh) precision range using quadruple precision arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Convergence of the non-relativistic and rela- tivistic energies with respect to the basis size is shown in Table S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' For selected systems and basis sizes, we continued the optimization of the ζi parameters by minimization of the no-pair DC(B) energy, and the computation remained variationally stable, the energy ‘converged from above’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (This variationally stable behaviour was absent during minimization of the relevant energy level of the bare DC Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=') We also note that there are no triplet contributions to the ground state (1 1Se 0) (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 419 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [41]), since even-parity 3P e states do not exist for a pseudo-one-particle 5 system (in contrast to helium-like systems [42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' In addition to variational no-pair DC and DCB computations, we computed the first-order perturbative Breit correction to the nth DC energy by [31, 32] EDC⟨B⟩,n = EDC,n + ⟨ΨDC,n|X†BXΨDC,n⟩ (28) where B is a sixteen-dimensional matrix with the B[4] blocks on its anti diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The second-order perturbative Breit correction is computed as EDCB2,n = EDC⟨B⟩,n + � i̸=n ��⟨ΨDC,i|X†BXΨDC,n⟩ ��2 EDC,i − EDC,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (29) The outlined algorithm has been implemented in the QUANTEN computer program, which is used as a molecular physics ‘platform’ for pre-Born–Oppenheimer, non-adiabatic, upper- and lower- bound, perturbative- and variational relativistic developments [29–32, 42–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Hartree atomic units are used, and the speed of light is c = α−1a0Eh/ℏ with α−1 = 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='035 999 084 [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' All computed no-pair energies are listed in Table S6, their change with the basis size can be used to assess their convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Further minimization tests for the DC(B) energy did not reveal major changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' For direct comparison of the computed no-pair energies with the current state-of-the-art nrQED values, we have (numerically) determined the α dependence of the no-pair energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' For this reason, we repeated the no-pair computations using the {α ∈ α0 ± n | n ∈ {−50, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=', 51}} series of the interaction constant, where α0 labels the value taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Then, we fitted the function F(α) = ε0 + α2ε2 + α3ε3 + α4 ln(α)ε′ 4 + α4ε4 (30) to the series of the no-pair energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Inclusion of higher-order, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=', α5 and α5 ln α, terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (30) did not make any visible difference at the current numerical precision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' A small fitting error was obtained, which had orders of magnitude smaller root-mean-squared deviation than the estimated energy convergence, Table S6), and smooth convergence of the fitted coefficients was observed with respect to the basis set size (Tables S1–S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' To obtain consistent results, it was essential to include also the α4 ln α term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (30), a simple α polynomial was insufficient to represent the high-precision no-pair energies (Table S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Table II shows the comparison of the α-dependence of the no-pair energies (fitted coefficients) and the nrQED corrections that were readily available to us or we could obtain with short calculation (Supplementary Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Excellent agreement is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The numerical deviation (uncertainty) of the coefficients fitted to the variational results is on the order of the convergence error of the no-pair energies that can be estimated from Table S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Regarding the large mass, m2 → ∞, limit and comparison with the one-electron Dirac energy, it is necessary to consider that the (bare) one-electron Dirac equation is with-pair (and correct for one electron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' At α3Eh order, the one-electron Dirac limit is recovered from our no-pair computations, by appending the no-pair energy with the (one) pair correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' For m2 → ∞, the one-pair Coulomb correction, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='9) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 54, is E(3) C1 (m1, ∞) = lim m2→∞ E(3) C1 (m1, m2) = lim m2→∞ 2µ3 3π � 2 m2 1 − 1 m1m2 + 2 m2 2 � = 4m1 3π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (31) In Table III, we can (numerically) observe that the large m2 limit of the ε3 coefficient, obtained from fitting F(α) to the no-pair energies, converges to −E(3) C1 (1, ∞), and hence cancel with the pair corrections (the two-pair contribution, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S12), vanishes) for m2 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Thereby, the one-particle Dirac limit is recovered at order α3Eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Furthermore, we also note that the Breit contribution vanishes as m2 → ∞ (Table S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 6 In this work, a computational relativistic quantum electrodynamics approach was put forward based on the exact equal-time Bethe–Salpeter equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' It is demonstrated that a relativistic reference state can be converged to a sub-parts-per-billion relative precision by variational solution of the no-pair Dirac–Coulomb(–Breit) wave equation including the dominant, instantaneous part of the electromagnetic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The α fine-structure dependence of the computed energies are in excellent agreement with the formal non-relativistic QED results corresponding to polynomial and logarithmic corrections in α, up to α6 ln α order in natural units (α4 ln αEh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Perturbative retardation, radiative, and pair corrections to the no-pair relativistic states had been formulated long ago, and their evaluation with the high-precision relativistic reference states computed in this work will be carried out in subsequent work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Financial support of the European Research Council through a Starting Grant (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 851421) is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' DF thanks a doctoral scholarship from the ´UNKP-22-4 New National Excellence Program of the Ministry for Innovation and Technology from the source of the National Research, Development, and Innovation Fund (´UNKP-22-4-I-ELTE-51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 7 Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Convergence of the no-pair Dirac–Coulomb(–Breit) energies, in Eh, computed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The spatial basis, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (27), used in the relativistic computation was parameterized by (numerical) minimization of the non-relativistic energy, Enr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The numerical value for the analytic (∞) non-relativistic energy is shown for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Nb Enr EDC EDC⟨B⟩ EDCB2 EDCB Ps (m2/m1 = 1): 10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='249 999 665 988 4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='249 997 227 989 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='250 016 969 603 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='250 016 992 755 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='250 016 992 809 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='249 999 999 919 4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='249 997 552 650 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='250 017 362 124 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='250 017 403 806 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='250 017 404 023 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='249 999 999 996 8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='249 997 552 766 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='250 017 362 426 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='250 017 404 153 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='250 017 404 371 40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='249 999 999 999 6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='249 997 552 778 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='250 017 362 470 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='250 017 404 205 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='250 017 404 425 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='249 999 999 999 9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='249 997 552 780 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='250 017 362 477 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='250 017 404 214 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='250 017 404 433 ∞ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='250 000 000 000 0 Mu (m2/m1 = 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='7682830): 10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 592 269 419 4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 598 739 220 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 599 489 904 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 599 489 917 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 599 489 918 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 593 472 285 4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 600 024 240 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 600 780 916 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 600 780 959 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 600 780 959 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 593 472 874 8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 600 025 977 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 600 782 839 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 600 782 891 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 600 782 891 40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 593 472 910 8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 600 026 241 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 600 783 176 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 600 783 235 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 600 783 235 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 593 472 915 7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 600 026 282 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 600 783 233 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 600 783 295 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 600 783 295 ∞ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 593 472 917 1 H (m2/m1 = 1836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='15267343): 10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 727 019 644 9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 733 723 658 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 733 809 460 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 733 809 460 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 733 809 460 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 727 839 067 5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 734 617 695 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 734 704 007 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 734 704 008 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 734 704 008 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 727 839 669 3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 734 619 508 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 734 705 842 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 734 705 843 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 734 705 843 40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 727 839 706 0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 734 619 795 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 734 706 138 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 734 706 139 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 734 706 140 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 727 839 710 9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 734 619 840 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 734 706 186 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 734 706 187 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 734 706 187 ∞ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 727 839 712 4 µH (m2/m1 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='88024337) : 10 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 263 579 73 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 730 693 26 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='923 396 814 39 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='923 397 816 36 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='923 397 817 07 20 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 416 825 53 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 890 799 40 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='923 572 907 75 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='923 575 558 50 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='923 575 566 96 30 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 417 297 88 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 891 278 83 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='923 573 403 13 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='923 576 058 44 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='923 576 066 97 40 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 417 310 07 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 891 312 69 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='923 573 493 64 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='923 576 164 19 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='923 576 173 06 50 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 417 311 03 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 891 313 65 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='923 573 494 58 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='923 576 165 15 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='923 576 174 01 ∞ −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 417 311 31 8 Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Comparison of variational no-pair results and nrQED corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The F(α) = ε0 +α2ε2 +α3ε3 + α4 ln(α)ε′ 4 + α4ε4 function was fitted to the no-pair energies to obtain the coefficients (var-fit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' All values correspond to hartree atomic units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (All fitted coefficients are listed in Table S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=') DC DC⟨B⟩ DCB ε2 ε3 ε′ 4 ε2 ε3 ε2 ε3 Ps = {e−, e+}: var-fit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='046 875 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='128 8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='063 4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='328 125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='280 2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='328 125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='189 9 nrQED a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='046 875 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='128 8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='062 5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='328 125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='280 3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='328 125 αn(δεn) b −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='5·10−12 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='2·10−12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='6·10−12 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='3·10−11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='5·10−11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='3·10−11 Mu = {e−, µ+}: var-fit −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='120 227 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='419 3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='967 2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='134 526 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='407 1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='134 526 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='407 2 nrQED a −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='120 227 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='419 3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='134 528 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='134 528 αn(δεn) b −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='7·10−11 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='2·10−11 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='0·10−10 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='1·10−10 H = {e−, p+}: var-fit −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='124 455 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='423 8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='983 7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='126 086 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='422 4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='126 086 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='422 4 nrQED a −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='124 456 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='423 8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='126 087 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='126 087 αn(δεn) b −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='1·10−11 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='0·10−11 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='8·10−11 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='8·10−11 µH = {µ−, p+}: var-fit −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='437 67 −67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='886 −130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='550 2 −59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='154 212 −18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='860 6 −59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='154 120 −24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='865 6 nrQED a −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='437 70 −67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='899 −59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='154 516 −59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='154 516 αn(δεn) b −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='7 · 10−9 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='4 · 10−9 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='6 · 10−8 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='1 · 10−8 a The nrQED corrections, with the corresponding references, are collected in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' b αn(δεn), in Eh, with the δεn = E(n) − εn difference of the nrQED value and the fitted coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Large mass, m2 → ∞, limit, of the α3Eh-order fitted coefficient of the no-pair DC energy, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (m1 = 1 corresponds to the electron mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=') m2 ε3 Ps = {e−, e+} 1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='128 8 Mu = {e−, µ+} 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='7682830 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='419 3 H = {e−, p+} 1836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='15267343 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='423 8 10H = {e−, 10p+} 18361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='5267343 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='424 3 −E(3) C1 (1, m2) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (31) ∞ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='424 413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 9 Supplementary Material Pre-Born–Oppenheimer Dirac–Coulomb–Breit computations for two-body systems D´avid Ferenc1 and Edit M´atyus1,∗ 1 ELTE, E¨otv¨os Lor´and University, Institute of Chemistry, P´azm´any P´eter s´et´any 1/A, Budapest, H-1117, Hungary ∗ edit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='matyus@ttk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='elte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='hu (Dated: January 31, 2022) Contents: S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Non-relativistic QED expressions compiled from the literature S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Matrix elements S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Expectation values and mass-dependent correction formulae S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Convergence tables S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Fitted coefficients References 10 NON-RELATIVISTIC QED EXPRESSIONS COMPILED FROM THE LITERATURE In the non-relativistic QED approach, the energy is obtained by evaluating E(n) terms for in- creasing powers of α as corrections to the E(0) nr non-relativistic energy (which is of α0 order in hartree atomic units), E = E(0) nr + α2E(2) + α3E(3) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S1) The Schr¨odinger equation of two-particle systems has a closed analytic solution, and the ground- state energy reads as E(0) nr = −µ 2 (S2) with the reduced mass µ = m1m2 m1 + m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S3) The next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (non-vanishing) α2Eh-order correction is the sum of two terms arising from the Coulomb and the Breit (non-retarded part of transverse) interactions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' α2E(2) DCB = α2E(2) DC + α2E(2) B ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S4) which is obtained by calculating the expectation value of the following operators [41],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' H(2) DC = −1 8 � 1 m3 1 + 1 m3 2 � (p2)2 − π 2 � 1 m2 1 + 1 m2 2 � δ(r) (S5) H(2) B = − 1 2m1m2r � p2 + r(rp)p r2 � − 2π m1m2 δ(r) (S6) with the non-relativistic ground-state wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' These expectation values can be written in a closed, analytic form (Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' and ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' For the 1 1Se state, they are E(2) DC = EMV + ED (S7) with EMV = −5 8µ4 � 1 m3 1 + 1 m3 2 � ED = µ3 2 � 1 m2 1 + 1 m2 2 � , (S8) from the first and second terms of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S5), respectively, and E(2) B = Eoo + Ess , (S9) where Eoo = − µ3 m1m2 and Ess = − 2µ3 m1m2 (S10) from the two terms of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Figure S1 shows the m2 dependence of the corrections for the m1 = 1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Interestingly, the relativistic DC correction vanishes for the m2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='209 and m2 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='791 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' In contrast to the non-relativistic energy, the mass-dependence of the corrections is not only through the reduced mass of the constituent particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The correction arising from non-crossed photons at α3Eh order has been reported by Fulton and 11 −20 0 20 −20 0 20 0 5 10 E [µEh] E(2) DC ED EMV E [µEh] m2/me E(2) DCB ED EMV Eoo Ess Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Dependence of the α2Eh-order Dirac–Coulomb (top) and Dirac–Coulomb–Breit (bottom) energy corrections of two-particle systems on the m2 particle mass with m1 = 1(me).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' For m1 = 1, the E(2) DC correction vanishes for the m2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='208 71 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='791 29 values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=', up to α2Eh the DC relativistic energy equals the non-relativistic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Martin (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='7) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [54]), E(3) C0,2(m1, m2) = −2µ3 3π � 2 m2 1 + 1 m1m2 + 2 m2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S11) We note that this expression contains the sum of the no-pair and the two-pair corrections (indicated by the ‘0,2’ subscript).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' To the best of our knowledge, the single and two-pair Coulomb corrections separately do not have any simple form for general m1, m2 masses, but can be calculated from the integral (two-pair part of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='1a)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='6) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 54): E(3) C2 (m1, m2) = −2µ3 π � ∞ 0 dk (E1(k) − m1)(E2(k) − m2) E1(k) + E2(k) + m1 + m2 with Ei(k) = � m2 i + k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S12) This integral can be evaluated by using (for example) a symbolic algebra program, and the resulting (lengthy) expression can be evaluated for the selected m1 and m2 masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' For the special case of m1 = m2 = 1, the integral simplifies to, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='26b) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [25], E(3) C2 (1, 1) = − 1 8π �5 3 − π 2 � , (S13) which together with using the simple expression for the non-crossed photon correction of Fulton and Martin, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S11), can be used to obtain the third-order perturbative no-pair Coulomb correction for unit masses, m1 = m2 = 1: E(3) C0 (1, 1) = E(3) C0,2(1, 1) − E(3) C2 (1, 1) = − 1 8π �π 2 + 5 3 � ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='128 815 , (S14) which is the same as the third-order no-pair Coulomb correction reported by Sucher (for the two electrons in helium, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='99) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The zero- plus two-pair contribution, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S11), for unit masses is E(3) C0,2(1, 1) = − 5 12π ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='132 629 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S15) The E(3) C0 no-pair contribution was not separately reported in the literature for non-unit masses, 12 Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Non-relativistic energy and perturbative correction values, in Hartree atomic units, calculated using the analytic expressions, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S1)–(S14), compiled from the literature [25, 41, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The m2/m1 mass ratios are taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' m2/m1 E(0) nr E(2) DC E(2) DCB E(3) C0,2 E(3) C2 E(3) C0 Ps = {e−, e+} 1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='250 000 000 000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='046 875 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='328 125 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='132 629 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='003 815 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='128 815a Mu= {e−, µ+} 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='7682830 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 593 472 917 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='120 227 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='134 528 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='419 336 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='6 · 10−6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='419 332 H = {e−, p+} 1836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='15267343 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 727 839 712 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='124 456 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='126 087 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='423 836 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='7 · 10−8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='423 836 µH= {µ−, p+} 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='88024337 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 417 311 307 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='437 699 −59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='154 516 −68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='110 857 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='210 9268 −67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='899 930 a Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S14) was used for Ps [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' and we calculated it for the relevant mass values using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S11) and (S12) (Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The α3Eh-order contribution from a single instantaneous Breit photon exchange including also the Coulomb ladder is known for unit masses (positronium) [25], E(3) B (1, 1) = 1 2π � 1 + π 2 � ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='409 155 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S16) The expansion coefficients of the bound-state energy also involve terms with logarithm of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' In the no-pair DC energy the α3 ln α terms are absent, yet there are logarithmic terms in the fourth order i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' α4 ln α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' For equal masses, a simple expression can be found for the DC correction in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [55] (and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' MATRIX ELEMENTS The spatial basis functions are fµ(r) = �2ζµ π �3/4 e−ζµr2 = Nµe−ζµr2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S17) The following notation is used, i, j, k, l ∈ {x, y, z} denote Cartesian components ζµν = ζµ + ζν (S18) � d3r = � ∞ −∞ dx dy dz (S19) � d3r e−ζr2 = �π ζ �3/2 (S20) r = � x2 + y2 + z2 (S21) Nµ = �2ζµ π �3/4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S22) In what follows, the Einstein summation convention is understood for the i, j, k, l Cartesian indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' The derivatives of the basis functions are ∂ie−ζµr2 = −2ζµrie−ζµr2 (S23) ∂j∂ie−ζµr2 = � −2ζµδij + 4ζ2 µrirj � e−ζµr2 (S24) ∂k∂j∂ie−ζµr2 = � 4ζ2 µ(δijrk + δikrj + δkjri) − 8ζ3 µrirjrk � e−ζµr2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S25) 13 A useful integrals for the calculation of the Coulomb matrix elements include � ∞ 0 dt (a + t2)−3/2 = 1 a (S26) � ∞ 0 dt (a + t2)−5/2 = 1 3a2 (S27) � ∞ 0 dt (a + t2)−7/2 = 1 15a3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='(S28) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='⟨fµ|fν⟩ = NµNν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='d3r e−(ζµ+ζν)r2 = (4ζµζν)3/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='(S29) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='⟨fµ|V |fν⟩ =NµNν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='d3r 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='r e−ζµνr2 = NµNν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='√π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='d3r 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='r e−(ζµν+t2)r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='= NµNν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='√π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='π3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='(ζµν + t2)3/2 = NµNν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='2π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζµν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='(ζµζν)3/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζµν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='(S30) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='⟨fµ|∇2|fν⟩ =NµNν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='d3r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='−2δiiζµ + 4ζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µriri ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='e−ζµνr2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='= −6ζµ(4ζµζν)3/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='+ 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='2ζ11/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ3/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ5/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='(S31) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='⟨fµ|∇2∇2|fν⟩ =NµNν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='d3r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='−2ζµδii + 4ζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µriri ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='−2ζνδjj + 4ζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='νrjrj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='e−ζµνr2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='= NµNν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='4ζµζνδiiδjj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='d3r e−ζµνr2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='− 8ζµζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='νδii ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='d3r rjrje−ζµνr2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='− 8ζνζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µδjj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='d3r ririe−ζµνr2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='+ 16ζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='νζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='d3r rjrjririe−ζµνr2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='(S32) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='36NµNνζµζν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='d3r e−ζµνr2 = 36NµNνζµζνπ3/2ζ−3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='(S33) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='−24NµNνζµζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='d3r rjrje−ζµνr2 = −36NµNνζµζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='νπ3/2ζ−5/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='(S34) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='The sum of the first three terms is zero: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='36NµNνζµζνπ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='− 36NµNνζµζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='νπ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ5/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='− 36NµNνζνζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µπ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ5/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='= 36NµNνζµζνπ3/2ζµν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν ζµν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='− 36NµNνζµζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='νπ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ5/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='− 36NµNνζνζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µπ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ5/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='= 36NµNνζµζνπ3/2(ζµ + ζν) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ5/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='− 36NµNνζµζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='νπ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ5/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='− 36NµNνζνζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µπ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ5/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='= 36NµNνζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µζνπ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ5/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='+ 36NµNνζµζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='νπ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ5/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='− 36NµNνζµζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='νπ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ5/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='− 36NµNνζνζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µπ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ5/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='= 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='(S35) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='16NµNνζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='νζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='d3r r2r2e−ζµνr2 = 60NµNνζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='νζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µπ3/2ζ−7/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='= 120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='2ζ11/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ11/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ7/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='(S36) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='⟨∂ifµ|V |∂jfν⟩ = NµNν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='8π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='3 ζµζνζ−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µν = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ7/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ7/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζ2µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='δij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='(S37) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='⟨∂i∂jfµ|V |∂k∂lfν⟩ =NµNν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� 8πζµζν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ζµν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='δijδkl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='− 16πζµζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='3ζ2µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='δijδkl − 16πζνζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='3ζ2µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='δijδkl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='+ 64πζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='kζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='ν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='15ζ3µν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='(δilδjk + δikδjl + δijδkl) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='(S38) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='It is convenient to introduce the following integrals ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='F1(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' j) = � d3r 1 r rirjfµfν (S39) = NµNν 1 3πδijζ−2 µν = δij 4 3 � 2 π ζ3/4 µ ζ3/4 ν ζ−2 µν (S40) F2(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' l) = � d3r 1 r rirjrkrlfµfν (S41) = NµNν 2 15πζ−3 µν (δilδjk + δikδjl + δijδkl) = 8 15 � 2 π ζ3/4 µ ζ3/4 ν ζ−3 µν (δilδjk + δikδjl + δijδkl) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S42) The Breit terms are expressed with the following functions I1(µ, ν, i, j, k, l) = � d3r (∂ifµ) rj r (∂k∂lfν) = 4ζµζνδklF1(i, j) − 8ζµζ2 νF2(i, j, k, l) (S43) I2(µ, ν, i, j, k, l) = � d3r fµ ri r (∂j∂k∂lfν) = 4ζ2 ν [δjkF1(i, l) + δjlF1(i, k) + δklF1(i, j)] − 8ζ3 νF2(i, j, k, l) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S44) 15 The matrix elements of the Breit operator are ⟨fµ|B1|fν⟩ = 1 4c2m1m2 ⟨fµ|{(σ1 · ∇)(σ2 · ∇)r} (σ1 · ∇)(σ2 · ∇)|fν⟩ = σ1iσ2jσ1kσ2l 4c2m1m2 ⟨fµ|{∂i∂jr} ∂k∂l|fν⟩ = σ1iσ2jσ1kσ2l 4c2m1m2 � − ⟨∂ifµ|rj r |∂k∂lfν⟩ − ⟨fµ|rj r |∂i∂k∂lfν⟩ � = σ1iσ2jσ1kσ2l 4c2m1m2 [−I1(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' l) − I2(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' l)] (S45) ⟨fµ|B2|fν⟩ = 1 4c2m1m2 ⟨fµ|(σ2 · ∇) {(σ1 · ∇)(σ2 · ∇)r} (σ1 · ∇)|fν⟩ = σ2iσ1jσ2kσ1l 4c2m1m2 ⟨fµ|∂i{∂j∂kr}∂l|fν⟩ = −σ2iσ1jσ2kσ1l 4c2m1m2 ⟨∂ifµ| � ∂j rk r � |∂lfν⟩ = σ2iσ1jσ2kσ1l 4c2m1m2 � ⟨∂j∂ifµ|rk r |∂lfν⟩ + ⟨∂ifµ|rk r |∂j∂lfν⟩ � = σ2iσ1jσ2kσ1l 4c2m1m2 [I1(ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' j) + I1(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' l)] (S46) ⟨fµ|B3|fν⟩ = 1 4c2m1m2 ⟨fµ|(σ1 · ∇) {(σ1 · ∇)(σ2 · ∇)r} (σ2 · ∇)|fν⟩ = σ1iσ1jσ2kσ2l 4c2m1m2 ⟨fµ|∂i{∂j∂kr}∂l|fν⟩ = −σ1iσ1jσ2kσ2l 4c2m1m2 ⟨∂ifµ| � ∂j rk r � |∂lfν⟩ = σ1iσ1jσ2kσ2l 4c2m1m2 � ⟨∂j∂ifµ|rk r |∂lfν⟩ + ⟨∂ifµ|rk r |∂j∂lfν⟩ � = σ1iσ1jσ2kσ2l 4c2m1m2 [I1(ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' j) + I1(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' l)] (S47) ⟨fµ|B4|fν⟩ = 1 4c2m1m2 ⟨fµ|(σ2 · ∇)(σ1 · ∇) {(σ1 · ∇)(σ2 · ∇)r}|fν⟩ = σ2iσ1jσ1kσ2l 4c2m1m2 ⟨fµ|∂i∂j {∂k∂lr}|fν⟩ = σ2iσ1jσ1kσ2l 4c2m1m2 ⟨∂i∂jfµ| � ∂k rl r � |fν⟩ = σ2iσ1jσ1kσ2l 4c2m1m2 � − ⟨∂k∂i∂jfµ|rl r |fν⟩ − ⟨∂i∂jfµ|rl r |∂kfν⟩ � = σ2iσ1jσ1kσ2l 4c2m1m2 [−I2(ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' k) − I1(ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' j)] (S48) EXPECTATION VALUES AND MASS-DEPENDENT CORRECTION FORMULAE The necessary expectation values with arbitrary reduced mass are En,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='l = − µ 2n2 (S49) ⟨n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' l|r−1|n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' l⟩ = µ n2 (S50) ⟨n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' l|r−2|n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' l⟩ = 2µ2 n3(2l + 1) (S51) ⟨n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' l|δ(r)|n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' l⟩ = |ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='l(0)|2 = µ3 πn3 δl0 (S52) 16 The following operators contribute to the α2Eh DC energy HMV = −1 8 � 1 m3 1 + 1 m3 2 � (p2)2 (S53) HD = −π 2 � 1 m2 1 + 1 m2 2 � δ(r) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S54) and the additional terms from the Breit interaction (for 1S states) are Hoo = − 1 2m1m2r � p2 + r(rp)p r2 � (S55) Hss = − 2π m1m2 δ(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S56) We use (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 421–422 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [41]) ⟨p2⟩ = 2µ �� E + 1 r �� (S57) ⟨(p2)2⟩ = 4µ2 �� E + 1 r �2� + 16µ2π|ψ(0)|2 − 16µ2π|ψ(0)|2δl0 (S58) �p2 r + r(rp)p r3 � = 4µ �1 r � E + 1 r �� − 4π|ψ(0)|2 − l(l + 1)⟨r−3⟩ (S59) to obtain the required expectation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' For l = 0, we get −1 8 � 1 m3 1 + 1 m3 2 � ⟨(p2)2⟩ = µ4 2 � 1 m3 1 + 1 m3 2 � � 3 4n4 − 2 n3 � (S60) π 2 � 1 m2 1 + 1 m2 2 � |ψ(0)|2 = µ3 2n3 � 1 m2 1 + 1 m2 2 � (S61) − 1 2m1m2 �1 r � p2 + r(rp)p r2 �� = 1 2m1m2 �2µ3 n4 − 8µ3 n3 + 4µ3 n3 � (S62) − 2π m1m2 ⟨δ(r)⟩ = − 2µ3 m1m2n3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (S63) 17 CONVERGENCE TABLES Convergence of the fitted ε0, ε2, ε3, ε4, and ε′ 4 coefficients of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' (30) with respect to the Nb number of basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Ps Nb ε0 ε2 ε3 ε′ 4 ε4 DC 10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='249 999 665 675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='045 946 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='048 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='734 6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='537 5 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='249 999 999 911 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='046 879 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='130 3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='135 4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='140 8 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='250 000 000 004 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='965 0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='662 8 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 593 472 904 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='134 526 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='407 2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='957 3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} 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443 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='425 7 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='040 9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='781 0 40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 727 839 687 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='124 453 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='424 1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} 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024 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='432 4 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='283 9 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='446 3 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 727 839 611 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='126 074 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='424 3 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 727 839 687 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='126 084 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='422 7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='990 5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='652 4 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 727 839 699 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='126 086 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='422 4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='982 6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='630 9 Table S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' µH Nb ε0 ε2 ε3 ε′ 4 ε4 DC 10 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 263 622 900 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='580 427 21.' metadata={'source': 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−79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='362 0 30 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 417 290 800 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='436 688 −68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='097 8 −138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='442 5 −67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='746 6 40 −92.' metadata={'source': 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30 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 417 286 500 −59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='150 638 −19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='459 2 −131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='205 7 −188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='305 9 40 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 417 307 300 −59.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='465 9 20 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 416 792 300 −59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='150 698 −25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='262 7 −103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='891 6 −194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='837 7 30 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 417 273 300 −59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='151 372 −25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='181 3 −101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='424 0 −187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='901 1 40 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 417 306 600 −59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='154 115 −24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='867 0 −94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='725 5 −172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='162 0 50 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 417 307 500 −59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='154 119 −24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='865 6 −94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='658 1 −171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='955 8 19 FITTED COEFFICIENTS Table S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Coefficients of the fitted F(α) = ε0 + α2ε2 + α3ε3 + α4 ln(α)ε′ 4 + α4ε4 polynomial to the no-pair energies evaluated for a series of α values using the largest basis sets generated in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' All values correspond to hartree atomic units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' ε0 ε2 ε3 ε′ 4 ε4 Ps (m2/m1 = 1): EDC −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='250 000 000 000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='046 875 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='128 8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='063 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='084 EDC⟨B⟩ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='249 999 999 999 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='328 125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='280 2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='202 6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='392 EDCB2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='249 999 999 993 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='328 126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='190 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='271 1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='425 EDCB −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='249 999 999 996 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='328 125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='189 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='232 9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='650 Mu (m2/m1 = 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='7682830): EDC −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 593 472 904 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='120 227 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='419 3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='967 2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='603 EDC⟨B⟩ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 593 472 904 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='134 526 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='407 1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='958 9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='644 EDCB2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 593 472 904 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='134 526 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='407 2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='957 3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='642 EDCB −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='497 593 472 904 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='134 526 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='407 2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='957 3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='642 H (m2/m1 = 1836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='15267343): EDC −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 727 839 699 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='124 455 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='423 8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='983 7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='626 EDC⟨B⟩ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 727 839 699 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='126 086 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='422 4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='982 6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='631 EDCB2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 727 839 699 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='126 086 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='422 4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='982 6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='631 EDCB −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='499 727 839 699 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='126 086 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='422 4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='982 6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='631 µH (m2/m1 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='88024337): EDC −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 417 310 141 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='437 667 −67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='886 2 −130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='550 2 −45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='021 EDC⟨B⟩ −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 417 308 281 −59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='154 212 −18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='860 6 −113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='969 5 −142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='977 EDCB2 −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 417 307 444 −59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='154 145 −24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='853 3 −93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='505 9 −164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='387 EDCB −92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='920 417 307 523 −59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='154 119 −24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='865 6 −94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='658 1 −171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='956 Table S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Relative importance, in ppm (10−6), of the Dirac–Coulomb and Breit contributions with respect to the mass ratio of the two fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' m2 m1 EDC−Enr |Enr| EDC⟨B⟩−EDC |EDC| EDCB2 −EDC⟨B⟩ |EDC⟨B⟩| EDCB−EDCB2 |EDCB2| Ps = {e−, e+} 1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='788 9 −79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='238 8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='166 9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='000 9 µH = {µ−, p+} 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='88024337 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='101 2 −28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='865 4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='028 7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='000 1 Mu = {e−, µ+} 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='768283 −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='170 1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='521 2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='000 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='000 0 H = {e−, p+} 1836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='15267343 −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='567 6a −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='172 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='000 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='000 0 µH∞= {µ−, p+ ∞} ∞ −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='313 2 0 0 0 H∞ = {e−, p+ ∞} ∞ −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='313 2 0 0 0 a By adding the α3Eh one-pair Coulomb correction to the no-pair DC energy, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 31, we obtain (EDC + E(3) C1 − Enr)/Enr = −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='237 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 20 ∗ edit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='matyus@ttk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='elte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content='hu [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Bethe and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Springer, Berlin, 1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Eides, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Grotch, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Shelyuto, Theory of light hydrogenlike atoms, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 342, 63 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Fee, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Mills, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Chu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Shaw, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Danzmann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Chichester, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Zucker- man, Measurement of the positronium 13s1-23s1 interval by continuous-wave two-photon excitation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 70, 1397 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [4] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' H¨ansch, Nobel lecture: Passion for precision, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 78, 1297 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [5] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Biraben, Spectroscopy of atomic hydrogen, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 172, 109 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Matveev, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Parthey, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Predehl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Alnis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Beyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Holzwarth, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Udem, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Wilken, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Kolachevsky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Abgrall, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Rovera, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Salomon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Laurent, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Grosche, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Terra, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Legero, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Schnatz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Weyers, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Altschul, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' H¨ansch, Precision measurement of the hydrogen 1S–2S frequency via a 920-km fiber link, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 110, 230801 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Ishida, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Namba, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Asai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Kobayashi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Saito, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Yoshida, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Tanaka, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Yamamoto, New precision measurement of hyperfine splitting of positronium, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' B 734, 338 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Frugiuele, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' P´erez-R´ıos, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Peset, Current and future perspectives of positronium and muonium spectroscopy as dark sectors probe, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' D 100, 015010 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [9] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Gurung, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Babij, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Hogan, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Cassidy, Precision microwave spectroscopy of the positronium n = 2 fine structure, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 125, 073002 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [10] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Ohayon, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Janka, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Cortinovis, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Burkley, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Borges, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Depero, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Golovizin, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Ni, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Salman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Suter, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Vigo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Prokscha, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Crivelli (Mu-MASS Collaboration), Precision measurement of the lamb shift in muonium, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 128, 011802 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [11] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Adkins, D.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Karshenboim, Precision physics of simple atoms: QED tests, nuclear structure and fundamental constants, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' 422, 1 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFRT4oBgHgl3EQfEzcl/content/2301.13477v1.pdf'} +page_content=' N.' metadata={'source': 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sha256:97cc8019751ebf9681d63bcdebfbc46acde40132424bd5f924f4b6701e10afdc +size 20447277 diff --git a/MdE2T4oBgHgl3EQfqQjZ/content/tmp_files/2301.04038v1.pdf.txt b/MdE2T4oBgHgl3EQfqQjZ/content/tmp_files/2301.04038v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..29e6ce23578ea0512a1bc45393f4cfe157a699ea --- /dev/null +++ b/MdE2T4oBgHgl3EQfqQjZ/content/tmp_files/2301.04038v1.pdf.txt @@ -0,0 +1,2103 @@ +Role of chemical potential at kinetic freeze-out using Tsallis non-extensive statistics in +proton-proton collisions at the Large Hadron Collider +Girija Shankar Pradhan1, Dushmanta Sahu1, Rutuparna Rath2, Raghunath Sahoo1∗, and Jean Cleymans3† +Department of Physics, Indian Institute of Technology Indore, Simrol, Indore 453552, India +2INFN - sezione di Bologna, via Irnerio 46, 40126 Bologna BO, Italy and +3UCT-CERN Research Centre and Physics Department, University of Cape Town, South Africa +(Dated: January 11, 2023) +The charged-particle transverse momentum spectra (pT-spectra) measured by ALICE collabora- +tion for pp collisions at √s = 7 and 13 TeV have been studied using a thermodynamically consistent +form of Tsallis non-extensive statistics. The Tsallis distribution function is fitted to the pT-spectra +and the results are analyzed as a function of final state charged-particle multiplicity for various +light flavor and strange particles, such as π±, K±, p + ¯p, φ, Λ + ¯Λ, Ξ + ¯Ξ, Ω + ¯Ω. At LHC energies, +particles and antiparticles are produced in equal numbers. However, there are various processes +that contribute to the asymmetry between hadrons and anti-hadrons at kinetic freeze-out, which +could contribute to make the chemical potential finite. This article emphasizes the importance of +the chemical potential of the system produced in pp collisions at the LHC energies using the Tsallis +distribution function. +PACS numbers: +I. +INTRODUCTION +It has been the ambitious force of research in high- +energy physics to use particle colliders like the Rela- +tivistic Heavy Ion Collider (RHIC) at Brookhaven Na- +tional Laboratory (BNL) and the Large Hadron Col- +lider (LHC) at CERN (European Council for Nuclear Re- +search), which fulfill the appetite for understanding the +matter formed during ultra-relativistic collisions. +The +primary objective of these experiments is to create ex- +treme conditions of high temperature and/or energy den- +sities through compression or heating in high-energy nu- +clear collisions [1–3], where a system of deconfined quarks +and gluons can be formed. These extreme conditions lead +to the asymptotic freedom of the QCD system, where +the quarks and gluons are no longer confined inside the +nucleon [4–7]. Afterwards, the produced system under- +goes expansion through cooling of the systems where the +quarks and gluons are combined to form hadrons. +In +this course of action, the inelastic collisions cease at the +chemical freeze-out boundary where the stable particle +yields get fixed at chemical freeze-out temperature (Tch). +Finally, at the kinetic freeze-out boundary the elastic +collisions among the final state particles no longer exist +and a stream of particles gets detected in our detectors. +This enables us to get the kinetic freeze-out tempera- +ture (Tth) of the system from the transverse momentum +spectra of the identified particles. The chemical freeze- +out stage is well understood and is strongly supported +by experimental results [8]. Furthermore, the results are +in agreement with a Lattice Quantum Chromodynam- +ics (LQCD) model based on first principles and a well- +∗Corresponding Author Email: Raghunath.Sahoo@cern.ch +†Deceased +established hydrodynamic technique [9]. Similarly, the +kinetic freeze-out stage has also been explored by sev- +eral phenomenological works, where information about +the system is extracted by fitting different fitting func- +tions to the transverse momentum spectra of the final +state particles. +Over the past several years, numerical studies of lattice +QCD have been successful at high temperatures and with +zero chemical potential. +However, studying the phase +structure of QCD at non-zero chemical potential is one of +the most exciting problems in contemporary physics [10– +14]. +It is noteworthy that on the theoretical side, the +color superconducting and superfluid phases have been +conjectured at high baryon densities [15]. Therefore, it +is necessary to investigate the QCD phase transition uti- +lizing lattice gauge theory simulations at the non-zero +chemical potential. As discussed in Ref. +[8], a conse- +quence of the vanishing baryon-chemical potential leads +to the vanishing of strangeness chemical potential µs, +which implies that the strange quantum number is no +longer relevant for particle production. +The yield of +strange and multi-strange mesons and baryons in the fire- +ball is solely determined by their mass, m, spin degen- +eracy, g, and the temperature, T. At the LHC energies +near chemical freeze-out, the baryochemical potential is +expected to be zero due to both particles and antiparti- +cles being produced in equal numbers. However, assum- +ing the same case at the kinetic freeze-out temperature +is not entirely trivial. Thus, there can be a finite total +chemical potential at the kinetic freeze-out stage, and its +implications cannot be ignored. Specifically, this study +considers the implications of chemical potentials at ki- +netic freeze-out. It is worth noting that like heavy-ion +collisions, in LHC pp collisions, one also observes a finite +hadronic phase lifetime [16–18]. This prompts us to look +into the effect of chemical potential in TeV pp collisions +at the kinetic freeze-out using the non-extensive Tsallis +arXiv:2301.04038v1 [hep-ph] 10 Jan 2023 + +2 +distribution function which describes the identified par- +ticle spectra up to several tens of GeV/c. +As discussed earlier, the pT-spectra measured in pp col- +lisions at the LHC energies can give information about +the kinetic freeze-out stage of the collision. We gener- +ally fit a Boltzmann-type distribution function to the +pT-spectra to extract useful information. However, the +Boltzmann distribution function can only explain the +low-pT part of the spectra. +As we know, transverse +momenta up to hundred of GeV have been measured +in high-energy hadronic collisions. +This suggests that +the high-pT regime is also very significant to understand +the system formed in such collisions and cannot be ig- +nored. Power-law type distribution functions can explain +the high-pT part of the spectra very well, which comes +from the perturbative QCD. But to understand the sys- +tem fully, a unique distribution function is needed, which +can explain both the low- and high-pT part of the spec- +tra. The thermodynamically consistent Tsallis distribu- +tion function has been widely used for this purpose [19– +28]. The non-extensive parameter, q, in the Tsallis distri- +bution function quantifies the degree of deviation of the +system from the equilibrium state. In addition, Tsallis +temperature (T) and volume (V ) can also be extracted +by fitting the Tsallis distribution to the transverse mo- +mentum spectra. +This article is organized as follows. In the next sec- +tion II, we explicitly discuss the single-particle Tsallis +distribution to fit the transverse momentum spectra of +the identified hadrons and determine the temperature T, +volume V , and q with the details of the formulation of +estimating the chemical potential from kinetic freeze-out. +In section III, we discuss our results, and finally, we sum- +marize our results and conclude in section IV. +II. +FORMULATION +The Tsallis distribution function that satisfies the ther- +modynamic consistency relations [29–31] is given by, +E d3N +d3p = gV E +1 +(2π)3 +� +1 + (q − 1)E − µ +T +�− +q +q−1 +. +(1) +Here, E represents particles’ energy, V is the volume +of the system, g is the degeneracy factor, q is referred to +as the non-extensive parameter, T is the corresponding +temperature, p denotes the momentum, and µ is the total +chemical potential (µ = BµB + SµS + QµQ). Here B is +the baryon number, S is the strangeness quantum num- +ber and Q is the electric charge. Physical interpretation +of q can be found in ref. [32]. At mid rapidity, i.e, y = 0, +Eq. 1 in terms of transverse momentum, pT, transverse +mass, mT = +� +p2 +T + m2 can be rewritten as, +d2N +dpT dy = gV pT mT +(2π)2 +� +1 + (q − 1)mT − µ +T +�− +q +q−1 +. +(2) +For extracting Tsallis parameters for the identified +non-strange, strange, and multi-strange particles, Eq. 2 +has been used. The degeneracy factor g = 2 × (2s + 1) is +taken to be 2, 2, 4, 3, 8, 4 and 8 for π±, K±, p + ¯p, φ, Λ + +¯Λ, Ξ+ ¯Ξ, Ω+ ¯Ω respectively. Here, s is the particle’s spin, +and factor 2 is for the antiparticles. Σ0 and Λ are not ex- +perimentally distinguishable. Thus the degeneracy factor +for the Λ particle is 8. It is worth noted that the four +parameters T, V, q and µ in Eq. 2 have a redundancy for +µ ̸= 0 in ref. [33–35]. Precisely for a fixed values of q, let +T = T0 and V = V0 at µ = 0. So, comparing Eq. 2 for +µ = 0 and for a finite value of µ, we obtain the following +transformation relations, +T0 = T +� +1 − (q − 1) µ +T +� +, with +µ ≤ +T +q − 1, +(3) +V0 = V +� +1 − (q − 1) µ +T +� +q +1−q . +(4) +Hence, the variables T and V are functions of µ at +fixed values of q and can be determined if the parame- +ters (T0, V0) and q are known. This redundancy is not +present when µ = 0. +Then the transverse momentum +distribution in terms of these modified variables can be +written as, +d2N +dpT dy = gV0 +pT mT +(2π)2 +� +1 + (q − 1)mT +T0 +�− +q +q−1 +(5) +Here, the system’s chemical potential (µ) does not ap- +pear explicitly. Analogous to the volumes V and V0 de- +fined in Eq. 1 and 4, we also introduce the corresponding +radii R and R0 +V += 4π +3 R3, +(6) +V0 = 4π +3 R3 +0. +(7) +It is to be noted that most of the former analyses +have confused the Eq. 2 with Eq. 5 and arrived at in- +correct conclusions, particularly that different hadrons, +e.g. π, K, p, φ..., cannot be described by the same val- +ues of T and V at the LHC energies. This study will +show that this is based on T0 and V0, not T and V , at +the LHC energies. It has been concluded that, at chemi- +cal equilibrium, one has µ = 0 for all quantum numbers +as the number of particles and antiparticles are equal. +However, the equality of particle-antiparticle at kinetic +freeze-out implies equal chemical potential but not nec- +essarily zero. We stress that Eqns. 2 and 5 carry different +meanings where neither T0 is not equal to T, nor is V0 +equal to V . It is worth noting that we do not have µ in +Eqn. 5. +The purpose of the current paper is to resolve this is- +sue. For this, we choose the following technique: + +3 +1. To determine the three parameters T0, q, and V0, +we use Eqn. 5 to fit the transverse momentum dis- +tribution keeping all the parameters free. +2. Fix the value of parameter q, which is obtained +from the previous step. +3. Then perform the fit to the transverse momentum +distributions using Eqn. 2, keeping q fixed as deter- +mined in the previous step, which determines the +parameters T, V and the chemical potential µ. +4. We show that the choice of q, which is particle +species dependent, appears to be independent of +the chemical potential of the system for all parti- +cles. +5. At last, check the consistency with Eqns. 3 and 4. +Each step of the fitting procedure includes only three +parameters to describe the transverse momentum dis- +tributions. +This method was presented in +[34, 35]. +The present work conveys that the chemical potential +at kinetic freeze-out is not identical to that at chemi- +cal freeze-out. +The chemical potentials are considered +zero at chemical freeze-out, where thermal and chemical +equilibrium has been established. At kinetic freeze-out, +the observed particle-antiparticle symmetry only signifies +that the chemical potentials for particles must be equal +to those for antiparticles. However, they do not have to +be zero due to the absence of chemical equilibrium at ki- +netic freeze-out. The only limitation is that they should +be equal for particles and antiparticles. +III. +RESULTS AND DISCUSSION +The pT-spectra for non-strange, strange, and multi- +strange particles such as π±, K±, p + ¯p, φ, Λ + ¯Λ, Ξ + ¯Ξ, +and Ω+ ¯Ω are fitted upto pT = 6 GeV/c with a thermody- +namically consistent form of Tsallis distribution function +for pp collisions at √s = 7 TeV and √s = 13 TeV for +different multiplicity classes. +The fitting is performed +utilizing the ROOT library. In the foremost step, we de- +termine non-extensive parameter (q), temperature (T0) +and radius (R0) at zero chemical potential (µ = 0). In +the subsequent step, we fix the non-extensive parame- +ter, which is obtained from the first step, to calculate all +the fitting parameters such as temperature (T), the ra- +dius of the system (R), and chemical potential (µ). The +quality of the fits indicated by the reduced χ2 is listed +in the tables and also plotted in fig. 8. This indicates +that the spectra are well described by the non-extensive +distribution function. +Fig. 1 shows the variation of non-extensive parameter +at (µ = 0) and (µ ̸= 0) for pp collisions at √s = 7 TeV +for different final state particles. As in the formulation +section of the article, we have mentioned that the value +of the non-extensive parameter q is kept fixed, and we +perform the fit to the transverse momentum distributions +1 +1.05 +1.1 +1.15 +1.2 + 0) +≠ + +µ +q ( +1 +1.05 +1.1 +1.15 +1.2 + = 0) +µ +q ( + = 7 TeV (ALICE) +s +pp, +-π + + ++ +π +- + + K ++ +K + p +p + +φ +Λ + + +Λ ++ +Ξ + + +- +Ξ ++ +Ω + + +- +Ω +FIG. 1: (Color online) Comparison of non-extensive parame- +ter (q) at (µ = 0) and (µ ̸= 0) for pp collisions at √s = 7 TeV +for different final state particles. +0 +5 +10 +15 +20 +25 +30 +> +η +/d +ch + +η +/d +ch + +η +/d +ch + +η +/d +ch + +η +/d +ch + +η +/d +ch + +η +/d +ch + +η +/d +ch + +η +/d +ch + +η +/d +ch + +η +/d +ch + +η +/d +ch + +η +/d +ch + +η +/d +ch + 0, and sh(x) = −∞ when x is unlimited, +x < 0. +All objects whose existence is postulated by BST are sets, sometimes +called internal sets for emphasis. The Separation axiom holds for ∈- +formulas only. But it is common practice in the literature based on +the internal axiomatic approach to introduce definable external sets as +convenient abbreviations (see eg. Vakil [32], Diener and Stroyan [6]). +One can enrich the language of the theory by names for extensions of +arbitrary st-∈-formulas and in this way talk about st-∈-definable sub- +classes of the universe of all (internal) sets. We note that (a) this does +not amount to a formalization of a new type of entity called “external +set,” which is a more complicated task; and (b) this does not amount +to informal use of the term “external set,” either (in the sense of relying +on a background formalization). It is similar to the way set theorists +routinely employ classes in ZFC (the class of all sets, the class of all +ordinals). Such classes serve as convenient shortcuts in mathematical +discourse because one can work with them “as if” they were objects, + +6 +KAREL HRBACEK AND MIKHAIL G. KATZ +but they can in principle be replaced by their defining formulas. Ex- +ample 2.1 below illustrates this familiar procedure. +Let φ(v) be an st-∈-formula with arbitrary parameters. We employ +dashed curly braces to denote the class +x | φ(x) . We emphasize that +this is merely a matter of convenience; the expression z ∈ x | φ(x) is +just another notation for φ(z). Usually we denote classes by boldface +characters. Those classes that are included in some set are external +sets. If there is a set A such that ∀x (x ∈ A ←→ φ(x)), then the class +x | φ(x) +can be identified with the set A.3 Monads and galaxies +are some familiar examples of external sets that are usually not sets. +Let (M, d) be a metric space: the monad of a ∈ M is M(a) = +x ∈ +M | d(x, a) ≃ 0 , and the galaxy of a ∈ M is G(a) = +x ∈ M | +d(x, a) is limited . Some useful proper classes (ie, classes that are not +external sets) are V = x | x = x (the universe of all (internal) sets),4 +and S = x | st(x) +(the universe of all standard sets). +Example 2.1. Let M, f : M → M and a ∈ M be standard. +A +convenient way of defining continuity is as follows: The function f is +continuous at a if f[M(a)] ⊆ M(f(a)). But the use of external sets in +this definition can be eliminated by rephrasing it as: The function f is +continuous at a if for all x, d(x, a) ≃ 0 implies d(f(x), f(a)) ≃ 0. +Definable external collections of internal sets are adequately handled +in BST in this manner; we refer to Vakil [32] for a thorough discussion. +Difficulties arise only when higher level constructs on external sets are +needed, such as quotient spaces and power sets. We show how to handle +such difficulties in Subsections 3.2, 4.1, 4.4, and 4.5. +3. Subuniverses of the universe of BST +3.1. w-standard sets. Let us fix a set w and a standard set I such +that w ∈ I.5 +Definition 3.1. A set x is called w-standard (notation: stw(x)) if +x = f(w) for some standard function f with domain I. Next, we let +3In the model-theoretic approach external sets are by definition the sets that are +not internal. In the axiomatic approach it is customary to view internal sets as a +special case of external sets. +4The symbol I is often used for this purpose in the literature, but in the context of +st-∈-theories, where all sets are internal, the notation V seems more appropriate. +5The Boundedness axiom guarantees that some such I exists. This is one of the +reasons we prefer to work with BST rather than IST. + +NONSTANDARD HULLS AND LOEB MEASURES +7 +Sw = x | stw(x) +be the universe of all w-standard sets. The notion +of w-standardness depends only on w, not on the choice of the standard +set I. +Proposition 3.2. (a) +∀x (st(x) → stw(x)). +(b) +∀x (stw(x) → st(x)) holds if and only if w is standard. +(c) +stw(f) ∧ stw(x) ∧ x ∈ dom f → stw(f(x)). +Proof. (a) Let x be standard; we have x = cx(w) where cx is the con- +stant function with value x. +(b) If w is standard, then every f(w) for standard f is standard. If +w is nonstandard, let f(i) = i be the identity function on I. Then +f(w) = w is w-standard but not standard. +(c) Assume stw(f), stw(x) and x ∈ dom f. Then there are standard +functions F, G on I such that f = F(w) and x = G(w). Define a +function H on I by +H(i) = F(i)(G(i)) when the right side is defined; H(i) = ∅ otherwise. +Then H is a standard function on I and H(w) = f(x). +□ +Definition 3.3. A set w is good if there is ν ∈ N such that ν is w- +standard but not standard. +In particular, if ν ∈ N is nonstandard, then w = ν is good and, more +generally, w = ⟨ν, z⟩ is good for any set z. +The following facts are immediate consequences of known results (see +Kanovei and Reeken [15], Sections 3.3, 6.1, 6.2, esp. Theorem 6.2.6). +For easy reference we give the proofs in Section 5. Quantifiers with the +superscript stw range over w-standard sets. +Proposition 3.4. +(1) (Transfer from w-standard sets) Let φ be an +∈-formula with w-standard parameters. Then +∀stwx φ(x) → ∀x φ(x). +(2) (Countable Idealization into w-standard sets) +Let φ be an ∈-formula with w-standard parameters. If w is +good, then +∀stn ∈ N ∃x ∀m ∈ N (m ≤ n → φ(m, x)) ←→ ∃stwx ∀stn ∈ N φ(n, x). +In other words, (Sw, S, ∈) satisfies Countable Idealization. +(3) (Representability) +Let φ(v) be an ∈-formula with standard parameters. +If I is +standard, w ∈ I, x is w-standard, and φ(x) holds, then there is +a standard function f with dom f = I such that x = f(w) and +φ(f(i)) holds for all i ∈ I. + +8 +KAREL HRBACEK AND MIKHAIL G. KATZ +An equivalent formulation of (1) is Transfer into w-standard sets: +∃x φ(x) → ∃stwx φ(x). +Another equivalent formulation for φ(v1, . . . , vr) with w-standard pa- +rameters is +∀stwx1, . . . , xr [ φ(x1, . . . , xr) ←→ φstw(x1, . . . , xr)], +where φstw is the formula obtained from φ by relativizing all quan- +tifiers to stw. +In yet other words, (V, Sw, ∈) satisfies Transfer. +It +also satisfies Boundedness and, for good w, Bounded Idealization (see +Kanovei and Reeken [15], Theorem 6.1.16 (i)), but not Standardization +(ibid, Theorem 6.1.15). Although we do not need these results in this +paper, together they show that (V, Sw, ∈) satisfies all the axioms of +BST except Standardization. Note that here Sw plays the role of a +new “thick standard universe.” On the other hand, (Sw, S, ∈) satisfies +Transfer, Boundedness, Standardization and Countable Idealization; +here Sw plays the role of a new “thin internal universe.” +Idealization can be strengthened from N to sets of cardinality κ if w +is chosen carefully. We leave the technical definition of κ+-good sets to +Subsection 5.2 (see Definition 5.7). For our applications we need only +to know that for every standard uncountable cardinal κ and every z +there exist κ+-good w so that z is w-standard, a result which is also +proved there. +Proposition 3.5. (Idealization into w-standard sets over sets of car- +dinality ≤ κ) Let φ be an ∈-formula with w-standard parameters. If w +is κ+-good, then for every standard set A of cardinality ≤ κ +∀stfina ⊆ A ∃y ∀x ∈ a φ(x, y) ←→ ∃stwy ∀stx ∈ A φ(x, y). +Propositions 3.4 and 3.5 do not exhaust the properties of the uni- +verses Sw; for a list of further useful principles see Kanovei and Reeken +[15], Theorem 3.3.7. +3.2. Coding external sets. Definition 3.1 provides a natural way to +represent w-standard sets by standard functions: A w-standard ξ ∈ Sw +is represented by any standard f ∈ VI such that f(w) = ξ. Note that +every ξ has a proper class of representations. This causes some techni- +cal difficulties (see Section 5), which for our purposes are best resolved +by fixing a universal standard set V so that all objects of interest are +subsets of V or relations on V ; usually one requires R ⊆ V . By Rep- +resentability, for every ξ ∈ V ∩ Sw there exists a standard f ∈ V I +such that f(w) = ξ. Moreover, if ψ is an ∈-formula with standard +parameters and ψ(ξ) holds, then f can be chosen so that ψ(f(i)) holds + +NONSTANDARD HULLS AND LOEB MEASURES +9 +for all i ∈ I. The representation makes possible a coding of the ex- +ternal subsets of V ∩ Sw by standard sets, and the coding process can +be continued to the putative higher levels of the external cumulative +hierarchy. This process is enabled by the principle of Standardization. +Definition 3.6. Let φ(v) be a formula in the st-∈-language, with ar- +bitrary parameters. We let +st{x ∈ A | φ(x)} +denote the standard set S such that ∀stx (x ∈ S ←→ x ∈ A ∧ φ(x)). +The principle of Standardization postulates the existence of this set +and Transfer guarantees its uniqueness. Also by Transfer, if ψ is any +∈-formula with standard parameters and ∀stx ∈ A (φ(x) → ψ(x)), +then ∀x ∈ S ψ(x). +In particular, if φ(u.v) is a formula in the st-∈-language with arbi- +trary parameters, A, B are standard, and for every standard x ∈ A +there is a unique standard y ∈ B such that φ(x, y), then there is a +unique standard function F : A → B such that ∀stx ∈ A φ(x, F(x)) +holds. It suffices to let F = st{⟨x, y⟩ ∈ A × B | φ(x, y)}. +Definition 3.7. The w, V -code of an external set X ⊆ V ∩ Sw is the +standard set +Ψw,V (X) = X = st{f ∈ V I | f(w) ∈ X}. +In words, Ψw,V (X) is the standard set whose standard elements are +precisely the standard f ∈ V I with f(w) ∈ X. +We omit the subscripts w and/or V when they are understood from +the context. +The coding is trivially seen to preserve elementary set-theoretic op- +erations. +Proposition 3.8. For any external sets X1, X2 ⊆ V ∩ Sw: +(1) Ψ(∅) = ∅, +X1 ⊆ X2 ←→ Ψ(X1) ⊆ Ψ(X2); +(2) Ψ(X1∪X2) = Ψ(X1)∪Ψ(X2), +Ψ(X1∩X2) = Ψ(X1)∩Ψ(X2); +(3) Ψ(X1 \ X2) = Ψ(X1) \ Ψ(X2). +The coding preserves infinite unions and intersections as well. An +externally countable sequence of external subsets of V can be viewed +in BST as an external subset X of N × V , with Xn = x | ⟨n, x⟩ ∈ X +for standard n ∈ N. Let ⟨Xn | n ∈ N⟩ be the standard sequence such +that Xn = Ψ(Xn) holds for all standard n (its existence follows from +Standardization). Then +Ψ( +� +n∈N∩S +Xn) = +� +n∈N +Xn + +10 +KAREL HRBACEK AND MIKHAIL G. KATZ +because if f ∈ � +n∈N Xn is standard, then the least n ∈ N such that +f ∈ Xn is standard. Similarly, +Ψ( +� +n∈N∩S +Xn) = +� +n∈N +Xn +because if a standard f ∈ Ψ(� +n∈N∩S Xn), then f ∈ Xn for all standard +n ∈ N, and hence f ∈ � +n∈N Xn by Transfer. +It is convenient to relax the definition of coding so that every stan- +dard S ⊆ V I is a code of some external X ⊆ V ∩ Sw. +Definition 3.9. A standard set S codes X ⊆ V ∩ Sw if f(w) ∈ X for +each standard f ∈ S and for each ξ ∈ X there is some standard f ∈ S +such that f(w) = ξ. +If S codes X, then +Ψ(X) = st{f ∈ V I | f(w) = g(w) for some standard g ∈ S}. +A code of X has to contain a representation for each ξ ∈ X, but not +necessarily all such representations. +We note that coding is independent of the choice of the universal +standard set in the following sense: If X ⊆ V1 ∩ Sw and V1 ⊆ V2, then +S codes X viewed as a subset of V1∩Sw iff S codes X viewed as a subset +of V2 ∩ Sw. Hence the exact choice of V is usually of little importance. +For any set x, let cx be the constant function with value x. The +informal identification of x with cx enables the identification of a stan- +dard set A with a code +st{cx | x ∈ A} = {cx | x ∈ A} for A ∩ S. On +the other hand, the standard set AI is a code for A ∩ Sw. +Every standard S ⊆ V I is a code of the unique external set +XS = ξ ∈ V ∩ Sw | ξ = f(w) for some standard fw ∈ S . +Such S codes a subset of an external set X iff ∀stf (f ∈ S → f(w) ∈ X) +iff S ⊆ Ψ(X). +Consequently, it would make sense to interpret the +power set of � +X = Ψ(X) as a code for the “non-existent” external power +set Pext(X) of X, and the standard subsets of P( � +X) as codes for the +“external subsets of Pext(X).” Since BST does not allow collections of +external sets, this last remark cannot be made rigorous in it, but one +can proceed “as if” such higher order external sets existed. +Intuitively, there is a hierarchy of external sets built up over V ∩Sw: +H1 = Pext(V ∩ Sw) and Hn+1 = Pext(Hn) for standard n ∈ N +(we stop here to avoid further complications). This hierarchy cannot +be formalized in BST. +But the corresponding hierarchy over V I is +well-defined: + +NONSTANDARD HULLS AND LOEB MEASURES +11 +H1 = P(V I) and Hn+1 = P(Hn) for standard n ∈ N. +In BST one can work legitimately in the latter hierarchy while keeping +in mind that the coding establishes a (many-one) correspondence be- +tween Hn and Hn ∩S. Both hierarchies and their relationship could be +formalized in HST, a conservative extension of BST to a theory that en- +compassess also external sets (Hrbacek [10], Kanovei and Reeken [15]). +Note that the coding process treats elements ξ of V ∩ Sw as individ- +uals; they are not coded by Ψ. Thus ξ is represented by any f with +f(w) = ξ, but Ψ(ξ) is undefined. The set {ξ} ⊆ V ∩ Sw is coded by +{f}, even when {ξ} ∈ V ∩ Sw. In particular, if R is a binary relation +on V ∩ Sw, then the code for R is the standard binary relation +Ψ(R) = R = st{⟨f, g⟩ | f, g ∈ V I ∧ ⟨f(w), g(w)⟩ ∈ X}. +Similarly for functions and relations of higher arity. +Another variant of coding represents each ξ ∈ X by a single object. +Definition 3.10. Let f ∈ V I be standard. We let +fw,V = fw = st{g ∈ V I | g(w) = f(w)}. +We define standard sets V I/w = st{fw | f ∈ V I} and, for X ⊆ +V ∩ Sw, +�Ψw(X) = st{fw ∈ V I/w | f(w) ∈ X} += st{F ∈ V I/w | ∃stf ∈ V I (F = fw ∧ f(w) ∈ X)}. +The coding �Ψw is one-one. For standard X ⊆ V I/w we let +�Ψ +−1 +w (X) = X = ξ ∈ V ∩ Sw | ∃stf ∈ V I (fw ∈ X ∧ f(w) = ξ) . +One can obtain �Ψw(X) from Ψw(X) and vice versa: +�Ψw(X) = st{fw | f ∈ Ψw(X)} and Ψw(X) = st{f ∈ V I | fw ∈ +�Ψw(X)}. +Proposition 3.8 and the subsequent paragraph hold with Ψ replaced +by �Ψ. +We define the hierarchy �H1 = P(V I/w) and �Hn+1 = P( �Hn) for +standard n ∈ N. The coding �Ψw maps H1 onto �H1 ∩ S. It extends +informally to higher levels by �Ψw(X) = st{�Ψw(Y) | Y ∈ X} and +provides a one-one correspondence between Hn and �Hn ∩ S. +4. Nonstandard hulls and Loeb measures in BST +4.1. Nonstandard hulls of standard metric spaces. Let R be the +field of real numbers and let (M, d) be a standard metric space, so that +M is a standard set and the distance function is a standard mapping + +12 +KAREL HRBACEK AND MIKHAIL G. KATZ +d : M × M → R. A point x ∈ M is finite if d(x, p) is limited for some +(equivalently, for all) standard p ∈ M. +Fix an unlimited integer w ∈ N. We define the standard set Bw by +Bw = st{f ∈ MN | f(w) is a finite point of M}. +The standard relation Ew on Bw is defined by +Ew = st{⟨f, g⟩ ∈ Bw × Bw | d(f(w), g(w)) ≃ 0}. +Clearly Ew is reflexive, symmetric and transitive on standard elements +of Bw; it follows by Transfer that Ew is an equivalence relation on Bw. +We denote the equivalence class of f modulo Ew by fEw. +By Standardization, there is a standard function Dw : Bw ×Bw → R +determined by the requirement that Dw(f, g) = sh(d(f(w), g(w))) for +all standard f, g ∈ Bw. +For standard f, g ∈ Bw the distance +d(f(w), g(w))) ≤ d(f(w), f(0)) + d(f(0), g(0)) + d(g(0), g(w)) +is limited, so sh(d(f(w), g(w))) ∈ R. +For standard f, g, h ∈ Bw clearly Dw(f, g) = Dw(g, f), +Dw(f, h) ≤ Dw(f, g) + Dw(g, h), and Dw(f, g) = 0 iff ⟨f, g⟩ ∈ Ew. +Also, for standard f, g, f ′, g′ ∈ Bw we have +� +⟨f, f ′⟩ ∈ Ew ∧ ⟨g, g′⟩ ∈ Ew +� +→ Dw(f, g) = Dw(f ′, g′). +By Transfer these properties hold for all f, g, h, f ′, g′ ∈ Bw. +We observe that, for the natural choice V = M ∪R, Bw is a code for +the external set +Bw = x ∈ M ∩ Sw | x is finite , +Ew is nothing but a code for the external equivalence relation +Ew = ⟨x, y⟩ ∈ Bw × Bw | d(x, y) ≃ 0 , +and Dw is a code for the external function Dw : Bw × Bw → R ∩ S +defined by Dw = ⟨x, y, r⟩ | r = sh(d(x, y)) . +The construction of the nonstandard hull of (M, d) by the usual ex- +ternal method would form the quotient space Bw/Ew of Bw modulo +Ew. This step cannot be carried out in BST. For x ∈ Bw, the equiv- +alence class of x modulo Ew is Mw(x) = z ∈ M ∩ Sw | d(x, z) ≃ 0 , +but the collection of the classes Mw(x) for all x ∈ Bw is not supported +by BST. However, for standard f ∈ Bw, the set fEw is a code of Mw(x) +when f(w) = x. Therefore Bw/Ew can be replaced by Bw/Ew, which +is a standard set in BST. The standard elements of Bw/Ew are pre- +cisely the codes of the monads Mw(x) for x ∈ Bw. Hence Bw/Ew is + +NONSTANDARD HULLS AND LOEB MEASURES +13 +a code (as discussed in Subsection 3.2) of the “non-existent” (in BST) +quotient space Bw/Ew. +We stress that while external sets serve as a motivation for our con- +structions, they are not actually used in them; the constructions deal +only with sets of BST. The same applies to the rest of this section. +We let � +Mw = Bw/Ew be the standard quotient space of Bw modulo +Ew. From now on we often omit the subscript w when it is understood +from the context. +The function D factors by E to a (standard) function �D = D/E on +� +M, defined by �D(fE, gE) = D(f, g) (as shown above, the value of �D is +independent of the choice of representatives f, g). It is clear that �D is +a metric on � +M. +The embedding c of M into � +M is via constant functions: for x ∈ M, +c(x) = (cx)E where cx is the constant function on N with value x. +Trivially, the embedding c preserves the metrics. We identify M with +its image in � +M under this embedding. +We emphasize that the structure (� +M, �D ) depends on the choice of +the parameter w, an unlimited integer. A metric space has a unique +completion up to isometry, but it may have many non-isometric non- +standard hulls. +Example 4.1. Let M = Q be the set of rationals and d(x, y) = |x−y| +be the usual metric on Q. We fix an unlimited w ∈ N. We note that, +for standard f, +f ∈ Bw ←→ f ∈ QN ∧ f(w) is limited, +and for standard f, g ∈ Bw +⟨f, g⟩ ∈ Ew ←→ f(w) ≃ g(w) +←→ f(w), g(w) ∈ M(a) for a = sh(f(w)) = sh(g(w)). +By Standardization, there is a standard function Γ : Bw/Ew → R +such that, for standard f, Γ(f/E) = sh(f(w)). It is easy to verify that +Γ ↾ (Bw/Ew) ∩ S is a one-one mapping of � +Mw ∩ S onto R ∩ S that +preserves the metrics: +�D(f/E, g/E) = D(f, g) = sh(|f(w) − g(w)|) += | sh(f(w)) − sh(g(w))| = |Γ(f) − Γ(g)|. +Moreover, Γ((cx)E) = x for any standard x ∈ Q. By Transfer, Γ is +an isometric isomorphism of � +Mw and R which is the identity on Q. In +particular, the nonstandard hull is independent of the choice of w. + +14 +KAREL HRBACEK AND MIKHAIL G. KATZ +Example 4.2. Let M = N and let d be the discrete metric on M; ie, +d(x, z) = 1 for all x, z ∈ M, x ̸= z. As all points of M are finite with +respect to this metric, we have Bw = NN. Also +Ew = st{⟨f, g⟩ ∈ NN × NN | f(w) = g(w)} +and, for standard f ∈ NN, f/E = st{g ∈ NN | g(w) = f(w)} = fw. +The space M is identified with a subset of � +Mw via Γ : (cx)w �→ x. +In Remark 5.5 we establish that � +Mw = Bw/Ew = NN/w is exactly +the ultrapower of M by Uw, an ultrafilter over N generated by w; in +particular, it has the cardinality of the continuum. +Also, �D is the +discrete metric on � +Mw. By replacing N with I as in Remark 4.6 one +can obtain nonstandard hulls of arbitrarily large cardinality. +Example 4.3. Approachable points play an impotant role in the study +of nonstandard hulls. We define the concept as follows. +Let (M, d) be a standard metric space. A point x ∈ M is approach- +able if for every standard ǫ > 0 there is a standard a ∈ M such that +d(x, a) ≤ ǫ. +The approachable points in M ∩ Sw become exactly the standard +points of the closure of M in its nonstandard hull � +Mw. Indeed, for +standard f ∈ Bw with f(w) = x ∈ M we have +x is approachable ←→ ∀stǫ > 0 ∃sta ∈ M (d(x, a) ≤ ǫ) ←→ +∀stǫ > 0 ∃sta ∈ M ( �D(f/E, a) ≤ ǫ) ←→ ∀ǫ > 0 ∃a ∈ M ( �D(f/E, a) ≤ ǫ), +where the last step is by Transfer. +Thus in Example 4.1 all finite +x ∈ Q are approachable and consequently all standard points in R are +in the closure of M = Q. By Transfer, this is true for all points in +R. In Example 4.2 all nonstandard points of N are inapproachable and +therefore M is closed in � +Mw. +Remark 4.4. In this and other constructions in this section we use +the coding based on Ψ. The advantage of this choice is that it produces +spaces of functions (see in particular Subsection 4.4). One can use �Ψ +instead, and define eg. +�Bw = +st{fw | f(w) is a finite point of M}. +The advantage here is that one gets an isomorphism of the external +structure (Bw, Ew, Dw) with (Bw ∩ S, Ew ∩ S, Dw ∩ S). It is thus +immediately apparent that (( �Bw/ �Ew) ∩ S, ( �Dw/ �Ew) ∩ S) would be iso- +morphic to (Bw/Ew, Dw/Ew) if the latter quotient could be formed in +BST. (It can be formed in HST and this claim is a theorem there.) +For the final result the choice of coding method does not matter. +Proposition 4.5. The structures ( �Bw/ �Ew, �Dw/ �Ew) and (Bw/Ew, Dw/Ew) +are isomorphic. + +NONSTANDARD HULLS AND LOEB MEASURES +15 +Proof. For standard f, g ∈ MN, f ∈ Bw iff fw ∈ �Bw, ⟨f, g⟩ ∈ Ew iff +⟨fw, gw⟩ ∈ �Ew and Dw(f, g) = �Dw(fw, gw). +□ +Remark 4.6. In the construction of (� +M, �D ) one can replace N by any +infinite set I, as long as w ∈ I is good. If Sw1 ⊆ Sw2, fix a stan- +dard function h ∈ II2 +1 +such that h(w2) = w1 and define the standard +mapping H : MI1 → MI2 by H(f) = f ◦ h. +Then H is an em- +bedding of (Bw1, Ew1, Dw1) into (Bw2, Ew2, Dw2) in an obvious sense, +and it factors to an isometric embedding of (Bw1/Ew1, Dw1/Ew1) into +(Bw2/Ew2, Dw2/Ew2). Given any good w1 and w2, let w = ⟨w1, w2⟩; +then both (� +Mw1, �Dw1) and (� +Mw2, �Dw2) embed into (� +Mw, �Dw). +4.2. Completeness of the nonstandard hull. This subsection il- +lustrates how one can work with the nonstandard hull as constructed +in Subsection 4.1. +Theorem 4.7. (� +Mw, �Dw ) is a complete metric space. +Proof. Let ⟨Fn | n ∈ N⟩ be a standard Cauchy sequence in (� +M, �D ); +we prove that it converges to some F ∈ � +M. +Using the Axiom of Countable Choice we obtain a standard sequence +⟨fn | n ∈ N⟩ such that Fn = (fn)E for all n ∈ N. +For k ∈ N let nk be the least element of N greater than or equal to +k such that +∀m, n +� +nk ≤ n ≤ m → �D(Fn, Fm) < +1 +k+1 +� +; +note that the sequence ⟨nk | k ∈ N⟩ is standard. From the definition +of �D we obtain, for standard k, +∀stm, n +� +nk ≤ n ≤ m → d(fn(w), fm(w)) < +1 +k+1 +� +. +Hence ∀stk ∃stm ∀ℓ ≤ k +� +nℓ ≤ m ∧ ∀n +� +nℓ ≤ n ≤ m → d(fn(w), fm(w)) < +1 +ℓ+1 +�� +. +By Countable Idealization into w-standard sets we get +∃stwm ∀stk +� +nk ≤ m ∧ ∀n +� +nk ≤ n ≤ m → d(fn(w), fm(w)) < +1 +k+1 +�� +. +Fix such an m; clearly it is unlimited. Consider the standard point +p = fn0(0). We have d(p, fm(w)) ≤ d(fn0(0), fn0(w))+d(fn0(w), fm(w)). +The first term on the right side of the inequality is limited because fn0 +is standard and fn0 ∈ B, and the second term is < 1 (take k = 0). +Hence fm(w) is a finite element of M. +We note that fm(w) is w-standard; hence there is a standard function +f ∈ MN such that f(w) = fm(w). It follows that f ∈ B. We let +F = fE ∈ � +M. + +16 +KAREL HRBACEK AND MIKHAIL G. KATZ +We have that, for all standard k, +∀stn +� +nk ≤ n → d(fn(w), f(w)) < +1 +k+1 +� +, +and hence +∀stn +� +nk ≤ n → �D(Fn, F) ≤ +1 +k+1 +� +. +This shows that the sequence ⟨Fn | n ∈ N⟩ converges to F. +□ +The proof of Theorem 4.2 goes through for any infinite set I in place +of N, as long as w ∈ I is good. +4.3. Nonstandard hulls of standard uniform spaces. We gener- +alize the construction of nonstandard hulls in BST to uniform spaces. +Let (M, ∆) be a standard uniform space. That is, M is a standard set +and ∆ is a standard family of pseudo-metrics on M which endows M +with a Hausdorff uniform structure. In a uniform space, x ∈ M is finite +if for all standard d ∈ ∆, d(x, p) is limited for some (equivalently: for +all) standard p ∈ M. Points x and y are infinitely close if d(x, y) ≃ 0 +for all standard d ∈ ∆. +We fix a standard infinite set I and a w ∈ I so that Idealization into +w-standard sets over standard sets of cardinality ≤ κ holds for κ = +max{|∆|, ℵ0} (see Proposition 3.5). A construction of the nonstandard +hull of (M, ∆) can now proceed much as in Subsection 4.1. We omit +the subscripts indicating its dependence on w. +We let B = st{f ∈ MI | f(w) is a finite point of M} and +E = st{⟨f, g⟩ ∈ B × B | d(f(w), g(w)) ≃ 0 for all standard d ∈ ∆}. +For each standard d ∈ ∆ the standard function D on B × B, as +well as � +M, �D and c, are defined as in Subsection 4.1. Each �D is a +pseudo-metric on � +M. We let �∆ = st{ �D | d ∈ ∆ ∩ S}. +Theorem 4.8. The structure (� +M, �∆) is a complete Hausdorff uniform +space and c embeds (M, ∆) into (� +M, �∆). +Proof. The proof follows the lines of the proof of Theorem 4.7; the main +difference is that Cauchy sequences have to be replaced by Cauchy nets. +Let ⟨Λ, ≤⟩ be a standard directed set and ⟨Fλ | λ ∈ Λ⟩ a standard +Cauchy net indexed by Λ. Using the Axiom of Choice one obtains a +standard net ⟨fλ | λ ∈ Λ⟩ such that fλ ∈ Fλ holds for all λ; then +⟨fλ(w) | λ ∈ Λ⟩ is a w-standard net. +The Cauchy property implies that for every standard d ∈ ∆ there is +a standard sequence ⟨λd +k | k ∈ N⟩ of elements of Λ such that λd +k ≤ λd +k+1 +holds for all k and +∀k ∀λ, µ ∈ Λ +� +λd +k ≤ λ ≤ µ → �d(Fλ, Fµ) < +1 +k+1 +� +. + +NONSTANDARD HULLS AND LOEB MEASURES +17 +Hence +∀stk ∀stλ, µ ∈ Λ +� +λd +k ≤ λ ≤ µ → d(fλ(w), fµ(w)) < +1 +k+1 +� +. +We conclude that for every standard k ∈ N, every standard finite +∆0 ⊆ ∆ and every standard finite Λ0 ⊆ Λ there is a standard µ ∈ Λ +such that for all ℓ ≤ k, all d ∈ ∆0 and all λ ∈ Λ0 +λd +ℓ ≤ µ ∧ +� +λd +ℓ ≤ λ → d(fλ(w), fµ(w)) < +1 +ℓ+1 +� +. +By Idealization into w-standard sets over sets of cardinality ≤ κ = +max{|∆|, ℵ0} we get a w-standard µ ∈ Λ such that for all standard k, +all standard d ∈ ∆ and all standard λ ∈ Λ +λd +k ≤ µ ∧ +� +λd +k ≤ λ → d(fλ(w), fµ(w)) < +1 +k+1 +� +. +Fix such a µ; as in Subsection 4.7 we have that d(p, fµ(w)) is limited +for every standard d ∈ ∆, ie, fµ(w) is a finite point of M. +By Representability there is a standard function f ∈ MI such that +f(w) = fµ(w). It follows that f ∈ B. Let F = f/E ∈ � +M. +We have that, for all standard d ∈ ∆ and k ∈ N, +∀stλ ∈ Λ +� +λd +k ≤ λ → d(fλ(w), f(w)) < +1 +k+1 +� +, +and hence +∀stλ ∈ Λ +� +λd +k ≤ λ → �D(Fλ, F) ≤ +1 +k+1 +� +. +This shows that the net ⟨Fλ | λ ∈ Λ⟩ converges to F. +□ +4.4. Internal normed vector spaces. Under many circumstances +the type of construction carried out in Subsections 4.1, 4.3 generalizes +to internal structures. We illustrate it in the case of normed vector +spaces. +Let M be an internal normed vector space over R. This means that +M, the operations of addition + on M × M and scalar multiplication +· on R × M, and the R-valued norm ∥.∥ on M, are (internal) sets and +satisfy the usual properties. In order to be able to apply our coding +technique we fix a standard set I and a good w ∈ I so that the set M, +the above operations and the norm belong to Sw. Other parameters +relevant to a particular investigation can also be made to belong to +Sw. By Transfer in (V, Sw, ∈), the properties of these objects that are +expressible by ∈-formulas continue to hold in Sw, so we can carry out +the desired construction “over Sw” rather than “over V.” +We first describe the external construction. Let +Bw = x ∈ M∩ Sw | ∥x∥ is limited +and Ew = x ∈ M∩ Sw | ∥x∥ ≃ 0 . +It is clear that Bw is an external vector space over the external field +R ∩ S and Ew is its subspace. Define an external equivalence relation + +18 +KAREL HRBACEK AND MIKHAIL G. KATZ +≈ on Bw by x ≈ y ←→ x − y ∈ Ew ←→ ∥x − y∥ ≃ 0. Obviously, +for x1, x2, y1, y2 ∈ Bw and c ∈ R ∩ S we have x1 ≈ y1 ∧ x2 ≈ y2 → +x1 + x2 ≈ y1 + y2, c · x1 ≈ c · y1 and ∥x1∥ ≃ ∥y1∥. If external collections +of external sets were available in BST, one could now form the quotient +space Bw/Ew with the norm ∥xEw∥ = sh(∥x∥), which would then be +an external normed vector space over the external field R ∩ S. This +construction is of course not possible in BST directly, but the coding +introduced in Subsection 3.2 enables us to carry it out and produce a +standard normed metric space which, when viewed from the standard +point of view, is (externally) isomorphic to Bw/Ew. +It follows from Representability that there is a standard function +⟨(Mi, +i, ·i) | i ∈ I⟩ such that (Mw, +w, ·w) = (M, +, ·) and, for all +i ∈ I, (Mi, +i, ·i) is a vector space over R. Let +Bw = st{f ∈ Πi∈IMi | ∥f(w)∥ is limited}; +Ew = st{f ∈ Πi∈IMi | ∥f(w)∥ ≃ 0}. +Addition and scalar multiplication on Bw are defined pointwise: +(f + g)(i) = f(i) +i g(i) +and +(c · f)(i) = c ·i f(i) +for f, g ∈ Bw, c ∈ R, and all i ∈ I. +By Standardization, there is +a standard function ∥.∥ such that for all standard f ∈ Bw we have +∥f∥ = sh(∥f(w)∥). +It is routine to verify that Bw is a standard vector space over R, +∥.∥ is a pseudo-norm on Bw, and f ∈ Ew ←→ ∥f∥ = 0. The quotient +�Ew = Bw/Ew is thus a well-defined standard normed vector space. The +proof given in Subsection 4.2 shows that �Ew = Bw/Ew is complete. +4.5. Loeb measures. Let (Ω, A, µ) be an internal finitely-additive +measure space, with Ω ⊆ O for a standard set O. As discussed in +Subsection 6.3, attempts to construct its Loeb extension “over V” are +only partially successful in BST. We fix a standard set I and a good +w ∈ I so that Ω, A, µ are w-standard. By Transfer, (Ω, A, µ) is an +internal finitely-additive measure space in the sense of Sw, and we con- +struct the Loeb extension “over Sw.” We usually do not indicate the +dependence on the choice of w. In this example it is convenient to +employ the variant of coding from Definition 3.10. +For every X ∈ A ∩ Sw let +(1) +[X] = st{fw | f ∈ OI ∧ f(w) ∈ X ∩ Sw}. +If X, X1, X2 ∈ A ∩ Sw, then the equivalences fw ∈ [X1 ∩ X2] iff fw ∈ +[X1] ∧ fw ∈ [X2] and fw ∈ [Ω \ X] iff fw ∈ [Ω] ∧ fw /∈ [X] hold + +NONSTANDARD HULLS AND LOEB MEASURES +19 +for all standard f ∈ OI. By Transfer, [X1 ∩ X2] = [X1] ∩ [X2] and +[Ω \ X] = [Ω] \ [X]. Let +B = st{A ∈ P(OI) | A = [X] for some X ∈ A ∩ Sw}. +Using Transfer again, it follows that B is a standard algebra of +subsets of [Ω]. +We note that X1, X2 ∈ A ∩ Sw, X1 ̸= X2, implies +[X1] ̸= [X2], so for standard A ∈ B there is a unique X ∈ A ∩ Sw with +A = [X]. +A standard finitely additive measure m on the algebra B with val- +ues in the interval [0, +∞] is determined by the requirement that for +standard A = [X] ∈ B +m(A) = sh(µ(X)). +The measure space ([Ω], B, m) satisfies Carath´eodory’s condition. +Lemma 4.9. If ⟨Ak | k ∈ N⟩ is a sequence of mutually disjoint sets +in B, A ∈ B, and A = � +k∈N Ak, then m(A) = Σk∈N m(Ak). +Proof. In view of Transfer, it suffices to prove this claim under the +assumption that ⟨Ak | k ∈ N⟩ and A are standard. +Suppose A = [X] where X ∈ A ∩ Sw and for each standard k, +Ak = [Xk] where Xk ∈ A ∩ Sw. Clearly Xk are mutually disjoint and +Xk ⊆ X for all standard k. +Assume that for every standard n there is a w-standard x ∈ X such +that x /∈ Xk holds for all k ≤ n. By Countable Idealization into Sw +there is a w-standard x ∈ X such that x /∈ Xk holds for all standard +k. By Representability, x = g(w) for some standard g ∈ OI. Then +gw ∈ A but ∀stk (gw /∈ Ak) and, by Transfer, ∀k (gw /∈ Ak). This is a +contradiction. +Therefore there is a standard n such that ∀stwx ∈ X ∃k ≤ n (x ∈ +Xk). +It follows that ∀stF ∈ A ∃k ≤ n (F ∈ Ak). +By Transfer, +� +k∈N Ak = � +k≤n Ak and, by finite additivity of m, we obtain m(A) = +Σk≤n m(Ak) = Σk∈N m(Ak). +□ +We conclude that m can be extended to a σ-additive measure m with +values in [0, +∞] on the σ-algebra B generated by B. The measure- +theoretic completion of ([Ω], B, m) is the desired Loeb measure space; +we denote it ([Ω], �B, �m) (of course, it depends on the choice of w). +Instead of an appeal to the Carath´eodory’s theorem, a direct proof +along the lines of [11] can be given; see also Albeverio et al. [1], Remark +3.1.5 and the references therein. +Remark 4.10. In order to explicate the relationship of ([Ω], �B, �m) to +the usual external Loeb measure space, we first recall that Sw satisfies + +20 +KAREL HRBACEK AND MIKHAIL G. KATZ +the statement that (Ω, A, µ) is a finitely-additive measure space. The +Loeb construction carried out over Sw would start with the external +finitely additive measure space (Ω, A, m), where Ω = Ω ∩ Sw, A = +X ∩ Sw | X ∈ A ∩ Sw , and m(X ∩ Sw) = sh(µ(X)) for X ∈ A. +This space would be extended to an external σ-additive measure space +using the Carath´eodory’s theorem, and then completed. We now take +V = Ω ∪ R, say, and note that for X ∈ A ∩ Sw, [X] is a w-code of +X ∩ Sw, and hence B is a w-code for A. It follows that ([Ω], B, m) is a +w-code for (Ω, A, m). The above construction cannot be carried out in +BST directly for the external measure space (Ω, A, m), but presents +no difficulties for its w-code ([Ω], B, m), a standard finitely-additive +measure space. +Remark 4.11. We compare Loeb measure spaces obtained from the +same (Ω, A, µ) for different choices of the parameter w. We use sub- +scripts to indicate dependence on this parameter and fix good w ∈ I, +z ∈ J where I, J are standard and Sw ⊆ Sz. +Proposition 4.12. There is a standard embedding �H = �Hw,z of �Bw +into �Bz that preserves complements and countable unions and restricts +to an isomorphism of Bw and Bz. If mw is σ-finite, then also �mw(B) = +�mz( �H(B)) for all B ∈ �Bw. +Proof. By Standardization, there is a standard function �H : �Bw → �Bz +such that �H(B) = �Ψz( �Ψ +−1 +w (B)) for standard B ∈ �Bw. ForA ∈ Bw ∩ S, +if A = [X]w for X ∈ A ∩ Sw then �H(A) = [X]z ∈ Bz ∩ S, and vice +versa. This shows that �H maps Bw ∩ S onto Bz ∩ S, and hence, by +Transfer, Bw onto Bz. +In Subsection 3.2 we point out that the coding �Ψ preserves comple- +ments and countable unions, so the same holds for �H. We give some de- +tails for the countable unions. Let B = � +n∈N Bn, where ⟨Bn | n ∈ N⟩ +is standard and B, Bn ∈ �Bw for all n ∈ N. Define the external set +X = ⟨n, x⟩ | x ∈ �Ψ +−1 +w (Bn) ⊆ (N∩S)×O, so that Bn = �Ψw(Xn). We +then have B = �Ψw(� +n∈N∩S Xn), so �Ψ +−1 +w (B) = � +n∈N∩S Xn and �H(B) = +�Ψz( �Ψ +−1 +w (B)) = �Ψz(� +n∈N∩S Xn) = � +n∈N �Ψz( �Ψ +−1 +w (Bn)) = � +n∈N �H(Bn). +It is clear from the definition of m(A) that mw(A) = mz( �H(A)) +holds for standard A ∈ Bw, and hence by Transfer, for all A ∈ Bw. +If mw is σ-finite, then the completed Carath´eodory’s measure �mw is +uniquely determined. If �mw(B) ̸= �mz( �H(B)) for some B ∈ �Bw, then +�mw and �mz ◦ �H would be two distinct extensions of mw from Bw to �Bw, +a contradiction. +□ + +NONSTANDARD HULLS AND LOEB MEASURES +21 +4.6. Lebesgue measure from Loeb measure. As is well known, +the Lebesgue measure can be obtained from a suitable Loeb measure.6 +We give the gist of the argument in our framework, for the interval +[0, 1]. More details can be found in Albeverio at al. [1] (using a model- +theoretic approach). +For n ∈ N let Tn = {i/n | 0 ≤ i ≤ n}. Fix a nonstandard integer +w ∈ N, and let T = Tw (in model-theoretic frameworks the equivalent +of T is called “hyperfinite time line”). Let O = [0, 1], Ω = T , A = +P(T ) and µ the counting measure on T , ie, µ(X) = |X|/|T | for all sets +X ⊆ T . The construction in the preceding subsection, with I = N, +yields the Loeb measure ([Ω], �B, �m). +For every standard A ⊆ [0, 1] define +•A = st{fw ∈ [Ω] | f(w) ≃ c for some standard c ∈ A}. +This just means that •A is a w-code for sh−1(A) ∩ T ∩ Sw. Standard +elements of the set •A are those fw ∈ [Ω] whose value “at infinity” (ie, +at w) is infinitely close to a standard real in A. +Let L = st{A ⊆ [0, 1] | •A ∈ �B } and let ℓ be the standard function +on L determined by the requirement that ℓ(A) = �m(•A) for all standard +A ∈ L. +Theorem 4.13. The triple ([0, 1], L, ℓ) is the Lebesgue measure space +on [0, 1]. +Proof. We prove that L is a σ-algebra containing all standard open +intervals (a, b) ⊆ [0, 1] and all singletons {c} for standard c ∈ [a, b]. +We also prove that ℓ is σ-additive and ℓ((a, b)) = b − a, ℓ({c}) = 0 for +all standard open intervals and singletons, respectively. This implies +that L contains all Lebesgue measurable subsets of [0, 1] and that ℓ +is the Lebesgue measure for such subsets. For a proof (in the model- +theoretic framework) that all sets in L are Lebesgue measurable see [1], +Proposition 3.2.5; it can be easily adapted to our framework. +Let ⟨Ak | k ∈ N⟩ be a standard sequence of elements of L and +let A = � +k∈N Ak. Then •Ak ∈ �B holds for all standard k ∈ N, and +we obtain that •A = � +k∈N +•Ak, because if f(w) ≃ a for some standard +a ∈ A, then a ∈ Ak for some standard k ∈ N by Transfer. +As �B +6A nonstandard construction of the Lebesgue measure can also be carried out with- +out using a Loeb measure as an intermediate step. In the axiomatic approach, one +method for doing so is developed by Lyantse and Kudryk [24], Appendix A, in the +framework of IST. Another way is implicit in Hrbacek [11] and explicitly presented +in Hrbacek and Katz [12] in the framework of SCOT, a subtheory of IST and BST +that conservatively extends ZF + Dependent Choice. For a “radically elementary” +approach see Cartier and Perrin [2, 3]. + +22 +KAREL HRBACEK AND MIKHAIL G. KATZ +is a σ-algebra, •A ∈ �B and we conclude that A ∈ L. Furthermore, +ℓ(A) = �m(•A) = Σk∈N �m(•Ak) = Σk∈N ℓ(Ak), establishing σ-additivity +of ℓ. +Given a standard open interval (a, b) ⊆ [0, 1], we let Xa,b = T ∩(a, b). +We have [Xa,b] ∈ B and m([Xa,b]) = sh(µ(Xa,b)) = b − a. +Let A = (a, b); it remains to observe that, for standard fw ∈ [Ω], fw ∈ +•A iff fw ∈ [Xa+1/m, b−1/m] for some standard m ∈ N\{0}. Standardiza- +tion gives the function ⟨Am | m ∈ N\{0}⟩ where Am = [Xa+1/m, b−1/m] +for standard m, and Transfer implies •A = � +m∈N\{0} Am. It follows that +•A ∈ B because the latter is a σ-algebra, and that m(A) = b−a because +m is a σ-additive extension of m. Hence A ∈ L and ℓ(A) = b − a. +The argument for A = {c} is similar, using the fact that •A = +� +m∈N\{0} Am with Am = [Xc−1/m, c+1/m] for standard m. +□ +4.7. Neutrices and external numbers. This is another application +of nonstandard analysis that extensively uses external sets; see Dinis +and van den Berg [7]. +A neutrix is a convex additive subgroup of +R. With the exception of {0} and R, neutrices are externals sets; the +monad M(0) and the galaxy G(0) are nontrivial examples. An external +number is an algebraic sum of a real number and a neutrix. Addition +and multiplication of external numbers are defined by the Minkowski +operations. Let N denote the collection of all neutrices and E denote +the collection of all external numbers. +Even in HST, N and E are +(definable) proper classes of external sets; they are “too large” to be +external sets (see Kanovei and Reeken [14]). +This difficulty can be remedied by relativizing these concepts to Sw +(for good w). In fact, Nw = X∩Sw | X ∈ N +and Ew = a+X∩Sw | +a ∈ R ∩ Sw ∧ X ∈ N . The collections Nw and Ew are external sets +(of external sets) in HST. +As the natural embedding of R ∩ Sw into Ew given by r �→ r + +{0} is crucial for applications of these concepts, it may be best for +most purposes to avoid coding as much as possible. For example, the +study of algebraic properties of operations + and × on Ew can be +carried out in BST while viewing external numbers as external subsets +of R ∩ Sw . However, work with external numbers often focuses on the +structure (Ew, +, ×), its subsets, functions with values in it, and so on. +Then one can use the techniques of Subsection 3.2; see in particular +Definition 3.10 with V = R, to code Nw and Ew by standard structures. +First, the external set R ∩ Sw is coded by the standard set RI/w. +As shown in Remark 5.5, RI/w = RI/Uw = ∗R, the hyperreals con- +structed as the standard ultrapower of R by the standard ultrafilter + +NONSTANDARD HULLS AND LOEB MEASURES +23 +Uw generated by w. Let ∗< , ∗+ and ∗× be the ordering, addition and +multiplication on the hyperreals. The coding provides an external iso- +morphism between (∗R∩S, ∗<, ∗+, ∗×) and (R∩Sw, <, +, ×), so we can +informally identify ∗R∩S and R∩Sw. Neutrices and external numbers +in the hyperreals (∗R, ∗<, ∗+, ∗×) can be defined the same way as in R. +External numbers in R ∩ Sw are coded by the standard external num- +bers in the hyperreals ∗R. The coding preserves algebraic operations on +external numbers. The collections Nw and Ew are coded respectively +by the standard sets N and E of all neutrices and external numbers in +∗R. +This approach is admittedly rather awkward. In HST the universes +Sw can be extended to “external universes” WF(Sw) (see Kanovei and +Reeken [15], Sections 6.3, 6.4 for details). Perhaps the most practical +way to handle external numbers would be to work with Nw and Ew in +these universes and in the end code the final results in BST, if desired. +5. Subuniverses and ultrapowers +5.1. Subuniverses Sw and ultrapowers. The universe Sw is closely +connected to the ultrapower of the standard universe by a standard +ultrafilter. +The principle of Standardization implies that there is a standard set +Uw such that +(2) +∀stX (X ∈ Uw ←→ X ∈ P(I) ∧ w ∈ X). +Clearly (i) ∅ /∈ Uw, and for standard X, Y ∈ P(I) +(ii) X ∈ Uw ∧ X ⊆ Y → Y ∈ Uw; +(iii) X, Y ∈ Uw → X ∩ Y ∈ Uw; +(iv) X ∈ Uw ∨ (I \ X) ∈ Uw. +By Transfer, (ii) – (iv) hold for all X, Y ∈ P(I), so Uw is an ultrafilter +over I. +One sees easily that Uw is nonprincipal if and only if w is +nonstandard. Conversely, Bounded Idealization of BST implies that +for every standard ultrafilter U over I there are (many) w ∈ I such +that U = Uw. +The ultrapower of the universe of all sets V by a standard ultrafilter +U is defined in the usual way. One defines an equivalence relation =U +on VI by +(3) +f =U g ←→ {i ∈ I | f(i) = g(i)} ∈ U, +and a membership relation +(4) +f ∈U g ←→ {i ∈ I | f(i) ∈ g(i)} ∈ U. + +24 +KAREL HRBACEK AND MIKHAIL G. KATZ +The usual procedure at this point is to form equivalence classes fU +of functions f ∈ VI modulo =U, using “Scott’s trick” of taking only +the functions of the minimal von Neumann rank to guarantee that the +equivalence classes are sets: Let +fU = {g ∈ VI | g =U f and ∀h ∈ VI (h =U f → rank h ≥ rank g)}; +see Jech [13], (9.3) and (28.15). One lets VI/U be the class of all fU +for f ∈ VI and defines +(5) +fU ∈U gU ←→ f ∈U g. +The ultrapower of V by U is the structure (VI/U, ∈U). The universe V +is embedded into VI/U via x �→ (cx)U where cx is the constant function +on I with value x. +We note that fU is standard iff fU = gU for some standard g ∈ VI. +We assume from now on that whenever the equivalence class fU is +standard, the representative function f is taken to be standard. +The key insight is that the standard elements of the ultrapower of V +by Uw are in equality-and-membership-preserving correspondence with +w-standard elements of V. It is expressed by the following proposition, +which is an immediate consequence of definitions (3) - (5). +Proposition 5.1. For any standard functions f, g ∈ VI: +f =Uw g ←→ fUw = gUw ←→ f(w) = g(w) +and +f ∈Uw g ←→ fUw ∈Uw gUw ←→ f(w) ∈ g(w). +The correspondence Φw is defined on Sw × (VI/Uw ∩ S) by +Φw(ξ, fUw) ←→ f(w) = ξ. +In this notation, Proposition 5.1 asserts the following. +Corollary 5.2. The class Φw is an isomorphism between the structures +(Sw, ∈) and (VI/Uw ∩ S, ∈Uw). +We note that (VI/Uw ∩ S, ∈Uw) is the ultrapower of the universe in +the sense of the standard universe S. If φ(v) is an ∈-formula such that +VI/Uw = F | φ(F) , then VI/Uw ∩ S = F ∈ S | φst(F) . If ψ(u, v) +is an ∈-formula such that F ∈Uw G ←→ ψ(F, G), then F ∈Uw G ←→ +ψst(F, G) holds for F, G ∈ S. +If Φw(ξ, fUw) holds, we write Φw(ξ) = fUw. We note that for x ∈ S, +Φw(x) = (cx)Uw where cx is the constant function on I with value x. As +is customary, we identify (cx)Uw with x. This gives a stronger version +of Corollary 5.2. + +NONSTANDARD HULLS AND LOEB MEASURES +25 +Corollary 5.3. The class Φw is an isomorphism between the structures +(Sw, S, ∈, ) and (VI/Uw ∩ S, S, ∈Uw). +We also note that Φw(w) = IdUw where Id(i) = i for all i ∈ I. +Recall that quantifiers with superscript stw range over w-standard +sets. +If φ is an ∈-formula, φstw is the formula obtained from φ by +relativizing all quantifiers to stw. �Lo´s’s Theorem for ∈-formulas takes +the following form: +Lemma 5.4. For standard f1, . . . , fr, +φstw(f1(w), . . . , fr(w)) ←→ {i ∈ I | φ(f1(i), . . . , fr(i))} ∈ Uw. +The structures (VI/Uw, ∈Uw) and (VI/Uw ∩ S, ∈Uw) are not mod- +els in the sense of model theory because their components are proper +classes; hence the satisfaction relation ⊨ is not available. Given an ∈- +formula φ with parameters from VI/Uw, we write “φ holds in (VI/Uw, ∈Uw +)” to stand for the formula obtained from φ by replacing all occurrences +of u ∈ v with u ∈Uw v and relativizing all quantifiers to VI/Uw; similarly +for “φ holds in (VI/Uw ∩ S, ∈Uw).” +Proof of Lemma 5.4. We have +φstw(f1(w), . . . , fr(w)) ←→ +φ((f1)Uw, . . . , (fr)Uw) holds in (VI/Uw ∩ S, ∈Uw) +[by Corollary 5.2] +←→ φ((f1)Uw, . . . , (fr)Uw) holds in (VI/Uw, ∈Uw) +[by Transfer] +←→ {i ∈ I | φ(f1(i), . . . , fr(i))} ∈ Uw +[by the usual �Lo´s’s Theorem]. +□ +Remark 5.5. In Subsection 3.2 we fix a universal standard set V and +for standard f ∈ V I define fw,V (see Definition 3.10). Clearly +fw,V = st{g ∈ V I | g(w) = f(w)} = st{g ∈ V I | g =Uw f} += {g ∈ V I | g =Uw f}, +where the last step is by Transfer. Thus, again by Transfer, V I/w is +nothing but the ultrapower V I/Uw. +5.2. Proofs of claims in Subsection 3.1. +Proposition 5.6. A set w is good iff Uw is countably incomplete. +Of course, ultrapowers by countably incomplete ultrafilters are the +ones of interest in nonstandard analysis. +Proof. Immediate from the isomorphism Φ between the universe of +w-internal sets (Sw, ∈) and the ultrapower ((VI/Uw) ∩ S, ∈Uw) (see +Chang-Keisler [4], Sec. 4.3). +□ + +26 +KAREL HRBACEK AND MIKHAIL G. KATZ +Proof of Proposition 3.4 +Proof of (1): Assume ∃x φ(x, p0) where p0 is (wlog. the only) pa- +rameter and stw(p0). Fix a standard set P such that p0 ∈ P (Bound- +edness). Use AC to obtain a standard function F on P such that ∀p ∈ +P (∃x φ(x, p) → φ(F(p), p)). Hence φ(F(p0), p0) holds. As F(p0) is w- +standard by Proposition 3.2 (c), we conclude that ∃stwx φ(x, p0). +□ +Proof of (2): It is well-known that every ultrapower by a countably +incomplete ultrafilter U is ω1-saturated (see Chang-Keisler [4], Theo- +rem 6.1.1). Hence Countable Idealization in the form +(6) +∀stn ∈ N ∃x ∀m ∈ N (m ≤ n → φ(m, x)) ←→ ∃x ∀stn ∈ N φ(n, x) +holds in (VI/Uw, V, ∈Uw). By BST Transfer, (6) holds in (VI/Uw ∩ +S, S, ∈Uw), and, by Corollary 5.3, it holds in (Sw, S, ∈). This translates +to ∀stn ∈ N ∃stwx ∀stwm ∈ N (m ≤ n → φstw(m, x)) ←→ ∃stwx ∀stn ∈ +N φstw(n, x). Using w-Transfer we get the desired form +∀stn ∈ N ∃x ∀m ∈ N (m ≤ n → φ(m, x)) ←→ ∃stwx ∀stn ∈ N φ(n, x). +□ +Proof of (3): Recall that stw(x) means that x = g(w) for some +standard g defined on I. +Let φ(v) be an ∈-formula with standard +parameters. Assume φ(g(w)); by w-Transfer then φstw(g(w)). From +the isomorphism between (Sw, ∈) and (VI/Uw) ∩ S, ∈Uw) and �Lo´s’s +Theorem it follows that X = {i ∈ I | φ(g(i))} ∈ Uw. Pick i0 ∈ X and +let f be the standard function defined by +f(i) = g(i) for i ∈ X; f(i) = f(i0) otherwise. +Then f(w) = x and φ(f(i)) holds for all i ∈ I. +□ +□ +Definition 5.7. Let κ be a standard infinite cardinal. The set w is κ+- +good if Uw is a countably incomplete κ+-good ultrafilter (In particular, +w is good iff it is ω1-good; see Proposition 5.6.) +We prove Proposition 3.5 in the following form. +Proposition 5.8. (Idealization into w-standard sets over sets of car- +dinality ≤ κ) Let φ be an ∈-formula with w-standard parameters. If w +is κ+-good, then for every standard set A of cardinality ≤ κ +∀stfina ⊆ A ∃y ∀x ∈ a φ(x, y) ←→ ∃stwy ∀stx ∈ A φ(x, y). +For every standard uncountable cardinal κ and every z there exist κ+- +good w so that z is w-standard. + +NONSTANDARD HULLS AND LOEB MEASURES +27 +Proof of Proposition 5.8 +It is well known (Chang and Keisler [4], Theorem 6.1.8) that any +ultrapower by a countably incomplete κ+-good ultrafilter U is κ+- +saturated. As in the proof of (2), it follows that Bounded Idealization +over sets of cardinality ≤ κ holds in (Sw, S, ∈) for κ+-good w. +To also obtain z ∈ Sw we use the following fact proved in Keisler [16]: +If U is a countably incomplete κ+-good ultrafilter over I and V is +an ultrafilter over J, then the ultrafilter U ⊗ V over I × J defined by +X ∈ U ⊗ V ←→ {i ∈ I | {j ∈ J | ⟨i, j⟩ ∈ X} ∈ V } ∈ U +is countably incomplete and κ+-good. +The definition of ⊗ implies that for every standard X ∈ U ⊗ Uz +there is a standard i ∈ I such that {j ∈ J | ⟨i, j⟩ ∈ X} ∈ Uz, and +hence ⟨i, z⟩ ∈ X. From Bounded Idealization one obtains w ∈ I such +that ⟨w, z⟩ ∈ X holds for all standard X ∈ U ⊗ Uz; in other words, +U⟨w,z⟩ = U ⊗ Uz. +Then z ∈ S⟨w,z⟩ and ⟨w, z⟩ is κ+-good, so S⟨w,z⟩ +satisfies Bounded Idealization over sets of cardinality ≤ κ. +□ +Remark 5.9. More general universes. The definition of w-standard +sets in Subsection 3.1 can be generalized. Let w : S → I where S, I +are standard sets. We let +Sw = f(w(s1), . . . , w(sk)) | k, f ∈ S, dom f = Ik and s1, . . . , sk ∈ S ∩ S . +It turns out that these universes correspond precisely to the standard +limit ultrapowers of the standard universe. The proof is similar to the +model-theoretic proof that every internal universe ∗V (X) is a bounded +limit ultrapower of the superstructure V (X); see Chang and Keisler [4], +Theorems 4.4.19 and 6.4.10. With suitable modifications, all results +described in this paper remain valid for this more general notion of +w-standard sets. +6. Some earlier constructions +There are several earlier publications where constructions of nonstan- +dard hulls and Loeb measures in the internal framework are discussed. +Below we summarize this work and provide some critical assessment. +6.1. The “full” nonstandard hull. Let (M, d) be a standard metric +space. A straightforward attempt to carry out the construction of the +nonstandard hull of (M, d) in BST can start as follows. Let +Bmax = x ∈ M | d(x, a) is limited for some standard a ∈ M . +Let Emax be the equivalence relation on Bmax defined by +Emax = ⟨x, y⟩ ∈ Bmax × Bmax | d(x, y) ≃ 0 . + +28 +KAREL HRBACEK AND MIKHAIL G. KATZ +Finally, let the function Dmax with standard real values be defined by +Dmax = ⟨x, y, r⟩ ∈ Bmax × Bmax × R | r = sh(d(x, y)) . +The classes Bmax, Emax and Dmax are (definable) external sets. +For +every x ∈ Bmax, the equivalence class M(x) = +z ∈ Bmax | d(x, z) ≃ +0 +is an external set. However, the final step in the construction of +the nonstandard hull of (M, d), to wit, the formation of the quotient +space (Bmax/Emax, Dmax/Emax), cannot be carried out in BST (it would +require a “class of classes” +M(x) | x ∈ Bmax ). +There are some ways around this difficulty. Perhaps the most straight- +forward is to forgo the formation of the quotient space and work with +the representatives of the equivalence classes (ie, the elements of Bmax), +and with the congruence Emax in place of the actual equality. This +would be similar to working with fractions rather than the rational +numbers. But this way does not produce the nonstandard hull as an +actual object of BST. +The quotient space (Bmax/Emax, Dmax/Emax) can be formed in HST +(using its axiom of Replacement for st-∈-formulas). An interpretation +of HST can be coded in BST (see [15], Definition 5.1.2, Theorem 5.1.4 +and Corollary 5.1.5), so in this indirect way the “full” nonstandard +hull can be coded in BST. Unfortunately, the coding involved is far +from being a “morphism” in any sense, so the resulting opaque code +is unsuitable for transferring nonstandard intuitions about the hull to +its coded version. One point of working with subuniverses is that they +have a natural coding (by standard sets). +Perhaps the most serious objection to this way of constructing non- +standard hulls is that (Bmax/Emax, Dmax/Emax) is just “too large.” +Trivially, every nonstandard hull (Bw/Ew, Dw/Ew) of (M, d) consid- +ered in Subsection 4.1 embeds isometrically into (Bmax/Emax, Dmax/Emax) +(note that Bw = Bmax∩Sw, Ew = Emax∩Sw, and Dw = Dmax∩Sw). In +many interesting cases, the metric space (M, d) has nonstandard hulls +of arbitrarily large cardinality, so Bmax/Emax is not of standard size. +It would be difficult to do further work with nonstandard hulls using +this method, such as compare them with other standard metric spaces, +take their products, or form the space of continuous functions on them. +They are analogous to the “universal group” that can be constructed +as a direct sum (or product) of all groups. This “object” is a proper +class in ZFC, hardly if ever used for more than bookkeeping purposes. +Another important point about working with subuniverses is that the +objects produced are standard sets (or external sets of standard size). + +NONSTANDARD HULLS AND LOEB MEASURES +29 +6.2. Vakil’s construction. In [31], Vakil presents a construction of +nonstandard hulls of uniform spaces in IST. +His method does not +require fixing a particular subuniverse, and Standardization is used +in a way similar to this paper. But Vakil’s method applies only to +a certain class of uniform spaces, the so-called Henson-Moore spaces. +Revealingly, these are precisely the spaces whose nonstandard hull is +independent of the choice of the nonstandard universe, ie, it is unique +up to isomorphism and of standard size; see Henson-Moore [9] and +Vakil [33]. +6.3. Loeb measures in IST. Diener and Stroyan [6], p. 274, outline +a possible construction of Loeb measures in IST, referencing Stroyan +and Bayod [30], Section 2.2, for further details. Here we briefly consider +this approach. +Let (Ω, A, µ) be an internal finitely-additive measure space, with Ω ⊆ +O for a standard set O. We take A = P(Ω) for simplicity, and analyze +Loeb’s construction from the point of view of BST. The first step is to +extend the algebra A to an external σ-algebra. Bounded Idealization +in BST implies that an externally countable union of (internal) sets is +either equal to a finite union or is not internal. So the construction +has to deal with external sets from the very beginning. The only way +to treat external sets as objects in BST is via some kind of coding +by sets. +For example, every external sequence +Xn | n ∈ N ∩ S +of (internal) sets has an extension ⟨Xn | n ∈ N⟩ to an (internal) +sequence,7 which can be regarded as its code (of course an external +sequence has many codes). This coding could be extended to higher +levels of the Borel hierarchy over the algebra of (internal) subsets of Ω. +A simpler solution, proposed in [6, 30], is to code Loeb measurable sets +with the help of Souslin schemata. One can define a Souslin schema in +BST as a function S : N<ω → P(Ω). Let F = Nω. The kernel of S is +the external set +kerS = +� +f∈F∩S +� +n∈N∩S +Sf↾n. +The external sets obtainable as kernels of Souslin schemata are Henson +sets and Henson sets whose complement in Ω is also Henson are the +Loeb sets. Loeb sets form the smallest external σ-algebra σ(A) contain- +ing P(Ω); see [30], Theorem 2.2.3 (Luzin Separation Theorem). Let L +be the external set of all pairs ⟨S1, S2⟩ such that kerS1∩kerS2 = ∅ and +kerS1 ∪ kerS2 = Ω. Then L can be viewed as a set of codes for Loeb +7This follows from the Extension principle of BST; see Kanovei and Reeken [15], +Section 3.2e. + +30 +KAREL HRBACEK AND MIKHAIL G. KATZ +sets. The algebraic operations (union, complement, externally count- +able union) can be coded by external relations on L, and the Loeb +measure itself can be coded by an external relation on L × (R ∩ S). +The objections raised in Subsection 6.1 against “full” nonstandard +hulls apply also to the above approach to Loeb measures. In particular, +one cannot form the quotient space of L modulo the relation in which +two codes are equivalent when they code the same external set, and +the coding of the set-theoretic operations is not a “morphism” (eg, the +code of the union of two sets is in no sense the union of their codes). It +would be awkward to work with random variables on L via codes, and +collections of random variables are even more challenging. In addition, +it is customary in the literature (see Albeverio et al. [1], Kanovei and +Reeken [15]) to regard not σ(A) but its completion L(A) as the Loeb +σ-algebra. It is not usually possible to extend L to an external set of +codes for L(A) (not even in HST, because the Power Set axiom for +external sets does not hold there). On these grounds, it is arguable +whether the claim that this method represents the Loeb measure space +is justified. +6.4. Bounded Internal Set Theory. Following upon ideas of Lind- +strøm [17], Diener and Stroyan [6] aim for “a description of the com- +mon ground between the two approaches to Robinson’s theory.” This +takes the form of working in a polysaturated8 nonstandard universe +N = (V (S), V (∗S), ∗) axiomatically. They formulate Bounded Internal +Set Theory (bIST) and show that its axioms hold in such universes. +This theory should not be confused with BST. The main differences +are: +• The language of bIST is closely tied to the structure N. +It +contains a constant symbol for every internal set in V (∗S) and +more; in particular, it is uncountable. +• The axiom schemata T, I and S are modified so as to apply only +to those formulas in which all quantifiers are bounded. +• The axioms of ZFC are not postulated (in fact, Replacement +fails in V (S)). +bIST proves many results familiar from IST or BST, but, just like in +these theories, its variables range over internal sets. It does not provide +access to higher order external sets without some coding. Of course, +the nonstandard universe N does provide external sets that can be used +to construct nonstandard hulls and Loeb measures in the usual way, +but then one is using the model-theoretic rather than the axiomatic +8N is polysaturated if it is κ-saturated for κ = |V (∗S)|. + +NONSTANDARD HULLS AND LOEB MEASURES +31 +approach. Overall, bIST is more a useful tool for work with super- +structures than a self-standing axiomatics for nonstandard analysis. +6.5. Measure and integration over finite sets. Nelson’s Radically +Elementary Probability Theory [26] works with (hyper)finite proba- +bility spaces only. It requires only very elementary axioms (see [26], +Chapter 4) that are easy consequences of the axioms of IST, or even of +the weaker and more effective theory SCOT (see footnote 6). Further +development of this approach can be found eg. in Cartier and Per- +rin [2, 3], who study both Nelson’s S-integral on finite measure spaces +(which they call Loeb-Nelson integral) and its refinement, which they +call Lebesgue integral. The “radically elementary” approach is simple +and elegant, and many interesting results have been obtained in this +way. +It is beyond the scope of this paper to try to compare the “radically +elementary” approach to the nonstandard measure theory based on +the work of Loeb. Nelson shows ([26], A.1) that for every stochastic +process there is a nearby elementary process, and that theorems of the +conventional theory of stochastic processes can be derived from their +elementary analogues. The works [26, 2, 3] do not address the question +whether (up to isomorphism) Loeb measure spaces can be obtained +from their constructions. +7. Conclusion +This paper is based on the fact that an internal set theory such as +BST provides a supply of structures that are analogous to the internal +part of model-theoretic nonstandard universes. Utilizing a natural cod- +ing and the principle of Standardization, we can mimic external sets by +standard ones. Nonstandard hulls and Loeb measures can be obtained +“up to an isomorphism” with their external counterparts. In BST they +are standard objects in a common framework, which makes it easy in +principle to compare them. In the end, they are the same structures +that could be obtained by the superstructure methods carried out in +BST, but our technique for obtaining them is more suitable for the +internal axiomatic framework. +The theory BST is the internal part of HST, a theory which ax- +iomatizes external sets, in addition to standard and internal ones. In +this theory one can carry out external constructions in the same way +as in nonstandard universes; no coding is needed. The coding can be +introduced afterwards in order to convert the external structures so +obtained to standard structures. + +32 +KAREL HRBACEK AND MIKHAIL G. KATZ +We conclude that the internal axiomatic approach based on BST can +implement virtually all techniques employed by the model-theoretic ap- +proach, with some advantages: It could make nonstandard hulls and +Loeb measures more easily accessible to working mathematicians who +have learned the nonstandard methods in IST or another axiomatic +framework, because it does not require a priori knowledge of ultrafil- +ters and ultrapowers or model theory. It establishes a single unified +framework in which both the standard “world” and the nonstandard +“world” are adequately axiomatized. +It allows a seamless extension +to a theory that encompasses also external sets. +Finally, it enables +reverse-mathematical analysis of the strength of the Axiom of Choice +(see eg. Hrbacek and Katz [12]) and other axioms used in the practice +of nonstandard analysis. +References +[1] S. Albeverio, R. Høegh-Krohn, J. Fenstad, T. 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Vakil, Real Analysis through Modern Infinitesimals, Cambridge University +Press, Cambridge, UK, 2011, xx+ 565 pp. +[33] N. Vakil, On uniform spaces with invariant nonstandard hulls, Journal of Logic +and Analysis 6:1 (2014), 1 - 13. doi: 10.4115/jla.2014.6.1 +[34] P. Zlatoˇs, Review of Kanovei–Reeken [14] for Mathematical Reviews, 1996. +https://mathscinet.ams.org/mathscinet-getitem?mr=1397485 + +34 +KAREL HRBACEK AND MIKHAIL G. KATZ +Department of Mathematics, City College of CUNY, New York, NY +10031, +Email address: khrbacek@icloud.com +Department of Mathematics, Bar Ilan University, Ramat Gan 5290002 +Israel, +Email address: katzmik@math.biu.ac.il + diff --git a/OdAyT4oBgHgl3EQfg_gX/content/tmp_files/load_file.txt b/OdAyT4oBgHgl3EQfg_gX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5d99a5cb425be46c9734dd556f7ff2e338fc4891 --- /dev/null +++ b/OdAyT4oBgHgl3EQfg_gX/content/tmp_files/load_file.txt @@ -0,0 +1,1191 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf,len=1190 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='00367v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='LO] 1 Jan 2023 CONSTRUCTING NONSTANDARD HULLS AND LOEB MEASURES IN INTERNAL SET THEORIES KAREL HRBACEK AND MIKHAIL G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' KATZ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Currently the two popular ways to practice Robin- son’s nonstandard analysis are the model-theoretic approach and the axiomatic/syntactic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It is sometimes claimed that the internal axiomatic approach is unable to handle constructions relying on external sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We show that internal frameworks pro- vide successful accounts of nonstandard hulls and Loeb measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The basic fact this work relies on is that the ultrapower of the standard universe by a standard ultrafilter is naturally isomorphic to a subuniverse of the internal universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Introduction 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The internal set theory BST 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Subuniverses of the universe of BST 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' w-standard sets 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Coding external sets 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Nonstandard hulls and Loeb measures in BST 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Nonstandard hulls of standard metric spaces 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Completeness of the nonstandard hull 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Nonstandard hulls of standard uniform spaces 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Internal normed vector spaces 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Loeb measures 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Lebesgue measure from Loeb measure 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Neutrices and external numbers 22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Subuniverses and ultrapowers 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Subuniverses Sw and ultrapowers 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Proofs of claims in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1 25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Some earlier constructions 27 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The “full” nonstandard hull 27 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Vakil’s construction 29 Date: December 4, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' nonstandard analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' internal set theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' external sets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' nonstandard hull;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Loeb measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 1 2 KAREL HRBACEK AND MIKHAIL G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' KATZ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Loeb measures in IST 29 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Bounded Internal Set Theory 30 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Measure and integration over finite sets 31 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Conclusion 31 References 32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Introduction Robinson named his theory “Non-standard Analysis since it involves and was, in part, inspired by the so-called Non-standard models of Arithmetic whose existence was first pointed out by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Skolem” (Robin- son [27], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' vii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Currently there are two popular ways to practice Robinson’s nonstandard analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The model-theoretic approach encompasses Robinson’s enlargements, obtained from the Compactness Theorem (Robinson [27]), ultrapowers (Luxemburg [22]), and nonstandard universes based on superstructures (Robinson and Zakon [28], Chang and Keisler [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The axiomatic/syntactic approach originated in Hrbacek [10] and Nelson [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Nelson’s IST is particularly well known (see for example Robert [29], Diener and Stroyan [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The monograph of Kanovei and Reeken [15] is a comprehensive reference for axiomatic nonstandard analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In the model-theoretic approach one works with various nonstan- dard universes in ZFC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' see Chang and Keisler [4], Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='4, for definitions and terminology associated with this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let N = (V (X), V (∗X), ∗) be a nonstandard universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The collection ∗V (X) of internal sets in N is isomorphic to some bounded ultrapower 1 of the superstructure V (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It is expanded to the superstructure V (∗X) that contains also external sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Axiomatic systems for the internal part of nonstandard set theory, formulated in the st-∈-language, are well suited for the development of infinitesimal analysis and much beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It is generally acknowl- edged that internal theories are easier to learn and to work with than the model-theoretic approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' However, it is sometimes claimed as a shortcoming of the internal approach that external sets are essen- tial for some of the most important new contributions of Robinsonian nonstandard analysis to mathematics, such as the constructions of non- standard hulls (Luxemburg [23]) and Loeb measures (Loeb [18]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' see for example Loeb and Wolff [20] p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' xiv and [21] p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' vii, Loeb [19] and 1More generally, some bounded limit ultrapower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' NONSTANDARD HULLS AND LOEB MEASURES 3 Zlatoˇs [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Such claims are not meant to be taken literally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' As IST includes ZFC among its axioms, nonstandard universes and the exter- nal constructions in them make sense in IST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' However, nonstandard universes have their own notions of standard and internal that are not immediately related to the analogous notions provided by IST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Such a duplication of concepts could be confusing and complicates attempts to work directly in superstructure frameworks inside IST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In any case, the issue in question is whether these constructions can be carried out in internal set theories using the notions that these theories axiomatize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In this paper we describe an approach to external constructions that is better suited to the conceptual framework provided by internal set theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The techniques we employ have been known for a long time, but their use for the purpose of implementing external methods in in- ternal set theories does not seem to explicitly appear in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We show that practically all objects whose construction in a superstruc- ture involves external sets can be obtained in the internal axiomatic setting by these techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We work in Bounded Set Theory BST (see Kanovei and Reeken [15], Chapter 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The axioms of BST are a slight modification of the more familiar axioms of IST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We state them in Section 2 and follow with a discussion of how definable external sets can be handled in BST as abbreviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The not-so-well known fact about BST is that its universe contains subuniverses isomorphic to any standard ultrapower (or even limit ul- trapower) of the standard universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For every internal set w there is a subuniverse Sw and a simply defined isomorphism of Sw with the ultrapower of the standard universe by a standard ultrafilter generated by w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Thus BST naturally provides an analog of the internal universe ∗V (X) of any superstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The subuniverses Sw satisfy some of the axioms of BST, as described in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' hence one can work with these subuniverses axiomatically, without any reference to their relationship to ultrafilters or ultrapowers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' External subsets of Sw and higher-order external sets built from them are not objects of BST, but the above-mentioned isomorphism, in com- bination with the powerful principle of Standardization, provides a nat- ural way to code these “non-existent” external sets by standard sets and to imitate the superstructure V (∗X) of external sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This idea is developed in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The heart of the paper is Section 4, where we show how to construct nonstandard hulls of standard metric and uniform spaces and Loeb measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We also consider analogous constructions on internal normed spaces, and outline how one can treat neutrices and external numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 4 KAREL HRBACEK AND MIKHAIL G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' KATZ From the practical point of view, the best way to use these techniques may be to work out the external constructions informally, and then code them up by standard sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In principle, any construction involving external sets can be carried out in the framework of BST by these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Section 5 develops the relationship between the subuniverses Sw and ultrapowers, and supplies proofs of the properties of Sw stated in Sub- section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In Section 6 we review some ways that have been proposed for doing external constructions in IST previously and discuss their shortcom- ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The principal difficulty is that they tend to produce objects that are “too large” to be suitable for further work (larger than the class of standard elements of any standard set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The internal set theory BST We work in the framework of BST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2 The theory BST is formulated in the st-∈-language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It is a conservative extension of ZFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Quantifiers with the superscript st range over standard sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Quan- tifiers with the superscript stfin range over standard finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The axioms of BST are, in addition to ZFC (the ZFC axiom schemata of Separation and Replacement apply to ∈-formulas only): B (Boundedness) ∀x ∃sty (x ∈ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' T (Transfer) Let φ(v) be an ∈-formula with standard parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Then ∀stx φ(x) → ∀x φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' S (Standardization) Let φ(v) be an st-∈-formula with arbitrary parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Then ∀A ∃stS ∀stx (x ∈ S ←→ x ∈ A ∧ φ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' BI (Bounded Idealization) Let φ(u, v) be an ∈-formula with arbitrary parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For every standard set A ∀stfina ⊆ A ∃y ∀x ∈ a φ(x, y) ←→ ∃y ∀stx ∈ A φ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' See the references Kanovei and Reeken [15] and Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' [8] for motivation and more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' An equivalent existential version of Transfer is ∃x φ(x) → ∃stx φ(x), 2The bounded sets of IST (those sets that are elements of standard sets) satisfy all of the axioms of BST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Hence all arguments in this paper can be carried out in IST, albeit less directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' NONSTANDARD HULLS AND LOEB MEASURES 5 for ∈-formulas φ with standard parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Yet another equivalent version, easily obtained by induction on the logical complexity of the ∈-formula φ(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' , vk) (with standard parameters), is ∀stx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' , xk [ φ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' , xk) ←→ φst(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' , xk)] where φst is the formula obtained from φ by relativizing all quantifiers to st (that is, by replacing each occurrence of ∃ by ∃st and each occurrence of ∀ by ∀st).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' As usual in mathematics, symbols N and R denote respectively the set of all natural numbers and the set of all reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In internal set theories there are two ways of thinking about them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In the “internal picture” R is viewed as the usual set of reals in which the predicate st singles out some elements as “standard”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' similarly for any infinite standard set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This is the view familiar from Nelson [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In the “standard picture” the usual set R is viewed as containing, in addition to its standard elements, also fictitious, ideal elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' See Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' [8] for further discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Mathematics in BST can be developed in the same way as in IST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In particular, real numbers r, s are infinitely close (notation: r ≃ s) if |r − s| < 1/n holds for all standard n ∈ N \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' A real number r is an infinitesimal if r ≃ 0, r ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It is limited if |r| < n for some standard n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We recall that for every limited x ∈ R there is a unique standard r ∈ R such that r ≃ x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' it is called the shadow (or standard part) of x and denoted sh(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' we also define sh(x) = +∞ when x is unlimited, x > 0, and sh(x) = −∞ when x is unlimited, x < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' All objects whose existence is postulated by BST are sets, sometimes called internal sets for emphasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The Separation axiom holds for ∈- formulas only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' But it is common practice in the literature based on the internal axiomatic approach to introduce definable external sets as convenient abbreviations (see eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Vakil [32], Diener and Stroyan [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' One can enrich the language of the theory by names for extensions of arbitrary st-∈-formulas and in this way talk about st-∈-definable sub- classes of the universe of all (internal) sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We note that (a) this does not amount to a formalization of a new type of entity called “external set,” which is a more complicated task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' and (b) this does not amount to informal use of the term “external set,” either (in the sense of relying on a background formalization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It is similar to the way set theorists routinely employ classes in ZFC (the class of all sets, the class of all ordinals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Such classes serve as convenient shortcuts in mathematical discourse because one can work with them “as if” they were objects, 6 KAREL HRBACEK AND MIKHAIL G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' KATZ but they can in principle be replaced by their defining formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Ex- ample 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1 below illustrates this familiar procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let φ(v) be an st-∈-formula with arbitrary parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We employ dashed curly braces to denote the class x | φ(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We emphasize that this is merely a matter of convenience;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' the expression z ∈ x | φ(x) is just another notation for φ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Usually we denote classes by boldface characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Those classes that are included in some set are external sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' If there is a set A such that ∀x (x ∈ A ←→ φ(x)), then the class x | φ(x) can be identified with the set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='3 Monads and galaxies are some familiar examples of external sets that are usually not sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let (M, d) be a metric space: the monad of a ∈ M is M(a) = x ∈ M | d(x, a) ≃ 0 , and the galaxy of a ∈ M is G(a) = x ∈ M | d(x, a) is limited .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Some useful proper classes (ie, classes that are not external sets) are V = x | x = x (the universe of all (internal) sets),4 and S = x | st(x) (the universe of all standard sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let M, f : M → M and a ∈ M be standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' A convenient way of defining continuity is as follows: The function f is continuous at a if f[M(a)] ⊆ M(f(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' But the use of external sets in this definition can be eliminated by rephrasing it as: The function f is continuous at a if for all x, d(x, a) ≃ 0 implies d(f(x), f(a)) ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Definable external collections of internal sets are adequately handled in BST in this manner;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' we refer to Vakil [32] for a thorough discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Difficulties arise only when higher level constructs on external sets are needed, such as quotient spaces and power sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We show how to handle such difficulties in Subsections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='4, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Subuniverses of the universe of BST 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' w-standard sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let us fix a set w and a standard set I such that w ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='5 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' A set x is called w-standard (notation: stw(x)) if x = f(w) for some standard function f with domain I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Next, we let 3In the model-theoretic approach external sets are by definition the sets that are not internal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In the axiomatic approach it is customary to view internal sets as a special case of external sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 4The symbol I is often used for this purpose in the literature, but in the context of st-∈-theories, where all sets are internal, the notation V seems more appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 5The Boundedness axiom guarantees that some such I exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This is one of the reasons we prefer to work with BST rather than IST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' NONSTANDARD HULLS AND LOEB MEASURES 7 Sw = x | stw(x) be the universe of all w-standard sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The notion of w-standardness depends only on w, not on the choice of the standard set I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' (a) ∀x (st(x) → stw(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' (b) ∀x (stw(x) → st(x)) holds if and only if w is standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' (c) stw(f) ∧ stw(x) ∧ x ∈ dom f → stw(f(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' (a) Let x be standard;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' we have x = cx(w) where cx is the con- stant function with value x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' (b) If w is standard, then every f(w) for standard f is standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' If w is nonstandard, let f(i) = i be the identity function on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Then f(w) = w is w-standard but not standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' (c) Assume stw(f), stw(x) and x ∈ dom f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Then there are standard functions F, G on I such that f = F(w) and x = G(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Define a function H on I by H(i) = F(i)(G(i)) when the right side is defined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' H(i) = ∅ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Then H is a standard function on I and H(w) = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' □ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' A set w is good if there is ν ∈ N such that ν is w- standard but not standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In particular, if ν ∈ N is nonstandard, then w = ν is good and, more generally, w = ⟨ν, z⟩ is good for any set z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The following facts are immediate consequences of known results (see Kanovei and Reeken [15], Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='3, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2, esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For easy reference we give the proofs in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Quantifiers with the superscript stw range over w-standard sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' (1) (Transfer from w-standard sets) Let φ be an ∈-formula with w-standard parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Then ∀stwx φ(x) → ∀x φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' (2) (Countable Idealization into w-standard sets) Let φ be an ∈-formula with w-standard parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' If w is good, then ∀stn ∈ N ∃x ∀m ∈ N (m ≤ n → φ(m, x)) ←→ ∃stwx ∀stn ∈ N φ(n, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In other words, (Sw, S, ∈) satisfies Countable Idealization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' (3) (Representability) Let φ(v) be an ∈-formula with standard parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' If I is standard, w ∈ I, x is w-standard, and φ(x) holds, then there is a standard function f with dom f = I such that x = f(w) and φ(f(i)) holds for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 8 KAREL HRBACEK AND MIKHAIL G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' KATZ An equivalent formulation of (1) is Transfer into w-standard sets: ∃x φ(x) → ∃stwx φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Another equivalent formulation for φ(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' , vr) with w-standard pa- rameters is ∀stwx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' , xr [ φ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' , xr) ←→ φstw(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' , xr)], where φstw is the formula obtained from φ by relativizing all quan- tifiers to stw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In yet other words, (V, Sw, ∈) satisfies Transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It also satisfies Boundedness and, for good w, Bounded Idealization (see Kanovei and Reeken [15], Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='16 (i)), but not Standardization (ibid, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Although we do not need these results in this paper, together they show that (V, Sw, ∈) satisfies all the axioms of BST except Standardization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Note that here Sw plays the role of a new “thick standard universe.” On the other hand, (Sw, S, ∈) satisfies Transfer, Boundedness, Standardization and Countable Idealization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' here Sw plays the role of a new “thin internal universe.” Idealization can be strengthened from N to sets of cardinality κ if w is chosen carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We leave the technical definition of κ+-good sets to Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2 (see Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For our applications we need only to know that for every standard uncountable cardinal κ and every z there exist κ+-good w so that z is w-standard, a result which is also proved there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' (Idealization into w-standard sets over sets of car- dinality ≤ κ) Let φ be an ∈-formula with w-standard parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' If w is κ+-good, then for every standard set A of cardinality ≤ κ ∀stfina ⊆ A ∃y ∀x ∈ a φ(x, y) ←→ ∃stwy ∀stx ∈ A φ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='5 do not exhaust the properties of the uni- verses Sw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' for a list of further useful principles see Kanovei and Reeken [15], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Coding external sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1 provides a natural way to represent w-standard sets by standard functions: A w-standard ξ ∈ Sw is represented by any standard f ∈ VI such that f(w) = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Note that every ξ has a proper class of representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This causes some techni- cal difficulties (see Section 5), which for our purposes are best resolved by fixing a universal standard set V so that all objects of interest are subsets of V or relations on V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' usually one requires R ⊆ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' By Rep- resentability, for every ξ ∈ V ∩ Sw there exists a standard f ∈ V I such that f(w) = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Moreover, if ψ is an ∈-formula with standard parameters and ψ(ξ) holds, then f can be chosen so that ψ(f(i)) holds NONSTANDARD HULLS AND LOEB MEASURES 9 for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The representation makes possible a coding of the ex- ternal subsets of V ∩ Sw by standard sets, and the coding process can be continued to the putative higher levels of the external cumulative hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This process is enabled by the principle of Standardization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let φ(v) be a formula in the st-∈-language, with ar- bitrary parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We let st{x ∈ A | φ(x)} denote the standard set S such that ∀stx (x ∈ S ←→ x ∈ A ∧ φ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The principle of Standardization postulates the existence of this set and Transfer guarantees its uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Also by Transfer, if ψ is any ∈-formula with standard parameters and ∀stx ∈ A (φ(x) → ψ(x)), then ∀x ∈ S ψ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In particular, if φ(u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='v) is a formula in the st-∈-language with arbi- trary parameters, A, B are standard, and for every standard x ∈ A there is a unique standard y ∈ B such that φ(x, y), then there is a unique standard function F : A → B such that ∀stx ∈ A φ(x, F(x)) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It suffices to let F = st{⟨x, y⟩ ∈ A × B | φ(x, y)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The w, V -code of an external set X ⊆ V ∩ Sw is the standard set Ψw,V (X) = X = st{f ∈ V I | f(w) ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In words, Ψw,V (X) is the standard set whose standard elements are precisely the standard f ∈ V I with f(w) ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We omit the subscripts w and/or V when they are understood from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The coding is trivially seen to preserve elementary set-theoretic op- erations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For any external sets X1, X2 ⊆ V ∩ Sw: (1) Ψ(∅) = ∅, X1 ⊆ X2 ←→ Ψ(X1) ⊆ Ψ(X2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' (2) Ψ(X1∪X2) = Ψ(X1)∪Ψ(X2), Ψ(X1∩X2) = Ψ(X1)∩Ψ(X2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' (3) Ψ(X1 \\ X2) = Ψ(X1) \\ Ψ(X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The coding preserves infinite unions and intersections as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' An externally countable sequence of external subsets of V can be viewed in BST as an external subset X of N × V , with Xn = x | ⟨n, x⟩ ∈ X for standard n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let ⟨Xn | n ∈ N⟩ be the standard sequence such that Xn = Ψ(Xn) holds for all standard n (its existence follows from Standardization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Then Ψ( � n∈N∩S Xn) = � n∈N Xn 10 KAREL HRBACEK AND MIKHAIL G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' KATZ because if f ∈ � n∈N Xn is standard, then the least n ∈ N such that f ∈ Xn is standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Similarly, Ψ( � n∈N∩S Xn) = � n∈N Xn because if a standard f ∈ Ψ(� n∈N∩S Xn), then f ∈ Xn for all standard n ∈ N, and hence f ∈ � n∈N Xn by Transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It is convenient to relax the definition of coding so that every stan- dard S ⊆ V I is a code of some external X ⊆ V ∩ Sw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' A standard set S codes X ⊆ V ∩ Sw if f(w) ∈ X for each standard f ∈ S and for each ξ ∈ X there is some standard f ∈ S such that f(w) = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' If S codes X, then Ψ(X) = st{f ∈ V I | f(w) = g(w) for some standard g ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' A code of X has to contain a representation for each ξ ∈ X, but not necessarily all such representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We note that coding is independent of the choice of the universal standard set in the following sense: If X ⊆ V1 ∩ Sw and V1 ⊆ V2, then S codes X viewed as a subset of V1∩Sw iff S codes X viewed as a subset of V2 ∩ Sw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Hence the exact choice of V is usually of little importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For any set x, let cx be the constant function with value x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The informal identification of x with cx enables the identification of a stan- dard set A with a code st{cx | x ∈ A} = {cx | x ∈ A} for A ∩ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' On the other hand, the standard set AI is a code for A ∩ Sw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Every standard S ⊆ V I is a code of the unique external set XS = ξ ∈ V ∩ Sw | ξ = f(w) for some standard fw ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Such S codes a subset of an external set X iff ∀stf (f ∈ S → f(w) ∈ X) iff S ⊆ Ψ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Consequently, it would make sense to interpret the power set of � X = Ψ(X) as a code for the “non-existent” external power set Pext(X) of X, and the standard subsets of P( � X) as codes for the “external subsets of Pext(X).” Since BST does not allow collections of external sets, this last remark cannot be made rigorous in it, but one can proceed “as if” such higher order external sets existed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Intuitively, there is a hierarchy of external sets built up over V ∩Sw: H1 = Pext(V ∩ Sw) and Hn+1 = Pext(Hn) for standard n ∈ N (we stop here to avoid further complications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This hierarchy cannot be formalized in BST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' But the corresponding hierarchy over V I is well-defined: NONSTANDARD HULLS AND LOEB MEASURES 11 H1 = P(V I) and Hn+1 = P(Hn) for standard n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In BST one can work legitimately in the latter hierarchy while keeping in mind that the coding establishes a (many-one) correspondence be- tween Hn and Hn ∩S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Both hierarchies and their relationship could be formalized in HST, a conservative extension of BST to a theory that en- compassess also external sets (Hrbacek [10], Kanovei and Reeken [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Note that the coding process treats elements ξ of V ∩ Sw as individ- uals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' they are not coded by Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Thus ξ is represented by any f with f(w) = ξ, but Ψ(ξ) is undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The set {ξ} ⊆ V ∩ Sw is coded by {f}, even when {ξ} ∈ V ∩ Sw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In particular, if R is a binary relation on V ∩ Sw, then the code for R is the standard binary relation Ψ(R) = R = st{⟨f, g⟩ | f, g ∈ V I ∧ ⟨f(w), g(w)⟩ ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Similarly for functions and relations of higher arity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Another variant of coding represents each ξ ∈ X by a single object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let f ∈ V I be standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We let fw,V = fw = st{g ∈ V I | g(w) = f(w)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We define standard sets V I/w = st{fw | f ∈ V I} and, for X ⊆ V ∩ Sw, �Ψw(X) = st{fw ∈ V I/w | f(w) ∈ X} = st{F ∈ V I/w | ∃stf ∈ V I (F = fw ∧ f(w) ∈ X)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The coding �Ψw is one-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For standard X ⊆ V I/w we let �Ψ −1 w (X) = X = ξ ∈ V ∩ Sw | ∃stf ∈ V I (fw ∈ X ∧ f(w) = ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' One can obtain �Ψw(X) from Ψw(X) and vice versa: �Ψw(X) = st{fw | f ∈ Ψw(X)} and Ψw(X) = st{f ∈ V I | fw ∈ �Ψw(X)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='8 and the subsequent paragraph hold with Ψ replaced by �Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We define the hierarchy �H1 = P(V I/w) and �Hn+1 = P( �Hn) for standard n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The coding �Ψw maps H1 onto �H1 ∩ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It extends informally to higher levels by �Ψw(X) = st{�Ψw(Y) | Y ∈ X} and provides a one-one correspondence between Hn and �Hn ∩ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Nonstandard hulls and Loeb measures in BST 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Nonstandard hulls of standard metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let R be the field of real numbers and let (M, d) be a standard metric space, so that M is a standard set and the distance function is a standard mapping 12 KAREL HRBACEK AND MIKHAIL G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' KATZ d : M × M → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' A point x ∈ M is finite if d(x, p) is limited for some (equivalently, for all) standard p ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Fix an unlimited integer w ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We define the standard set Bw by Bw = st{f ∈ MN | f(w) is a finite point of M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The standard relation Ew on Bw is defined by Ew = st{⟨f, g⟩ ∈ Bw × Bw | d(f(w), g(w)) ≃ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Clearly Ew is reflexive, symmetric and transitive on standard elements of Bw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' it follows by Transfer that Ew is an equivalence relation on Bw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We denote the equivalence class of f modulo Ew by fEw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' By Standardization, there is a standard function Dw : Bw ×Bw → R determined by the requirement that Dw(f, g) = sh(d(f(w), g(w))) for all standard f, g ∈ Bw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For standard f, g ∈ Bw the distance d(f(w), g(w))) ≤ d(f(w), f(0)) + d(f(0), g(0)) + d(g(0), g(w)) is limited, so sh(d(f(w), g(w))) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For standard f, g, h ∈ Bw clearly Dw(f, g) = Dw(g, f), Dw(f, h) ≤ Dw(f, g) + Dw(g, h), and Dw(f, g) = 0 iff ⟨f, g⟩ ∈ Ew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Also, for standard f, g, f ′, g′ ∈ Bw we have � ⟨f, f ′⟩ ∈ Ew ∧ ⟨g, g′⟩ ∈ Ew � → Dw(f, g) = Dw(f ′, g′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' By Transfer these properties hold for all f, g, h, f ′, g′ ∈ Bw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We observe that, for the natural choice V = M ∪R, Bw is a code for the external set Bw = x ∈ M ∩ Sw | x is finite , Ew is nothing but a code for the external equivalence relation Ew = ⟨x, y⟩ ∈ Bw × Bw | d(x, y) ≃ 0 , and Dw is a code for the external function Dw : Bw × Bw → R ∩ S defined by Dw = ⟨x, y, r⟩ | r = sh(d(x, y)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The construction of the nonstandard hull of (M, d) by the usual ex- ternal method would form the quotient space Bw/Ew of Bw modulo Ew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This step cannot be carried out in BST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For x ∈ Bw, the equiv- alence class of x modulo Ew is Mw(x) = z ∈ M ∩ Sw | d(x, z) ≃ 0 , but the collection of the classes Mw(x) for all x ∈ Bw is not supported by BST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' However, for standard f ∈ Bw, the set fEw is a code of Mw(x) when f(w) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Therefore Bw/Ew can be replaced by Bw/Ew, which is a standard set in BST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The standard elements of Bw/Ew are pre- cisely the codes of the monads Mw(x) for x ∈ Bw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Hence Bw/Ew is NONSTANDARD HULLS AND LOEB MEASURES 13 a code (as discussed in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2) of the “non-existent” (in BST) quotient space Bw/Ew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We stress that while external sets serve as a motivation for our con- structions, they are not actually used in them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' the constructions deal only with sets of BST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The same applies to the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We let � Mw = Bw/Ew be the standard quotient space of Bw modulo Ew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' From now on we often omit the subscript w when it is understood from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The function D factors by E to a (standard) function �D = D/E on � M, defined by �D(fE, gE) = D(f, g) (as shown above, the value of �D is independent of the choice of representatives f, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It is clear that �D is a metric on � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The embedding c of M into � M is via constant functions: for x ∈ M, c(x) = (cx)E where cx is the constant function on N with value x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Trivially, the embedding c preserves the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We identify M with its image in � M under this embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We emphasize that the structure (� M, �D ) depends on the choice of the parameter w, an unlimited integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' A metric space has a unique completion up to isometry, but it may have many non-isometric non- standard hulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let M = Q be the set of rationals and d(x, y) = |x−y| be the usual metric on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We fix an unlimited w ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We note that, for standard f, f ∈ Bw ←→ f ∈ QN ∧ f(w) is limited, and for standard f, g ∈ Bw ⟨f, g⟩ ∈ Ew ←→ f(w) ≃ g(w) ←→ f(w), g(w) ∈ M(a) for a = sh(f(w)) = sh(g(w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' By Standardization, there is a standard function Γ : Bw/Ew → R such that, for standard f, Γ(f/E) = sh(f(w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It is easy to verify that Γ ↾ (Bw/Ew) ∩ S is a one-one mapping of � Mw ∩ S onto R ∩ S that preserves the metrics: �D(f/E, g/E) = D(f, g) = sh(|f(w) − g(w)|) = | sh(f(w)) − sh(g(w))| = |Γ(f) − Γ(g)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Moreover, Γ((cx)E) = x for any standard x ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' By Transfer, Γ is an isometric isomorphism of � Mw and R which is the identity on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In particular, the nonstandard hull is independent of the choice of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 14 KAREL HRBACEK AND MIKHAIL G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' KATZ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let M = N and let d be the discrete metric on M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' ie, d(x, z) = 1 for all x, z ∈ M, x ̸= z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' As all points of M are finite with respect to this metric, we have Bw = NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Also Ew = st{⟨f, g⟩ ∈ NN × NN | f(w) = g(w)} and, for standard f ∈ NN, f/E = st{g ∈ NN | g(w) = f(w)} = fw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The space M is identified with a subset of � Mw via Γ : (cx)w �→ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='5 we establish that � Mw = Bw/Ew = NN/w is exactly the ultrapower of M by Uw, an ultrafilter over N generated by w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' in particular, it has the cardinality of the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Also, �D is the discrete metric on � Mw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' By replacing N with I as in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='6 one can obtain nonstandard hulls of arbitrarily large cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Approachable points play an impotant role in the study of nonstandard hulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We define the concept as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let (M, d) be a standard metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' A point x ∈ M is approach- able if for every standard ǫ > 0 there is a standard a ∈ M such that d(x, a) ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The approachable points in M ∩ Sw become exactly the standard points of the closure of M in its nonstandard hull � Mw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Indeed, for standard f ∈ Bw with f(w) = x ∈ M we have x is approachable ←→ ∀stǫ > 0 ∃sta ∈ M (d(x, a) ≤ ǫ) ←→ ∀stǫ > 0 ∃sta ∈ M ( �D(f/E, a) ≤ ǫ) ←→ ∀ǫ > 0 ∃a ∈ M ( �D(f/E, a) ≤ ǫ), where the last step is by Transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Thus in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1 all finite x ∈ Q are approachable and consequently all standard points in R are in the closure of M = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' By Transfer, this is true for all points in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2 all nonstandard points of N are inapproachable and therefore M is closed in � Mw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In this and other constructions in this section we use the coding based on Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The advantage of this choice is that it produces spaces of functions (see in particular Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' One can use �Ψ instead, and define eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' �Bw = st{fw | f(w) is a finite point of M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The advantage here is that one gets an isomorphism of the external structure (Bw, Ew, Dw) with (Bw ∩ S, Ew ∩ S, Dw ∩ S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It is thus immediately apparent that (( �Bw/ �Ew) ∩ S, ( �Dw/ �Ew) ∩ S) would be iso- morphic to (Bw/Ew, Dw/Ew) if the latter quotient could be formed in BST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' (It can be formed in HST and this claim is a theorem there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=') For the final result the choice of coding method does not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The structures ( �Bw/ �Ew, �Dw/ �Ew) and (Bw/Ew, Dw/Ew) are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' NONSTANDARD HULLS AND LOEB MEASURES 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For standard f, g ∈ MN, f ∈ Bw iff fw ∈ �Bw, ⟨f, g⟩ ∈ Ew iff ⟨fw, gw⟩ ∈ �Ew and Dw(f, g) = �Dw(fw, gw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In the construction of (� M, �D ) one can replace N by any infinite set I, as long as w ∈ I is good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' If Sw1 ⊆ Sw2, fix a stan- dard function h ∈ II2 1 such that h(w2) = w1 and define the standard mapping H : MI1 → MI2 by H(f) = f ◦ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Then H is an em- bedding of (Bw1, Ew1, Dw1) into (Bw2, Ew2, Dw2) in an obvious sense, and it factors to an isometric embedding of (Bw1/Ew1, Dw1/Ew1) into (Bw2/Ew2, Dw2/Ew2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Given any good w1 and w2, let w = ⟨w1, w2⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' then both (� Mw1, �Dw1) and (� Mw2, �Dw2) embed into (� Mw, �Dw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Completeness of the nonstandard hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This subsection il- lustrates how one can work with the nonstandard hull as constructed in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' (� Mw, �Dw ) is a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let ⟨Fn | n ∈ N⟩ be a standard Cauchy sequence in (� M, �D );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' we prove that it converges to some F ∈ � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Using the Axiom of Countable Choice we obtain a standard sequence ⟨fn | n ∈ N⟩ such that Fn = (fn)E for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For k ∈ N let nk be the least element of N greater than or equal to k such that ∀m, n � nk ≤ n ≤ m → �D(Fn, Fm) < 1 k+1 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' note that the sequence ⟨nk | k ∈ N⟩ is standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' From the definition of �D we obtain, for standard k, ∀stm, n � nk ≤ n ≤ m → d(fn(w), fm(w)) < 1 k+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Hence ∀stk ∃stm ∀ℓ ≤ k � nℓ ≤ m ∧ ∀n � nℓ ≤ n ≤ m → d(fn(w), fm(w)) < 1 ℓ+1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' By Countable Idealization into w-standard sets we get ∃stwm ∀stk � nk ≤ m ∧ ∀n � nk ≤ n ≤ m → d(fn(w), fm(w)) < 1 k+1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Fix such an m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' clearly it is unlimited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Consider the standard point p = fn0(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We have d(p, fm(w)) ≤ d(fn0(0), fn0(w))+d(fn0(w), fm(w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The first term on the right side of the inequality is limited because fn0 is standard and fn0 ∈ B, and the second term is < 1 (take k = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Hence fm(w) is a finite element of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We note that fm(w) is w-standard;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' hence there is a standard function f ∈ MN such that f(w) = fm(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It follows that f ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We let F = fE ∈ � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 16 KAREL HRBACEK AND MIKHAIL G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' KATZ We have that, for all standard k, ∀stn � nk ≤ n → d(fn(w), f(w)) < 1 k+1 � , and hence ∀stn � nk ≤ n → �D(Fn, F) ≤ 1 k+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This shows that the sequence ⟨Fn | n ∈ N⟩ converges to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' □ The proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2 goes through for any infinite set I in place of N, as long as w ∈ I is good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Nonstandard hulls of standard uniform spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We gener- alize the construction of nonstandard hulls in BST to uniform spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let (M, ∆) be a standard uniform space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' That is, M is a standard set and ∆ is a standard family of pseudo-metrics on M which endows M with a Hausdorff uniform structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In a uniform space, x ∈ M is finite if for all standard d ∈ ∆, d(x, p) is limited for some (equivalently: for all) standard p ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Points x and y are infinitely close if d(x, y) ≃ 0 for all standard d ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We fix a standard infinite set I and a w ∈ I so that Idealization into w-standard sets over standard sets of cardinality ≤ κ holds for κ = max{|∆|, ℵ0} (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' A construction of the nonstandard hull of (M, ∆) can now proceed much as in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We omit the subscripts indicating its dependence on w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We let B = st{f ∈ MI | f(w) is a finite point of M} and E = st{⟨f, g⟩ ∈ B × B | d(f(w), g(w)) ≃ 0 for all standard d ∈ ∆}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For each standard d ∈ ∆ the standard function D on B × B, as well as � M, �D and c, are defined as in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Each �D is a pseudo-metric on � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We let �∆ = st{ �D | d ∈ ∆ ∩ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The structure (� M, �∆) is a complete Hausdorff uniform space and c embeds (M, ∆) into (� M, �∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The proof follows the lines of the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' the main difference is that Cauchy sequences have to be replaced by Cauchy nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let ⟨Λ, ≤⟩ be a standard directed set and ⟨Fλ | λ ∈ Λ⟩ a standard Cauchy net indexed by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Using the Axiom of Choice one obtains a standard net ⟨fλ | λ ∈ Λ⟩ such that fλ ∈ Fλ holds for all λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' then ⟨fλ(w) | λ ∈ Λ⟩ is a w-standard net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The Cauchy property implies that for every standard d ∈ ∆ there is a standard sequence ⟨λd k | k ∈ N⟩ of elements of Λ such that λd k ≤ λd k+1 holds for all k and ∀k ∀λ, µ ∈ Λ � λd k ≤ λ ≤ µ → �d(Fλ, Fµ) < 1 k+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' NONSTANDARD HULLS AND LOEB MEASURES 17 Hence ∀stk ∀stλ, µ ∈ Λ � λd k ≤ λ ≤ µ → d(fλ(w), fµ(w)) < 1 k+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We conclude that for every standard k ∈ N, every standard finite ∆0 ⊆ ∆ and every standard finite Λ0 ⊆ Λ there is a standard µ ∈ Λ such that for all ℓ ≤ k, all d ∈ ∆0 and all λ ∈ Λ0 λd ℓ ≤ µ ∧ � λd ℓ ≤ λ → d(fλ(w), fµ(w)) < 1 ℓ+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' By Idealization into w-standard sets over sets of cardinality ≤ κ = max{|∆|, ℵ0} we get a w-standard µ ∈ Λ such that for all standard k, all standard d ∈ ∆ and all standard λ ∈ Λ λd k ≤ µ ∧ � λd k ≤ λ → d(fλ(w), fµ(w)) < 1 k+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Fix such a µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' as in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='7 we have that d(p, fµ(w)) is limited for every standard d ∈ ∆, ie, fµ(w) is a finite point of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' By Representability there is a standard function f ∈ MI such that f(w) = fµ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It follows that f ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let F = f/E ∈ � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We have that, for all standard d ∈ ∆ and k ∈ N, ∀stλ ∈ Λ � λd k ≤ λ → d(fλ(w), f(w)) < 1 k+1 � , and hence ∀stλ ∈ Λ � λd k ≤ λ → �D(Fλ, F) ≤ 1 k+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This shows that the net ⟨Fλ | λ ∈ Λ⟩ converges to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Internal normed vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Under many circumstances the type of construction carried out in Subsections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='3 generalizes to internal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We illustrate it in the case of normed vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let M be an internal normed vector space over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This means that M, the operations of addition + on M × M and scalar multiplication on R × M, and the R-valued norm ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='∥ on M, are (internal) sets and satisfy the usual properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In order to be able to apply our coding technique we fix a standard set I and a good w ∈ I so that the set M, the above operations and the norm belong to Sw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Other parameters relevant to a particular investigation can also be made to belong to Sw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' By Transfer in (V, Sw, ∈), the properties of these objects that are expressible by ∈-formulas continue to hold in Sw, so we can carry out the desired construction “over Sw” rather than “over V.” We first describe the external construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let Bw = x ∈ M∩ Sw | ∥x∥ is limited and Ew = x ∈ M∩ Sw | ∥x∥ ≃ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It is clear that Bw is an external vector space over the external field R ∩ S and Ew is its subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Define an external equivalence relation 18 KAREL HRBACEK AND MIKHAIL G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' KATZ ≈ on Bw by x ≈ y ←→ x − y ∈ Ew ←→ ∥x − y∥ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Obviously, for x1, x2, y1, y2 ∈ Bw and c ∈ R ∩ S we have x1 ≈ y1 ∧ x2 ≈ y2 → x1 + x2 ≈ y1 + y2, c · x1 ≈ c · y1 and ∥x1∥ ≃ ∥y1∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' If external collections of external sets were available in BST, one could now form the quotient space Bw/Ew with the norm ∥xEw∥ = sh(∥x∥), which would then be an external normed vector space over the external field R ∩ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This construction is of course not possible in BST directly, but the coding introduced in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2 enables us to carry it out and produce a standard normed metric space which, when viewed from the standard point of view, is (externally) isomorphic to Bw/Ew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It follows from Representability that there is a standard function ⟨(Mi, +i, ·i) | i ∈ I⟩ such that (Mw, +w, ·w) = (M, +, ·) and, for all i ∈ I, (Mi, +i, ·i) is a vector space over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let Bw = st{f ∈ Πi∈IMi | ∥f(w)∥ is limited};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Ew = st{f ∈ Πi∈IMi | ∥f(w)∥ ≃ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Addition and scalar multiplication on Bw are defined pointwise: (f + g)(i) = f(i) +i g(i) and (c · f)(i) = c ·i f(i) for f, g ∈ Bw, c ∈ R, and all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' By Standardization, there is a standard function ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='∥ such that for all standard f ∈ Bw we have ∥f∥ = sh(∥f(w)∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It is routine to verify that Bw is a standard vector space over R, ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='∥ is a pseudo-norm on Bw, and f ∈ Ew ←→ ∥f∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The quotient �Ew = Bw/Ew is thus a well-defined standard normed vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The proof given in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2 shows that �Ew = Bw/Ew is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Loeb measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let (Ω, A, µ) be an internal finitely-additive measure space, with Ω ⊆ O for a standard set O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' As discussed in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='3, attempts to construct its Loeb extension “over V” are only partially successful in BST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We fix a standard set I and a good w ∈ I so that Ω, A, µ are w-standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' By Transfer, (Ω, A, µ) is an internal finitely-additive measure space in the sense of Sw, and we con- struct the Loeb extension “over Sw.” We usually do not indicate the dependence on the choice of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In this example it is convenient to employ the variant of coding from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For every X ∈ A ∩ Sw let (1) [X] = st{fw | f ∈ OI ∧ f(w) ∈ X ∩ Sw}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' If X, X1, X2 ∈ A ∩ Sw, then the equivalences fw ∈ [X1 ∩ X2] iff fw ∈ [X1] ∧ fw ∈ [X2] and fw ∈ [Ω \\ X] iff fw ∈ [Ω] ∧ fw /∈ [X] hold NONSTANDARD HULLS AND LOEB MEASURES 19 for all standard f ∈ OI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' By Transfer, [X1 ∩ X2] = [X1] ∩ [X2] and [Ω \\ X] = [Ω] \\ [X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let B = st{A ∈ P(OI) | A = [X] for some X ∈ A ∩ Sw}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Using Transfer again, it follows that B is a standard algebra of subsets of [Ω].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We note that X1, X2 ∈ A ∩ Sw, X1 ̸= X2, implies [X1] ̸= [X2], so for standard A ∈ B there is a unique X ∈ A ∩ Sw with A = [X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' A standard finitely additive measure m on the algebra B with val- ues in the interval [0, +∞] is determined by the requirement that for standard A = [X] ∈ B m(A) = sh(µ(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The measure space ([Ω], B, m) satisfies Carath´eodory’s condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' If ⟨Ak | k ∈ N⟩ is a sequence of mutually disjoint sets in B, A ∈ B, and A = � k∈N Ak, then m(A) = Σk∈N m(Ak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In view of Transfer, it suffices to prove this claim under the assumption that ⟨Ak | k ∈ N⟩ and A are standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Suppose A = [X] where X ∈ A ∩ Sw and for each standard k, Ak = [Xk] where Xk ∈ A ∩ Sw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Clearly Xk are mutually disjoint and Xk ⊆ X for all standard k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Assume that for every standard n there is a w-standard x ∈ X such that x /∈ Xk holds for all k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' By Countable Idealization into Sw there is a w-standard x ∈ X such that x /∈ Xk holds for all standard k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' By Representability, x = g(w) for some standard g ∈ OI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Then gw ∈ A but ∀stk (gw /∈ Ak) and, by Transfer, ∀k (gw /∈ Ak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Therefore there is a standard n such that ∀stwx ∈ X ∃k ≤ n (x ∈ Xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It follows that ∀stF ∈ A ∃k ≤ n (F ∈ Ak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' By Transfer, � k∈N Ak = � k≤n Ak and, by finite additivity of m, we obtain m(A) = Σk≤n m(Ak) = Σk∈N m(Ak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' □ We conclude that m can be extended to a σ-additive measure m with values in [0, +∞] on the σ-algebra B generated by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The measure- theoretic completion of ([Ω], B, m) is the desired Loeb measure space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' we denote it ([Ω], �B, �m) (of course, it depends on the choice of w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Instead of an appeal to the Carath´eodory’s theorem, a direct proof along the lines of [11] can be given;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' see also Albeverio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' [1], Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='5 and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In order to explicate the relationship of ([Ω], �B, �m) to the usual external Loeb measure space, we first recall that Sw satisfies 20 KAREL HRBACEK AND MIKHAIL G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' KATZ the statement that (Ω, A, µ) is a finitely-additive measure space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The Loeb construction carried out over Sw would start with the external finitely additive measure space (Ω, A, m), where Ω = Ω ∩ Sw, A = X ∩ Sw | X ∈ A ∩ Sw , and m(X ∩ Sw) = sh(µ(X)) for X ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This space would be extended to an external σ-additive measure space using the Carath´eodory’s theorem, and then completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We now take V = Ω ∪ R, say, and note that for X ∈ A ∩ Sw, [X] is a w-code of X ∩ Sw, and hence B is a w-code for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It follows that ([Ω], B, m) is a w-code for (Ω, A, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The above construction cannot be carried out in BST directly for the external measure space (Ω, A, m), but presents no difficulties for its w-code ([Ω], B, m), a standard finitely-additive measure space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We compare Loeb measure spaces obtained from the same (Ω, A, µ) for different choices of the parameter w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We use sub- scripts to indicate dependence on this parameter and fix good w ∈ I, z ∈ J where I, J are standard and Sw ⊆ Sz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' There is a standard embedding �H = �Hw,z of �Bw into �Bz that preserves complements and countable unions and restricts to an isomorphism of Bw and Bz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' If mw is σ-finite, then also �mw(B) = �mz( �H(B)) for all B ∈ �Bw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' By Standardization, there is a standard function �H : �Bw → �Bz such that �H(B) = �Ψz( �Ψ −1 w (B)) for standard B ∈ �Bw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' ForA ∈ Bw ∩ S, if A = [X]w for X ∈ A ∩ Sw then �H(A) = [X]z ∈ Bz ∩ S, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This shows that �H maps Bw ∩ S onto Bz ∩ S, and hence, by Transfer, Bw onto Bz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2 we point out that the coding �Ψ preserves comple- ments and countable unions, so the same holds for �H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We give some de- tails for the countable unions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let B = � n∈N Bn, where ⟨Bn | n ∈ N⟩ is standard and B, Bn ∈ �Bw for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Define the external set X = ⟨n, x⟩ | x ∈ �Ψ −1 w (Bn) ⊆ (N∩S)×O, so that Bn = �Ψw(Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We then have B = �Ψw(� n∈N∩S Xn), so �Ψ −1 w (B) = � n∈N∩S Xn and �H(B) = �Ψz( �Ψ −1 w (B)) = �Ψz(� n∈N∩S Xn) = � n∈N �Ψz( �Ψ −1 w (Bn)) = � n∈N �H(Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It is clear from the definition of m(A) that mw(A) = mz( �H(A)) holds for standard A ∈ Bw, and hence by Transfer, for all A ∈ Bw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' If mw is σ-finite, then the completed Carath´eodory’s measure �mw is uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' If �mw(B) ̸= �mz( �H(B)) for some B ∈ �Bw, then �mw and �mz ◦ �H would be two distinct extensions of mw from Bw to �Bw, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' □ NONSTANDARD HULLS AND LOEB MEASURES 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Lebesgue measure from Loeb measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' As is well known, the Lebesgue measure can be obtained from a suitable Loeb measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='6 We give the gist of the argument in our framework, for the interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' More details can be found in Albeverio at al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' [1] (using a model- theoretic approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For n ∈ N let Tn = {i/n | 0 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Fix a nonstandard integer w ∈ N, and let T = Tw (in model-theoretic frameworks the equivalent of T is called “hyperfinite time line”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let O = [0, 1], Ω = T , A = P(T ) and µ the counting measure on T , ie, µ(X) = |X|/|T | for all sets X ⊆ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The construction in the preceding subsection, with I = N, yields the Loeb measure ([Ω], �B, �m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For every standard A ⊆ [0, 1] define A = st{fw ∈ [Ω] | f(w) ≃ c for some standard c ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This just means that •A is a w-code for sh−1(A) ∩ T ∩ Sw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Standard elements of the set •A are those fw ∈ [Ω] whose value “at infinity” (ie, at w) is infinitely close to a standard real in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let L = st{A ⊆ [0, 1] | •A ∈ �B } and let ℓ be the standard function on L determined by the requirement that ℓ(A) = �m(•A) for all standard A ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The triple ([0, 1], L, ℓ) is the Lebesgue measure space on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We prove that L is a σ-algebra containing all standard open intervals (a, b) ⊆ [0, 1] and all singletons {c} for standard c ∈ [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We also prove that ℓ is σ-additive and ℓ((a, b)) = b − a, ℓ({c}) = 0 for all standard open intervals and singletons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This implies that L contains all Lebesgue measurable subsets of [0, 1] and that ℓ is the Lebesgue measure for such subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For a proof (in the model- theoretic framework) that all sets in L are Lebesgue measurable see [1], Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' it can be easily adapted to our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let ⟨Ak | k ∈ N⟩ be a standard sequence of elements of L and let A = � k∈N Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Then •Ak ∈ �B holds for all standard k ∈ N, and we obtain that •A = � k∈N Ak, because if f(w) ≃ a for some standard a ∈ A, then a ∈ Ak for some standard k ∈ N by Transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' As �B 6A nonstandard construction of the Lebesgue measure can also be carried out with- out using a Loeb measure as an intermediate step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In the axiomatic approach, one method for doing so is developed by Lyantse and Kudryk [24], Appendix A, in the framework of IST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Another way is implicit in Hrbacek [11] and explicitly presented in Hrbacek and Katz [12] in the framework of SCOT, a subtheory of IST and BST that conservatively extends ZF + Dependent Choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For a “radically elementary” approach see Cartier and Perrin [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 22 KAREL HRBACEK AND MIKHAIL G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' KATZ is a σ-algebra, •A ∈ �B and we conclude that A ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Furthermore, ℓ(A) = �m(•A) = Σk∈N �m(•Ak) = Σk∈N ℓ(Ak), establishing σ-additivity of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Given a standard open interval (a, b) ⊆ [0, 1], we let Xa,b = T ∩(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We have [Xa,b] ∈ B and m([Xa,b]) = sh(µ(Xa,b)) = b − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let A = (a, b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' it remains to observe that, for standard fw ∈ [Ω], fw ∈ A iff fw ∈ [Xa+1/m, b−1/m] for some standard m ∈ N\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Standardiza- tion gives the function ⟨Am | m ∈ N\\{0}⟩ where Am = [Xa+1/m, b−1/m] for standard m, and Transfer implies •A = � m∈N\\{0} Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It follows that A ∈ B because the latter is a σ-algebra, and that m(A) = b−a because m is a σ-additive extension of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Hence A ∈ L and ℓ(A) = b − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The argument for A = {c} is similar, using the fact that •A = � m∈N\\{0} Am with Am = [Xc−1/m, c+1/m] for standard m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Neutrices and external numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This is another application of nonstandard analysis that extensively uses external sets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' see Dinis and van den Berg [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' A neutrix is a convex additive subgroup of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' With the exception of {0} and R, neutrices are externals sets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' the monad M(0) and the galaxy G(0) are nontrivial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' An external number is an algebraic sum of a real number and a neutrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Addition and multiplication of external numbers are defined by the Minkowski operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let N denote the collection of all neutrices and E denote the collection of all external numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Even in HST, N and E are (definable) proper classes of external sets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' they are “too large” to be external sets (see Kanovei and Reeken [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This difficulty can be remedied by relativizing these concepts to Sw (for good w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In fact, Nw = X∩Sw | X ∈ N and Ew = a+X∩Sw | a ∈ R ∩ Sw ∧ X ∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The collections Nw and Ew are external sets (of external sets) in HST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' As the natural embedding of R ∩ Sw into Ew given by r �→ r + {0} is crucial for applications of these concepts, it may be best for most purposes to avoid coding as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For example, the study of algebraic properties of operations + and × on Ew can be carried out in BST while viewing external numbers as external subsets of R ∩ Sw .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' However, work with external numbers often focuses on the structure (Ew, +, ×), its subsets, functions with values in it, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Then one can use the techniques of Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' see in particular Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='10 with V = R, to code Nw and Ew by standard structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' First, the external set R ∩ Sw is coded by the standard set RI/w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' As shown in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='5, RI/w = RI/Uw = ∗R, the hyperreals con- structed as the standard ultrapower of R by the standard ultrafilter NONSTANDARD HULLS AND LOEB MEASURES 23 Uw generated by w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let ∗< , ∗+ and ∗× be the ordering, addition and multiplication on the hyperreals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The coding provides an external iso- morphism between (∗R∩S, ∗<, ∗+, ∗×) and (R∩Sw, <, +, ×), so we can informally identify ∗R∩S and R∩Sw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Neutrices and external numbers in the hyperreals (∗R, ∗<, ∗+, ∗×) can be defined the same way as in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' External numbers in R ∩ Sw are coded by the standard external num- bers in the hyperreals ∗R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The coding preserves algebraic operations on external numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The collections Nw and Ew are coded respectively by the standard sets N and E of all neutrices and external numbers in ∗R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This approach is admittedly rather awkward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In HST the universes Sw can be extended to “external universes” WF(Sw) (see Kanovei and Reeken [15], Sections 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='3, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='4 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Perhaps the most practical way to handle external numbers would be to work with Nw and Ew in these universes and in the end code the final results in BST, if desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Subuniverses and ultrapowers 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Subuniverses Sw and ultrapowers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The universe Sw is closely connected to the ultrapower of the standard universe by a standard ultrafilter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The principle of Standardization implies that there is a standard set Uw such that (2) ∀stX (X ∈ Uw ←→ X ∈ P(I) ∧ w ∈ X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Clearly (i) ∅ /∈ Uw, and for standard X, Y ∈ P(I) (ii) X ∈ Uw ∧ X ⊆ Y → Y ∈ Uw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' (iii) X, Y ∈ Uw → X ∩ Y ∈ Uw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' (iv) X ∈ Uw ∨ (I \\ X) ∈ Uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' By Transfer, (ii) – (iv) hold for all X, Y ∈ P(I), so Uw is an ultrafilter over I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' One sees easily that Uw is nonprincipal if and only if w is nonstandard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Conversely, Bounded Idealization of BST implies that for every standard ultrafilter U over I there are (many) w ∈ I such that U = Uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The ultrapower of the universe of all sets V by a standard ultrafilter U is defined in the usual way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' One defines an equivalence relation =U on VI by (3) f =U g ←→ {i ∈ I | f(i) = g(i)} ∈ U, and a membership relation (4) f ∈U g ←→ {i ∈ I | f(i) ∈ g(i)} ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 24 KAREL HRBACEK AND MIKHAIL G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' KATZ The usual procedure at this point is to form equivalence classes fU of functions f ∈ VI modulo =U, using “Scott’s trick” of taking only the functions of the minimal von Neumann rank to guarantee that the equivalence classes are sets: Let fU = {g ∈ VI | g =U f and ∀h ∈ VI (h =U f → rank h ≥ rank g)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' see Jech [13], (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='3) and (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' One lets VI/U be the class of all fU for f ∈ VI and defines (5) fU ∈U gU ←→ f ∈U g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The ultrapower of V by U is the structure (VI/U, ∈U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The universe V is embedded into VI/U via x �→ (cx)U where cx is the constant function on I with value x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We note that fU is standard iff fU = gU for some standard g ∈ VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We assume from now on that whenever the equivalence class fU is standard, the representative function f is taken to be standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The key insight is that the standard elements of the ultrapower of V by Uw are in equality-and-membership-preserving correspondence with w-standard elements of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It is expressed by the following proposition, which is an immediate consequence of definitions (3) - (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For any standard functions f, g ∈ VI: f =Uw g ←→ fUw = gUw ←→ f(w) = g(w) and f ∈Uw g ←→ fUw ∈Uw gUw ←→ f(w) ∈ g(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The correspondence Φw is defined on Sw × (VI/Uw ∩ S) by Φw(ξ, fUw) ←→ f(w) = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In this notation, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1 asserts the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The class Φw is an isomorphism between the structures (Sw, ∈) and (VI/Uw ∩ S, ∈Uw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We note that (VI/Uw ∩ S, ∈Uw) is the ultrapower of the universe in the sense of the standard universe S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' If φ(v) is an ∈-formula such that VI/Uw = F | φ(F) , then VI/Uw ∩ S = F ∈ S | φst(F) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' If ψ(u, v) is an ∈-formula such that F ∈Uw G ←→ ψ(F, G), then F ∈Uw G ←→ ψst(F, G) holds for F, G ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' If Φw(ξ, fUw) holds, we write Φw(ξ) = fUw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We note that for x ∈ S, Φw(x) = (cx)Uw where cx is the constant function on I with value x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' As is customary, we identify (cx)Uw with x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This gives a stronger version of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' NONSTANDARD HULLS AND LOEB MEASURES 25 Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The class Φw is an isomorphism between the structures (Sw, S, ∈, ) and (VI/Uw ∩ S, S, ∈Uw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We also note that Φw(w) = IdUw where Id(i) = i for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Recall that quantifiers with superscript stw range over w-standard sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' If φ is an ∈-formula, φstw is the formula obtained from φ by relativizing all quantifiers to stw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' �Lo´s’s Theorem for ∈-formulas takes the following form: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For standard f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' , fr, φstw(f1(w), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' , fr(w)) ←→ {i ∈ I | φ(f1(i), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' , fr(i))} ∈ Uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The structures (VI/Uw, ∈Uw) and (VI/Uw ∩ S, ∈Uw) are not mod- els in the sense of model theory because their components are proper classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' hence the satisfaction relation ⊨ is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Given an ∈- formula φ with parameters from VI/Uw, we write “φ holds in (VI/Uw, ∈Uw )” to stand for the formula obtained from φ by replacing all occurrences of u ∈ v with u ∈Uw v and relativizing all quantifiers to VI/Uw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' similarly for “φ holds in (VI/Uw ∩ S, ∈Uw).” Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We have φstw(f1(w), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' , fr(w)) ←→ φ((f1)Uw, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' , (fr)Uw) holds in (VI/Uw ∩ S, ∈Uw) [by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2] ←→ φ((f1)Uw, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' , (fr)Uw) holds in (VI/Uw, ∈Uw) [by Transfer] ←→ {i ∈ I | φ(f1(i), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' , fr(i))} ∈ Uw [by the usual �Lo´s’s Theorem].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2 we fix a universal standard set V and for standard f ∈ V I define fw,V (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Clearly fw,V = st{g ∈ V I | g(w) = f(w)} = st{g ∈ V I | g =Uw f} = {g ∈ V I | g =Uw f}, where the last step is by Transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Thus, again by Transfer, V I/w is nothing but the ultrapower V I/Uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Proofs of claims in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' A set w is good iff Uw is countably incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Of course, ultrapowers by countably incomplete ultrafilters are the ones of interest in nonstandard analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Immediate from the isomorphism Φ between the universe of w-internal sets (Sw, ∈) and the ultrapower ((VI/Uw) ∩ S, ∈Uw) (see Chang-Keisler [4], Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' □ 26 KAREL HRBACEK AND MIKHAIL G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' KATZ Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='4 Proof of (1): Assume ∃x φ(x, p0) where p0 is (wlog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' the only) pa- rameter and stw(p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Fix a standard set P such that p0 ∈ P (Bound- edness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Use AC to obtain a standard function F on P such that ∀p ∈ P (∃x φ(x, p) → φ(F(p), p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Hence φ(F(p0), p0) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' As F(p0) is w- standard by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2 (c), we conclude that ∃stwx φ(x, p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' □ Proof of (2): It is well-known that every ultrapower by a countably incomplete ultrafilter U is ω1-saturated (see Chang-Keisler [4], Theo- rem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Hence Countable Idealization in the form (6) ∀stn ∈ N ∃x ∀m ∈ N (m ≤ n → φ(m, x)) ←→ ∃x ∀stn ∈ N φ(n, x) holds in (VI/Uw, V, ∈Uw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' By BST Transfer, (6) holds in (VI/Uw ∩ S, S, ∈Uw), and, by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='3, it holds in (Sw, S, ∈).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This translates to ∀stn ∈ N ∃stwx ∀stwm ∈ N (m ≤ n → φstw(m, x)) ←→ ∃stwx ∀stn ∈ N φstw(n, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Using w-Transfer we get the desired form ∀stn ∈ N ∃x ∀m ∈ N (m ≤ n → φ(m, x)) ←→ ∃stwx ∀stn ∈ N φ(n, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' □ Proof of (3): Recall that stw(x) means that x = g(w) for some standard g defined on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let φ(v) be an ∈-formula with standard parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Assume φ(g(w));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' by w-Transfer then φstw(g(w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' From the isomorphism between (Sw, ∈) and (VI/Uw) ∩ S, ∈Uw) and �Lo´s’s Theorem it follows that X = {i ∈ I | φ(g(i))} ∈ Uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Pick i0 ∈ X and let f be the standard function defined by f(i) = g(i) for i ∈ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' f(i) = f(i0) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Then f(w) = x and φ(f(i)) holds for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' □ □ Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let κ be a standard infinite cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The set w is κ+- good if Uw is a countably incomplete κ+-good ultrafilter (In particular, w is good iff it is ω1-good;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=') We prove Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='5 in the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' (Idealization into w-standard sets over sets of car- dinality ≤ κ) Let φ be an ∈-formula with w-standard parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' If w is κ+-good, then for every standard set A of cardinality ≤ κ ∀stfina ⊆ A ∃y ∀x ∈ a φ(x, y) ←→ ∃stwy ∀stx ∈ A φ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For every standard uncountable cardinal κ and every z there exist κ+- good w so that z is w-standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' NONSTANDARD HULLS AND LOEB MEASURES 27 Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='8 It is well known (Chang and Keisler [4], Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='8) that any ultrapower by a countably incomplete κ+-good ultrafilter U is κ+- saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' As in the proof of (2), it follows that Bounded Idealization over sets of cardinality ≤ κ holds in (Sw, S, ∈) for κ+-good w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' To also obtain z ∈ Sw we use the following fact proved in Keisler [16]: If U is a countably incomplete κ+-good ultrafilter over I and V is an ultrafilter over J, then the ultrafilter U ⊗ V over I × J defined by X ∈ U ⊗ V ←→ {i ∈ I | {j ∈ J | ⟨i, j⟩ ∈ X} ∈ V } ∈ U is countably incomplete and κ+-good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The definition of ⊗ implies that for every standard X ∈ U ⊗ Uz there is a standard i ∈ I such that {j ∈ J | ⟨i, j⟩ ∈ X} ∈ Uz, and hence ⟨i, z⟩ ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' From Bounded Idealization one obtains w ∈ I such that ⟨w, z⟩ ∈ X holds for all standard X ∈ U ⊗ Uz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' in other words, U⟨w,z⟩ = U ⊗ Uz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Then z ∈ S⟨w,z⟩ and ⟨w, z⟩ is κ+-good, so S⟨w,z⟩ satisfies Bounded Idealization over sets of cardinality ≤ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' More general universes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The definition of w-standard sets in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1 can be generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let w : S → I where S, I are standard sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We let Sw = f(w(s1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' , w(sk)) | k, f ∈ S, dom f = Ik and s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' , sk ∈ S ∩ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It turns out that these universes correspond precisely to the standard limit ultrapowers of the standard universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The proof is similar to the model-theoretic proof that every internal universe ∗V (X) is a bounded limit ultrapower of the superstructure V (X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' see Chang and Keisler [4], Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='19 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' With suitable modifications, all results described in this paper remain valid for this more general notion of w-standard sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Some earlier constructions There are several earlier publications where constructions of nonstan- dard hulls and Loeb measures in the internal framework are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Below we summarize this work and provide some critical assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The “full” nonstandard hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let (M, d) be a standard metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' A straightforward attempt to carry out the construction of the nonstandard hull of (M, d) in BST can start as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let Bmax = x ∈ M | d(x, a) is limited for some standard a ∈ M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let Emax be the equivalence relation on Bmax defined by Emax = ⟨x, y⟩ ∈ Bmax × Bmax | d(x, y) ≃ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 28 KAREL HRBACEK AND MIKHAIL G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' KATZ Finally, let the function Dmax with standard real values be defined by Dmax = ⟨x, y, r⟩ ∈ Bmax × Bmax × R | r = sh(d(x, y)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The classes Bmax, Emax and Dmax are (definable) external sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For every x ∈ Bmax, the equivalence class M(x) = z ∈ Bmax | d(x, z) ≃ 0 is an external set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' However, the final step in the construction of the nonstandard hull of (M, d), to wit, the formation of the quotient space (Bmax/Emax, Dmax/Emax), cannot be carried out in BST (it would require a “class of classes” M(x) | x ∈ Bmax ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' There are some ways around this difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Perhaps the most straight- forward is to forgo the formation of the quotient space and work with the representatives of the equivalence classes (ie, the elements of Bmax), and with the congruence Emax in place of the actual equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This would be similar to working with fractions rather than the rational numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' But this way does not produce the nonstandard hull as an actual object of BST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The quotient space (Bmax/Emax, Dmax/Emax) can be formed in HST (using its axiom of Replacement for st-∈-formulas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' An interpretation of HST can be coded in BST (see [15], Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='4 and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='5), so in this indirect way the “full” nonstandard hull can be coded in BST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Unfortunately, the coding involved is far from being a “morphism” in any sense, so the resulting opaque code is unsuitable for transferring nonstandard intuitions about the hull to its coded version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' One point of working with subuniverses is that they have a natural coding (by standard sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Perhaps the most serious objection to this way of constructing non- standard hulls is that (Bmax/Emax, Dmax/Emax) is just “too large.” Trivially, every nonstandard hull (Bw/Ew, Dw/Ew) of (M, d) consid- ered in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1 embeds isometrically into (Bmax/Emax, Dmax/Emax) (note that Bw = Bmax∩Sw, Ew = Emax∩Sw, and Dw = Dmax∩Sw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In many interesting cases, the metric space (M, d) has nonstandard hulls of arbitrarily large cardinality, so Bmax/Emax is not of standard size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It would be difficult to do further work with nonstandard hulls using this method, such as compare them with other standard metric spaces, take their products, or form the space of continuous functions on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' They are analogous to the “universal group” that can be constructed as a direct sum (or product) of all groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This “object” is a proper class in ZFC, hardly if ever used for more than bookkeeping purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Another important point about working with subuniverses is that the objects produced are standard sets (or external sets of standard size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' NONSTANDARD HULLS AND LOEB MEASURES 29 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Vakil’s construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In [31], Vakil presents a construction of nonstandard hulls of uniform spaces in IST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' His method does not require fixing a particular subuniverse, and Standardization is used in a way similar to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' But Vakil’s method applies only to a certain class of uniform spaces, the so-called Henson-Moore spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Revealingly, these are precisely the spaces whose nonstandard hull is independent of the choice of the nonstandard universe, ie, it is unique up to isomorphism and of standard size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' see Henson-Moore [9] and Vakil [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Loeb measures in IST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Diener and Stroyan [6], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 274, outline a possible construction of Loeb measures in IST, referencing Stroyan and Bayod [30], Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2, for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Here we briefly consider this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let (Ω, A, µ) be an internal finitely-additive measure space, with Ω ⊆ O for a standard set O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' We take A = P(Ω) for simplicity, and analyze Loeb’s construction from the point of view of BST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The first step is to extend the algebra A to an external σ-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Bounded Idealization in BST implies that an externally countable union of (internal) sets is either equal to a finite union or is not internal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' So the construction has to deal with external sets from the very beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The only way to treat external sets as objects in BST is via some kind of coding by sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' For example, every external sequence Xn | n ∈ N ∩ S of (internal) sets has an extension ⟨Xn | n ∈ N⟩ to an (internal) sequence,7 which can be regarded as its code (of course an external sequence has many codes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This coding could be extended to higher levels of the Borel hierarchy over the algebra of (internal) subsets of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' A simpler solution, proposed in [6, 30], is to code Loeb measurable sets with the help of Souslin schemata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' One can define a Souslin schema in BST as a function S : N<ω → P(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let F = Nω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The kernel of S is the external set kerS = � f∈F∩S � n∈N∩S Sf↾n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The external sets obtainable as kernels of Souslin schemata are Henson sets and Henson sets whose complement in Ω is also Henson are the Loeb sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Loeb sets form the smallest external σ-algebra σ(A) contain- ing P(Ω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' see [30], Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='3 (Luzin Separation Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Let L be the external set of all pairs ⟨S1, S2⟩ such that kerS1∩kerS2 = ∅ and kerS1 ∪ kerS2 = Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Then L can be viewed as a set of codes for Loeb 7This follows from the Extension principle of BST;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' see Kanovei and Reeken [15], Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 30 KAREL HRBACEK AND MIKHAIL G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' KATZ sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The algebraic operations (union, complement, externally count- able union) can be coded by external relations on L, and the Loeb measure itself can be coded by an external relation on L × (R ∩ S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The objections raised in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1 against “full” nonstandard hulls apply also to the above approach to Loeb measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In particular, one cannot form the quotient space of L modulo the relation in which two codes are equivalent when they code the same external set, and the coding of the set-theoretic operations is not a “morphism” (eg, the code of the union of two sets is in no sense the union of their codes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It would be awkward to work with random variables on L via codes, and collections of random variables are even more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In addition, it is customary in the literature (see Albeverio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' [1], Kanovei and Reeken [15]) to regard not σ(A) but its completion L(A) as the Loeb σ-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It is not usually possible to extend L to an external set of codes for L(A) (not even in HST, because the Power Set axiom for external sets does not hold there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' On these grounds, it is arguable whether the claim that this method represents the Loeb measure space is justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Bounded Internal Set Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Following upon ideas of Lind- strøm [17], Diener and Stroyan [6] aim for “a description of the com- mon ground between the two approaches to Robinson’s theory.” This takes the form of working in a polysaturated8 nonstandard universe N = (V (S), V (∗S), ∗) axiomatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' They formulate Bounded Internal Set Theory (bIST) and show that its axioms hold in such universes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' This theory should not be confused with BST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The main differences are: The language of bIST is closely tied to the structure N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It contains a constant symbol for every internal set in V (∗S) and more;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' in particular, it is uncountable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The axiom schemata T, I and S are modified so as to apply only to those formulas in which all quantifiers are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The axioms of ZFC are not postulated (in fact, Replacement fails in V (S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' bIST proves many results familiar from IST or BST, but, just like in these theories, its variables range over internal sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It does not provide access to higher order external sets without some coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Of course, the nonstandard universe N does provide external sets that can be used to construct nonstandard hulls and Loeb measures in the usual way, but then one is using the model-theoretic rather than the axiomatic 8N is polysaturated if it is κ-saturated for κ = |V (∗S)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' NONSTANDARD HULLS AND LOEB MEASURES 31 approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Overall, bIST is more a useful tool for work with super- structures than a self-standing axiomatics for nonstandard analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Measure and integration over finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Nelson’s Radically Elementary Probability Theory [26] works with (hyper)finite proba- bility spaces only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It requires only very elementary axioms (see [26], Chapter 4) that are easy consequences of the axioms of IST, or even of the weaker and more effective theory SCOT (see footnote 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Further development of this approach can be found eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' in Cartier and Per- rin [2, 3], who study both Nelson’s S-integral on finite measure spaces (which they call Loeb-Nelson integral) and its refinement, which they call Lebesgue integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The “radically elementary” approach is simple and elegant, and many interesting results have been obtained in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' It is beyond the scope of this paper to try to compare the “radically elementary” approach to the nonstandard measure theory based on the work of Loeb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Nelson shows ([26], A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='1) that for every stochastic process there is a nearby elementary process, and that theorems of the conventional theory of stochastic processes can be derived from their elementary analogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The works [26, 2, 3] do not address the question whether (up to isomorphism) Loeb measure spaces can be obtained from their constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Conclusion This paper is based on the fact that an internal set theory such as BST provides a supply of structures that are analogous to the internal part of model-theoretic nonstandard universes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Utilizing a natural cod- ing and the principle of Standardization, we can mimic external sets by standard ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' Nonstandard hulls and Loeb measures can be obtained “up to an isomorphism” with their external counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In BST they are standard objects in a common framework, which makes it easy in principle to compare them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In the end, they are the same structures that could be obtained by the superstructure methods carried out in BST, but our technique for obtaining them is more suitable for the internal axiomatic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The theory BST is the internal part of HST, a theory which ax- iomatizes external sets, in addition to standard and internal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' In this theory one can carry out external constructions in the same way as in nonstandard universes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' no coding is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' The coding can be introduced afterwards in order to convert the external structures so obtained to standard structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' 32 KAREL HRBACEK AND MIKHAIL G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content=' KATZ We conclude that the internal axiomatic approach based on BST can implement virtually all techniques employed by the model-theoretic ap- proach, with some advantages: It could make nonstandard hulls and Loeb measures more easily accessible to working mathematicians who have learned the nonstandard methods in IST or another axiomatic framework, because it does not require a priori knowledge of ultrafil- ters and ultrapowers or model theory.' 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York, NY 10031, Email address: khrbacek@icloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='com Department of Mathematics, Bar Ilan University, Ramat Gan 5290002 Israel, Email address: katzmik@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='biu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} +page_content='il' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQfg_gX/content/2301.00367v1.pdf'} diff --git a/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf b/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf new file mode 100644 index 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b/TdAyT4oBgHgl3EQfuflV/content/tmp_files/2301.00613v1.pdf.txt @@ -0,0 +1,1338 @@ +CMB power spectrum for emergent scenario and slow expansion +in scalar-tensor theory of gravity +Qihong Huang1∗, He Huang2 and Bing Xu3 +1 School of Physics and Electronic Science, +Zunyi Normal University, Zunyi 563006, China +2 Institute of Applied Mechanics, Zhejiang University, Zhejiang 310058, China +3 School of Electrical and Electronic Engineering, +Anhui Science and Technology University, Bengbu, Anhui 233030, China +Abstract +We analyze the stability of the Einstein static universe in scalar-tensor theory of gravity, and +find the Einstein static universe can be stable against both scalar and tensor perturbations under +certain conditions. By assuming the emergent scenario originating from an Einstein static state, +followed by an instantaneous transition to an inflationary phase, we study and obtain the analytical +approximations of the primordial power spectrum for the emergent scenario. Then, we plot the +primordial power spectrum and CMB TT-spectrum of the emergent scenario and the slow expan- +sion. These figures show that both of these spectra for the slow expansion are the same as that for +ΛCDM, and the spectra of the emergent scenario are suppressed at large scales. +PACS numbers: 98.80.Cq +∗ Corresponding author: huangqihongzynu@163.com +1 +arXiv:2301.00613v1 [gr-qc] 2 Jan 2023 + +I. +INTRODUCTION +Inflation [1–3] posits an epoch very early in the universe, during which the scale factor +grows exponentially with time. It can solve most of problems in the standard cosmology. +The primordial scalar perturbations originating from quantum fluctuations during the infla- +tionary epoch not only explain the cosmic microwave background radiation anisotropy but +also seed the large-scale structure of the universe [4, 5]. Although it achieves great success, +it still suffers from the big bang singularity problem. To solve this intractable problem, +some scenarios had been proposed and suggested to construct non-singular or past eternal +cosmological models, such as the emergent scenario [6, 7] and the slow expansion [8]. +In the emergent scenario, the universe is assumed to start from an Einstein static universe +and then evolves into an inflationary era [6, 7]. Since the universe stems from an Einstein +static universe in the emergent scenario, the big bang singularity is avoided naturally. In +addition, the e-folding number and the nearly scale-invariant spectral index can also be pro- +duced by the inflation of the emergent scenario [7]. Thus, the emergent scenario has drew +lots of attention after it was proposed [9–16]. For the other model, namely the slow expan- +sion, the universe originates from an Einstein static universe and then enters into an epoch +in which the universe expands very slowly. During this epoch, the nearly scale-invariant +primordial power spectrum is provided [8, 17–23]. After this epoch, the universe evolves +into the big bang epoch. It was found that general relativity can be recovered and the uni- +verse will evolve in accordance with the standard cosmology after the slow expansion ends +[22, 23]. Similar to the emergent scenario, the big bang singularity is also avoided since the +universe originates from an Einstein static universe. Since both the emergent scenario and +the slow expansion assume that the universe originates from an Einstein static universe and +that a stable Einstein static universe must be stable against both the scalar perturbations +and the tensor perturbations, to find a stable Einstein static universe becomes a crucial +issue. Fortunately, it was found that a stable Einstein static universe exists in Mimetic grav- +ity [16], scalar-fluid theory [24], non-minimal derivative coupling model [25, 26], braneworld +model [27], Jordan-Brans-Dicke theory [28], Eddington-inspired Born-Infeld theory [29], hy- +2 + +brid metric-Palatini gravity [30], GUP theory [31], f(R,T) gravity [32], f(R,T,Q) gravity [33] +and massive gravity [34] and so on. Thus, the big bang singularity can be solved in the +theories of modified gravity by using the emergent scenario and the slow expansion. +It is notable that, except for avoiding the big bang singularity, both the emergent scenario +and the slow expansion can produce a nearly scale-invariant primordial power spectrum. +Recently, by assuming the Einstein static state as a superinflating phase [35] or a static state +phase [36, 37], the CMB TT-spectrum of the emergent scenario was studied in the framework +of general relativity, and the results show that the CMB TT-spectrum is suppressed at large +scales. However, it is quite unclear that whether the CMB TT-spectrum of the slow expansion +is also suppressed at large scales, and whether the CMB TT-spectrum can be utilized to +discriminate the emergent scenario from the slow expansion. To answer these questions, we +will study the CMB TT-spectrum of the emergent scenario and the slow expansion in the +scalar-tensor theory of gravity. +The paper is organized as follows. In section II, we briefly review the field equations of +the scalar-tensor theory of gravity. In section III, we will give the derivation of equations +of motion for perturbations. In section IV, we study the stability conditions of the Einstein +static universe. In section V, we plot the primordial power spectrum and the CMB TT- +spectrum of the emergent scenario and the slow expansion. Finally, our main conclusions +are shown in Section VI. +II. +FIELD EQUATIONS +In this paper, we consider the following scalar-tensor theory of gravity, whose action takes +the following form [38–40] +S = +� +d4x√−g +�1 +2f(ϕ)R − 1 +2ω(ϕ)gαβ(∂αϕ)(∂βϕ) − V (ϕ) + Lm +� +, +(1) +where R is the Ricci curvature scalar, ϕ is the scalar field of scalar-tensor theory, f(ϕ) +and ω(ϕ) are coupling function of scalar field, V (ϕ) is the potential, and Lm represents the +Lagrangian density of a perfect fluid. Here, the coupling function f(ϕ) needs to be positive +for the gravitons to carry positive energy. +3 + +Varying the action (1) with respect to the metric tensor gαβ and the scalar field ϕ, we +obtain +f(Rαβ − 1 +2gαβR) − ∇α∇βf + gαβ∇σ∇σf − ω +� +∇αϕ∇βϕ − 1 +2gαβ∇σϕ∇σϕ +� ++ gαβV = Tαβ,(2) +and +fϕR + ωϕ∇σϕ∇σϕ + 2ω∇σ∇σϕ − 2Vϕ = 0, +(3) +where fϕ denotes df +dϕ. +The field equations (2) can be expressed as the standard form of general relativity Gαβ = +T eff +αβ +with a modification in the energy-momentum tensor +T eff +αβ = 1 +f +� +∇α∇βf − gαβ(∇σ∇σf + V ) + ω +� +∇αϕ∇βϕ − 1 +2gαβ∇σϕ∇σϕ +� ++ Tαβ +� +. +(4) +We consider a homogeneous and isotropic universe described by FLRW metric +ds2 = −a2(η)dt2 + a2(η)γijdxidxj, +γijdxidxj = +dr2 +1 − Kr2 + r2(dθ2 + sin2 θdϕ2), +(5) +where K = 1, 0, −1 corresponds to a closed, flat and open universe, respectively. +The +background equation can be obtained by substituting this metric into the field equations (2), +the 0 − 0 component gives the Friedmann equation +3H2 + 3K + 3Hf ′ +f = 1 +f +�1 +2ωϕ′2 + a2V + a2ρ +� +, +(6) +and the i − i component gives +2H′ + H2 + f ′′ +f + Hf ′ +f + K = − 1 +f +�1 +2ωϕ′2 − a2V + a2p +� +, +(7) +where ′ denotes a derivative with respect to the conformal time η, and ρ and p denote the +energy density and pressure of the perfect fluid with p = (γ − 1)ρ. Combining Eqs. (6) and +(7), and eliminating ρ, one obtain +2H′ + f ′′ +f + (3γ − 2) +� +H2 + K + Hf ′ +f +� += 1 +f +�1 +2(γ − 2)ωϕ′2 + γa2V +� +. +(8) +The field equation (3) can be expressed as +ϕ′′ + 2Hϕ′ + 1 +2ω +� +ωϕϕ′2 + 2a2Vϕ − 6(H′ + H2 + K)fϕ +� += 0. +(9) +4 + +III. +PERTURBATIONS +To analyze the power spectrum of the emergent scenario and the slow expansion in scalar- +tensor theory of gravity, we are required to obtain the equation of motion for perturbation. +In this section, we will derive this equation. +A. +Conformal transformation to Einstein gravity +The scalar-tensor theory of gravity can be transformed into Einstein gravity by performing +a conformal transformation on the metric [41–45] +˜gαβ = Ω2gαβ, +(10) +and the corresponding action becomes +˜S = +� +d˜td3x +� +−˜g +�1 +2 +˜R − 1 +2 +˜∇σ ˜ϕ ˜∇σ ˜ϕ − ˜V +� +, +(11) +which is the action for a minimal coupling single scalar field ˜ϕ and the conformal factor is +Ω = √f. The corresponding variables are defined as +d˜t = Ωdt, +˜a = Ωa, +˜Φ = Φ + δΩ, +˜Ψ = Ψ − δΩ, +d ˜ϕ = +� +ω +f + 3 +2 +f 2 +,ϕ +f 2 dϕ, +˜V = V +f 2.(12) +Even though the action (1) and (11) are fully equivalent [41–48], cosmological perturbations +are easier to analyze in action (11). Therefore, we shall analyze the perturbation in the +action (11). The background equation and the equation of motion for the field ˜ϕ can be +written as follows +3 ˜H2 + 3K +˜a2 = 1 +2 +˙˜ϕ2 + ˜V , +(13) +3 ˜H2 + 2 ˙˜H + K +˜a2 = − +�1 +2 +˙˜ϕ2 − ˜V +� +, +(14) +and +¨˜ϕ + 3 ˜H ˙˜ϕ + ˜V ˜ϕ = 0. +(15) +5 + +Here, the no ghost condition ˙˜ϕ2 > 0 gives ωf + 3 +2 +� df +dϕ +�2 > 0 which is contained in expres- +sion (30), and a dot represents the derivative with respect to the cosmic time t. +In order to obtain the perturbation equations with curvature K, we use the method +developed by Garriga and Mukhanov [49, 50]. Then, the perturbation equations for 0 − 0 +and 0 − i components can be written as [50, 51] +2 +� 1 +˜a2∆˜Ψ − 3 ˜H ˙˜Ψ + 3 +�K +˜a2 ˜Ψ − ˜H2 ˜Φ +�� += ˙˜ϕ ˙δ ˜ϕ − ˜Φ ˙˜ϕ2 + ˜V ˜ϕδ ˜ϕ, +(16) +2∇i +� ˙˜Ψ + ˜H ˜Φ +� += ˙˜ϕ∇iδ ˜ϕ, +(17) +where ∆ = ∇2 is Laplace operator, and the component for i ̸= j is +1 +˜a2∇j∇i�˜Φ − ˜Ψ +� += 0, +(18) +which gives ˜Φ = ˜Ψ. After introducing two new variables +˜ξ = 2˜a +˜H +˜Ψ, +(19) +˜ζ = ˜Ψ + ˜H δ ˜ϕ +˙˜ϕ − 2K +˜a2 ˙˜ϕ2 ˜Ψ, +(20) +the equations (17) and (16) can be simplified as +˙˜ξ = ˜a +˙˜ϕ2 +˜H2 ˜ζ, +(21) +˙˜ζ = +˜H2 +˜a3 ˙˜ϕ2 +� +∆ + Y K +�˜ξ, +(22) +where +Y = −2 +˜V ˜ϕ +˜H ˙˜ϕ +. +(23) +The detailed process is given in Appendix. +In order to obtain the amplitude of quantum fluctuations, one needs to expand the action +for the gravitational and scalar fields to second order in perturbations which are cumbersome. +Since the second order perturbation action can be inferred directly from the equations of +6 + +motion (21) and (22), these cumbersome steps can be avoided. The detailed steps are given +in Ref. [49, 50]. Thus, the action reproducing the perturbation equations (21) and (22) can +be written as +˜S = +� � +˜ξ ˜O ˙˜ζ − 1 +2 +˜H2 +˜a3 ˙˜ϕ2 ˜ξ ˜O(∆ + Y K)˜ξ + 1 +2 +˜a ˙˜ϕ2 +˜H2 ˜ζ ˜O˜ζ +� +d˜td3x, +(24) +where ˜O = ∆ + 3K is a time-independent operator. Expressing ˜ξ in terms of ˙˜ζ by equa- +tion (22), the action can be reduced to +˜S = +� +z2�˜ζ′2 + ˜ζ(∆ + Y K)˜ζ +� +dηd3x, +(25) +where prime denotes the derivative with respect to the conformal time η, and the variable z +is +z = ˜a ˜ϕ′ +˜H +� +˜O +∆ + Y K . +(26) +The Laplacian ∆ should be understood as a c − number and represents the corresponding +eigenvalue. +B. +Equation of motion for perturbation +The second perturbation action in the scalar-tensor theory of gravity can be obtained by +a transformation for the action (25). Introducing the canonical quantization variable v = z˜ζ +and utilizing the corresponding transform equations (12), the action (25) can be rewritten +as follows +S = 1 +2 +� � +v′2 + v(∆ + Y K)v + z′′ +z v2� +dηd3x. +(27) +Here, z and Y are +z = a2ϕ′√ +E +(a√f)′ +� +∆ + 3K +∆ + Y K , +(28) +and +Y = 2(a2ϕ′√ +E)′/(a2ϕ′√ +E) +(a√f)′/(a√f) +, +(29) +7 + +where +E = f +� +ω + 3 +2 +f 2 +ϕ +f +� +. +(30) +Varying the action (27) with respect to v, one can straightforwardly get the equation of +motion for the variable as +v′′ − (∆ + Y K)v − z′′ +z v = 0. +(31) +Expressing ˜ζ and ˙˜ζ in the action (24) in terms of ˜˙ξ and ˜ξ, we obtain +u′′ − (∆ + Y K)u − Z′′ +Z u = 0, +(32) +where +u = a +� +f 3 +ϕ′√ +E +˜ψ, +Z = (a√f)′ +a2ϕ′√ +E +. +(33) +The equation of motion for the variable u was obtained in Ref. [52, 53] by doing some +calculations. +IV. +EINSTEIN STATIC UNIVERSE +Both in the emergent scenario and in the slow expansion, the universe stems from an +Einstein static universe. However, it was found that the Einstein static solution is unstable +in scalar-tensor theory of gravity when the perfect fluid is pressureless matter(γ = 1) or +radiation(γ = 4 +3) [54]. So, to find a stable Einstein static solution becomes crucial for the +emergent scenario and the slow expansion in this theory. In order to find a stable Einstein +static solution, we reanalyze the stability of the Einstein static solution by considering 0 ≤ +γ ≤ 2 in scalar-tensor theory of gravity. +A. +Static solutions +The Einstein static solution requires a = a0 = constant and a′ +0 = a′′ +0 = 0 which indicates +H0 = H′ +0 = 0. And Eqs. (6) and (7) indicate that a constant a requires ϕ = ϕ0 = constant +8 + +and ϕ′ +0 = ϕ′′ +0 = 0. Thus, for the Einstein static solution, Eqs. (8) and (9) show +K +a2 +0 += +γV0 +(3γ − 2)f0 +, +(34) +and +K +a2 +0 += V0ϕ +3f0ϕ +, +(35) +where 0 denotes the corresponding static state value, and 0ϕ = +� +d +dϕ +� +ϕ=ϕ0. In addition, the +energy density ρ0 is given by Eq. (6) +ρ0 = +2V0 +3γ − 2. +(36) +The existence conditions of Einstein static solutions require a2 +0 > 0 and ρ2 +0 > 0 which mean +f0 > 0, +V0 > 0, +2 +3 < γ ≤ 2, +V0ϕ +f0ϕ +> 0, +(37) +for K = 1 and +f0 < 0, +V0 > 0, +2 +3 < γ ≤ 2, +V0ϕ +f0ϕ +< 0. +(38) +for K = −1. +Since a stable Einstein static universe is required to be stable against both scalar pertur- +bations and tensor perturbations, we will analyze the stability in the following subsections. +B. +Tensor perturbations +Since tensor perturbations are easy to analyze, we will analyze it at first and the perturbed +metric is given as [55] +ds2 = −a2(η)dη2 + a2(η)(γij + 2hij)dxidxj. +(39) +Performing a harmonic decomposition for the perturbed variable hij, we obtain +hij = HT,klm(t)Yij,klm(θn). +(40) +9 + +Because the quantum numbers m and l do not enter the perturbed differential equations, +the harmonic function Yk = Yklm(θn) satisfies [56] +∆Yk = −K2Yk = +� −k(k + 2)Yk, +k = 0, 1, 2, ..., +K = +1 +−k2Yk, +k2 ≥ 0, +K = 0 +−(k2 + 1)Yk, +k2 ≥ 0, +K = −1 +(41) +where ∆ represents the three-dimensional spatial Laplacian operator. Following Ref. [57, 58], +considering the static conditions and then substituting the perturbed metric (39) into the +field equations (2), the equation of tensor perturbations becomes +H′′ +T + (k2 + 2K)HT = 0. +(42) +According to this equation, we can find that k2 + 2K > 0 must be satisfied for any k to +obtain a stable solution. As a result, the Einstein static solutions are stable against the +tensor perturbations for the case K = 1. +C. +Scalar perturbations +Since the Einstein static solutions can be stable against the tensor perturbation for K = 1, +we will analyze the stability of the static solutions under the scalar perturbations in the closed +spacetime in this subsection. According to the results in Ref [54], the stability of the static +solutions against the scalar perturbations in scalar-tensor theory of gravity are determined +by the eigenvalues of the matrix of N, which is given as +N = +� +N11 +N12 +N21 +N22 +� +(43) +10 + +where +N11 = [2ω0f0 + (3γ − 1)f 2 +0ϕ]K2 + 2a2 +0f0V0ϕϕ − 6f0f0ϕϕ − 6f 2 +0ϕ +2ω0f0 + 3f 2 +0ϕ +, +(44) +N12 = 2(3γ − 4)(K2 − 3)f0ϕ +2ω0f0 + 3f 2 +0ϕ +, +(45) +N21 = [(γ − 2)ω0f0f0ϕ − f 3 +0ϕ]K2 + (3f0ϕϕ − a2 +0V0ϕϕ − 2ω0)f0f0ϕ +2ω0f0 + 3f 2 +0ϕ +, +(46) +N22 = [f 2 +0ϕ + 2(γ − 1)ω0f0]K2 + 2(2 − 3γ)ω0f0 − 6f 2 +0ϕ +2ω0f0 + 3f 2 +0ϕ +, +(47) +and the eigenvalues of can be expressed as +µ2 +1,2 = M ± +√ +N +2 +, +(48) +with +M = N11 + N22, +N = (N11 + N22)2 + 4N12N21 − 4N11N22. +(49) +If the imaginary components of µ1 and µ2 are nonzero, the corresponding Einstein static +solutions are unstable. So, the stability conditions can be rewritten as +M > 0, +N > 0, +M 2 − N > 0. +(50) +Since the homogeneous scalar perturbation corresponds to the case K2 = k(k +2) = 0 and +the inhomogeneous ones correspond to the other case, the stable Einstein static solutions +are required to be stable for all values of k. So, by considering the existence conditions of +Einstein static solution Eq. (37) and solving the inequalities (50), we find the Einstein static +solutions can be stable under the conditions f0 > 0, ω0 < 0, V0 > 0 with +V0ϕ < 0, 19 + +√ +201 +24 +< γ ≤ 7 +5, − +� +−(2 − 3γ)2f0ω0 +18(γ − 1) +< f0ϕ < −(2 − 9γ)f0ω0 +12γ − 1 +, A < V0ϕϕ < B, +(51) +V0ϕ > 0, 19 + +√ +201 +24 +< γ ≤ 7 +5, (2 − 9γ)f0ω0 +12γ − 1 +< f0ϕ < +� +−(2 − 3γ)2f0ω0 +18(γ − 1) +, A < V0ϕϕ < B, +(52) +11 + +where +A = [3f0f0ϕϕ + (12 − 9γ)f 2 +0ϕ + (2 − 3γ)f0ω0]V0ϕ +3f0f0ϕ ++ +� +(4 − 3γ)2(2f0ω0 + 3f 2 +0ϕ)V 2 +0ϕ +3f 2 +0 +, +B = V0ϕf0ϕϕ +f0ϕ ++ +2f0ϕV0ϕ +(3γ − 2)f0 +. +From these conditions, we see that the range of value γ is extremely narrow. In Fig. (1), we +have plotted contours of the stable regions of the homogeneous and inhomogeneous scalar +perturbations. In this figure, k = 0 denotes the homogeneous scalar perturbations, and k = +2, 3, 4, 5 correspond to the inhomogeneous ones. For the inhomogeneous scalar perturbations, +the stable region becomes larger and larger with the increase of k, and the stable region of +k = 0 overlaps with the case k = 5. As a result, the region for k = 2 is the smallest stable +region which represents the stable region for the Einstein static solutions. Thus, a stable +Einstein static universe can exist in the scalar-tensor theory of gravity and an example is +depicted in Fig. (2) by solving Eqs. (8) and (9) numerically under the stability conditions. +1.011 +1.012 +1.013 +1.014 +1.015 +1.016 +1.402 +1.403 +1.404 +1.405 +1.406 +1.407 +f0 φ +V0 φφ +k=0 +1.011 +1.012 +1.013 +1.014 +1.015 +1.016 +1.402 +1.403 +1.404 +1.405 +1.406 +1.407 +f0 φ +V0 φφ +k=5 +k=4 +k=3 +k=2 +FIG. 1: +Stability regions in (V0ϕϕ, f0ϕ) plane under homogeneous and inhomogeneous scalar per- +turbations. k = 0 represents the homogeneous scalar perturbations while k = 2, 3, 4, 5 correspond +to the inhomogeneous ones. These figures are plotted for f0 = 1.532, ω0 = −1, V0ϕ = 0.9052, +γ = 1.4 and f0ϕϕ = 0.964. +12 + +-10 +-5 +0 +5 +10 +1.70 +1.75 +1.80 +1.85 +1.90 +η +a +-10 +-5 +0 +5 +10 +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +η +φ +FIG. 2: +Evolutionary curves of scale factor a and scalar field ϕ under the stability conditions. +These figures are plotted for f = 1 + 0.05ϕ + 0.482ϕ2, ω = − 1 +ϕ, V = 0.5138 − 0.5ϕ + 0.7026ϕ2 and +γ = 1.4. +V. +CMB POWER SPECTRUM +In order to discuss whether the CMB TT-spectrum can discriminate the emergent sce- +nario from the slow expansion, we study the primordial power spectrum and the CMB +TT-spectrum of the emergent scenario and the slow expansion in this section. +A. +Emergent scenario +In emergent scenario, the universe stems from an Einstein static state, and then evolves +into an inflationary epoch [6, 7]. To realize this transition, there exist two different ap- +proaches: (i) assuming the Einstein static state defined by a′ = 0 and then invoking an +instantaneous transition to the inflationary epoch [10, 13]. (ii) considering the evolution of +the scale factor as a(t) = a0 + AeH0t [6, 7]. In our previous work [36], we found that both +approaches produce the same CMB TT-spectra. So, in this section, we will adopt the first +13 + +approach. +Following Ref. [36, 59, 60], considering the Einstein static conditions, the scale factor in +Einstein static state a0 is given by Eq. (34). During this epoch, the variable Y and z in +Eqs. (29) and (28) reduce to +Y ≈ 0, +z ≈ 0. +(53) +So, the equation of motion for the variable v (Eq. (31)) can be written as +v′′ +k + k2 +−vk = 0, +k2 +− = k(k + 2), +(54) +which has the solution +vk(η) = +� +1 +2ke−ik−η, +(55) +where the normalization conditions vkv∗′ +k − v +′ +kv∗ +k = i and the Bunch-Davies vacuum are +considered. +In slow-roll region, using the slow-roll conditions f ′′ ≪ Hf ′ ≪ H2f and ϕ′2 ≪ a2V +[61], +Eq. (8) reduces to +H′ − H − 1 ≈ 0, +(56) +which has the solution +a = +a0 +cos(η − ηt). +(57) +Thus, the scale factor for the emergent scenario can be expressed as +a(η) = +� +a0, +η < ηt +a0 +cos(η−ηt), ηt ≤ η < ηt + π +2, +(58) +which is also obtained in general relativity [36, 37]. +With η approaching to ηt + π +2, the +universe freezes out into the inflationary phase. The evolutionary curve of scale factor a0 is +shown in Fig. (3), the purple point denotes the transition point. To realize this transition, +one can break the stability conditions by considering the scalar potential or the equation of +14 + +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +0 +1 +2 +3 +4 +5 +η +a(η) +Static state +Inflation +FIG. 3: +Evolutionary curve of scale factor a. The scale factor in the Einstein static state has been +chosen as a0 = 1 and the purple point denotes the transition point. +state varying with conformal time η slowly [10, 13, 16]. Once η evolves to a critical point, +the stability condition will break down automatically. +During the slow-roll region, the variable Y and z reduce to +Y ≈ 4, +z ≈ aϕ′√ω +H +� +k(k + 2) + 3 +k(k + 2) + 4. +(59) +So, the equation of motion for the variable v (Eq. (31)) becomes +v′′ +k + +� +k2 ++ − +2 +� +η − (ηt + π +2) +�2 +� +vk = 0, +k2 ++ = k(k + 2) − 17 +3 , +(60) +which has the solution taking the form +vk(η) = +�π +4 +�� +ηt + π +2 +� +− η +� +CkH(1) +3/2 +� +k+ +� +(ηt + π +2 ) − η +�� ++ DkH(2) +3/2 +� +k+ +� +(ηt + π +2 ) − η +��� +,(61) +where H(1) and H(2) are the Hankel functions of the first and second kinds. +To determine Ck and Dk, we use the continuity condition of vk and v′ +k to match Eqs. (55) +15 + +and (61) at the transition time ηt and obtain +Ck = 1 +4e−ik−ηt +� +1 +k− +� +iπk+H(2) +1/2 +�π +2 k+ +� ++ (−2i + πk−)H(2) +3/2 +�π +2 k+ +�� +, +(62) +Dk = −1 +4e−ik−ηt +� +1 +k− +� +iπk+H(1) +1/2 +�π +2 k+ +� ++ (−2i + πk−)H(1) +3/2 +�π +2 k+ +�� +. +(63) +The curved primordial power spectrum of the comoving curvature perturbation R is de- +fined as +PR = k3 +2π2 |Rk|2 = k3 +2π2 +���� +vk +zk +���� +2 +. +(64) +Then, substituting Eq. (61) into Eq. (64), we obtain the curved primordial power spectrum +of R +PR += k3 +2π2 |Rk|2 ≈ +lim +η→ηt+ π +2 +1 +4π2 +1 +a2ϕ′2ω +H +k(k+2)+3 +k(k+2)+4 +k3 +k3 ++ +1 +� +η − (ηt + π +2) +�2 |Ck − Dk|2 += As +k3 +k3 ++ +k(k + 2) + 3 +k(k + 2) + 4 |Ck − Dk|2 . +(65) +And the analytical primordial power spectrum can be parameterized as +PR = As +� k +k∗ +�ns−1 k3 +k3 ++ +k(k + 2) + 3 +k(k + 2) + 4 |Ck − Dk|2 , +(66) +where k∗ = 0.05Mpc−1 corresponds to the pivot perturbation mode. +B. +Slow expansion +In slow expansion, the universe also stems from an Einstein static state, and then evolves +into a slowly expanded epoch which can generate the scale invariant primordial power spec- +trum [8]. In scalar-tensor theory of gravity, by considering f(ϕ) = 1−ξϕ+λϕ2, ω(ϕ) = ω0ϕ−1 +and V (ϕ) = −V0ϕ +3 +2, the slow expansion was analyzed and it was found that the analytical +primordial power spectrum is scale invariant and has the form [23] +PR = k3 +2π2 +���� +vk +zk +���� +2 +≈ V0ξ +5 +3 +128π2 = As. +(67) +16 + +Parameterizing this primordial power spectrum, it can be written as +PR = As +� k +k∗ +�ns−1 +, +(68) +which is the same as that in ΛCDM model. +C. +Power spectrum +To plot the primordial power spectrum in the closed universe, we use the Planck 2018 +results in the curved universes best-fit data (TT,TE,EE+lowl+lowE+lensing) As = 2.0771± +0.1017 × 10−9 and ns = 0.9699 ± 0.0090. For the flat universe, the Planck 2018 results in +Ref. [62] is adopted. In the left panel of Fig. (4), we have plotted the primordial power +spectrum for ΛCDM, KΛCDM, the emergent scenario and the slow expansion. This figure +shows that the primordial power spectra of the slow expansion and ΛCDM are overlapped. +Comparing to ΛCDM, KΛCDM and the slow expansion, the primordial power spectrum of +emergent scenario oscillates and is suppressed during the region k < 100. +Then, using CLASS code [63], we have depicted the CMB TT-spectrum in the right panel +of Fig. (4). From this figure, we can see that the CMB TT-spectrum of the slow expansion is +the same as that in ΛCDM, and the CMB TT-spectrum of emergent scenario is suppressed +for l < 10. Thus, comparing to the slow expansion, the emergent scenario can explain the +suppression of the CMB TT-spectrum at large scales. +VI. +CONCLUSION +The scalar-tensor theory is an extension of general relativity by coupling a scalar field ϕ +to the Ricci scalar R with terms f(ϕ)R, and it can be expressed as general relativity with a +modified energy-momentum tensor. In this paper, we study the primordial power spectrum +and CMB TT-spectrum of the emergent scenario and the slow expansion in the scalar-tensor +theory of gravity. Since both in the emergent scenario and in the slow expansion, the universe +stems from an Einstein static universe, we analyze the stability of the Einstein static universe +17 + +10 +100 +1000 +104 +2.0 +2.2 +2.4 +2.6 +2.8 +3.0 +k +Log(1010PR) +Slow expansion +Emergent scenario +ΛCDM +KΛCDM +2 +10 +100 +1000 2000 +ℓ +0 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +DTT +ℓ +[µK2] +KΛCDM +Emergent scenario +ΛCDM +Slow expansion +Planck 2018 +FIG. 4: +Primordial power spectrum and CMB TT-spectrum for the emergent scenario and the +slow expansion. +in scalar-tensor theory of gravity at first, and find the Einstein static universe can be stable +against both scalar and tensor perturbations under the certain conditions. +Assuming the emergent scenario starts from an Einstein static universe followed by an +instantaneous transition to an inflationary phase, we study the primordial power spectrum +for the emergent scenario and obtain the analytical approximations of this spectrum. To +comparing the primordial power spectrum and CMB TT-spectrum of the emergent scenario +and the slow expansion, we have plotted these spectra by using Planck 2018 results. These +figures show that both of these spectra for the slow expansion are the same as the one for +ΛCDM, and the spectra of the emergent scenario are suppressed slightly at large scales. +Thus, comparing to the slow expansion, the emergent scenario can explain the suppression +of the CMB TT-spectrum at large scales. +VII. +APPENDIX +Equation (17) can be written as +(˜a˜Ψ). = 1 +2˜a ˙˜ϕ2�δ ˜ϕ +˙˜ϕ +� +, +(69) +18 + +and expressing δ ˜ϕ +˙˜ϕ and ˜Ψ in terms of ˜ζ and ˜ξ, one obtains +˙˜H ˜ξ + ˜H ˙˜ξ = ˜a ˙˜ϕ2� ˜ζ +˜H +− +� 1 +2˜a − +K +˜a3 ˙˜ϕ2 +� +˜ξ +� += ˜a ˙˜ϕ2 ˜ζ +˜H +− +� ˙˜ϕ2 +2 − K +˜a2 +� +˜ξ. +(70) +Then, using the equation ˙˜H − K +˜a2 = − 1 +2 ˙˜ϕ2 obtained from the background equation (13) and +(14), one gets +˙˜ξ = ˜a +˙˜ϕ2 +˜H2 ˜ζ, +(71) +Equation (16) can be written as +1 +˜a2(∆ + 3K)˜Ψ − 3 ˜H( ˙˜Ψ + ˜H ˜Φ) = 1 +2 +� +˙˜ϕ ˙δ ˜ϕ − ˜Φ ˙˜ϕ2 − +� +˙˜ϕ ¨˜ϕ + 3 ˜H ˙˜ϕ2� 1 +˙˜ϕδ ˜ϕ +� +(72) += 1 +2 +� +− ˙˜ϕ2� +˜Φ − ˙˜ϕ +˙δ ˜ϕ +˙˜ϕ2 + ¨˜ϕδ ˜ϕ +˙˜ϕ2 +� +− 3 ˜H ˙˜ϕ2δ ˜ϕ +˙˜ϕ +� += −1 +2 +� +˙˜ϕ2� +˜Φ − +�δ ˜ϕ +˙˜ϕ +�.� ++ 3 ˜H ˙˜ϕ2�δ ˜ϕ +˙˜ϕ +�� +, +1 +˜a2(∆ + 3K)˜Ψ = 1 +2 +˙˜ϕ2��δ ˜ϕ +˙˜ϕ +�. +− ˜Ψ +� +− 3 +2 +˜H ˙˜ϕ2�δ ˜ϕ +˙˜ϕ +� ++ 3 ˜H( ˙˜Ψ + ˜H ˜Ψ) +(73) += 1 +2 +˙˜ϕ2��δ ˜ϕ +˙˜ϕ +�. +− ˜Ψ +� +− 3 +2 +˜H ˙˜ϕ2�δ ˜ϕ +˙˜ϕ +� ++ 3 ˜H +�1 +2 +˙˜ϕ2�δ ˜ϕ +˙˜ϕ +�� += 1 +2 +˙˜ϕ2��δ ˜ϕ +˙˜ϕ +�. +− ˜Ψ +� +, +which gives +�δ ˜ϕ +˙˜ϕ +�. += +� +2 +˜a2 ˙˜ϕ2(∆ + 3K) + 1 +� +˜Ψ. +(74) +Expressing δ ˜ϕ +˙˜ϕ and ˜Ψ in terms of ˜ζ and ˜ξ, one obtains +� ˜ζ +˜H +− +� 1 +2˜a − +K +˜a3 ˙˜ϕ2 +� +˜ξ +�. += +� +2 +˜a2 ˙˜ϕ2(∆ + 3K) + 1 +� +˜Ψ, +(75) +˙˜ζ ˜H − ˜ζ ˙˜H +˜H2 ++ 1 +2 +˜H +˜a +˜ξ + K +� 3 ˜H +˙˜ϕ2˜a3 + 2 +˜V ˜ϕ +˙˜ϕ3˜a3 +� +˜ξ − +� 1 +2˜a − +K +˙˜ϕ3˜a2 +� ˙˜ξ = +� +2 +˜a2 ˙˜ϕ2(∆ + 3K) + 1 +� +˜Ψ, (76) +19 + +˙˜ζ +˜H +− ˙˜H +˜ζ +˜H2 − +�1 +2 +˙˜ϕ2 − K +˜a2 +� ˜ζ +˜H2 + 1 +2 +˜H +˜a +˜ξ + K +� 3 ˜H +˙˜ϕ2˜a3 + 2 +˜V ˜ϕ +˙˜ϕ3˜a3 +� +˜ξ = +� +˜H +˜a3 ˙˜ϕ2(∆ + 3K) + +˜H +2˜a +� +˜ξ,(77) +which reduces to +˙˜ζ +˜H ++ 2 +˜V ˜ϕ +˙˜ϕ3˜a3K ˜ξ = +˜H +˜a3 ˙˜ϕ2∆˜ξ, +(78) +and can be rewritten as +˙˜ζ = H2 +˜a3 ˙˜ϕ2 +� +∆ − 2 +˜V ˜ϕ +˜H ˙˜ϕ +K +� +˜ξ = +˜H2 +˜a3 ˙˜ϕ2(∆ + Y K)˜ξ. +(79) +where Y = −2 +˜V ˜ +ϕ +˜H ˙˜ϕ. +Acknowledgments +This work was supported by the National Natural Science Foundation of China under +Grants Nos. 11865018, 12265019, the Natural Science Research Project of Education Depart- +ment of Anhui Province of China under Grants No.2022AH051634, the Doctoral Foundation +of Zunyi Normal University of China under Grants No. BS[2017]07. +[1] A. 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Tram, JCAP 07, 034 (2011). +22 + diff --git a/TdAyT4oBgHgl3EQfuflV/content/tmp_files/load_file.txt b/TdAyT4oBgHgl3EQfuflV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..93af736dd71d81d79a5129bd846b4477e80db0f3 --- /dev/null +++ b/TdAyT4oBgHgl3EQfuflV/content/tmp_files/load_file.txt @@ -0,0 +1,633 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf,len=632 +page_content='CMB power spectrum for emergent scenario and slow expansion in scalar-tensor theory of gravity Qihong Huang1∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' He Huang2 and Bing Xu3 1 School of Physics and Electronic Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Zunyi Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Zunyi 563006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' China 2 Institute of Applied Mechanics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Zhejiang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Zhejiang 310058,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' China 3 School of Electrical and Electronic Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Anhui Science and Technology University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Bengbu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Anhui 233030,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' China Abstract We analyze the stability of the Einstein static universe in scalar-tensor theory of gravity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' and find the Einstein static universe can be stable against both scalar and tensor perturbations under certain conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' By assuming the emergent scenario originating from an Einstein static state, followed by an instantaneous transition to an inflationary phase, we study and obtain the analytical approximations of the primordial power spectrum for the emergent scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Then, we plot the primordial power spectrum and CMB TT-spectrum of the emergent scenario and the slow expan- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' These figures show that both of these spectra for the slow expansion are the same as that for ΛCDM, and the spectra of the emergent scenario are suppressed at large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' PACS numbers: 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='Cq ∗ Corresponding author: huangqihongzynu@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='com 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='00613v1 [gr-qc] 2 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' INTRODUCTION Inflation [1–3] posits an epoch very early in the universe, during which the scale factor grows exponentially with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' It can solve most of problems in the standard cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' The primordial scalar perturbations originating from quantum fluctuations during the infla- tionary epoch not only explain the cosmic microwave background radiation anisotropy but also seed the large-scale structure of the universe [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Although it achieves great success, it still suffers from the big bang singularity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' To solve this intractable problem, some scenarios had been proposed and suggested to construct non-singular or past eternal cosmological models, such as the emergent scenario [6, 7] and the slow expansion [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' In the emergent scenario, the universe is assumed to start from an Einstein static universe and then evolves into an inflationary era [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Since the universe stems from an Einstein static universe in the emergent scenario, the big bang singularity is avoided naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' In addition, the e-folding number and the nearly scale-invariant spectral index can also be pro- duced by the inflation of the emergent scenario [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Thus, the emergent scenario has drew lots of attention after it was proposed [9–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' For the other model, namely the slow expan- sion, the universe originates from an Einstein static universe and then enters into an epoch in which the universe expands very slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' During this epoch, the nearly scale-invariant primordial power spectrum is provided [8, 17–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' After this epoch, the universe evolves into the big bang epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' It was found that general relativity can be recovered and the uni- verse will evolve in accordance with the standard cosmology after the slow expansion ends [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Similar to the emergent scenario, the big bang singularity is also avoided since the universe originates from an Einstein static universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Since both the emergent scenario and the slow expansion assume that the universe originates from an Einstein static universe and that a stable Einstein static universe must be stable against both the scalar perturbations and the tensor perturbations, to find a stable Einstein static universe becomes a crucial issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Fortunately, it was found that a stable Einstein static universe exists in Mimetic grav- ity [16], scalar-fluid theory [24], non-minimal derivative coupling model [25, 26], braneworld model [27], Jordan-Brans-Dicke theory [28], Eddington-inspired Born-Infeld theory [29], hy- 2 brid metric-Palatini gravity [30], GUP theory [31], f(R,T) gravity [32], f(R,T,Q) gravity [33] and massive gravity [34] and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Thus, the big bang singularity can be solved in the theories of modified gravity by using the emergent scenario and the slow expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' It is notable that, except for avoiding the big bang singularity, both the emergent scenario and the slow expansion can produce a nearly scale-invariant primordial power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Recently, by assuming the Einstein static state as a superinflating phase [35] or a static state phase [36, 37], the CMB TT-spectrum of the emergent scenario was studied in the framework of general relativity, and the results show that the CMB TT-spectrum is suppressed at large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' However, it is quite unclear that whether the CMB TT-spectrum of the slow expansion is also suppressed at large scales, and whether the CMB TT-spectrum can be utilized to discriminate the emergent scenario from the slow expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' To answer these questions, we will study the CMB TT-spectrum of the emergent scenario and the slow expansion in the scalar-tensor theory of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' In section II, we briefly review the field equations of the scalar-tensor theory of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' In section III, we will give the derivation of equations of motion for perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' In section IV, we study the stability conditions of the Einstein static universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' In section V, we plot the primordial power spectrum and the CMB TT- spectrum of the emergent scenario and the slow expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Finally, our main conclusions are shown in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' FIELD EQUATIONS In this paper, we consider the following scalar-tensor theory of gravity, whose action takes the following form [38–40] S = � d4x√−g �1 2f(ϕ)R − 1 2ω(ϕ)gαβ(∂αϕ)(∂βϕ) − V (ϕ) + Lm � , (1) where R is the Ricci curvature scalar, ϕ is the scalar field of scalar-tensor theory, f(ϕ) and ω(ϕ) are coupling function of scalar field, V (ϕ) is the potential, and Lm represents the Lagrangian density of a perfect fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Here, the coupling function f(ϕ) needs to be positive for the gravitons to carry positive energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' 3 Varying the action (1) with respect to the metric tensor gαβ and the scalar field ϕ, we obtain f(Rαβ − 1 2gαβR) − ∇α∇βf + gαβ∇σ∇σf − ω � ∇αϕ∇βϕ − 1 2gαβ∇σϕ∇σϕ � + gαβV = Tαβ,(2) and fϕR + ωϕ∇σϕ∇σϕ + 2ω∇σ∇σϕ − 2Vϕ = 0, (3) where fϕ denotes df dϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' The field equations (2) can be expressed as the standard form of general relativity Gαβ = T eff αβ with a modification in the energy-momentum tensor T eff αβ = 1 f � ∇α∇βf − gαβ(∇σ∇σf + V ) + ω � ∇αϕ∇βϕ − 1 2gαβ∇σϕ∇σϕ � + Tαβ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (4) We consider a homogeneous and isotropic universe described by FLRW metric ds2 = −a2(η)dt2 + a2(η)γijdxidxj, γijdxidxj = dr2 1 − Kr2 + r2(dθ2 + sin2 θdϕ2), (5) where K = 1, 0, −1 corresponds to a closed, flat and open universe, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' The background equation can be obtained by substituting this metric into the field equations (2), the 0 − 0 component gives the Friedmann equation 3H2 + 3K + 3Hf ′ f = 1 f �1 2ωϕ′2 + a2V + a2ρ � , (6) and the i − i component gives 2H′ + H2 + f ′′ f + Hf ′ f + K = − 1 f �1 2ωϕ′2 − a2V + a2p � , (7) where ′ denotes a derivative with respect to the conformal time η, and ρ and p denote the energy density and pressure of the perfect fluid with p = (γ − 1)ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (6) and (7), and eliminating ρ, one obtain 2H′ + f ′′ f + (3γ − 2) � H2 + K + Hf ′ f � = 1 f �1 2(γ − 2)ωϕ′2 + γa2V � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (8) The field equation (3) can be expressed as ϕ′′ + 2Hϕ′ + 1 2ω � ωϕϕ′2 + 2a2Vϕ − 6(H′ + H2 + K)fϕ � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (9) 4 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' PERTURBATIONS To analyze the power spectrum of the emergent scenario and the slow expansion in scalar- tensor theory of gravity, we are required to obtain the equation of motion for perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' In this section, we will derive this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Conformal transformation to Einstein gravity The scalar-tensor theory of gravity can be transformed into Einstein gravity by performing a conformal transformation on the metric [41–45] ˜gαβ = Ω2gαβ, (10) and the corresponding action becomes ˜S = � d˜td3x � −˜g �1 2 ˜R − 1 2 ˜∇σ ˜ϕ ˜∇σ ˜ϕ − ˜V � , (11) which is the action for a minimal coupling single scalar field ˜ϕ and the conformal factor is Ω = √f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' The corresponding variables are defined as d˜t = Ωdt, ˜a = Ωa, ˜Φ = Φ + δΩ, ˜Ψ = Ψ − δΩ, d ˜ϕ = � ω f + 3 2 f 2 ,ϕ f 2 dϕ, ˜V = V f 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (12) Even though the action (1) and (11) are fully equivalent [41–48], cosmological perturbations are easier to analyze in action (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Therefore, we shall analyze the perturbation in the action (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' The background equation and the equation of motion for the field ˜ϕ can be written as follows 3 ˜H2 + 3K ˜a2 = 1 2 ˙˜ϕ2 + ˜V , (13) 3 ˜H2 + 2 ˙˜H + K ˜a2 = − �1 2 ˙˜ϕ2 − ˜V � , (14) and ¨˜ϕ + 3 ˜H ˙˜ϕ + ˜V ˜ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (15) 5 Here, the no ghost condition ˙˜ϕ2 > 0 gives ωf + 3 2 � df dϕ �2 > 0 which is contained in expres- sion (30), and a dot represents the derivative with respect to the cosmic time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' In order to obtain the perturbation equations with curvature K, we use the method developed by Garriga and Mukhanov [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Then, the perturbation equations for 0 − 0 and 0 − i components can be written as [50, 51] 2 � 1 ˜a2∆˜Ψ − 3 ˜H ˙˜Ψ + 3 �K ˜a2 ˜Ψ − ˜H2 ˜Φ �� = ˙˜ϕ ˙δ ˜ϕ − ˜Φ ˙˜ϕ2 + ˜V ˜ϕδ ˜ϕ, (16) 2∇i � ˙˜Ψ + ˜H ˜Φ � = ˙˜ϕ∇iδ ˜ϕ, (17) where ∆ = ∇2 is Laplace operator, and the component for i ̸= j is 1 ˜a2∇j∇i�˜Φ − ˜Ψ � = 0, (18) which gives ˜Φ = ˜Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' After introducing two new variables ˜ξ = 2˜a ˜H ˜Ψ, (19) ˜ζ = ˜Ψ + ˜H δ ˜ϕ ˙˜ϕ − 2K ˜a2 ˙˜ϕ2 ˜Ψ, (20) the equations (17) and (16) can be simplified as ˙˜ξ = ˜a ˙˜ϕ2 ˜H2 ˜ζ, (21) ˙˜ζ = ˜H2 ˜a3 ˙˜ϕ2 � ∆ + Y K �˜ξ, (22) where Y = −2 ˜V ˜ϕ ˜H ˙˜ϕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (23) The detailed process is given in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' In order to obtain the amplitude of quantum fluctuations, one needs to expand the action for the gravitational and scalar fields to second order in perturbations which are cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Since the second order perturbation action can be inferred directly from the equations of 6 motion (21) and (22), these cumbersome steps can be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' The detailed steps are given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Thus, the action reproducing the perturbation equations (21) and (22) can be written as ˜S = � � ˜ξ ˜O ˙˜ζ − 1 2 ˜H2 ˜a3 ˙˜ϕ2 ˜ξ ˜O(∆ + Y K)˜ξ + 1 2 ˜a ˙˜ϕ2 ˜H2 ˜ζ ˜O˜ζ � d˜td3x, (24) where ˜O = ∆ + 3K is a time-independent operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Expressing ˜ξ in terms of ˙˜ζ by equa- tion (22), the action can be reduced to ˜S = � z2�˜ζ′2 + ˜ζ(∆ + Y K)˜ζ � dηd3x, (25) where prime denotes the derivative with respect to the conformal time η, and the variable z is z = ˜a ˜ϕ′ ˜H � ˜O ∆ + Y K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (26) The Laplacian ∆ should be understood as a c − number and represents the corresponding eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Equation of motion for perturbation The second perturbation action in the scalar-tensor theory of gravity can be obtained by a transformation for the action (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Introducing the canonical quantization variable v = z˜ζ and utilizing the corresponding transform equations (12), the action (25) can be rewritten as follows S = 1 2 � � v′2 + v(∆ + Y K)v + z′′ z v2� dηd3x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (27) Here, z and Y are z = a2ϕ′√ E (a√f)′ � ∆ + 3K ∆ + Y K , (28) and Y = 2(a2ϕ′√ E)′/(a2ϕ′√ E) (a√f)′/(a√f) , (29) 7 where E = f � ω + 3 2 f 2 ϕ f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (30) Varying the action (27) with respect to v, one can straightforwardly get the equation of motion for the variable as v′′ − (∆ + Y K)v − z′′ z v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (31) Expressing ˜ζ and ˙˜ζ in the action (24) in terms of ˜˙ξ and ˜ξ, we obtain u′′ − (∆ + Y K)u − Z′′ Z u = 0, (32) where u = a � f 3 ϕ′√ E ˜ψ, Z = (a√f)′ a2ϕ′√ E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (33) The equation of motion for the variable u was obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' [52, 53] by doing some calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' EINSTEIN STATIC UNIVERSE Both in the emergent scenario and in the slow expansion, the universe stems from an Einstein static universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' However, it was found that the Einstein static solution is unstable in scalar-tensor theory of gravity when the perfect fluid is pressureless matter(γ = 1) or radiation(γ = 4 3) [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' So, to find a stable Einstein static solution becomes crucial for the emergent scenario and the slow expansion in this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' In order to find a stable Einstein static solution, we reanalyze the stability of the Einstein static solution by considering 0 ≤ γ ≤ 2 in scalar-tensor theory of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Static solutions The Einstein static solution requires a = a0 = constant and a′ 0 = a′′ 0 = 0 which indicates H0 = H′ 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' And Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (6) and (7) indicate that a constant a requires ϕ = ϕ0 = constant 8 and ϕ′ 0 = ϕ′′ 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Thus, for the Einstein static solution, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (8) and (9) show K a2 0 = γV0 (3γ − 2)f0 , (34) and K a2 0 = V0ϕ 3f0ϕ , (35) where 0 denotes the corresponding static state value, and 0ϕ = � d dϕ � ϕ=ϕ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' In addition, the energy density ρ0 is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (6) ρ0 = 2V0 3γ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (36) The existence conditions of Einstein static solutions require a2 0 > 0 and ρ2 0 > 0 which mean f0 > 0, V0 > 0, 2 3 < γ ≤ 2, V0ϕ f0ϕ > 0, (37) for K = 1 and f0 < 0, V0 > 0, 2 3 < γ ≤ 2, V0ϕ f0ϕ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (38) for K = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Since a stable Einstein static universe is required to be stable against both scalar pertur- bations and tensor perturbations, we will analyze the stability in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Tensor perturbations Since tensor perturbations are easy to analyze, we will analyze it at first and the perturbed metric is given as [55] ds2 = −a2(η)dη2 + a2(η)(γij + 2hij)dxidxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (39) Performing a harmonic decomposition for the perturbed variable hij, we obtain hij = HT,klm(t)Yij,klm(θn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (40) 9 Because the quantum numbers m and l do not enter the perturbed differential equations, the harmonic function Yk = Yklm(θn) satisfies [56] ∆Yk = −K2Yk = � −k(k + 2)Yk, k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=', K = +1 −k2Yk, k2 ≥ 0, K = 0 −(k2 + 1)Yk, k2 ≥ 0, K = −1 (41) where ∆ represents the three-dimensional spatial Laplacian operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' [57, 58], considering the static conditions and then substituting the perturbed metric (39) into the field equations (2), the equation of tensor perturbations becomes H′′ T + (k2 + 2K)HT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (42) According to this equation, we can find that k2 + 2K > 0 must be satisfied for any k to obtain a stable solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' As a result, the Einstein static solutions are stable against the tensor perturbations for the case K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Scalar perturbations Since the Einstein static solutions can be stable against the tensor perturbation for K = 1, we will analyze the stability of the static solutions under the scalar perturbations in the closed spacetime in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' According to the results in Ref [54],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' the stability of the static solutions against the scalar perturbations in scalar-tensor theory of gravity are determined by the eigenvalues of the matrix of N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' which is given as N = � N11 N12 N21 N22 � (43) 10 where N11 = [2ω0f0 + (3γ − 1)f 2 0ϕ]K2 + 2a2 0f0V0ϕϕ − 6f0f0ϕϕ − 6f 2 0ϕ 2ω0f0 + 3f 2 0ϕ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (44) N12 = 2(3γ − 4)(K2 − 3)f0ϕ 2ω0f0 + 3f 2 0ϕ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (45) N21 = [(γ − 2)ω0f0f0ϕ − f 3 0ϕ]K2 + (3f0ϕϕ − a2 0V0ϕϕ − 2ω0)f0f0ϕ 2ω0f0 + 3f 2 0ϕ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (46) N22 = [f 2 0ϕ + 2(γ − 1)ω0f0]K2 + 2(2 − 3γ)ω0f0 − 6f 2 0ϕ 2ω0f0 + 3f 2 0ϕ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (47) and the eigenvalues of can be expressed as µ2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='2 = M ± √ N 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (48) with M = N11 + N22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' N = (N11 + N22)2 + 4N12N21 − 4N11N22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (49) If the imaginary components of µ1 and µ2 are nonzero, the corresponding Einstein static solutions are unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' So, the stability conditions can be rewritten as M > 0, N > 0, M 2 − N > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (50) Since the homogeneous scalar perturbation corresponds to the case K2 = k(k +2) = 0 and the inhomogeneous ones correspond to the other case, the stable Einstein static solutions are required to be stable for all values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' So, by considering the existence conditions of Einstein static solution Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (37) and solving the inequalities (50),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' we find the Einstein static solutions can be stable under the conditions f0 > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' ω0 < 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' V0 > 0 with V0ϕ < 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' 19 + √ 201 24 < γ ≤ 7 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' − � −(2 − 3γ)2f0ω0 18(γ − 1) < f0ϕ < −(2 − 9γ)f0ω0 12γ − 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' A < V0ϕϕ < B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (51) V0ϕ > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' 19 + √ 201 24 < γ ≤ 7 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (2 − 9γ)f0ω0 12γ − 1 < f0ϕ < � −(2 − 3γ)2f0ω0 18(γ − 1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' A < V0ϕϕ < B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (52) 11 where A = [3f0f0ϕϕ + (12 − 9γ)f 2 0ϕ + (2 − 3γ)f0ω0]V0ϕ 3f0f0ϕ + � (4 − 3γ)2(2f0ω0 + 3f 2 0ϕ)V 2 0ϕ 3f 2 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' B = V0ϕf0ϕϕ f0ϕ + 2f0ϕV0ϕ (3γ − 2)f0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' From these conditions, we see that the range of value γ is extremely narrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (1), we have plotted contours of the stable regions of the homogeneous and inhomogeneous scalar perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' In this figure, k = 0 denotes the homogeneous scalar perturbations, and k = 2, 3, 4, 5 correspond to the inhomogeneous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' For the inhomogeneous scalar perturbations, the stable region becomes larger and larger with the increase of k, and the stable region of k = 0 overlaps with the case k = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' As a result, the region for k = 2 is the smallest stable region which represents the stable region for the Einstein static solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Thus, a stable Einstein static universe can exist in the scalar-tensor theory of gravity and an example is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (2) by solving Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (8) and (9) numerically under the stability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='011 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='012 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='013 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='014 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='015 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='016 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='402 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='403 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='404 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='405 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='406 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='407 f0 φ V0 φφ k=0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='011 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='012 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='013 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='014 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='015 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='016 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='402 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='403 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='404 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='405 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='406 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='407 f0 φ V0 φφ k=5 k=4 k=3 k=2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' 1: Stability regions in (V0ϕϕ, f0ϕ) plane under homogeneous and inhomogeneous scalar per- turbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' k = 0 represents the homogeneous scalar perturbations while k = 2, 3, 4, 5 correspond to the inhomogeneous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' These figures are plotted for f0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='532, ω0 = −1, V0ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='9052, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='4 and f0ϕϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' 12 10 5 0 5 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='90 η a 10 5 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='2 η φ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' 2: Evolutionary curves of scale factor a and scalar field ϕ under the stability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' These figures are plotted for f = 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='05ϕ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='482ϕ2, ω = − 1 ϕ, V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='5138 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='5ϕ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='7026ϕ2 and γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' CMB POWER SPECTRUM In order to discuss whether the CMB TT-spectrum can discriminate the emergent sce- nario from the slow expansion, we study the primordial power spectrum and the CMB TT-spectrum of the emergent scenario and the slow expansion in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Emergent scenario In emergent scenario, the universe stems from an Einstein static state, and then evolves into an inflationary epoch [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' To realize this transition, there exist two different ap- proaches: (i) assuming the Einstein static state defined by a′ = 0 and then invoking an instantaneous transition to the inflationary epoch [10, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (ii) considering the evolution of the scale factor as a(t) = a0 + AeH0t [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' In our previous work [36], we found that both approaches produce the same CMB TT-spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' So, in this section, we will adopt the first 13 approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' [36, 59, 60], considering the Einstein static conditions, the scale factor in Einstein static state a0 is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' During this epoch, the variable Y and z in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (29) and (28) reduce to Y ≈ 0, z ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (53) So, the equation of motion for the variable v (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (31)) can be written as v′′ k + k2 −vk = 0, k2 − = k(k + 2), (54) which has the solution vk(η) = � 1 2ke−ik−η, (55) where the normalization conditions vkv∗′ k − v ′ kv∗ k = i and the Bunch-Davies vacuum are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' In slow-roll region, using the slow-roll conditions f ′′ ≪ Hf ′ ≪ H2f and ϕ′2 ≪ a2V [61], Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (8) reduces to H′ − H − 1 ≈ 0, (56) which has the solution a = a0 cos(η − ηt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (57) Thus, the scale factor for the emergent scenario can be expressed as a(η) = � a0, η < ηt a0 cos(η−ηt), ηt ≤ η < ηt + π 2, (58) which is also obtained in general relativity [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' With η approaching to ηt + π 2, the universe freezes out into the inflationary phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' The evolutionary curve of scale factor a0 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (3), the purple point denotes the transition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' To realize this transition, one can break the stability conditions by considering the scalar potential or the equation of 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='5 0 1 2 3 4 5 η a(η) Static state Inflation FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' 3: Evolutionary curve of scale factor a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' The scale factor in the Einstein static state has been chosen as a0 = 1 and the purple point denotes the transition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' state varying with conformal time η slowly [10, 13, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Once η evolves to a critical point, the stability condition will break down automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' During the slow-roll region, the variable Y and z reduce to Y ≈ 4, z ≈ aϕ′√ω H � k(k + 2) + 3 k(k + 2) + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (59) So, the equation of motion for the variable v (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (31)) becomes v′′ k + � k2 + − 2 � η − (ηt + π 2) �2 � vk = 0, k2 + = k(k + 2) − 17 3 , (60) which has the solution taking the form vk(η) = �π 4 �� ηt + π 2 � − η � CkH(1) 3/2 � k+ � (ηt + π 2 ) − η �� + DkH(2) 3/2 � k+ � (ηt + π 2 ) − η ��� ,(61) where H(1) and H(2) are the Hankel functions of the first and second kinds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' To determine Ck and Dk, we use the continuity condition of vk and v′ k to match Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (55) 15 and (61) at the transition time ηt and obtain Ck = 1 4e−ik−ηt � 1 k− � iπk+H(2) 1/2 �π 2 k+ � + (−2i + πk−)H(2) 3/2 �π 2 k+ �� , (62) Dk = −1 4e−ik−ηt � 1 k− � iπk+H(1) 1/2 �π 2 k+ � + (−2i + πk−)H(1) 3/2 �π 2 k+ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (63) The curved primordial power spectrum of the comoving curvature perturbation R is de- fined as PR = k3 2π2 |Rk|2 = k3 2π2 ���� vk zk ���� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (64) Then, substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (61) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (64), we obtain the curved primordial power spectrum of R PR = k3 2π2 |Rk|2 ≈ lim η→ηt+ π 2 1 4π2 1 a2ϕ′2ω H k(k+2)+3 k(k+2)+4 k3 k3 + 1 � η − (ηt + π 2) �2 |Ck − Dk|2 = As k3 k3 + k(k + 2) + 3 k(k + 2) + 4 |Ck − Dk|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (65) And the analytical primordial power spectrum can be parameterized as PR = As � k k∗ �ns−1 k3 k3 + k(k + 2) + 3 k(k + 2) + 4 |Ck − Dk|2 , (66) where k∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='05Mpc−1 corresponds to the pivot perturbation mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Slow expansion In slow expansion, the universe also stems from an Einstein static state, and then evolves into a slowly expanded epoch which can generate the scale invariant primordial power spec- trum [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' In scalar-tensor theory of gravity, by considering f(ϕ) = 1−ξϕ+λϕ2, ω(ϕ) = ω0ϕ−1 and V (ϕ) = −V0ϕ 3 2, the slow expansion was analyzed and it was found that the analytical primordial power spectrum is scale invariant and has the form [23] PR = k3 2π2 ���� vk zk ���� 2 ≈ V0ξ 5 3 128π2 = As.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (67) 16 Parameterizing this primordial power spectrum, it can be written as PR = As � k k∗ �ns−1 , (68) which is the same as that in ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Power spectrum To plot the primordial power spectrum in the closed universe, we use the Planck 2018 results in the curved universes best-fit data (TT,TE,EE+lowl+lowE+lensing) As = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='0771± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='1017 × 10−9 and ns = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='9699 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='0090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' For the flat universe, the Planck 2018 results in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' [62] is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' In the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (4), we have plotted the primordial power spectrum for ΛCDM, KΛCDM, the emergent scenario and the slow expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' This figure shows that the primordial power spectra of the slow expansion and ΛCDM are overlapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Comparing to ΛCDM, KΛCDM and the slow expansion, the primordial power spectrum of emergent scenario oscillates and is suppressed during the region k < 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Then, using CLASS code [63], we have depicted the CMB TT-spectrum in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' From this figure, we can see that the CMB TT-spectrum of the slow expansion is the same as that in ΛCDM, and the CMB TT-spectrum of emergent scenario is suppressed for l < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Thus, comparing to the slow expansion, the emergent scenario can explain the suppression of the CMB TT-spectrum at large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' CONCLUSION The scalar-tensor theory is an extension of general relativity by coupling a scalar field ϕ to the Ricci scalar R with terms f(ϕ)R, and it can be expressed as general relativity with a modified energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' In this paper, we study the primordial power spectrum and CMB TT-spectrum of the emergent scenario and the slow expansion in the scalar-tensor theory of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Since both in the emergent scenario and in the slow expansion, the universe stems from an Einstein static universe, we analyze the stability of the Einstein static universe 17 10 100 1000 104 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='0 k Log(1010PR) Slow expansion Emergent scenario ΛCDM KΛCDM 2 10 100 1000 2000 ℓ 0 1000 2000 3000 4000 5000 6000 7000 DTT ℓ [µK2] KΛCDM Emergent scenario ΛCDM Slow expansion Planck 2018 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' 4: Primordial power spectrum and CMB TT-spectrum for the emergent scenario and the slow expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' in scalar-tensor theory of gravity at first, and find the Einstein static universe can be stable against both scalar and tensor perturbations under the certain conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Assuming the emergent scenario starts from an Einstein static universe followed by an instantaneous transition to an inflationary phase, we study the primordial power spectrum for the emergent scenario and obtain the analytical approximations of this spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' To comparing the primordial power spectrum and CMB TT-spectrum of the emergent scenario and the slow expansion, we have plotted these spectra by using Planck 2018 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' These figures show that both of these spectra for the slow expansion are the same as the one for ΛCDM, and the spectra of the emergent scenario are suppressed slightly at large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Thus, comparing to the slow expansion, the emergent scenario can explain the suppression of the CMB TT-spectrum at large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' APPENDIX Equation (17) can be written as (˜a˜Ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' = 1 2˜a ˙˜ϕ2�δ ˜ϕ ˙˜ϕ � , (69) 18 and expressing δ ˜ϕ ˙˜ϕ and ˜Ψ in terms of ˜ζ and ˜ξ, one obtains ˙˜H ˜ξ + ˜H ˙˜ξ = ˜a ˙˜ϕ2� ˜ζ ˜H − � 1 2˜a − K ˜a3 ˙˜ϕ2 � ˜ξ � = ˜a ˙˜ϕ2 ˜ζ ˜H − � ˙˜ϕ2 2 − K ˜a2 � ˜ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (70) Then, using the equation ˙˜H − K ˜a2 = − 1 2 ˙˜ϕ2 obtained from the background equation (13) and (14), one gets ˙˜ξ = ˜a ˙˜ϕ2 ˜H2 ˜ζ, (71) Equation (16) can be written as 1 ˜a2(∆ + 3K)˜Ψ − 3 ˜H( ˙˜Ψ + ˜H ˜Φ) = 1 2 � ˙˜ϕ ˙δ ˜ϕ − ˜Φ ˙˜ϕ2 − � ˙˜ϕ ¨˜ϕ + 3 ˜H ˙˜ϕ2� 1 ˙˜ϕδ ˜ϕ � (72) = 1 2 � − ˙˜ϕ2� ˜Φ − ˙˜ϕ ˙δ ˜ϕ ˙˜ϕ2 + ¨˜ϕδ ˜ϕ ˙˜ϕ2 � − 3 ˜H ˙˜ϕ2δ ˜ϕ ˙˜ϕ � = −1 2 � ˙˜ϕ2� ˜Φ − �δ ˜ϕ ˙˜ϕ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='� + 3 ˜H ˙˜ϕ2�δ ˜ϕ ˙˜ϕ �� , 1 ˜a2(∆ + 3K)˜Ψ = 1 2 ˙˜ϕ2��δ ˜ϕ ˙˜ϕ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' − ˜Ψ � − 3 2 ˜H ˙˜ϕ2�δ ˜ϕ ˙˜ϕ � + 3 ˜H( ˙˜Ψ + ˜H ˜Ψ) (73) = 1 2 ˙˜ϕ2��δ ˜ϕ ˙˜ϕ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' − ˜Ψ � − 3 2 ˜H ˙˜ϕ2�δ ˜ϕ ˙˜ϕ � + 3 ˜H �1 2 ˙˜ϕ2�δ ˜ϕ ˙˜ϕ �� = 1 2 ˙˜ϕ2��δ ˜ϕ ˙˜ϕ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' − ˜Ψ � , which gives �δ ˜ϕ ˙˜ϕ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' = � 2 ˜a2 ˙˜ϕ2(∆ + 3K) + 1 � ˜Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (74) Expressing δ ˜ϕ ˙˜ϕ and ˜Ψ in terms of ˜ζ and ˜ξ, one obtains � ˜ζ ˜H − � 1 2˜a − K ˜a3 ˙˜ϕ2 � ˜ξ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' = � 2 ˜a2 ˙˜ϕ2(∆ + 3K) + 1 � ˜Ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (75) ˙˜ζ ˜H − ˜ζ ˙˜H ˜H2 + 1 2 ˜H ˜a ˜ξ + K � 3 ˜H ˙˜ϕ2˜a3 + 2 ˜V ˜ϕ ˙˜ϕ3˜a3 � ˜ξ − � 1 2˜a − K ˙˜ϕ3˜a2 � ˙˜ξ = � 2 ˜a2 ˙˜ϕ2(∆ + 3K) + 1 � ˜Ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (76) 19 ˙˜ζ ˜H − ˙˜H ˜ζ ˜H2 − �1 2 ˙˜ϕ2 − K ˜a2 � ˜ζ ˜H2 + 1 2 ˜H ˜a ˜ξ + K � 3 ˜H ˙˜ϕ2˜a3 + 2 ˜V ˜ϕ ˙˜ϕ3˜a3 � ˜ξ = � ˜H ˜a3 ˙˜ϕ2(∆ + 3K) + ˜H 2˜a � ˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='(77) which reduces to ˙˜ζ ˜H + 2 ˜V ˜ϕ ˙˜ϕ3˜a3K ˜ξ = ˜H ˜a3 ˙˜ϕ2∆˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (78) and can be rewritten as ˙˜ζ = H2 ˜a3 ˙˜ϕ2 � ∆ − 2 ˜V ˜ϕ ˜H ˙˜ϕ K � ˜ξ = ˜H2 ˜a3 ˙˜ϕ2(∆ + Y K)˜ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' (79) where Y = −2 ˜V ˜ ϕ ˜H ˙˜ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' 11865018, 12265019, the Natural Science Research Project of Education Depart- ment of Anhui Province of China under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content='2022AH051634, the Doctoral Foundation of Zunyi Normal University of China under Grants No.' metadata={'source': 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Ziaie, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' C 79, 270 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' [16] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Huang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} +page_content=' Xu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdAyT4oBgHgl3EQfuflV/content/2301.00613v1.pdf'} 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Aguado,3,4 Federico Sestito,5 Jonay I. González Hernández,6,7 +Nicolas F. Martin,2,8 Else Starkenburg,9 Pascale Jablonka10,11 and Zhen Yuan2 +1 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK +2 Université de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg, France +3 Dipartimento di Fisica e Astronomia, Universitá degli Studi di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino, Italy +4 INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy. +5 Department of Physics and Astronomy, University of Victoria, PO Box 3055, STN CSC, Victoria BC V8W 3P6, Canada +6 Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain +7 Universidad de La Laguna, Dpto. Astrofísica, E-38206 La Laguna, Tenerife, Spain +8 Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany +9 Kapteyn Astronomical Institute, University of Groningen, Postbus 800, 9700 AV, Groningen, the Netherlands +10 Laboratoire d’astrophysique, École Polytechnique Fédérale de Lausanne (EPFL), Observatoire, CH-1290 Versoix, Switzerland +11 GEPI, Observatoire de Paris, Université PSL, CNRS, Place Jules Janssen, F-92195 Meudon, France +Accepted 2023 January 05. Received 2023 January 05; in original form 2022 November 03 +ABSTRACT +Ultra metal-poor stars ([Fe/H] < −4.0) are very rare, and finding them is a challenging task. Both narrow-band photometry and +low-resolution spectroscopy have been useful tools for identifying candidates, and in this work we combine both approaches. +We cross-matched metallicity-sensitive photometry from the Pristine survey with the low-resolution spectroscopic LAMOST +database, and re-analysed all LAMOST spectra with [Fe/H]Pristine < −2.5. We find that ∼1/3rd of this sample (selected without +[Fe/H]Pristine quality cuts) also have spectroscopic [Fe/H] < −2.5. From this sample, containing many low signal-to-noise +(S/N) spectra, we selected eleven stars potentially having [Fe/H] < −4.0 or [Fe/H] < −3.0 with very high carbon abundances, +and we performed higher S/N medium-resolution spectroscopic follow-up with OSIRIS on the 10.4m Gran Telescopio Canarias +(GTC). We confirm their extremely low metallicities, with a mean of [Fe/H] = −3.4 and the most metal-poor star having +[Fe/H] = −3.8. Three of these are clearly carbon-enhanced metal-poor (CEMP) stars with +1.65 < [C/Fe] < +2.45. The two +most carbon-rich stars are either among the most metal-poor CEMP-s stars or the most carbon-rich CEMP-no stars known, the +third is likely a CEMP-no star. We derived orbital properties for the OSIRIS sample and find that only one of our targets can be +confidently associated with known substructures/accretion events, and that three out of four inner halo stars have prograde orbits. +Large spectroscopic surveys may contain many hidden extremely and ultra metal-poor stars, and adding additional information +from e.g. photometry as in this work can uncover them more efficiently and confidently. +Key words: stars: Population II – Galaxy: halo – stars: chemically peculiar – techniques: spectroscopic +1 INTRODUCTION +The most metal-poor stars still present in the Milky Way today are +valuable portals to the early Universe and the pristine environments +these stars were born in. They are thought to have formed from +material enriched by the first generation(s) of stars, and their chemical +abundances can be used to constrain the properties of the stars that +came before them. Additionally, the dynamical properties of the most +metal-poor stars teach us about the early formation of the Milky Way. +Much can be, and has been, learned from very/extremely/ultra metal- +poor stars with [Fe/H] < −2.0 (VMP)/−3.0 (EMP)/−4.0 (UMP) +(e.g. Beers & Christlieb 2005; Frebel & Norris 2015), although they +are exceedingly rare. +★ Email: anke.arentsen@ast.cam.ac.uk +The metal-poor halo has been found to be a melting pot of many +accreted structures. It is populated by the remnants of the larger +mergers that the Galaxy experienced across its history, such as Gaia- +Sausage/Enceladus (GSE, e.g., Belokurov et al. 2018; Helmi et al. +2018), Sequoia (e.g., Barbá et al. 2019; Myeong et al. 2019), Tham- +nos (e.g., Koppelman et al. 2019), and Sagittarius (e.g., Ibata et al. +1994). The plethora of recently discovered stellar streams are in- +dicative of part of the later accretion events from dwarf/ultra faint +galaxies and globular clusters (e.g., Ibata et al. 2021; Li et al. 2022; +Martin et al. 2022a,b). Additionally, as much as half of the stars in +the halo appears to be born in-situ, likely consisting of both an 𝛼-rich +splashed disk component (e.g., Bonaca et al. 2017; Haywood et al. +2018; Di Matteo et al. 2019; Gallart et al. 2019; Belokurov et al. +2020) and stars that formed in a hot and disordered pre-disk state +(e.g., Belokurov & Kravtsov 2022; Conroy et al. 2022). +© 2023 The Authors +arXiv:2301.02265v1 [astro-ph.GA] 5 Jan 2023 + +2 +Arentsen et al. +The common picture from various cosmological simulations sug- +gests that the VMP stars that inhabit the spatial inner region of the +Milky Way, i.e., the bulge and the disk, are amongst the oldest stars +(e.g., Starkenburg et al. 2017a; El-Badry et al. 2018; Sestito et al. +2021). These stars are therefore great tracers of the early Galactic +assembly. On the observational point of view, many VMP stars have +been observed with such kinematics, focusing on the bulge (e.g., +Howes et al. 2014, 2015, 2016; Arentsen et al. 2020; Lucey et al. +2022; Sestito et al. 2023) and the disk (e.g., Sestito et al. 2019, 2020; +Di Matteo et al. 2020; Carter et al. 2021; Cordoni et al. 2021). The +chemical properties of these populations indicate that the building +blocks of the inner Galaxy consisted of a variety of objects – some +stars appear to have formed in systems very similar to ultra faint +dwarf galaxies, while others are consistent with being born in globu- +lar cluster-like systems (e.g., Schiavon et al. 2017; Sestito et al. 2023, +and references therein), and finally there may also be a significant +contribution of in-situ VMP stars in the inner Galaxy (Belokurov & +Kravtsov 2022; Rix et al. 2022). +Many low-metallicity stars have been found to be carbon-enhanced +metal-poor (CEMP) stars, with frequencies of the order of 30 − 50% +among stars with [Fe/H] < −3.0 (Beers & Christlieb 2005; Yong +et al. 2013; Placco et al. 2014). There are two main types of CEMP +stars. CEMP-s stars are thought to have become carbon-rich later +in their life due to mass-transfer from a (former) asymptotic giant +branch (AGB) star companion – these are typically in binary systems +(e.g. Hansen et al. 2016b), are enhanced in s-process elements as well +as carbon (a signature of AGB star nucleosynthesis), and are more +frequent for [Fe/H] > −3.0. The CEMP-no stars are hypothesised +to have been born from carbon-enhanced gas in the early Universe +– they do not have s-process over-abundances, are less frequently +found to be in binary systems (e.g. Hansen et al. 2016a, although +still more than expected, see Arentsen et al. 2019), and mostly occur +at [Fe/H] < −3.0. The exact frequencies of CEMP-no and CEMP-s +stars as function of metallicity is still under debate (Arentsen et al. +2022), and may also vary with Galactic environment (e.g. inner vs. +outer halo, bulge, dwarf galaxies, globular clusters). +To build large samples of extremely metal-poor stars, many ded- +icated searches have happened in the past 40 years. Several differ- +ent techniques have been used to identify metal-poor stars, such +as following up high-proper motion stars with ultraviolet excesses +(Ryan & Norris 1991), identifying objects with small Ca II H & K +lines in large objective-prism surveys (Beers et al. 1985; Christlieb +et al. 2008), or using metallicity-sensitive (narrow-band) photometry +(Schlaufman & Casey 2014; Starkenburg et al. 2017b; Da Costa et al. +2019; Galarza et al. 2022; Placco et al. 2022). Very and extremely +metal-poor stars have also been identified in greater numbers in large +scale spectroscopic surveys such as the Sloan Digital Sky Survey +(SDSS, York et al. 2000), the Large sky Area Multi-Object fiber +Spectroscopic Telescope (LAMOST1, Deng et al. 2012), RAdial Ve- +locity Experiment (RAVE, Steinmetz et al. 2006) and the GALactic +Archaeology with HERMES spectroscopic survey (GALAH, Buder +et al. 2021), see e.g. Lee et al. (2013), Li et al. (2018), Matijevič +et al. (2017) and Hughes et al. (2022). These are often paired with +dedicated follow-up efforts (Caffau et al. 2013; Allende Prieto et al. +2015; Bonifacio et al. 2015; Aguado et al. 2016; Placco et al. 2018; +Li et al. 2022; Da Costa et al. 2022). +In this work, we combine the strengths of metallicity-sensitive pho- +tometry and large spectroscopic surveys by cross-matching metal- +poor candidates from the photometric Pristine survey (Starkenburg +1 http://www.lamost.org/public/?locale=en +et al. 2017b) with the large database of spectra from LAMOST, with +the goal of identifying new extremely or even ultra metal-poor stars. +The Pristine survey uses metallicity-sensitive narrow-band CaHK +photometry to derive photometric metallicities of millions of stars +towards the Galactic halo, which is very efficient even for extremely +metal-poor stars (Youakim et al. 2017; Aguado et al. 2019). How- +ever, the selection still suffers from some more metal-rich contami- +nation. In this work we alleviate this by adding an extra step, namely +by cross-matching candidates with [Fe/H]Pristine < −2.5 with the +LAMOST spectroscopic database, and doing a dedicated analysis +of all these (often low signal-to-noise) spectra. We select exciting +candidates from this analysis, and follow them up using the OSIRIS +spectrograph at the 10.4m Gran Telescopio Canarias (GTC) (Cepa +et al. 2000) to obtain higher S/N observations, from which we can de- +rive high-quality metallicities to confirm their extremely metal-poor +nature. +We describe our initial candidate selection from Pristine and +LAMOST in Section 2, including some discussion about the suc- +cess rates. The OSIRIS observations for 11 stars and the derivation +of their radial velocities, stellar parameters, distances and orbits is +described in Section 3. We present results for the OSIRIS sample in +Section 4, discussing the presence of carbon-enhanced metal-poor +(CEMP) stars, the orbital properties for the sample and a compar- +ison with a new value-added LAMOST catalogue. We conclude in +Section 5. +2 SELECTION OF EMP CANDIDATES USING PRISTINE +AND LAMOST +The LAMOST archive contains low-resolution spectra (R∼1800) for +millions of stars, but not all spectra have stellar parameters in the +standard LAMOST catalogue tables. We discovered that many of the +most metal-poor stars ([Fe/H] < −2.5) are missed by their standard +pipeline (Wu et al. 2014), and also by the dedicated very metal-poor +pipeline of Li et al. (2018). This is particularly severe for hotter stars +and stars with lower signal-to-noise (S/N). Other dedicated analyses +might be able to deal better with these spectra, and identify promising +extremely metal-poor stars. +At the time our selection was made (February 2021), the latest +LAMOST release was DR6. To avoid having to analyse the full +data release, which contains almost 10 million spectra, we made +a pre-selection of promising extremely metal-poor candidates us- +ing photometric metallicities from the Pristine survey. We used the +internal Pristine data release containing all CaHK observations un- +til Semester 2020A, and adopted the CaHK + SDSS photometric +metallicities (Starkenburg et al. 2017b). We queried the LAMOST +archive for all stars in the Pristine survey with photometric metal- +licities [Fe/H]Pristine < −2.5 (from using either 𝑔 − 𝑖 or 𝑔 − 𝑟) and +𝑔sdss < 18, and found ∼ 7500 cross-matches for ∼ 6000 unique tar- +gets. No other quality cuts were applied, which usually are included +when we do dedicated target selection for Pristine follow-up imme- +diately from the photometry (Youakim et al. 2017), to be as inclusive +as possible. +2.1 Preliminary ULySS analysis +A first-pass analysis of these candidates was done with the ULySS2 +code (Koleva et al. 2009). ULySS is a full-spectrum fitting package +2 ULySS is available from http://ulyss.univ-lyon1.fr/ +MNRAS 000, 1–13 (2023) + +GTC follow-up of Pristine/LAMOST +3 +that employs empirical spectral libraries to determine stellar param- +eters (𝑇eff, log 𝑔, [Fe/H], radial velocities, spectral broadening), and +can be applied to stars of a wide range of stellar parameters and +metallicities. We employed this code because we were interested in +the types of contamination in the Pristine selection, which one cannot +study with the dedicated metal-poor analysis described in the next +sub-section. +For the models, we adopted the empirical MILES library (Sánchez- +Blázquez et al. 2006; Falcón-Barroso et al. 2011) and used the ULySS +MILES polynomial interpolator originally built by Prugniel et al. +(2011) and updated for cool stars by Sharma et al. (2016). The library +has a resolving power of 𝑅 ∼ 2200, and the interpolator extends down +to [Fe/H] = −2.8 (with the possibility to extrapolate, at ones own +risk). The LAMOST spectra were fitted between 3750 and 5500 +Å using a multiplicative Legendre polynomial of degree 15 for the +normalisation. This degree is large enough to absorb some of the large +mismatches between models and observations for carbon-enhanced +metal-poor stars in regions of carbon-related molecular bands, which +is necessary since the ULySS models do not include [C/Fe] as a free +parameter, and large carbon features could mess up the normalisation. +There is also an automatic masking routine in ULySS, which excludes +outlier pixels iteratively and typically masks the wavelength regions +of the largest carbon features in CEMP stars. +The resulting Kiel-diagram and metallicity histogram from our +ULySS analysis are shown in Figure 1, for all exposures of the ∼ 4900 +unique stars that remain after removing fits with signal-to-residual +ratios < 8, broadening > 400 km s−1 (which usually indicates a very +bad fit), and duplicate LAMOST spectra for the same star. The metal- +poor stars show a clear red giant branch (RGB) and main-sequence +turn-off sequence, except for a small cloud of stars to the left of the +RGB, which mostly consists of stars in the low S/N-tail of the sample +without good fits. +Most of the stars in our selection are indeed very metal-poor (keep- +ing only the fit with the highest signal-to-residual ratio per star): 71% +have [Fe/H]ULySS < −2.0 and 25% have [Fe/H]ULySS < −2.5. The +latter goes up to 34% when using the FERRE metallicities described +later in this section, which perform better in this regime than the +ULySS metallicities, because the MILES library does not have many +stars in this [Fe/H] range (especially for the turn-off region). There +is a contamination of metal-rich stars with [Fe/H]ULySS > −1.0 of +16%. +ULySS is also the main software used by the LAMOST team for +the parameters in their public data releases (Wu et al. 2014). They use +the interpolator based on the ELODIE library (Prugniel & Soubiran +2001; Wu et al. 2011), which has a more limited coverage of the +parameter space compared to MILES, and extends only down to +[Fe/H] = −2.5. Of the stars that have [Fe/H]ULySS < −2.0/−2.5 +in our analysis, only 30%/17% have stellar parameters in the public +LAMOST DR7 catalogue. This is likely partly due to the ELODIE +library being less good at low metallicities, and partly due to more +stringent quality cuts being applied for stars to make it into the +LAMOST data releases. +2.2 Success rates +In our original selection we did not make any additional photomet- +ric quality cuts. The Pristine team developed several quality cuts to +remove metal-rich outliers and improve the success rates of the spec- +troscopic follow-up of extremely metal-poor candidates. The cuts +applied for the main Pristine follow-up campaign are discussed in +Section 4.1 of Youakim et al. (2017). We apply very similar cuts to the +4000 +6000 +9000 +Teff (ULySS) +0 +1 +2 +3 +4 +5 +log g (ULySS) +−6 +−4 +−2 +0 +[Fe/H] +0 +1000 +Nr of stars +ULySS +FERRE VMP +−3.5 +−3.0 +−2.5 +−2.0 +−1.5 +−1.0 +−0.5 +0.0 +[Fe/H] (ULySS) +Figure 1. Top: Kiel diagram for all exposures of the 4900 unique Pristine- +selected stars in LAMOST analysed with ULySS, colour-coded by metallicity. +No quality cuts were applied to the photometric metallicities in the selection. +The results for the eleven stars that were followed up with OSIRIS (see +Section 3) are highlighted with larger symbols (the two high and low log 𝑔 +outliers are CEMP stars). Bottom: ULySS metallicity histogram of the same +sample in black, and FERRE metallicity histogram for the VMP sub-sample +in red. +Pristine +LAMOST sample to see how that changes the metallicity +distribution, keeping only the stars that have: +• CASU flag = −1 or 1 +• young stars flag = 0 +• (𝑢0 − 𝑔0) > 0.6 +• 0.25 < (𝑔0 − 𝑖0) < 1.5 and 0.15 < (𝑔0 − 𝑟0) < 1.2 +• [Fe/H]Pristine < −2.5 (from using either SDSS 𝑔 − 𝑖 or 𝑔 − 𝑟) +and ≠ −99 (−99 is assigned if the star falls outside of the parameter +space for which the photometric metallicity assignment has a valid +calibration) +• instead of the PanSTARRS variability catalogue as in Youakim +et al. (2017), we use the Gaia photometric variability to remove +variable stars as in Fernández-Alvar et al. (2021) +The uncertainties on [Fe/H]Pristine are not taken into account +here, whereas they were in Youakim et al. (2017). After applying +the above cuts, the sample goes from 4900 stars to 4100 stars again +keeping the highest signal-to-residual spectrum per star). Of these, +78% have [Fe/H]ULySS < −2.0, and 28% have [Fe/H]ULySS < −2.5 +(the latter goes up to 38% for the FERRE metallicities), compared to +the previous 71% and 25% (and 34% for FERRE), respectively. The +metal-rich contamination goes down to 12%. Doing the same only +for stars with signal-to-residual ratios > 20 instead of our initial cut +at > 8, the results are very similar. We conclude that, for the Pristine ++LAMOST sample, the photometric quality cuts slightly improve +the selection efficiency, but not by a lot. +The +success +rate +of +previous +Pristine +follow-up +for +[Fe/H]Pristine < −2.5 was found to be 56% (Aguado et al. 2019). +The lower fraction in this work (38% when applying the photo- +metric quality cuts and adopting the FERRE metallicities) could be +due to various reasons. For example, the dedicated Pristine follow- +MNRAS 000, 1–13 (2023) + +4 +Arentsen et al. +Table 1. List of 481 EMP candidates (533 spectra) with FERRE spectroscopic parameters used for target selection. No quality cuts have been applied. The first +few lines of the table are shown here for guidance, the full table and figures showing the best fits are available as online supplementary material. +LAMOST spectrum name +Gaia DR3 source_id +ra +dec +𝑇eff +log 𝑔 +[Fe/H] +[C/Fe] +S/N +log(𝜒2) +[deg] +[deg] +[K] +cgs +spec-56746-HD121251N314746M01_sp02-087 +4014278062082321152 +181.215811 +30.331285 +6232 ± 325 +5.0 ± 1.4 +−3.7 ± 15.1 +−0.4 ± 90.4 +6 +−0.338 +spec-57308-EG000023N024031M01_sp03-076 +2739551594198812160 +358.800705 +2.493245 +6521 ± 161 +4.7 ± 0.4 +−3.2 ± 0.8 +0.1 ± 6.9 +24 +−0.214 +spec-57754-HD122624N271605M02_sp03-108 +4009964020835772288 +185.239126 +27.687022 +5675 ± 57 +4.3 ± 0.3 +−3.4 ± 0.1 +0.3 ± 0.3 +34 +−0.123 +... +... +... +... +... +... +... +... +... +... +up presented in Youakim et al. (2017) and Aguado et al. (2019) +focused on the most metal-poor stars, and did not homogeneously +observe all stars with [Fe/H]Pristine < −2.5. Additionally, extra qual- +ity cuts were sometimes implemented for subsets of the follow-up +presented in Aguado et al. (2019), most notably the consistency of +the photometric temperatures derived from SDSS between (𝑔 − 𝑖) +and (𝑔 − 𝑟). If we select only stars with [Fe/H]Pristine < −2.7 and +|𝑇eff (𝑔−𝑖) − 𝑇eff (𝑔−𝑟) | < 200 K, the success rate in this work for +[Fe/H]FERRE < −2.5 goes up to 50%, and the contamination of stars +with [Fe/H]ULySS > −1.0 is reduced from 12% to 2%. +The numbers in this section are not meant to override the previ- +ously published Pristine success rates by Youakim et al. (2017) and +Aguado et al. (2019). Our results confirm that the success rates are +high, and highlight some of the subtleties in deriving such success +rates. Overall we conclude that our methodology to find hidden very +and extremely metal-poor stars in the large LAMOST database is +extremely efficient. +2.3 Dedicated VMP FERRE analysis +A dedicated very metal-poor analysis was performed for the sub- +sample of LAMOST spectra with [Fe/H]ULySS < −2.0, with the +aim of deriving better metallicities and carbon abundances for the +most metal-poor stars and identifying potential ultra metal-poor can- +didates. We followed a similar methodology as in Aguado et al. +(2017a,b), using the FERRE3 code (Allende Prieto et al. 2006). The +code interpolates between the nodes of a library of synthetic spectra +and derives simultaneously the set of best stellar parameters (𝑇eff, +log 𝑔, [Fe/H], [C/Fe]). For this preliminary analysis, we used the +default Nelder-Mead search algorithm and linear interpolation. The +dedicated very metal-poor synthetic models were computed with the +ASSET code (Koesterke et al. 2008) and published in Aguado et al. +(2017b) with the following parameter coverage: +• 4750 K < 𝑇eff < 7000 K, Δ𝑇eff = 250 K +• 1.0 < log 𝑔 < 5.0, Δ log 𝑔 = 0.5 +• −6.0 < [Fe/H] < −2.0, Δ[Fe/H] = 0.5 +• −1.0 < [C/Fe] < +5.0, Δ[C/Fe] = 1.0 +and a fixed [𝛼/Fe] = +0.4 and [N/Fe] = 0. Both the data and +the models were continuum normalised with a running mean filter +with a 30 pixel window. We limited the fit to the wavelength range +3700 − 5500 Å, where most of the features for extremely metal-poor +stars are present. The spectra were shifted to rest-wavelength using +the ULySS radial velocities. +The resulting metallicity distribution is shown in red in the bottom +panel of Figure 1, without any additional quality cuts applied. The +hard limit at [Fe/H]FERRE = −2.0 is due to the limit of the grid. +3 FERRE is available from http://github.com/callendeprieto/ferre +The ULySS and FERRE distributions peak at roughly the same metal- +licities, but the FERRE distribution has a larger tail towards lower +metallicities – as expected. +We inspected the > 500 fits in the resulting FERRE-analysed sam- +ple with [Fe/H]FERRE < −3.0 by eye, and identified a number of +(previously unknown) stars of interest that could potentially have +[Fe/H] < −4.0 or that looked very carbon-rich and extremely metal- +poor ([Fe/H] < −3.0). Practically none of our candidates had pa- +rameters in the public DR6 catalogue. Most of our candidates had +relatively low S/N, so follow-up spectroscopy was necessary to con- +firm their extremely or even ultra metal-poor nature. +The full list of EMP candidates that we inspected is given in Ta- +ble 1, with figures for all the spectral fits provided in the online +supplementary materials.4 This candidate list with its derived pa- +rameters should not be used blindly since no quality cuts have been +applied (on e.g. S/N, log(𝜒2) or parameter uncertainties), but it could +be used in combination with the figures to select other EMP stars +for follow-up. Stars may occur multiple times in this list if they have +more than one LAMOST spectrum. +3 OSIRIS FOLLOW-UP OF EMP CANDIDATES +We obtained GTC/OSIRIS observations for 11 of our most promising +candidates (16.9 < 𝑔 < 17.9) in Semester 2021A. We used OSIRIS +in longslit mode with the 2500U grating, a 1 arcsec slit and 2x2 +binning, resulting in spectra covering 3440 − 4610 Å at a resolving +power 𝑅 ∼ 2400 (providing an instrument profile with a FWHM of +∼ 125 km s−1). We aimed for a S/N of 40 at 4000 Å, corresponding to +exposure times of 3000s for stars of magnitude 𝑔 ∼ 17.5. A summary +of the observations is presented in Table 2. Individual exposures of +1400, 1600 and 1800s were executed for different targets. +3.1 Radial velocities +Radial velocities (RVs) are derived using the cross-correlation +technique. We have a high-quality GTC/OSIRIS spectrum of a +bright extremely metal-poor star G64-12 (𝑇eff = 6463K, log 𝑔 = +4.26, [Fe/H] = −3.29, [C/Fe] = +1.07, Placco et al. 2016 and ref- +erences therein) from previous campaigns acquired with the same +setup (Aguado et al. 2017a, 2018), that we use as a cross-correlation +template. The OSIRIS spectra of both our targets and the template +star are normalized with the same method, using a running mean +filter with a width of 30 pixels. We built the cross-correlation func- +tion (CCF) with our own IDL-based automated code in the spectral +4 The table and figures of FERRE fits for EMP candidates can temporar- +ily be found at the following URL: https://drive.google.com/drive/ +folders/1qLypkgyG6m-XbwPGpmNrBnXFQf70-0se?usp=sharing, +and +they will be part of the online material with the published paper. +MNRAS 000, 1–13 (2023) + +GTC follow-up of Pristine/LAMOST +5 +Table 2. Summary of the OSIRIS observations: our reference for each star, the LAMOST and Gaia DR3 designations, positions, SDSS magnitudes in 𝑢 and 𝑔, +total exposure time, signal to noise at two different wavelengths, number of observations and date that the spectra were observed. +Star # +LAMOST +Gaia DR3 source_id +ra +dec +u_mag +g_mag +total exp +S/N (@392/ +Nobs +date observed +designation +[mag] +[mag] +time [s] +450 nm) +(DD-MM-2021) +LP1 +J002953.07+320229.9 +2862648994739368704 +00:29:53.07 +32:02:29.9 +18.49 +17.58 +2800 +33/79 +2 +14-07 +LP2 +J131532.41+121107.4 +3736550805114696192 +13:15:32.41 +12:11:07.4 +18.33 +17.44 +2800 +57/127 +2 +11-04 +LP3 +J134510.95+424910.8 +1500794652785646976 +13:45:10.96 +42:49:10.9 +17.75 +16.94 +1400 +74/180 +1 +15-06 +LP4 +J142055.86+075308.7 +3673778479398720000 +14:20:55.87 +07:53:08.7 +17.96 +17.04 +1400 +62/100 +1 +15-06 +LP5 +J144714.22+163425.4 +1186662458446883328 +14:47:14.22 +16:34:25.5 +18.73 +17.92 +2800 +27/88 +2 +10-06 +LP6 +J145214.98+160357.6 +1186397549159479424 +14:52:14.99 +16:03:57.7 +18.25 +16.88 +1400 +12/97 +1 +10-06 +LP7 +J145611.30+161925.7 +1187873604865113472 +14:56:11.31 +16:19:25.8 +18.77 +17.82 +4600 +34/116 +3 +08-04 +LP8 +J161021.42+451247.5 +1386190837835580288 +16:10:21.43 +45:12:47.6 +18.50 +17.63 +5600 +30/79 +4 +10-06/17-07 +LP9 +J162359.32+303740.8 +1318369490300720000 +16:23:59.32 +30:37:40.9 +18.46 +17.52 +5600 +24/78 +4 +11-04/14-07 +LP10 +J212109.07+151328.7 +1783524305407324672 +21:21:09.06 +15:13:29.0 +18.15 +17.15 +4800 +38/145 +3 +15-06/13-07/14-07 +LP11 +J230209.39+302100.6 +1886140596052059392 +23:02:09.39 +30:21:00.7 +18.01 +17.09 +3600 +39/192 +2 +23-05/13-07 +Table 3. Radial velocities, heliocentric distance, probability of the distance within 3𝜎 around the maximum of the distance PDF, maximum height from the +plane, apocentric and pericentric distances, eccentricity, energy, the action vector are reported. [updated] +Star # +RV +D +P𝐷 +𝑍max +Rapo +Rperi +𝜖 +E/104 +J𝜙 +Jr +JZ +[km s−1] +[kpc] +[kpc] +[kpc] +[kpc] +[kpc km2 s−2] +[kpc km s−1] +[kpc km s−1] +[kpc km s−1] +LP1 +−18 ± 15 +8.62 ± 0.47 +0.14 +15.6+4.9 +−4.4 +16.2+6.8 +−3.1 +2.4+1.3 +−0.8 +0.75+0.04 +−0.03 +−4.79+1.29 +−0.86 +−140+365 +−417 +829+177 +−142 +780+215 +−329 +LP2 +211 ± 15 +6.88 ± 0.15 +0.66 +15.5+12.6 +−4.7 +24.3+31.4 +−9.4 +8.5+0.8 +−0.7 +0.52+0.19 +−0.19 +−2.76+2.17 +−1.63 +−1821+405 +−434 +17+1 +−1 +905+306 +−153 +LP3 +−162 ± 15 +2.35 ± 0.12 +0.99 +8.0+1.6 +−1.5 +9.8+0.5 +−0.4 +2.4+0.7 +−0.7 +0.60+0.10 +−0.10 +−6.62+0.15 +−0.13 +368+197 +−223 +294+87 +−76 +501+188 +−145 +LP4 +−70 ± 15 +6.03 ± 0.10 +1.0 +7.5+0.3 +−1.3 +7.8+0.2 +−0.5 +1.5+0.9 +−0.6 +0.66+0.12 +−0.13 +−7.58+0.36 +−0.22 +−15+207 +−195 +200+63 +−60 +556+159 +−175 +LP5 +−126 ± 15 +3.25 ± 0.16 +0.87 +5.3+1.4 +−0.8 +10.1+0.4 +−0.3 +2.9+0.5 +−0.5 +0.54+0.06 +−0.06 +−6.50+0.14 +−0.14 +750+151 +−151 +269+56 +−49 +259+76 +−64 +LP6 +53 ± 15 +22.7 ± 1.4 +1.0 +21.2+3.3 +−3.6 +21.7+5.7 +−3.6 +11.8+6.3 +−3.5 +0.30+0.06 +−0.08 +−3.02+1.16 +−0.81 +576+407 +−203 +198+43 +−61 +2480+770 +−527 +LP7 +−136 ± 15 +8.05 ± 0.18 +0.92 +36.2+48.7 +−22.3 +52.8+135.8 +−35.2 +7.7+1.1 +−0.7 +0.74+0.16 +−0.27 +−0.85+4.15 +−3.23 +−1583+189 +−99 +38+1 +−1 +1382+889 +−575 +LP8 +−56 ± 15 +3.62 ± 0.23 +0.54 +2.8+0.6 +−0.5 +9.2+0.4 +−0.3 +4.6+0.5 +−0.5 +0.33+0.05 +−0.05 +−6.48+0.18 +−0.15 +1215+107 +−114 +109+38 +−27 +112+25 +−23 +LP9 +−254 ± 15 +7.25 ± 0.14 +0.99 +31.8+50.0 +−19.8 +39.9+90.7 +−20.9 +3.6+1.8 +−0.7 +0.83+0.08 +−0.09 +−1.61+3.56 +−2.59 +−726+196 +−115 +269+10 +−10 +903+943 +−528 +LP10 +−361 ± 15 +5.79 ± 0.10 +0.93 +23.3+10.4 +−11.3 +52.4+115.2 +−28.8 +5.1+0.8 +−0.4 +0.83+0.10 +−0.13 +−0.70+3.49 +−2.44 +−1185+234 +−182 +191+10 +−10 +957+769 +−500 +LP11 +−68 ± 15 +6.09 ± 0.10 +0.99 +12.5+6.6 +−3.9 +15.7+8.4 +−3.9 +10.5+0.7 +−0.8 +0.21+0.16 +−0.09 +−3.89+1.17 +−0.78 +1467+141 +−116 +58+255 +−57 +1084+465 +−336 +range 3755-4455 Å with a window of 3000 km s−1. The main fea- +tures in the template are the CaII H&K lines, the HI lines of the +Balmer series, and the G-band in carbon-enhanced stars (see Fig. 2). +The normalization method produces a shape of the CCF profile that +mimics the shape of all balmer lines in the warm template EMP star, +which does not resemble a gaussian shape. We thus fit the CCF profile +with a parabolic fit using the closest 6 points to the CCF peak. The +statistical uncertainty of the centroid of the parabolic fit is typically +below 1 km s−1, significantly below the pixel size of ∼ 0.57 Å/pixel +(∼ 42 km s−1/pixel). The results of the OSIRIS spectra show intra- +night RV variations with standard deviations below ∼ 7 km s−1, but +RV variations from different nights with standard deviations in the +range 3 − 20 km s−1. +We also derive the RV for the same stars from their LAMOST spec- +tra (which typically have much lower S/N than the OSIRIS spectra), +using the same technique to check the consistency with our OSIRIS +RVs. The LAMOST spectrum of G64-12 is used as cross-correlation +template and all spectra are normalized using a running mean filter +with a width of 15 pixels of ∼ 1.38 Å/pixel (∼ 81 km s−1/pixel). The +CCF is built from the spectral range 3755-6755 Å, which includes +H𝛼 and H𝛽, providing more stability to the CCF profile given the +lower quality LAMOST spectra. We find a reasonable consistency +when comparing to the OSIRIS results, with a mean difference of +−4.4 km s−1 and a standard deviation of 15.9 km s−1. +For each target we adopt the weighted mean of the OSIRIS RVs +derived from each individual spectrum and the corresponding error +of the mean as the final RV. We apply an quadratically added uncer- +tainty floor of 15 km s−1 to the RV uncertainties, which seems more +realistic than the CCF uncertainties given the RV variations within +and between different nights and the differences with the LAMOST +RVs. This floor reflects the systematic RV uncertainty due to possible +instrument flexures, pointing, guiding RV drifts, etc. +3.2 Distances +It has been widely demonstrated that simply inverting the parallax to +infer the distance can lead to wrong results, and including additional +priors and/or data improves distance estimates (e.g., Bailer-Jones +et al. 2018, 2021; Anders et al. 2022). This is especially the case when +the parallax has poor measurements, i.e., 𝜛 < 0 and/or 𝜛/𝜎𝜛 < 20. +We therefore use a Bayesian approach to infer the distances for the +stars in our sample. The probability distribution function (PDF), or +posterior, is inferred following the method fully described in Sestito +et al. (2019). Briefly, the likelihood is the product of the Gaussian +distributions for the parallax and photometry. The prior takes into +MNRAS 000, 1–13 (2023) + +6 +Arentsen et al. +Table 4. Table with adopted stellar parameters, with 𝑇eff, [Fe/H] and [C/Fe] from FERRE and photometric log 𝑔. +Star # +𝑇eff +log 𝑔 +[Fe/H] +[C/Fe] +corr.2 +[K] +cgs +A(C)⊙ = 8.391 +LP1 +5790 ± 101 +4.83 ± 0.22 +−3.52 ± 0.11 ++1.65 ± 0.21 +LP2 +6419 ± 108 +3.85 ± 0.30 +−3.43 ± 0.12 +< +1.90 +LP3 +6365 ± 102 +4.57 ± 0.08 +−3.32 ± 0.11 +< +1.5 +LP4 +5993 ± 103 +3.65 ± 0.28 +−3.30 ± 0.11 +< +0.70 +LP5 +6134 ± 102 +4.59 ± 0.37 +−3.42 ± 0.11 +< +0.89 +LP6 +4575 ± 103 +1.00 ± 0.20 +−3.30 ± 0.11 ++2.21 ± 0.21 ++0.24 +LP7 +6413 ± 103 +3.85 ± 0.39 +−3.50 ± 0.11 +< +1.0 +LP8 +6363 ± 103 +4.47 ± 0.19 +−2.90 ± 0.11 +< +0.96 +LP9 +6018 ± 103 +3.70 ± 0.21 +−3.83 ± 0.12 +< +0.70 +LP10 +6304 ± 102 +3.75 ± 0.14 +−3.47 ± 0.11 +< +1.02 +LP11 +5730 ± 102 +3.53 ± 0.24 +−3.12 ± 0.13 ++2.17 ± 0.22 +1Solar abundance adopted from Asplund et al. (2005) +2Evolutionary [C/Fe] correction following Placco et al. (2014) +account a power-law stellar distribution (see the halo prior in Sestito +et al. 2019), and, through a set of VMP ([M/H]= −2.5) MESA/MIST +isochrones5 (Dotter 2016; Choi et al. 2016), the knowledge that VMP +stars are old (11 − 13.8 Gyr), low-mass (< 1 M⊙), and distributed +with a given IMF-based luminosity function in the CMD diagram. +The zero-point offset has been applied to the Gaia EDR3 parallaxes +(Lindegren et al. 2021) using the python gaiadr3_zeropoint6 pack- +age. This method, widely used for chemo-dynamical investigations +of VMP stars (e.g., Sestito et al. 2019, 2020; Venn et al. 2020), +produces low uncertainties on the distances even in case of large par- +allax uncertainties. This is because the isochrones limit the possible +distances for a star with a given colour to two different solutions, +a dwarf and a giant solution, and nothing in between. The parallax +would then typically prefer one of the two, or, in case of a very poor +parallax measurement, the two peaks would be given a different prob- +ability. We calculate the probabilities following Sestito et al. (2019). +For seven of the OSIRIS stars the probability of the main peak is +larger than 92 per cent. For two stars it is 86 per cent (LP1, although +for this star we adopt the less probable solution, see Section 4.1) and +87 per cent (LP5), while for the remaining two stars it is 54 per cent +(LP8) and 66 per cent (LP2). +3.3 Orbital parameters +The orbital parameters are inferred using Galpy7 (Bovy 2015). +The code requires as input the inferred distances, the RVs, and +the proper motions and coordinates from Gaia (E)DR3. The total +fixed gravitational potential that we adopt is the sum of a Navarro- +Frenk-White dark matter halo (Navarro et al. 1997, NFWPoten- +tial), a Miyamoto-Nagai potential disc (Miyamoto & Nagai 1975, +MiyamotoNagaiPotential) and an exponentially cut-off bulge +(PowerSphericalPotentialwCutoff). All of the aforementioned +potentials are usually invoked by the MWPotential14 package. +However, we adopt a more massive and up-to-date halo (Bland- +Hawthorn & Gerhard 2016), with a mass of 1.2 × 1012 M⊙ (vs. +0.8 × 1012 M⊙ for MWPotential14). +For each star, we perform a Monte Carlo simulation with 1000 +5 https://waps.cfa.harvard.edu/MIST/ +6 https://gitlab.com/icc-ub/public/gaiadr3_zeropoint +7 http://github.com/jobovy/galpy +random draws on the input parameters to infer the orbital parameters +and their uncertainties. In case of the proper motion components, we +consider their correlation given the coefficients from Gaia EDR3, +drawing randomly with a multivariate Gaussian function. The RVs +(from the OSIRIS spectra) and coordinates are treated as a Gaussian. +In order to account for possible systematics on the distances (e.g. due +to the adopted isochrones and other assumptions), we assume a 15 +per cent uncertainty on the distances. The integration time is set to 1 +Gyr. The orbital parameters are inferred for both of the peaks in the +distance PDFs. +The output orbital parameters are the Galactocentric Cartesian +coordinates (X, Y, Z), the maximum distance from the Milky Way +plane Zmax, the apocentric and pericentric distances (Rapo, Rperi), +the eccentricity 𝜖, the energy E, and the spherical actions coordinates +(J𝜙, Jr, JZ). Table 3 reports the main orbital parameters from the +most probable distance, except for star LP1 where we adopt the +less probable distance (see Section 4.1). The orbital parameters are +discussed in Section 4.1. +3.4 Stellar parameters +The OSIRIS data were analysed with FERRE in a similar manner as +the LAMOST spectra. For this analysis we use the more sophisticated +Boender-Timmer-Rinnoy Kan (BTRK, Boender et al. 1982) global +search algorithm and Bézier cubic interpolation. We use the same +grid, except for the coolest star in the sample, for which we employ +a similar grid that has been extended down to 4500 K (as used +e.g. in Arentsen et al. 2021). Again we used a 30 pixel window +for the running mean normalisation, suitable for OSIRIS resolution +(𝑅 = 𝜆/𝛿𝜆 ∼ 2400). To avoid problems in the noisy blue region we +only analyse the spectra in the range (3750 − 4500 Å). +We found that for the warm stars in the sample (with𝑇eff > 5500 K, +which is all stars except for LP6), the log 𝑔 values that FERRE finds +are typically at the edges of the FERRE grid, e.g. at log 𝑔 = 5.0 or +log 𝑔 < 2.0, see the black points in Figure 3. This is likely the re- +sult of not much log 𝑔 information being present in these extremely +metal-poor stars in the available wavelength range. Previous work +on metal-poor stars with FERRE has shown that systematically off- +set log 𝑔 values strongly impact the derived [C/Fe] (Aguado et al. +2019; Arentsen et al. 2021). Therefore we decided to adopt photo- +metric log 𝑔 values for the warm stars, shown by the magenta points +in Figure 3. These were inferred from the Stefan-Boltzmann equa- +MNRAS 000, 1–13 (2023) + +GTC follow-up of Pristine/LAMOST +7 +Figure 2. OSIRIS/GTC spectra (3750 Å-4500 Å) of our stellar sample (black line) and the best fits calculated with FERRE, colour-coded by 𝑇eff (the bluer the +hotter) and sorted by decreasing [Fe/H]. The Balmer lines (yellow) and main metallic absorption features (purple) are high-lighted. Above each spectrum the +metallicity, effective temperature and carbonicity are displayed. +MNRAS 000, 1–13 (2023) + +8 +Arentsen et al. +5000 +6000 +Teff +1 +2 +3 +4 +5 +log g +[Fe/H] = −3.0 +[Fe/H] = −3.5 +FERRE +adopted +Figure 3. Kiel diagram showing the pure FERRE stellar parameters (black) +and the adopted stellar parameters and uncertainties (magenta). See the text for +details. Also shown are Yonsei-Yale isochrones for two different metallicities +(both with age = 12 Gyr, [𝛼/Fe] = +0.4). +tion, which needs as input the dereddened absolute G magnitude +(derived using the Gaia G-band magnitude, the 3D extinction map +from Green et al. 2019 and the distances from Table 3), an estimate +of the effective temperature, and the bolometric corrections on the +flux (Andrae et al. 2018). We adopt the FERRE effective temperature +and its inflated uncertainty (see last paragraph of this subsection) in +the calculation. We perform a Monte Carlo iteration with 1000 ran- +dom draws on the input parameters. Each of them is described by a +Gaussian distribution. +We run FERRE again for the warm stars, fixing the 𝑇eff to the +previously derived FERRE value and log 𝑔 to the photometric values +values, while letting [Fe/H] and [C/Fe] free. The final spectral fits +are shown in Figure 2 and a summary of the results is provided in +Table 4. The differences between the original FERRE run and the +run with fixed 𝑇eff and log 𝑔 are small for the metallicities, with the +adopted [Fe/H] being higher by 0.07 dex with a standard deviation of +0.06 dex. The differences for [C/Fe] are also small for the stars with +original log 𝑔 > 4 and measured [C/Fe] (see next section), they are +0.05 on average, with a standard deviation of 0.09 dex. However, for +the one star with measured carbon and FERRE log 𝑔 < 3 (LP11), the +new [C/Fe] is 0.7 dex lower. +There are three stars (LP4, LP7 and LP9) that have very high +FERRE internal [Fe/H] uncertainties of 0.5 − 1.0 dex when calcu- +lated by inverting the covariance matrix (our original approach). This +could be attributed to some negative/zero fluxes in blue end of the +OSIRIS data. To avoid this issue we recalculated the internal FERRE +uncertainties using a Monte Carlo simulation. We performed 50 ex- +periments and use the dispersion on the derived [Fe/H] and [C/Fe] as +the uncertainty following Aguado et al. (2017a). As a result of that the +issue with the large uncertainties was fixed for the three problematic +stars, and the uncertainties for the other stars remain the same within +0.01−0.02 dex. We adopt the Monte Carlo internal uncertainties. +To provide the final uncertainties for the stellar parameters, we +add estimates of the external uncertainties from a previous analysis +of EMP stars with FERRE (Aguado et al. 2017a) to our internal FERRE +uncertainties. These are 100 K, 0.1 and 0.2 dex for 𝑇eff, [Fe/H] and +[C/Fe], respectively. For [Fe/H] and [C/Fe] we adopt the internal +uncertainties from the first FERRE run, because the second run does +not properly reflect the real uncertainties since it fits only two of +the four parameters. For log 𝑔, we adopted the uncertainties from +the photometric determination for the warm stars, and for the coolest +star we quadratically added 0.2 dex of external uncertainties (Aguado +et al. 2017a) to the internal FERRE uncertainty. The results are shown +in Table 4. +3.5 Carbon determination +Deriving carbon abundance from low-resolution data of EMP stars is +non-trivial. Our employed grid is suitable for the analysis of CEMP +stars, since carbon-enhancement was not only considered in the spec- +tral synthesis step but also in the ATLAS stellar models (Sbordone +et al. 2007). This is crucial because high carbon abundances can sig- +nificantly impact the stellar atmospheres. The grid of models has been +used successfully to derive carbon abundances in several works (e.g. +Aguado et al. 2017b,a, 2019; Arentsen et al. 2021, 2022), although +there are some differences with other synthetic grids that can lead to +systematic differences in derived carbon abundances (Arentsen et al. +2022). This is likely related to the use of different codes, line lists +and assumptions (e.g. different [N/Fe] abundances). +The ability of the FERRE code to detect – and successfully fit – +carbon absorption features from low-resolution data strongly depends +on a)𝑇eff (and log 𝑔 to a lesser extent), b) the carbon abundance, +and c) the SNR of the spectra. In our sample there are three stars +(LP1, LP6, and LP11) that fulfil the sensitivity criteria derived by +Aguado et al. (2019) based on these parameters, all of them have +𝑇eff < 6000 K and show strong CH absorption features. For these +objects we derived [C/Fe] = +1.65/+2.21/+2.17 respectively, with +reasonable uncertainties (∼ 0.2 dex). For the other stars we can only +provide upper limits on the carbon abundances. The carbon results +are summarised in Table 4. +The object with the lowest 𝑇eff in our sample, LP6, shows clear +CN features at ∼ 3885 Å that our best fit is not able to reproduce, +although the CH & G-band fit is good (see Fig. 2, red spectrum). +The reason for this is that our FERRE synthetic spectral library +assumes [N/Fe] = 0.0 for all stellar models. Querying the high- +resolution spectroscopy compilation in the JINAbase (Abohalima & +Frebel 2018) for stars with −3.5 < [Fe/H] < −3.0, we find that all of +those with measured nitrogen abundances have [N/Fe] > 0, and stars +with [C/Fe] > +2.0 typically have 1.5 < [N/Fe] < 3.0. This is very +different from the assumed [N/Fe] in the FERRE grid, and can explain +why the CN band for LP6 is much stronger in the data than in the +model fit. However, the fit reproduces quite well the Ca ii at 3933 Å +and several other Fe i, Ti ii, and Sr ii lines in the 4040−4080 Å region. +Additionally, the majority of the carbon information is significantly +concentrated around the G−band (4200-4330 Å) and our fit is good +in that area. Therefore, we conclude that the CN absorption features +in the blue are not significantly affecting the best fit for this object. +The carbon abundance of evolved giants decreases with decreasing +log 𝑔 due to mixing processes, especially in metal-poor stars (Gratton +et al. 2000; Placco et al. 2014). We estimate the evolutionary carbon +correction for the most evolved star in our sample (LP6, the only star +that should be affected by this effect) using the web calculator8 by +V.M. Placco, and find it to be +0.24 dex. +8 https://vplacco.pythonanywhere.com/ +MNRAS 000, 1–13 (2023) + +GTC follow-up of Pristine/LAMOST +9 +Figure 4. Orbital parameters. Three left panels: pericenter, eccentricity, and maximum distance from the Milky Way plane as a function of the apocentric +distance. The grey-shaded areas denote the forbidden region in which the Zmax > Rapo or Rperi > Rapo. Upper right panel: Energy vs. rotational component of +the action, J𝜙. Bottom right panel: Action space; the y-axis is the difference between the vertical and radial component of the action, while the x-axis is the +rotational component; axes are normalised by Jtot = |J𝜙 | + Jr + JZ. The inner (Rapo < 11 kpc) and the outer (Rapo > 15 kpc) groups are squares and circles, +respectively. Green and magenta solid lines in the bottom right panel denotes the regions of Gaia-Sausage/Enceladus (Belokurov et al. 2018; Helmi et al. 2018) +and Sequoia (Barbá et al. 2019; Myeong et al. 2019), respectively. Grey small dots in the background of all panels are VMP stars studied in Sestito et al. (2020), +in which the orbital parameters have been inferred with the same potential as this work. +4 OSIRIS SAMPLE RESULTS +The derived properties for our 11 OSIRIS stars are summarised in +Tables 3 and 4. In this section, we will use these parameters to study +the Galactic orbital properties of our sample, to study the carbon- +enhanced metal-poor stars in our sample, and to make a comparison +with a recent LAMOST catalogue that includes VMP stars. +4.1 Orbital properties +Here we discuss the orbital parameters for our EMP OSIRIS sample. +We adopted the results for the most probable distance solution (see +Section 3.2), except for LP1 for which the most probable solution +leads to an unbound orbit – we therefore prefer the less probable +distance solution for this star. The five panels in Figure 4 display +the main orbital parameters typically used to classify the kinematic +properties of stars. The three panels on the left-hand side show the +pericentric distance, the eccentricity, and the maximum height from +the plane as a function of the apocentric distance. The right-hand +two panels display the energy vs. the rotational component of the +action (top) and the action space (bottom). The sample appears to +split into two broad populations in the Zmax vs. apocenter and the +E vs J𝜙 panels – one that inhabits the inner region of the Milky +Way (Rapo ≲ 10 kpc) and one that reaches the outer part Milky Way +halo (Rapo ≳ 15 kpc). We mark these with black squares and circles, +respectively. +The first group is composed of four stars with apocentric distances +MNRAS 000, 1–13 (2023) + +X104 +0 +-2 +s +9- +E +-8 +Retrograde +Prograde +-10 +-3000 -2000 -1000 +0 +1000 2000 3000 +J (kpc km s-1) +GSE Gaia-Sequoia +Polar +1 +0.5 +Retrograde +tot +Prograde +0 +N +-0.5 +-1 +Radial +-1 +-0.5 +0 +0.5 +1 +tot13 +11 +9 +5 +3 +0.8 +Icity +0.6 +0.2 +30 +20 +15 +7 +5 +8 +9.10 +12 +15 +20 +2530 +40 +50 +60 +Apocentre (kpc)10 +Arentsen et al. +of ∼ 7−10 kpc. Three of them (LP3, LP4, LP5) have pericentres that +bring them into the spatial region of the Milky Way bulge (Rperi < 3 +kpc). The remaining one, LP8, has a higher pericenter (Rperi ∼ 4.5 +kpc) and is among the lowest eccentricity stars in the sample (𝜖 ∼ 0.3) +– its Zmax < 3.0 kpc and positive angular momentum indicate the +star is moving in a prograde orbit relatively close to the plane of the +Milky Way. All stars in this group are prograde, with the exception +of LP4, which has a very high eccentric orbit (𝜖 ∼ 0.7), and almost +no rotation (𝐽𝜙/𝐽tot ∼ 0). These extremely metal-poor inner halo +stars may be connected to very first Milky Way halo building blocks, +the ancient Galactic disk and/or the chaotic (but slightly rotating) +pre-disk Milky Way. +The second group is composed of the remaining seven stars with +orbits compatible with outer halo stars. Three of them, LP1, LP9 +and LP10, have pericentric distances in the range 2.0 < Rperi < 5.5 +kpc, the other four, LP2, LP6, LP7 and LP11, have larger pericentric +distances. From the action space of Figure 4, it is evident that none +of our targets is clearly kinematically associated with GSE (green +box) or Sequoia (magenta box). One of the stars, LP1 (sitting near +the centre of the action diamond), could still have belonged to the +GSE progenitor since it has high eccentricity (∼ 0.75) and is not +far out of the GSE box. Previous works have associated some stars +in this region with GSE (e.g. Yuan et al. 2020) or shown that in +simulations there are GSE stars on a variety of orbits larger than +the typical selection boxes (e.g. Naidu et al. 2021; Amarante et al. +2022). A possible association of LP11 (the most prograde star in +the outer halo group) can be made with the Helmi stream (Helmi +et al. 1999), as it is sits in a similar region of the action diamond and +the E-𝐽𝜙 space (see e.g. Yuan et al. 2020) and has strong vertical +motion (𝐽𝑧 = 1084 kpc km s−1), consistent with the very polar orbit +of the Helmi stream. Association with other halo-substructures (such +as the dynamically tagged groups of VMP stars by Yuan et al. 2020 +and others) is difficult due to the relatively large uncertainties on +the orbital parameters for most stars. The majority of our stars were +likely brought into the Milky Way in smaller accretion events. +High-resolution spectroscopic observations would be needed to +determine the detailed chemo-dynamical properties of the stars in +this work. They would provide better RVs to derive more precise +orbital parameters and more importantly detailed chemical abun- +dances, from different nucleosynthetic production channels, which +are needed to better characterise the formation sites and origins of +the stars in our sample. +4.2 CEMP stars +Following the Aoki et al. (2007) definition of CEMP stars ([C/Fe] > ++0.7), three of our stars can be classified as carbon-enhanced: LP1, +LP6 and LP11. For two other objects (LP4 and LP9, with 𝑇eff ∼ +6000 K but no clear features within the G band), we were able to +provide an informative upper limit of [C/Fe] < +0.7, making these +carbon-normal stars. The other six targets (LP2, LP3, LP5, LP7, LP8, +and LP10) are relatively warm (𝑇eff > 6100 K) and the absence of +CH absorption features only allow us to provide upper limits that are +larger than [C/Fe] = +0.7, according to the sensitivity criteria from +Aguado et al. (2019). We do not derive the fraction of CEMP stars +in our sample, since the preselection was strongly biased. +Since we do not have estimates of any s-process element abun- +dances for our sample9, we cannot constrain the types of CEMP stars +in our sample using that method. However, CEMP-s and CEMP-no +9 There are two relatively strong lines of Sr and Ba in our wavelength cover- +−5 +−4 +−3 +−2 +[Fe/H] +5 +6 +7 +8 +9 +A(C) +Y16 CEMP-s +Y16 CEMP-no +Figure 5. [Fe/H] versus A(C) (corrected for evolutionary effects) for the stars +in our sample (large yellow symbols, and grey symbols for upper limits) and +the CEMP stars in the Yoon et al. (2016) compilation (small symbols colour- +coded by CEMP type). The uncertainties on A(C) are the quadratic sum of +the adopted uncertainties on [Fe/H] and [C/Fe]. The black and grey dashed +lines indicate the limits of [C/Fe] = +0.7 and +1.0, respectively. +stars also have different distributions in their metallicities and car- +bon abundances (e.g. Spite et al. 2013; Bonifacio et al. 2015; Yoon +et al. 2016). We can use this to make a preliminary classification +of CEMP stars. Figure 5 presents the [Fe/H] − A(C)10 diagram of +the stars in our sample, together with a compilation of CEMP stars +from Yoon et al. (2016). The two most carbon-rich CEMP stars in +our sample (LP6 and LP11) are on the border between the CEMP-no +and CEMP-s regions. The third (LP1) lies in the CEMP-no region of +the diagram, as well as the other stars with [C/Fe] upper limits. +All three CEMP stars have large apocentres (> 20 kpc), and the +two most carbon-rich CEMP stars also have the highest pericentres +in our sample (> 8 kpc). As discussed above, these are indications +that they likely came into the Milky Way in a relatively small dwarf +galaxy. Previous work has suggested that the fraction of CEMP-no +compared to CEMP-s stars is larger in the outer halo than in the +inner halo (Yoon et al. 2018; Lee et al. 2019), as well as in smaller +halo building blocks (Yoon et al. 2019; Zepeda et al. 2022). This is +additional indirect evidence that the two most carbon-rich stars in +our sample are more likely to be CEMP-no. +If LP6 and LP11 are CEMP-s stars, they are among the lowest +metallicity CEMP-s stars known. If they are CEMP-no stars, they are +among the highest-A(C) CEMP-no stars known. There are not that +many literature stars in this region, so it would be interesting to do +further higher resolution follow-up of these two stars to investigate +their nature. +4.3 LAMOST DR8 VaC comparison +A new analysis of the LAMOST DR8 spectra was published in a +value-added-catalogue (VaC) by Wang et al. (2022), employing neu- +ral networks to derive stellar parameters (𝑇eff, log 𝑔 and [Fe/H]). +age, but the combination of resolution, S/N and extremely low metallicities +of the stars do not permit their detection. +10 𝐴(C) = log 𝜖 (𝐶) = log(𝑁𝐶/𝑁𝐻)+12, with A(C)⊙ = 8.39 from Asplund +et al. (2005) +MNRAS 000, 1–13 (2023) + +00 +中 +00 +8 +0 ++ +11 +000 +88 +00GTC follow-up of Pristine/LAMOST +11 +−4.0 +−3.5 +−3.0 +−2.5 +−2.0 +[Fe/H] OSIRIS +−4 +−3 +−2 +−1 +0 +[Fe/H] LAMOST VaC DR8 (W22) +DR8 VMP +DR8 PASTEL +Figure 6. Comparison between our derived metallicities from the OSIRIS +spectra and those from the Wang et al. (2022) LAMOST DR8 value-added- +catalogue. The points are colour-coded by the version of the neural network +applied to the DR8 data, and the cool CEMP star in our sample (LP6) has been +highlighted with a large circle. The bisector is indicated with a grey-dashed +line. +They train one of the neural networks on stars of all metallicities in +the PASTEL catalogue (Soubiran et al. 2010), and another network +only on metal-poor stars ([Fe/H] < −1.5) to improve their [Fe/H] +estimates for VMP stars. They claim that the metallicities in their +VMP catalogue are reliable down to [Fe/H] ∼ −3.5. +Ten out of our eleven OSIRIS stars have stellar parameters in the +DR8 VaC (the only star absent is our most metal-rich star, LP8, +with [Fe/H]FERRE = −2.9). We present the comparison between the +DR8 VaC metallicities and the metallicities derived in this work +in Figure 6. The very carbon-enhanced cool star LP6 has extreme +metallicities in both the PASTEL and VMP catalogues, which is not +unexpected since the spectrum is dominated by carbon features and +this is not taken into account in the Wang et al. (2022) analysis. +Focusing on the [Fe/H]VMP estimates, the other stars are all found +to have systematically higher metallicities compared to our analysis, +mostly between −3.0 < [Fe/H]W22 < −2.3. Since we are using +spectra of much higher SNR and we are employing a dedicated +analysis method for extremely metal-poor (and/or carbon-enhanced) +stars, we conclude that some caution should be taken with the Wang +et al. (2022) VMP catalogues for [Fe/H]W22 < −2.5. We further note +that more EMP stars may be hidden in large catalogues, especially +among stars with low S/N spectra. +5 SUMMARY +In this work, we employed the combination of metallicity-sensitive +photometry from the Pristine survey (Starkenburg et al. 2017b) and +the large low-resolution spectroscopic LAMOST database to identify +promising ultra metal-poor and/or carbon-enhanced extremely metal- +poor candidates. We analysed ∼ 7500 LAMOST spectra for targets +with [Fe/H]Pristine < −2.5 and 𝑔 < 18, finding success rates of +stars with [Fe/H]spec < −2.5 between 34%−50%, depending on the +applied quality cuts. We inspected all the fits with [Fe/H]spec < −3.0 +to identify candidates for follow-up, and we release this full list +together with figures of the best fits (see Section 2.3). +We observed eleven of the most exciting candidates (mostly with +low LAMOST S/N) using OSIRIS at the GTC. We analysed the +higher S/N medium-resolution OSIRIS spectra (𝑅 ∼ 2400) using +the FERRE code to derive 𝑇eff, [Fe/H] and [C/Fe], adopting log 𝑔 +from photometry. The metallicities for the eleven stars range from +[Fe/H] = −2.9 ± 0.1 to −3.8 ± 0.2, with a mean [Fe/H] = −3.4. +We set out to identify UMP stars, but none of the targets had +[Fe/H] < −4.0 – such stars are indeed incredibly rare. Our selection +of (carbon-enhanced) extremely metal-poor stars, however, was still +very efficient. +For three out of the eleven stars we were able to derive carbon +abundances, for the others we derived upper limits – two of which +are constraining and classify the stars as carbon-normal. Given their +[Fe/H], A(C) and orbital properties, all three CEMP stars are likely +part of the CEMP-no category, although the two most carbon-rich ob- +jects lie in an underpopulated region, where there are both CEMP-no +and CEMP-s stars in the literature. Further follow-up is necessary to +understand the physical processes causing the carbon-enhancement +in these stars. +We derive orbital properties using the OSIRIS radial velocities, +Gaia proper motions and distances based on photometry and paral- +laxes from Gaia combined with MIST isochrones, integrating orbits +in the MW-Potential14 with a more massive halo. We find that four +of the stars have inner halo kinematics, with three of them on pro- +grade orbits. The other seven stars have orbits more consistent with +the outer halo. None of the stars in our sample are confidently asso- +ciated with previously known substructures/accretion events, partly +due to uncertainties on the orbital parameters. +Ongoing and upcoming spectroscopic surveys are so large that +it is crucial to have general automatic analyses of the spectra, but +doing this well for extremely metal-poor stars is a challenge. They +are only a small subset, hence pipelines are often not optimised +for them, and their spectra are challenging to analyse due to weak +spectral features and/or peculiar chemical abundances. It will remain +important to do dedicated metal-poor analyses in the future. Adding +additional information like metallicity-sensitive photometry as in +this work could uncover hidden promising candidates at the lowest +metallicities. +ACKNOWLEDGEMENTS +We thank the reviewer for their valuable comments, which helped im- +prove the paper. The authors thank Carlos Allende Prieto, Carmela +Lardo and Lyudmila Mashonkina, as well as the rest of the Pris- +tine collaboration, for their support of this paper and/or their useful +comments. +AA, NFM and ZY gratefully acknowledge support from the Eu- +ropean Research Council (ERC) under the European Unions Hori- +zon 2020 research and innovation programme (grant agreement No. +834148). AA acknowledges support from the Herchel Smith Fel- +lowship at the University of Cambridge and the Fitzwilliam College +Isaac Newton Trust Research Fellowship. DA acknowledges support +from the European Research Council (ERC) Starting Grant NEFER- +TITI H2020/808240. FS thanks the Dr. Margaret "Marmie" Perkins +Hess postdoctoral fellowship for funding his work at the University +of Victoria. JIGH acknowledges financial support from the Spanish +Ministry of Science and Innovation (MICINN) project PID2020- +117493GB-I00 and also from the Spanish MICINN under 2013 +Ramón y Cajal program RYC-2013-14875. NFM and ZY acknowl- +edge support from the French National Research Agency (ANR) +funded project “Pristine” (ANR-18-CE31-0017), NFM also acknowl- +MNRAS 000, 1–13 (2023) + +12 +Arentsen et al. +edges funding from CNRS/INSU through the Programme National +Galaxies et Cosmologie and through the CNRS grant PICS07708. +ES acknowledges funding through VIDI grant “Pushing Galactic Ar- +chaeology to its limits” (with project number VI.Vidi.193.093) which +is funded by the Dutch Research Council (NWO). PJ acknowlegdes +support from the Swiss National Foundation. +Based on observations made with the Gran Telescopio Canarias +(GTC), installed at the SpanishObservatoriodelRoquedelos Mucha- +chos of the Instituto de Astrofísica de Canarias, on the island of La +Palma. +Based on observations obtained with MegaPrime/MegaCam, a +joint project of CFHT and CEA/DAPNIA, at the Canada-France- +Hawaii Telescope (CFHT) which is operated by the National Re- +search Council (NRC) of Canada, the Institut National des Sciences +de l’Univers of the Centre National de la Recherche Scientifique of +France, and the University of Hawaii. +Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber +Spectroscopic Telescope LAMOST) is a National Major Scientific +Project built by the Chinese Academy of Sciences. Funding for the +project has been provided by the National Development and Reform +Commission. LAMOST is operated and managed by the National +Astronomical Observatories, Chinese Academy of Sciences. +Funding for the Sloan Digital Sky Survey IV has been provided by +the Alfred P. Sloan Foundation, the U.S. Department of Energy Office +of Science, and the Participating Institutions. SDSS-IV acknowledges +support and resources from the Center for High Performance Com- +puting at the University of Utah. The SDSS website is www.sdss.org. +SDSS-IV is managed by the Astrophysical Research Consortium for +the Participating Institutions of the SDSS Collaboration including the +Brazilian Participation Group, the Carnegie Institution for Science, +Carnegie Mellon University, Center for Astrophysics | Harvard & +Smithsonian, the Chilean Participation Group, the French Participa- +tion Group, Instituto de Astrofísica de Canarias, The Johns Hopkins +University, Kavli Institute for the Physics and Mathematics of the Uni- +verse (IPMU) / University of Tokyo, the Korean Participation Group, +Lawrence Berkeley National Laboratory, Leibniz Institut für Astro- +physik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA +Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), +Max-Planck-Institut für Extraterrestrische Physik (MPE), National +Astronomical Observatories of China, New Mexico State Univer- +sity, New York University, University of Notre Dame, Observatário +Nacional / MCTI, The Ohio State University, Pennsylvania State +University, Shanghai Astronomical Observatory, United Kingdom +Participation Group, Universidad Nacional Autónoma de México, +University of Arizona, University of Colorado Boulder, University +of Oxford, University of Portsmouth, University of Utah, Univer- +sity of Virginia, University of Washington, University of Wisconsin, +Vanderbilt University, and Yale University. +This work has made use of data from the European Space +Agency (ESA) mission Gaia (https://www.cosmos.esa.int/ +gaia), processed by the Gaia Data Processing and Analysis Consor- +tium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/ +consortium). Funding for the DPAC has been provided by national +institutions, in particular the institutions participating in the Gaia +Multilateral Agreement. +DATA AVAILABILITY +The LAMOST spectra used in this work are public. Our EMP candi- +date list is available in Table 1, and all relevant data for the OSIRIS +stars is available in Tables 2 − 4. These tables will also be available +at the CDS. The OSIRIS spectra will be shared on reasonable request +to the authors. +REFERENCES +Abohalima A., Frebel A., 2018, ApJS, 238, 36 +Aguado D. S., Allende Prieto C., González Hernández J. I., Carrera R., Rebolo +R., Shetrone M., Lambert D. L., Fernández-Alvar E., 2016, A&A, 593, +A10 +Aguado D. S., Allende Prieto C., González Hernández J. I., Rebolo R., Caffau +E., 2017a, A&A, 604, A9 +Aguado D. S., González Hernández J. I., Allende Prieto C., Rebolo R., 2017b, +A&A, 605, A40 +Aguado D. S., González Hernández J. I., Allende Prieto C., Rebolo R., 2018, +ApJ, 852, L20 +Aguado D. 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G., et al., 2000, AJ, 120, 1579 +Youakim K., et al., 2017, MNRAS, 472, 2963 +Yuan Z., et al., 2020, ApJ, 891, 39 +Zepeda J., et al., 2022, arXiv e-prints, p. arXiv:2209.12224 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–13 (2023) + diff --git a/VtE0T4oBgHgl3EQfVgCD/content/tmp_files/load_file.txt b/VtE0T4oBgHgl3EQfVgCD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..68de55a5e0ca240695dd5ac04b2cdc76e9252a10 --- /dev/null +++ b/VtE0T4oBgHgl3EQfVgCD/content/tmp_files/load_file.txt @@ -0,0 +1,1798 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf,len=1797 +page_content='MNRAS 000, 1–13 (2023) Preprint 9 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 The Pristine survey – XX: GTC follow-up observations of extremely metal-poor stars identified from Pristine and LAMOST Anke Arentsen,1,2 ★ David S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Aguado,3,4 Federico Sestito,5 Jonay I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' González Hernández,6,7 Nicolas F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Martin,2,8 Else Starkenburg,9 Pascale Jablonka10,11 and Zhen Yuan2 1 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK 2 Université de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg, France 3 Dipartimento di Fisica e Astronomia, Universitá degli Studi di Firenze, Via G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Sansone 1, I-50019 Sesto Fiorentino, Italy 4 INAF-Osservatorio Astrofisico di Arcetri, Largo E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Fermi 5, I-50125 Firenze, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 5 Department of Physics and Astronomy, University of Victoria, PO Box 3055, STN CSC, Victoria BC V8W 3P6, Canada 6 Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain 7 Universidad de La Laguna, Dpto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Astrofísica, E-38206 La Laguna, Tenerife, Spain 8 Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany 9 Kapteyn Astronomical Institute, University of Groningen, Postbus 800, 9700 AV, Groningen, the Netherlands 10 Laboratoire d’astrophysique, École Polytechnique Fédérale de Lausanne (EPFL), Observatoire, CH-1290 Versoix, Switzerland 11 GEPI, Observatoire de Paris, Université PSL, CNRS, Place Jules Janssen, F-92195 Meudon, France Accepted 2023 January 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Received 2023 January 05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' in original form 2022 November 03 ABSTRACT Ultra metal-poor stars ([Fe/H] < −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0) are very rare, and finding them is a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Both narrow-band photometry and low-resolution spectroscopy have been useful tools for identifying candidates, and in this work we combine both approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We cross-matched metallicity-sensitive photometry from the Pristine survey with the low-resolution spectroscopic LAMOST database, and re-analysed all LAMOST spectra with [Fe/H]Pristine < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We find that ∼1/3rd of this sample (selected without [Fe/H]Pristine quality cuts) also have spectroscopic [Fe/H] < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' From this sample, containing many low signal-to-noise (S/N) spectra, we selected eleven stars potentially having [Fe/H] < −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 or [Fe/H] < −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 with very high carbon abundances, and we performed higher S/N medium-resolution spectroscopic follow-up with OSIRIS on the 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4m Gran Telescopio Canarias (GTC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We confirm their extremely low metallicities, with a mean of [Fe/H] = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4 and the most metal-poor star having [Fe/H] = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Three of these are clearly carbon-enhanced metal-poor (CEMP) stars with +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='65 < [C/Fe] < +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The two most carbon-rich stars are either among the most metal-poor CEMP-s stars or the most carbon-rich CEMP-no stars known, the third is likely a CEMP-no star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We derived orbital properties for the OSIRIS sample and find that only one of our targets can be confidently associated with known substructures/accretion events, and that three out of four inner halo stars have prograde orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Large spectroscopic surveys may contain many hidden extremely and ultra metal-poor stars, and adding additional information from e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' photometry as in this work can uncover them more efficiently and confidently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Key words: stars: Population II – Galaxy: halo – stars: chemically peculiar – techniques: spectroscopic 1 INTRODUCTION The most metal-poor stars still present in the Milky Way today are valuable portals to the early Universe and the pristine environments these stars were born in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' They are thought to have formed from material enriched by the first generation(s) of stars, and their chemical abundances can be used to constrain the properties of the stars that came before them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Additionally, the dynamical properties of the most metal-poor stars teach us about the early formation of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Much can be, and has been, learned from very/extremely/ultra metal- poor stars with [Fe/H] < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 (VMP)/−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 (EMP)/−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 (UMP) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Beers & Christlieb 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Frebel & Norris 2015), although they are exceedingly rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' ★ Email: anke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='arentsen@ast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='uk The metal-poor halo has been found to be a melting pot of many accreted structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' It is populated by the remnants of the larger mergers that the Galaxy experienced across its history, such as Gaia- Sausage/Enceladus (GSE, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=', Belokurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Helmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2018), Sequoia (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=', Barbá et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Myeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2019), Tham- nos (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=', Koppelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2019), and Sagittarius (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=', Ibata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The plethora of recently discovered stellar streams are in- dicative of part of the later accretion events from dwarf/ultra faint galaxies and globular clusters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=', Ibata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2022a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Additionally, as much as half of the stars in the halo appears to be born in-situ, likely consisting of both an 𝛼-rich splashed disk component (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=', Bonaca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Haywood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Di Matteo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Gallart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Belokurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2020) and stars that formed in a hot and disordered pre-disk state (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=', Belokurov & Kravtsov 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Conroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='02265v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='GA] 5 Jan 2023 2 Arentsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The common picture from various cosmological simulations sug- gests that the VMP stars that inhabit the spatial inner region of the Milky Way, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=', the bulge and the disk, are amongst the oldest stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=', Starkenburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' El-Badry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' These stars are therefore great tracers of the early Galactic assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' On the observational point of view, many VMP stars have been observed with such kinematics, focusing on the bulge (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=', Howes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2014, 2015, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Arentsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Lucey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2023) and the disk (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=', Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Di Matteo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Carter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Cordoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The chemical properties of these populations indicate that the building blocks of the inner Galaxy consisted of a variety of objects – some stars appear to have formed in systems very similar to ultra faint dwarf galaxies, while others are consistent with being born in globu- lar cluster-like systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=', Schiavon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2023, and references therein), and finally there may also be a significant contribution of in-situ VMP stars in the inner Galaxy (Belokurov & Kravtsov 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Rix et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Many low-metallicity stars have been found to be carbon-enhanced metal-poor (CEMP) stars, with frequencies of the order of 30 − 50% among stars with [Fe/H] < −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 (Beers & Christlieb 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Yong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Placco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' There are two main types of CEMP stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' CEMP-s stars are thought to have become carbon-rich later in their life due to mass-transfer from a (former) asymptotic giant branch (AGB) star companion – these are typically in binary systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2016b), are enhanced in s-process elements as well as carbon (a signature of AGB star nucleosynthesis), and are more frequent for [Fe/H] > −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The CEMP-no stars are hypothesised to have been born from carbon-enhanced gas in the early Universe – they do not have s-process over-abundances, are less frequently found to be in binary systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2016a, although still more than expected, see Arentsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2019), and mostly occur at [Fe/H] < −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The exact frequencies of CEMP-no and CEMP-s stars as function of metallicity is still under debate (Arentsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2022), and may also vary with Galactic environment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' inner vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' outer halo, bulge, dwarf galaxies, globular clusters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' To build large samples of extremely metal-poor stars, many ded- icated searches have happened in the past 40 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Several differ- ent techniques have been used to identify metal-poor stars, such as following up high-proper motion stars with ultraviolet excesses (Ryan & Norris 1991), identifying objects with small Ca II H & K lines in large objective-prism surveys (Beers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Christlieb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2008), or using metallicity-sensitive (narrow-band) photometry (Schlaufman & Casey 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Starkenburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2017b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Da Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Galarza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Placco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Very and extremely metal-poor stars have also been identified in greater numbers in large scale spectroscopic surveys such as the Sloan Digital Sky Survey (SDSS, York et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2000), the Large sky Area Multi-Object fiber Spectroscopic Telescope (LAMOST1, Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2012), RAdial Ve- locity Experiment (RAVE, Steinmetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2006) and the GALactic Archaeology with HERMES spectroscopic survey (GALAH, Buder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2021), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2013), Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2018), Matijevič et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2017) and Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' These are often paired with dedicated follow-up efforts (Caffau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Allende Prieto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Bonifacio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Aguado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Placco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Da Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' In this work, we combine the strengths of metallicity-sensitive pho- tometry and large spectroscopic surveys by cross-matching metal- poor candidates from the photometric Pristine survey (Starkenburg 1 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='lamost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='org/public/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='locale=en et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2017b) with the large database of spectra from LAMOST, with the goal of identifying new extremely or even ultra metal-poor stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The Pristine survey uses metallicity-sensitive narrow-band CaHK photometry to derive photometric metallicities of millions of stars towards the Galactic halo, which is very efficient even for extremely metal-poor stars (Youakim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Aguado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' How- ever, the selection still suffers from some more metal-rich contami- nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' In this work we alleviate this by adding an extra step, namely by cross-matching candidates with [Fe/H]Pristine < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 with the LAMOST spectroscopic database, and doing a dedicated analysis of all these (often low signal-to-noise) spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We select exciting candidates from this analysis, and follow them up using the OSIRIS spectrograph at the 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4m Gran Telescopio Canarias (GTC) (Cepa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2000) to obtain higher S/N observations, from which we can de- rive high-quality metallicities to confirm their extremely metal-poor nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We describe our initial candidate selection from Pristine and LAMOST in Section 2, including some discussion about the suc- cess rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The OSIRIS observations for 11 stars and the derivation of their radial velocities, stellar parameters, distances and orbits is described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We present results for the OSIRIS sample in Section 4, discussing the presence of carbon-enhanced metal-poor (CEMP) stars, the orbital properties for the sample and a compar- ison with a new value-added LAMOST catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We conclude in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2 SELECTION OF EMP CANDIDATES USING PRISTINE AND LAMOST The LAMOST archive contains low-resolution spectra (R∼1800) for millions of stars, but not all spectra have stellar parameters in the standard LAMOST catalogue tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We discovered that many of the most metal-poor stars ([Fe/H] < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5) are missed by their standard pipeline (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2014), and also by the dedicated very metal-poor pipeline of Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' This is particularly severe for hotter stars and stars with lower signal-to-noise (S/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Other dedicated analyses might be able to deal better with these spectra, and identify promising extremely metal-poor stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' At the time our selection was made (February 2021), the latest LAMOST release was DR6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' To avoid having to analyse the full data release, which contains almost 10 million spectra, we made a pre-selection of promising extremely metal-poor candidates us- ing photometric metallicities from the Pristine survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We used the internal Pristine data release containing all CaHK observations un- til Semester 2020A, and adopted the CaHK + SDSS photometric metallicities (Starkenburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We queried the LAMOST archive for all stars in the Pristine survey with photometric metal- licities [Fe/H]Pristine < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 (from using either 𝑔 − 𝑖 or 𝑔 − 𝑟) and 𝑔sdss < 18, and found ∼ 7500 cross-matches for ∼ 6000 unique tar- gets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' No other quality cuts were applied, which usually are included when we do dedicated target selection for Pristine follow-up imme- diately from the photometry (Youakim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2017), to be as inclusive as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='1 Preliminary ULySS analysis A first-pass analysis of these candidates was done with the ULySS2 code (Koleva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' ULySS is a full-spectrum fitting package 2 ULySS is available from http://ulyss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='univ-lyon1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='fr/ MNRAS 000, 1–13 (2023) GTC follow-up of Pristine/LAMOST 3 that employs empirical spectral libraries to determine stellar param- eters (𝑇eff, log 𝑔, [Fe/H], radial velocities, spectral broadening), and can be applied to stars of a wide range of stellar parameters and metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We employed this code because we were interested in the types of contamination in the Pristine selection, which one cannot study with the dedicated metal-poor analysis described in the next sub-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' For the models, we adopted the empirical MILES library (Sánchez- Blázquez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Falcón-Barroso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2011) and used the ULySS MILES polynomial interpolator originally built by Prugniel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2011) and updated for cool stars by Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The library has a resolving power of 𝑅 ∼ 2200, and the interpolator extends down to [Fe/H] = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='8 (with the possibility to extrapolate, at ones own risk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The LAMOST spectra were fitted between 3750 and 5500 Å using a multiplicative Legendre polynomial of degree 15 for the normalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' This degree is large enough to absorb some of the large mismatches between models and observations for carbon-enhanced metal-poor stars in regions of carbon-related molecular bands, which is necessary since the ULySS models do not include [C/Fe] as a free parameter, and large carbon features could mess up the normalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' There is also an automatic masking routine in ULySS, which excludes outlier pixels iteratively and typically masks the wavelength regions of the largest carbon features in CEMP stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The resulting Kiel-diagram and metallicity histogram from our ULySS analysis are shown in Figure 1, for all exposures of the ∼ 4900 unique stars that remain after removing fits with signal-to-residual ratios < 8, broadening > 400 km s−1 (which usually indicates a very bad fit), and duplicate LAMOST spectra for the same star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The metal- poor stars show a clear red giant branch (RGB) and main-sequence turn-off sequence, except for a small cloud of stars to the left of the RGB, which mostly consists of stars in the low S/N-tail of the sample without good fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Most of the stars in our selection are indeed very metal-poor (keep- ing only the fit with the highest signal-to-residual ratio per star): 71% have [Fe/H]ULySS < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 and 25% have [Fe/H]ULySS < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The latter goes up to 34% when using the FERRE metallicities described later in this section, which perform better in this regime than the ULySS metallicities, because the MILES library does not have many stars in this [Fe/H] range (especially for the turn-off region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' There is a contamination of metal-rich stars with [Fe/H]ULySS > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 of 16%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' ULySS is also the main software used by the LAMOST team for the parameters in their public data releases (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' They use the interpolator based on the ELODIE library (Prugniel & Soubiran 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2011), which has a more limited coverage of the parameter space compared to MILES, and extends only down to [Fe/H] = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Of the stars that have [Fe/H]ULySS < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0/−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 in our analysis, only 30%/17% have stellar parameters in the public LAMOST DR7 catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' This is likely partly due to the ELODIE library being less good at low metallicities, and partly due to more stringent quality cuts being applied for stars to make it into the LAMOST data releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='2 Success rates In our original selection we did not make any additional photomet- ric quality cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The Pristine team developed several quality cuts to remove metal-rich outliers and improve the success rates of the spec- troscopic follow-up of extremely metal-poor candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The cuts applied for the main Pristine follow-up campaign are discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='1 of Youakim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We apply very similar cuts to the 4000 6000 9000 Teff (ULySS) 0 1 2 3 4 5 log g (ULySS) −6 −4 −2 0 [Fe/H] 0 1000 Nr of stars ULySS FERRE VMP −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 [Fe/H] (ULySS) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Top: Kiel diagram for all exposures of the 4900 unique Pristine- selected stars in LAMOST analysed with ULySS, colour-coded by metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' No quality cuts were applied to the photometric metallicities in the selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The results for the eleven stars that were followed up with OSIRIS (see Section 3) are highlighted with larger symbols (the two high and low log 𝑔 outliers are CEMP stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Bottom: ULySS metallicity histogram of the same sample in black, and FERRE metallicity histogram for the VMP sub-sample in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Pristine +LAMOST sample to see how that changes the metallicity distribution, keeping only the stars that have: CASU flag = −1 or 1 young stars flag = 0 (𝑢0 − 𝑔0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='25 < (𝑔0 − 𝑖0) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='15 < (𝑔0 − 𝑟0) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='2 [Fe/H]Pristine < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 (from using either SDSS 𝑔 − 𝑖 or 𝑔 − 𝑟) and ≠ −99 (−99 is assigned if the star falls outside of the parameter space for which the photometric metallicity assignment has a valid calibration) instead of the PanSTARRS variability catalogue as in Youakim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2017), we use the Gaia photometric variability to remove variable stars as in Fernández-Alvar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2021) The uncertainties on [Fe/H]Pristine are not taken into account here, whereas they were in Youakim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' After applying the above cuts, the sample goes from 4900 stars to 4100 stars again keeping the highest signal-to-residual spectrum per star).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Of these, 78% have [Fe/H]ULySS < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0, and 28% have [Fe/H]ULySS < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 (the latter goes up to 38% for the FERRE metallicities), compared to the previous 71% and 25% (and 34% for FERRE), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The metal-rich contamination goes down to 12%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Doing the same only for stars with signal-to-residual ratios > 20 instead of our initial cut at > 8, the results are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We conclude that, for the Pristine +LAMOST sample, the photometric quality cuts slightly improve the selection efficiency, but not by a lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The success rate of previous Pristine follow-up for [Fe/H]Pristine < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 was found to be 56% (Aguado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The lower fraction in this work (38% when applying the photo- metric quality cuts and adopting the FERRE metallicities) could be due to various reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' For example, the dedicated Pristine follow- MNRAS 000, 1–13 (2023) 4 Arentsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' List of 481 EMP candidates (533 spectra) with FERRE spectroscopic parameters used for target selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' No quality cuts have been applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The first few lines of the table are shown here for guidance, the full table and figures showing the best fits are available as online supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' LAMOST spectrum name Gaia DR3 source_id ra dec 𝑇eff log 𝑔 [Fe/H] [C/Fe] S/N log(𝜒2) [deg] [deg] [K] cgs spec-56746-HD121251N314746M01_sp02-087 4014278062082321152 181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='215811 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='331285 6232 ± 325 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='7 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4 ± 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4 6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='338 spec-57308-EG000023N024031M01_sp03-076 2739551594198812160 358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='800705 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='493245 6521 ± 161 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='1 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='9 24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='214 spec-57754-HD122624N271605M02_sp03-108 4009964020835772288 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='239126 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='687022 5675 ± 57 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='3 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='3 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='123 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' up presented in Youakim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2017) and Aguado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2019) focused on the most metal-poor stars, and did not homogeneously observe all stars with [Fe/H]Pristine < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Additionally, extra qual- ity cuts were sometimes implemented for subsets of the follow-up presented in Aguado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2019), most notably the consistency of the photometric temperatures derived from SDSS between (𝑔 − 𝑖) and (𝑔 − 𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' If we select only stars with [Fe/H]Pristine < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='7 and |𝑇eff (𝑔−𝑖) − 𝑇eff (𝑔−𝑟) | < 200 K, the success rate in this work for [Fe/H]FERRE < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 goes up to 50%, and the contamination of stars with [Fe/H]ULySS > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 is reduced from 12% to 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The numbers in this section are not meant to override the previ- ously published Pristine success rates by Youakim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2017) and Aguado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Our results confirm that the success rates are high, and highlight some of the subtleties in deriving such success rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Overall we conclude that our methodology to find hidden very and extremely metal-poor stars in the large LAMOST database is extremely efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='3 Dedicated VMP FERRE analysis A dedicated very metal-poor analysis was performed for the sub- sample of LAMOST spectra with [Fe/H]ULySS < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0, with the aim of deriving better metallicities and carbon abundances for the most metal-poor stars and identifying potential ultra metal-poor can- didates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We followed a similar methodology as in Aguado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2017a,b), using the FERRE3 code (Allende Prieto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The code interpolates between the nodes of a library of synthetic spectra and derives simultaneously the set of best stellar parameters (𝑇eff, log 𝑔, [Fe/H], [C/Fe]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' For this preliminary analysis, we used the default Nelder-Mead search algorithm and linear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The dedicated very metal-poor synthetic models were computed with the ASSET code (Koesterke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2008) and published in Aguado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2017b) with the following parameter coverage: 4750 K < 𝑇eff < 7000 K, Δ𝑇eff = 250 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 < log 𝑔 < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0, Δ log 𝑔 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 < [Fe/H] < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0, Δ[Fe/H] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 < [C/Fe] < +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0, Δ[C/Fe] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 and a fixed [𝛼/Fe] = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4 and [N/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Both the data and the models were continuum normalised with a running mean filter with a 30 pixel window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We limited the fit to the wavelength range 3700 − 5500 Å, where most of the features for extremely metal-poor stars are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The spectra were shifted to rest-wavelength using the ULySS radial velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The resulting metallicity distribution is shown in red in the bottom panel of Figure 1, without any additional quality cuts applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The hard limit at [Fe/H]FERRE = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 is due to the limit of the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 3 FERRE is available from http://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='com/callendeprieto/ferre The ULySS and FERRE distributions peak at roughly the same metal- licities, but the FERRE distribution has a larger tail towards lower metallicities – as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We inspected the > 500 fits in the resulting FERRE-analysed sam- ple with [Fe/H]FERRE < −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 by eye, and identified a number of (previously unknown) stars of interest that could potentially have [Fe/H] < −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 or that looked very carbon-rich and extremely metal- poor ([Fe/H] < −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Practically none of our candidates had pa- rameters in the public DR6 catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Most of our candidates had relatively low S/N, so follow-up spectroscopy was necessary to con- firm their extremely or even ultra metal-poor nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The full list of EMP candidates that we inspected is given in Ta- ble 1, with figures for all the spectral fits provided in the online supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4 This candidate list with its derived pa- rameters should not be used blindly since no quality cuts have been applied (on e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' S/N, log(𝜒2) or parameter uncertainties), but it could be used in combination with the figures to select other EMP stars for follow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Stars may occur multiple times in this list if they have more than one LAMOST spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 3 OSIRIS FOLLOW-UP OF EMP CANDIDATES We obtained GTC/OSIRIS observations for 11 of our most promising candidates (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='9 < 𝑔 < 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='9) in Semester 2021A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We used OSIRIS in longslit mode with the 2500U grating, a 1 arcsec slit and 2x2 binning, resulting in spectra covering 3440 − 4610 Å at a resolving power 𝑅 ∼ 2400 (providing an instrument profile with a FWHM of ∼ 125 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We aimed for a S/N of 40 at 4000 Å, corresponding to exposure times of 3000s for stars of magnitude 𝑔 ∼ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' A summary of the observations is presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Individual exposures of 1400, 1600 and 1800s were executed for different targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='1 Radial velocities Radial velocities (RVs) are derived using the cross-correlation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We have a high-quality GTC/OSIRIS spectrum of a bright extremely metal-poor star G64-12 (𝑇eff = 6463K, log 𝑔 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='26, [Fe/H] = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='29, [C/Fe] = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='07, Placco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2016 and ref- erences therein) from previous campaigns acquired with the same setup (Aguado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2017a, 2018), that we use as a cross-correlation template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The OSIRIS spectra of both our targets and the template star are normalized with the same method, using a running mean filter with a width of 30 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We built the cross-correlation func- tion (CCF) with our own IDL-based automated code in the spectral 4 The table and figures of FERRE fits for EMP candidates can temporar- ily be found at the following URL: https://drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='com/drive/ folders/1qLypkgyG6m-XbwPGpmNrBnXFQf70-0se?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='usp=sharing, and they will be part of the online material with the published paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) GTC follow-up of Pristine/LAMOST 5 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Summary of the OSIRIS observations: our reference for each star, the LAMOST and Gaia DR3 designations, positions, SDSS magnitudes in 𝑢 and 𝑔, total exposure time, signal to noise at two different wavelengths, number of observations and date that the spectra were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Star # LAMOST Gaia DR3 source_id ra dec u_mag g_mag total exp S/N (@392/ Nobs date observed designation [mag] [mag] time [s] 450 nm) (DD-MM-2021) LP1 J002953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='07+320229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='9 2862648994739368704 00:29:53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='07 32:02:29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='9 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='49 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='58 2800 33/79 2 14-07 LP2 J131532.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='41+121107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4 3736550805114696192 13:15:32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='41 12:11:07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='94 1400 74/180 1 15-06 LP4 J142055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='86+075308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='7 3673778479398720000 14:20:55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='87 07:53:08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='7 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='96 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='04 1400 62/100 1 15-06 LP5 J144714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='22+163425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4 1186662458446883328 14:47:14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='22 16:34:25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='73 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='92 2800 27/88 2 10-06 LP6 J145214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='98+160357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='6 1186397549159479424 14:52:14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='99 16:03:57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='7 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='25 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='88 1400 12/97 1 10-06 LP7 J145611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='30+161925.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='7 1187873604865113472 14:56:11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='31 16:19:25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='77 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='82 4600 34/116 3 08-04 LP8 J161021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='42+451247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 1386190837835580288 16:10:21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='43 45:12:47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='50 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='63 5600 30/79 4 10-06/17-07 LP9 J162359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='32+303740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='8 1318369490300720000 16:23:59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='32 30:37:40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='9 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='46 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='52 5600 24/78 4 11-04/14-07 LP10 J212109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='07+151328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='7 1783524305407324672 21:21:09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='06 15:13:29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='15 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='15 4800 38/145 3 15-06/13-07/14-07 LP11 J230209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='39+302100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='6 1886140596052059392 23:02:09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='39 30:21:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='7 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='01 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='09 3600 39/192 2 23-05/13-07 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Radial velocities, heliocentric distance, probability of the distance within 3𝜎 around the maximum of the distance PDF, maximum height from the plane, apocentric and pericentric distances, eccentricity, energy, the action vector are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' [updated] Star # RV D P𝐷 𝑍max Rapo Rperi 𝜖 E/104 J𝜙 Jr JZ [km s−1] [kpc] [kpc] [kpc] [kpc] [kpc km2 s−2] [kpc km s−1] [kpc km s−1] [kpc km s−1] LP1 −18 ± 15 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='14 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='6+4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='75+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='03 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='79+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='86 −140+365 −417 829+177 −142 780+215 −329 LP2 211 ± 15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='66 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5+12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='6 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='7 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='3+31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='52+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='19 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='76+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='17 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='63 −1821+405 −434 17+1 −1 905+306 −153 LP3 −162 ± 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} 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112+25 −23 LP9 −254 ± 15 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='99 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='8+50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 −19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='8 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='09 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='61+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='56 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='59 −726+196 −115 269+10 −10 903+943 −528 LP10 −361 ± 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='93 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='3+10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4 −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='3 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4+115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='2 −28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='83+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='70+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='49 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='44 −1185+234 −182 191+10 −10 957+769 −500 LP11 −68 ± 15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='99 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='6 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='9 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='7+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='21+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='09 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='89+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='78 1467+141 −116 58+255 −57 1084+465 −336 range 3755-4455 Å with a window of 3000 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The main fea- tures in the template are the CaII H&K lines, the HI lines of the Balmer series, and the G-band in carbon-enhanced stars (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The normalization method produces a shape of the CCF profile that mimics the shape of all balmer lines in the warm template EMP star, which does not resemble a gaussian shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We thus fit the CCF profile with a parabolic fit using the closest 6 points to the CCF peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The statistical uncertainty of the centroid of the parabolic fit is typically below 1 km s−1, significantly below the pixel size of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='57 Å/pixel (∼ 42 km s−1/pixel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The results of the OSIRIS spectra show intra- night RV variations with standard deviations below ∼ 7 km s−1, but RV variations from different nights with standard deviations in the range 3 − 20 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We also derive the RV for the same stars from their LAMOST spec- tra (which typically have much lower S/N than the OSIRIS spectra), using the same technique to check the consistency with our OSIRIS RVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The LAMOST spectrum of G64-12 is used as cross-correlation template and all spectra are normalized using a running mean filter with a width of 15 pixels of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='38 Å/pixel (∼ 81 km s−1/pixel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The CCF is built from the spectral range 3755-6755 Å, which includes H𝛼 and H𝛽, providing more stability to the CCF profile given the lower quality LAMOST spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We find a reasonable consistency when comparing to the OSIRIS results, with a mean difference of −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4 km s−1 and a standard deviation of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='9 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' For each target we adopt the weighted mean of the OSIRIS RVs derived from each individual spectrum and the corresponding error of the mean as the final RV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We apply an quadratically added uncer- tainty floor of 15 km s−1 to the RV uncertainties, which seems more realistic than the CCF uncertainties given the RV variations within and between different nights and the differences with the LAMOST RVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' This floor reflects the systematic RV uncertainty due to possible instrument flexures, pointing, guiding RV drifts, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='2 Distances It has been widely demonstrated that simply inverting the parallax to infer the distance can lead to wrong results, and including additional priors and/or data improves distance estimates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=', Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2018, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Anders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' This is especially the case when the parallax has poor measurements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=', 𝜛 < 0 and/or 𝜛/𝜎𝜛 < 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We therefore use a Bayesian approach to infer the distances for the stars in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The probability distribution function (PDF), or posterior, is inferred following the method fully described in Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Briefly, the likelihood is the product of the Gaussian distributions for the parallax and photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The prior takes into MNRAS 000, 1–13 (2023) 6 Arentsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Table with adopted stellar parameters, with 𝑇eff, [Fe/H] and [C/Fe] from FERRE and photometric log 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Star # 𝑇eff log 𝑔 [Fe/H] [C/Fe] corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='2 [K] cgs A(C)⊙ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='391 LP1 5790 ± 101 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='22 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='11 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='21 LP2 6419 ± 108 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='30 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='12 < +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='90 LP3 6365 ± 102 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='08 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='11 < +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 LP4 5993 ± 103 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='28 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='11 < +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='70 LP5 6134 ± 102 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='37 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='11 < +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='89 LP6 4575 ± 103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='20 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='11 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='21 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='24 LP7 6413 ± 103 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='39 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='11 < +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 LP8 6363 ± 103 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='19 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='11 < +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='96 LP9 6018 ± 103 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='21 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='12 < +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='70 LP10 6304 ± 102 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='14 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='11 < +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='02 LP11 5730 ± 102 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='24 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='13 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='22 1Solar abundance adopted from Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2005) 2Evolutionary [C/Fe] correction following Placco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2014) account a power-law stellar distribution (see the halo prior in Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2019), and, through a set of VMP ([M/H]= −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5) MESA/MIST isochrones5 (Dotter 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2016), the knowledge that VMP stars are old (11 − 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='8 Gyr), low-mass (< 1 M⊙), and distributed with a given IMF-based luminosity function in the CMD diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The zero-point offset has been applied to the Gaia EDR3 parallaxes (Lindegren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2021) using the python gaiadr3_zeropoint6 pack- age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' This method, widely used for chemo-dynamical investigations of VMP stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=', Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Venn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2020), produces low uncertainties on the distances even in case of large par- allax uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' This is because the isochrones limit the possible distances for a star with a given colour to two different solutions, a dwarf and a giant solution, and nothing in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The parallax would then typically prefer one of the two, or, in case of a very poor parallax measurement, the two peaks would be given a different prob- ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We calculate the probabilities following Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' For seven of the OSIRIS stars the probability of the main peak is larger than 92 per cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' For two stars it is 86 per cent (LP1, although for this star we adopt the less probable solution, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='1) and 87 per cent (LP5), while for the remaining two stars it is 54 per cent (LP8) and 66 per cent (LP2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='3 Orbital parameters The orbital parameters are inferred using Galpy7 (Bovy 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The code requires as input the inferred distances, the RVs, and the proper motions and coordinates from Gaia (E)DR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The total fixed gravitational potential that we adopt is the sum of a Navarro- Frenk-White dark matter halo (Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 1997, NFWPoten- tial), a Miyamoto-Nagai potential disc (Miyamoto & Nagai 1975, MiyamotoNagaiPotential) and an exponentially cut-off bulge (PowerSphericalPotentialwCutoff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' All of the aforementioned potentials are usually invoked by the MWPotential14 package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' However, we adopt a more massive and up-to-date halo (Bland- Hawthorn & Gerhard 2016), with a mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='2 × 1012 M⊙ (vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='8 × 1012 M⊙ for MWPotential14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' For each star, we perform a Monte Carlo simulation with 1000 5 https://waps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='cfa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='edu/MIST/ 6 https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='com/icc-ub/public/gaiadr3_zeropoint 7 http://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='com/jobovy/galpy random draws on the input parameters to infer the orbital parameters and their uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' In case of the proper motion components, we consider their correlation given the coefficients from Gaia EDR3, drawing randomly with a multivariate Gaussian function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The RVs (from the OSIRIS spectra) and coordinates are treated as a Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' In order to account for possible systematics on the distances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' due to the adopted isochrones and other assumptions), we assume a 15 per cent uncertainty on the distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The integration time is set to 1 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The orbital parameters are inferred for both of the peaks in the distance PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The output orbital parameters are the Galactocentric Cartesian coordinates (X, Y, Z), the maximum distance from the Milky Way plane Zmax, the apocentric and pericentric distances (Rapo, Rperi), the eccentricity 𝜖, the energy E, and the spherical actions coordinates (J𝜙, Jr, JZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Table 3 reports the main orbital parameters from the most probable distance, except for star LP1 where we adopt the less probable distance (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The orbital parameters are discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4 Stellar parameters The OSIRIS data were analysed with FERRE in a similar manner as the LAMOST spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' For this analysis we use the more sophisticated Boender-Timmer-Rinnoy Kan (BTRK, Boender et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 1982) global search algorithm and Bézier cubic interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We use the same grid, except for the coolest star in the sample, for which we employ a similar grid that has been extended down to 4500 K (as used e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' in Arentsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Again we used a 30 pixel window for the running mean normalisation, suitable for OSIRIS resolution (𝑅 = 𝜆/𝛿𝜆 ∼ 2400).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' To avoid problems in the noisy blue region we only analyse the spectra in the range (3750 − 4500 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We found that for the warm stars in the sample (with𝑇eff > 5500 K, which is all stars except for LP6), the log 𝑔 values that FERRE finds are typically at the edges of the FERRE grid, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' at log 𝑔 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 or log 𝑔 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0, see the black points in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' This is likely the re- sult of not much log 𝑔 information being present in these extremely metal-poor stars in the available wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Previous work on metal-poor stars with FERRE has shown that systematically off- set log 𝑔 values strongly impact the derived [C/Fe] (Aguado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Arentsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Therefore we decided to adopt photo- metric log 𝑔 values for the warm stars, shown by the magenta points in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' These were inferred from the Stefan-Boltzmann equa- MNRAS 000, 1–13 (2023) GTC follow-up of Pristine/LAMOST 7 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' OSIRIS/GTC spectra (3750 Å-4500 Å) of our stellar sample (black line) and the best fits calculated with FERRE, colour-coded by 𝑇eff (the bluer the hotter) and sorted by decreasing [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The Balmer lines (yellow) and main metallic absorption features (purple) are high-lighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Above each spectrum the metallicity, effective temperature and carbonicity are displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) 8 Arentsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 5000 6000 Teff 1 2 3 4 5 log g [Fe/H] = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 [Fe/H] = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 FERRE adopted Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Kiel diagram showing the pure FERRE stellar parameters (black) and the adopted stellar parameters and uncertainties (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' See the text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Also shown are Yonsei-Yale isochrones for two different metallicities (both with age = 12 Gyr, [𝛼/Fe] = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' tion, which needs as input the dereddened absolute G magnitude (derived using the Gaia G-band magnitude, the 3D extinction map from Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2019 and the distances from Table 3), an estimate of the effective temperature, and the bolometric corrections on the flux (Andrae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We adopt the FERRE effective temperature and its inflated uncertainty (see last paragraph of this subsection) in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We perform a Monte Carlo iteration with 1000 ran- dom draws on the input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Each of them is described by a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We run FERRE again for the warm stars, fixing the 𝑇eff to the previously derived FERRE value and log 𝑔 to the photometric values values, while letting [Fe/H] and [C/Fe] free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The final spectral fits are shown in Figure 2 and a summary of the results is provided in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The differences between the original FERRE run and the run with fixed 𝑇eff and log 𝑔 are small for the metallicities, with the adopted [Fe/H] being higher by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='07 dex with a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='06 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The differences for [C/Fe] are also small for the stars with original log 𝑔 > 4 and measured [C/Fe] (see next section), they are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='05 on average, with a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='09 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' However, for the one star with measured carbon and FERRE log 𝑔 < 3 (LP11), the new [C/Fe] is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='7 dex lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' There are three stars (LP4, LP7 and LP9) that have very high FERRE internal [Fe/H] uncertainties of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 dex when calcu- lated by inverting the covariance matrix (our original approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' This could be attributed to some negative/zero fluxes in blue end of the OSIRIS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' To avoid this issue we recalculated the internal FERRE uncertainties using a Monte Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We performed 50 ex- periments and use the dispersion on the derived [Fe/H] and [C/Fe] as the uncertainty following Aguado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' As a result of that the issue with the large uncertainties was fixed for the three problematic stars, and the uncertainties for the other stars remain the same within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='01−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='02 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We adopt the Monte Carlo internal uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' To provide the final uncertainties for the stellar parameters, we add estimates of the external uncertainties from a previous analysis of EMP stars with FERRE (Aguado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2017a) to our internal FERRE uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' These are 100 K, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='2 dex for 𝑇eff, [Fe/H] and [C/Fe], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' For [Fe/H] and [C/Fe] we adopt the internal uncertainties from the first FERRE run, because the second run does not properly reflect the real uncertainties since it fits only two of the four parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' For log 𝑔, we adopted the uncertainties from the photometric determination for the warm stars, and for the coolest star we quadratically added 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='2 dex of external uncertainties (Aguado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2017a) to the internal FERRE uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The results are shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 Carbon determination Deriving carbon abundance from low-resolution data of EMP stars is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Our employed grid is suitable for the analysis of CEMP stars, since carbon-enhancement was not only considered in the spec- tral synthesis step but also in the ATLAS stellar models (Sbordone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' This is crucial because high carbon abundances can sig- nificantly impact the stellar atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The grid of models has been used successfully to derive carbon abundances in several works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Aguado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2017b,a, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Arentsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2021, 2022), although there are some differences with other synthetic grids that can lead to systematic differences in derived carbon abundances (Arentsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' This is likely related to the use of different codes, line lists and assumptions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' different [N/Fe] abundances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The ability of the FERRE code to detect – and successfully fit – carbon absorption features from low-resolution data strongly depends on a)𝑇eff (and log 𝑔 to a lesser extent), b) the carbon abundance, and c) the SNR of the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' In our sample there are three stars (LP1, LP6, and LP11) that fulfil the sensitivity criteria derived by Aguado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2019) based on these parameters, all of them have 𝑇eff < 6000 K and show strong CH absorption features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' For these objects we derived [C/Fe] = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='65/+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='21/+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='17 respectively, with reasonable uncertainties (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='2 dex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' For the other stars we can only provide upper limits on the carbon abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The carbon results are summarised in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The object with the lowest 𝑇eff in our sample, LP6, shows clear CN features at ∼ 3885 Å that our best fit is not able to reproduce, although the CH & G-band fit is good (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2, red spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The reason for this is that our FERRE synthetic spectral library assumes [N/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 for all stellar models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Querying the high- resolution spectroscopy compilation in the JINAbase (Abohalima & Frebel 2018) for stars with −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 < [Fe/H] < −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0, we find that all of those with measured nitrogen abundances have [N/Fe] > 0, and stars with [C/Fe] > +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 typically have 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 < [N/Fe] < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' This is very different from the assumed [N/Fe] in the FERRE grid, and can explain why the CN band for LP6 is much stronger in the data than in the model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' However, the fit reproduces quite well the Ca ii at 3933 Å and several other Fe i, Ti ii, and Sr ii lines in the 4040−4080 Å region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Additionally, the majority of the carbon information is significantly concentrated around the G−band (4200-4330 Å) and our fit is good in that area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Therefore, we conclude that the CN absorption features in the blue are not significantly affecting the best fit for this object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The carbon abundance of evolved giants decreases with decreasing log 𝑔 due to mixing processes, especially in metal-poor stars (Gratton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Placco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We estimate the evolutionary carbon correction for the most evolved star in our sample (LP6, the only star that should be affected by this effect) using the web calculator8 by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Placco, and find it to be +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='24 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 8 https://vplacco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='pythonanywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='com/ MNRAS 000, 1–13 (2023) GTC follow-up of Pristine/LAMOST 9 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Orbital parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Three left panels: pericenter, eccentricity, and maximum distance from the Milky Way plane as a function of the apocentric distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The grey-shaded areas denote the forbidden region in which the Zmax > Rapo or Rperi > Rapo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Upper right panel: Energy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' rotational component of the action, J𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Bottom right panel: Action space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' the y-axis is the difference between the vertical and radial component of the action, while the x-axis is the rotational component;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' axes are normalised by Jtot = |J𝜙 | + Jr + JZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The inner (Rapo < 11 kpc) and the outer (Rapo > 15 kpc) groups are squares and circles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Green and magenta solid lines in the bottom right panel denotes the regions of Gaia-Sausage/Enceladus (Belokurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Helmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2018) and Sequoia (Barbá et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Myeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2019), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Grey small dots in the background of all panels are VMP stars studied in Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2020), in which the orbital parameters have been inferred with the same potential as this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 4 OSIRIS SAMPLE RESULTS The derived properties for our 11 OSIRIS stars are summarised in Tables 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' In this section, we will use these parameters to study the Galactic orbital properties of our sample, to study the carbon- enhanced metal-poor stars in our sample, and to make a comparison with a recent LAMOST catalogue that includes VMP stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='1 Orbital properties Here we discuss the orbital parameters for our EMP OSIRIS sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We adopted the results for the most probable distance solution (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='2), except for LP1 for which the most probable solution leads to an unbound orbit – we therefore prefer the less probable distance solution for this star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The five panels in Figure 4 display the main orbital parameters typically used to classify the kinematic properties of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The three panels on the left-hand side show the pericentric distance, the eccentricity, and the maximum height from the plane as a function of the apocentric distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The right-hand two panels display the energy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' the rotational component of the action (top) and the action space (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The sample appears to split into two broad populations in the Zmax vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' apocenter and the E vs J𝜙 panels – one that inhabits the inner region of the Milky Way (Rapo ≲ 10 kpc) and one that reaches the outer part Milky Way halo (Rapo ≳ 15 kpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We mark these with black squares and circles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The first group is composed of four stars with apocentric distances MNRAS 000, 1–13 (2023) X104 0 2 s 9- E 8 Retrograde Prograde 10 3000 -2000 -1000 0 1000 2000 3000 J (kpc km s-1) GSE Gaia-Sequoia Polar 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 Retrograde tot Prograde 0 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 1 Radial 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 1 tot13 11 9 5 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='8 Icity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='2 30 20 15 7 5 8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='10 12 15 20 2530 40 50 60 Apocentre (kpc)10 Arentsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' of ∼ 7−10 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Three of them (LP3, LP4, LP5) have pericentres that bring them into the spatial region of the Milky Way bulge (Rperi < 3 kpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The remaining one, LP8, has a higher pericenter (Rperi ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 kpc) and is among the lowest eccentricity stars in the sample (𝜖 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='3) – its Zmax < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 kpc and positive angular momentum indicate the star is moving in a prograde orbit relatively close to the plane of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' All stars in this group are prograde, with the exception of LP4, which has a very high eccentric orbit (𝜖 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='7), and almost no rotation (𝐽𝜙/𝐽tot ∼ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' These extremely metal-poor inner halo stars may be connected to very first Milky Way halo building blocks, the ancient Galactic disk and/or the chaotic (but slightly rotating) pre-disk Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The second group is composed of the remaining seven stars with orbits compatible with outer halo stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Three of them, LP1, LP9 and LP10, have pericentric distances in the range 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 < Rperi < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 kpc, the other four, LP2, LP6, LP7 and LP11, have larger pericentric distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' From the action space of Figure 4, it is evident that none of our targets is clearly kinematically associated with GSE (green box) or Sequoia (magenta box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' One of the stars, LP1 (sitting near the centre of the action diamond), could still have belonged to the GSE progenitor since it has high eccentricity (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='75) and is not far out of the GSE box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Previous works have associated some stars in this region with GSE (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2020) or shown that in simulations there are GSE stars on a variety of orbits larger than the typical selection boxes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Amarante et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' A possible association of LP11 (the most prograde star in the outer halo group) can be made with the Helmi stream (Helmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 1999), as it is sits in a similar region of the action diamond and the E-𝐽𝜙 space (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2020) and has strong vertical motion (𝐽𝑧 = 1084 kpc km s−1), consistent with the very polar orbit of the Helmi stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Association with other halo-substructures (such as the dynamically tagged groups of VMP stars by Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2020 and others) is difficult due to the relatively large uncertainties on the orbital parameters for most stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The majority of our stars were likely brought into the Milky Way in smaller accretion events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' High-resolution spectroscopic observations would be needed to determine the detailed chemo-dynamical properties of the stars in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' They would provide better RVs to derive more precise orbital parameters and more importantly detailed chemical abun- dances, from different nucleosynthetic production channels, which are needed to better characterise the formation sites and origins of the stars in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='2 CEMP stars Following the Aoki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2007) definition of CEMP stars ([C/Fe] > +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='7), three of our stars can be classified as carbon-enhanced: LP1, LP6 and LP11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' For two other objects (LP4 and LP9, with 𝑇eff ∼ 6000 K but no clear features within the G band), we were able to provide an informative upper limit of [C/Fe] < +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='7, making these carbon-normal stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The other six targets (LP2, LP3, LP5, LP7, LP8, and LP10) are relatively warm (𝑇eff > 6100 K) and the absence of CH absorption features only allow us to provide upper limits that are larger than [C/Fe] = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='7, according to the sensitivity criteria from Aguado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We do not derive the fraction of CEMP stars in our sample, since the preselection was strongly biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Since we do not have estimates of any s-process element abun- dances for our sample9, we cannot constrain the types of CEMP stars in our sample using that method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' However, CEMP-s and CEMP-no 9 There are two relatively strong lines of Sr and Ba in our wavelength cover- −5 −4 −3 −2 [Fe/H] 5 6 7 8 9 A(C) Y16 CEMP-s Y16 CEMP-no Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' [Fe/H] versus A(C) (corrected for evolutionary effects) for the stars in our sample (large yellow symbols, and grey symbols for upper limits) and the CEMP stars in the Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2016) compilation (small symbols colour- coded by CEMP type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The uncertainties on A(C) are the quadratic sum of the adopted uncertainties on [Fe/H] and [C/Fe].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The black and grey dashed lines indicate the limits of [C/Fe] = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='7 and +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' stars also have different distributions in their metallicities and car- bon abundances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Spite et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Bonifacio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We can use this to make a preliminary classification of CEMP stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Figure 5 presents the [Fe/H] − A(C)10 diagram of the stars in our sample, together with a compilation of CEMP stars from Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The two most carbon-rich CEMP stars in our sample (LP6 and LP11) are on the border between the CEMP-no and CEMP-s regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The third (LP1) lies in the CEMP-no region of the diagram, as well as the other stars with [C/Fe] upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' All three CEMP stars have large apocentres (> 20 kpc), and the two most carbon-rich CEMP stars also have the highest pericentres in our sample (> 8 kpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' As discussed above, these are indications that they likely came into the Milky Way in a relatively small dwarf galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Previous work has suggested that the fraction of CEMP-no compared to CEMP-s stars is larger in the outer halo than in the inner halo (Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2019), as well as in smaller halo building blocks (Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Zepeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' This is additional indirect evidence that the two most carbon-rich stars in our sample are more likely to be CEMP-no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' If LP6 and LP11 are CEMP-s stars, they are among the lowest metallicity CEMP-s stars known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' If they are CEMP-no stars, they are among the highest-A(C) CEMP-no stars known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' There are not that many literature stars in this region, so it would be interesting to do further higher resolution follow-up of these two stars to investigate their nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='3 LAMOST DR8 VaC comparison A new analysis of the LAMOST DR8 spectra was published in a value-added-catalogue (VaC) by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2022), employing neu- ral networks to derive stellar parameters (𝑇eff, log 𝑔 and [Fe/H]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' age, but the combination of resolution, S/N and extremely low metallicities of the stars do not permit their detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 10 𝐴(C) = log 𝜖 (𝐶) = log(𝑁𝐶/𝑁𝐻)+12, with A(C)⊙ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='39 from Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2005) MNRAS 000, 1–13 (2023) 00 中 00 8 0 + 11 000 88 00GTC follow-up of Pristine/LAMOST 11 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 [Fe/H] OSIRIS −4 −3 −2 −1 0 [Fe/H] LAMOST VaC DR8 (W22) DR8 VMP DR8 PASTEL Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Comparison between our derived metallicities from the OSIRIS spectra and those from the Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2022) LAMOST DR8 value-added- catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The points are colour-coded by the version of the neural network applied to the DR8 data, and the cool CEMP star in our sample (LP6) has been highlighted with a large circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The bisector is indicated with a grey-dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' They train one of the neural networks on stars of all metallicities in the PASTEL catalogue (Soubiran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2010), and another network only on metal-poor stars ([Fe/H] < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5) to improve their [Fe/H] estimates for VMP stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' They claim that the metallicities in their VMP catalogue are reliable down to [Fe/H] ∼ −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Ten out of our eleven OSIRIS stars have stellar parameters in the DR8 VaC (the only star absent is our most metal-rich star, LP8, with [Fe/H]FERRE = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We present the comparison between the DR8 VaC metallicities and the metallicities derived in this work in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The very carbon-enhanced cool star LP6 has extreme metallicities in both the PASTEL and VMP catalogues, which is not unexpected since the spectrum is dominated by carbon features and this is not taken into account in the Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2022) analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Focusing on the [Fe/H]VMP estimates, the other stars are all found to have systematically higher metallicities compared to our analysis, mostly between −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 < [Fe/H]W22 < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Since we are using spectra of much higher SNR and we are employing a dedicated analysis method for extremely metal-poor (and/or carbon-enhanced) stars, we conclude that some caution should be taken with the Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' (2022) VMP catalogues for [Fe/H]W22 < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We further note that more EMP stars may be hidden in large catalogues, especially among stars with low S/N spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 5 SUMMARY In this work, we employed the combination of metallicity-sensitive photometry from the Pristine survey (Starkenburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 2017b) and the large low-resolution spectroscopic LAMOST database to identify promising ultra metal-poor and/or carbon-enhanced extremely metal- poor candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We analysed ∼ 7500 LAMOST spectra for targets with [Fe/H]Pristine < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 and 𝑔 < 18, finding success rates of stars with [Fe/H]spec < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='5 between 34%−50%, depending on the applied quality cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We inspected all the fits with [Fe/H]spec < −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 to identify candidates for follow-up, and we release this full list together with figures of the best fits (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We observed eleven of the most exciting candidates (mostly with low LAMOST S/N) using OSIRIS at the GTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We analysed the higher S/N medium-resolution OSIRIS spectra (𝑅 ∼ 2400) using the FERRE code to derive 𝑇eff, [Fe/H] and [C/Fe], adopting log 𝑔 from photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The metallicities for the eleven stars range from [Fe/H] = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='1 to −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='2, with a mean [Fe/H] = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We set out to identify UMP stars, but none of the targets had [Fe/H] < −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='0 – such stars are indeed incredibly rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Our selection of (carbon-enhanced) extremely metal-poor stars, however, was still very efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' For three out of the eleven stars we were able to derive carbon abundances, for the others we derived upper limits – two of which are constraining and classify the stars as carbon-normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Given their [Fe/H], A(C) and orbital properties, all three CEMP stars are likely part of the CEMP-no category, although the two most carbon-rich ob- jects lie in an underpopulated region, where there are both CEMP-no and CEMP-s stars in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Further follow-up is necessary to understand the physical processes causing the carbon-enhancement in these stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We derive orbital properties using the OSIRIS radial velocities, Gaia proper motions and distances based on photometry and paral- laxes from Gaia combined with MIST isochrones, integrating orbits in the MW-Potential14 with a more massive halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' We find that four of the stars have inner halo kinematics, with three of them on pro- grade orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The other seven stars have orbits more consistent with the outer halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' None of the stars in our sample are confidently asso- ciated with previously known substructures/accretion events, partly due to uncertainties on the orbital parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Ongoing and upcoming spectroscopic surveys are so large that it is crucial to have general automatic analyses of the spectra, but doing this well for extremely metal-poor stars is a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' They are only a small subset, hence pipelines are often not optimised for them, and their spectra are challenging to analyse due to weak spectral features and/or peculiar chemical abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' It will remain important to do dedicated metal-poor analyses in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Adding additional information like metallicity-sensitive photometry as in this work could uncover hidden promising candidates at the lowest metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank the reviewer for their valuable comments, which helped im- prove the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The authors thank Carlos Allende Prieto, Carmela Lardo and Lyudmila Mashonkina, as well as the rest of the Pris- tine collaboration, for their support of this paper and/or their useful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' AA, NFM and ZY gratefully acknowledge support from the Eu- ropean Research Council (ERC) under the European Unions Hori- zon 2020 research and innovation programme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' 834148).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' AA acknowledges support from the Herchel Smith Fel- lowship at the University of Cambridge and the Fitzwilliam College Isaac Newton Trust Research Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' DA acknowledges support from the European Research Council (ERC) Starting Grant NEFER- TITI H2020/808240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' FS thanks the Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Margaret "Marmie" Perkins Hess postdoctoral fellowship for funding his work at the University of Victoria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' JIGH acknowledges financial support from the Spanish Ministry of Science and Innovation (MICINN) project PID2020- 117493GB-I00 and also from the Spanish MICINN under 2013 Ramón y Cajal program RYC-2013-14875.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' NFM and ZY acknowl- edge support from the French National Research Agency (ANR) funded project “Pristine” (ANR-18-CE31-0017), NFM also acknowl- MNRAS 000, 1–13 (2023) 12 Arentsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' edges funding from CNRS/INSU through the Programme National Galaxies et Cosmologie and through the CNRS grant PICS07708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' ES acknowledges funding through VIDI grant “Pushing Galactic Ar- chaeology to its limits” (with project number VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='Vidi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='093) which is funded by the Dutch Research Council (NWO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' PJ acknowlegdes support from the Swiss National Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Based on observations made with the Gran Telescopio Canarias (GTC), installed at the SpanishObservatoriodelRoquedelos Mucha- chos of the Instituto de Astrofísica de Canarias, on the island of La Palma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France- Hawaii Telescope (CFHT) which is operated by the National Re- search Council (NRC) of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Funding for the project has been provided by the National Development and Reform Commission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Sloan Foundation, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Department of Energy Office of Science, and the Participating Institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' SDSS-IV acknowledges support and resources from the Center for High Performance Com- puting at the University of Utah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The SDSS website is www.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Center for Astrophysics | Harvard & Smithsonian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' the Chilean Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' the French Participa- tion Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Instituto de Astrofísica de Canarias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The Johns Hopkins University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Kavli Institute for the Physics and Mathematics of the Uni- verse (IPMU) / University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' the Korean Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Leibniz Institut für Astro- physik Potsdam (AIP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Max-Planck-Institut für Astronomie (MPIA Heidelberg),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Max-Planck-Institut für Astrophysik (MPA Garching),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Max-Planck-Institut für Extraterrestrische Physik (MPE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' National Astronomical Observatories of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' New Mexico State Univer- sity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' New York University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' University of Notre Dame,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Observatário Nacional / MCTI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Shanghai Astronomical Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' United Kingdom Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Universidad Nacional Autónoma de México,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' University of Arizona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' University of Colorado Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' University of Portsmouth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' University of Utah,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Univer- sity of Virginia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' University of Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' University of Wisconsin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Vanderbilt University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' and Yale University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='int/ gaia), processed by the Gaia Data Processing and Analysis Consor- tium (DPAC, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content='int/web/gaia/dpac/ consortium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' DATA AVAILABILITY The LAMOST spectra used in this work are public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' Our EMP candi- date list is available in Table 1, and all relevant data for the OSIRIS stars is available in Tables 2 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' These tables will also be available at the CDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' The OSIRIS spectra will be shared on reasonable request to the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' REFERENCES Abohalima A.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQfVgCD/content/2301.02265v1.pdf'} diff --git a/XtE4T4oBgHgl3EQfNgyO/content/tmp_files/2301.04957v1.pdf.txt b/XtE4T4oBgHgl3EQfNgyO/content/tmp_files/2301.04957v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..881e0151eef20eae1bf6c49713782a21aaa4907f --- /dev/null +++ b/XtE4T4oBgHgl3EQfNgyO/content/tmp_files/2301.04957v1.pdf.txt @@ -0,0 +1,2900 @@ +Fusion of two stable elastic structures resulting in an +unstable system +MARCO ROSSI1, ANDREA PICCOLROAZ1, AND DAVIDE BIGONI*1 +1DICAM, University of Trento, via Mesiano 77, Trento, Italy +Abstract +It is shown that a compound elastic structure, which displays a dynamic instability, may be de- +signed as the union (or ‘fusion’) of two structures which are stable when separately analyzed. The +compound elastic structure has two degrees of freedom and is made up of a rigid rod connected with +two springs to a smooth support, which evidences a jump in the curvature at the equilibrium configura- +tion. Instability is proven in a linearized context and is related to the application of a non-conservative +load of the follower type, so that the instability disappears under dead loads. In the fully nonlinear +range, the instability is also confirmed through numerical simulations. The obtained results may be +useful in the design of new mechanical sensors, or devices for energy harvesting, or architected mate- +rials. In addition, our findings have conceptual implications on piecewise-linear theories of mechanics +such as for instance plasticity or frictional contact. +Keywords: Flutter instability; Tensile instability +1 +Introduction +Consider two elastic structures loaded within their stable range and imagine that these represent two +parts of a third elastic structure obtained through the ‘fusion’ of the initial structures, becoming the +‘constituents’ of the new structure. Stability of the compound structure is expected when the latter is +subject to the same load at which the two component structures are stable. This intuitive belief is true +for dead loading and smooth systems, but it is shown in the present article to be false for follower loads +and non-smooth mechanical behaviour. From a purely mathematical point of view, the singularity of +a non-smooth behaviour obtained as combination of two stable dynamical systems may be expected, +as was advocated through a purely mathematical example by Branicky [1] and Carmona et al. [2] for +abstract systems of non-smooth differential equations. However, an elastic structure exhibiting this +peculiar kind of instability has never been discovered so far. The purpose of the present article is to +fill this gap through the invention and the theoretical and numerical analysis of a structure designed to +demonstrate instability of a simple two d.o.f. non-smooth mechanical system that is composed of two +stable smooth subsystems. This finding brings a mathematical result into the realm of mechanics. +The elastic structures considered here are of the type shown in Fig. 1, consisting of a rigid rod with an +end sliding on a smooth profile, while the other end is subject to a tangentially follower load, remaining +parallel to the bar. The deformed configuration of the structure is defined by two degrees of freedom, +specifically the arc length distance characterizing the roller position along the profile and the angle of +rotation of the bar with respect to the vertical direction. The structure is stiffened by a longitudinal +spring connecting the roller to a fixed point and a rotational spring interposed betweeen the rod and +the roller. +*Corresponding author: e-mail: bigoni@ing.unitn.it; phone: +39 0461 282507. +1 +arXiv:2301.04957v1 [physics.class-ph] 12 Jan 2023 + +0 +20 +40 +60 +80 +0 +1 +2 +τ +ξ +0 +20 +40 +60 +80 +0 +1 +2 +τ +ξ +0 +20 +40 +60 +80 +0 +1 +2 +τ +ξ +Figure 1: Two stable smooth subsystems with positive and negative curvature of a sliding constraint (upper part: +left and center) and the fusion of these two structures, namely, a compound non-smooth structure dis- +playing instability (upper part: right), although the two ‘components’ are stable. The trajectories of the +end of the structures is also reported for vibrational motion, together with the corresponding arc-length +ξ vs time τ behaviours, showing sinusoidal (i.e. stable, lower part: left and center) and unstable (lower +part: right) oscillations. The tensile force acting at the free end of the rods is tangentially follower and the +same for all the three structures, laying well below the critical load for instability in the case of the two +smooth ‘component systems’, both displaying motions confined in a neighborhood of the trivial equilib- +rium configuration. Differently, a flutter-like instability is observed for the composite structure (upper +part: right), as evidenced by the unstable and exponentially growing oscillations of the loaded end. +The first two structures (from left) shown in Fig. 1 are characterized by a circular profile (with posi- +tive and negative curvature, respectively). These structures are described by smooth dynamical systems +and suffer flutter and divergence instabilities. Assuming that the magnitude of the follower force is well +below the critical values, the vertical trivial equilibrium configuration is stable, so that a small perturba- +tion in the initial conditions generates a motion which remains confined within a small neighborhood +of the fixed point, as shown in the lower part of Fig. 1 (obtained through numerical integration of the +nonlinear equations of motion and representing the arc-length distance ξ traveled by the roller as a +function of the elapsed time τ). +The third elastic structure shown on the right in Fig. 1 is also described by two degrees of freedom +and is obtained as the ‘fusion’ of the two structures sketched on the left and center of the same figure. +This new structure is characterized by a smooth sliding profile, which evidences a jump in the curvature +so that the dynamics is characterized by piecewise smooth differential equations. The tangent to the +profile at the junction is horizontal and at this point the longitudinal spring is unloaded, so that the +vertical configuration of the rigid rod is the trivial equilibrium configuration. +If this structure is subject to a load smaller than the critical loads of the two ‘generating structures’, it +might be expected that the structure would be stable. This becomes true when the load is conservative, +but now the load is follower, so that: +it is shown in this article that the non-smooth structure (Fig. 1 right), obtained as the fusion of two +smooth structures (Fig. 1 left and center), may be dynamically unstable at a load well inside the +stability domains of both the generating structures. +Indeed, at a load well below both critical loads of the smooth constituent structures, the compound +2 + +non-smooth structure exhibits the exponentially growing oscillation illustrated in Fig. 1 (lower part, +obtained through numerical integration of the nonlinear equations of motion), resembling the flutter +instability, occurring in smooth systems, for instance the celebrated Ziegler double pendulum [3]. +Note that the structures shown in Fig. 1 are similar to those investigated in [4] and [5], but now the +load is non-conservative and follower, so that the instability landscape results completely changed and, +in particular, the possibility arises of finding instabilities unrelated to the instabilities of the component +structures. +The above-stated result, referred to the structure shown in Fig. 1 on the right, follows from the com- +bination of two features, namely, the presence of (i.) a non-conservative follower load and (ii.) a jump +in curvature in the sliding constraint. The latter feature implies that the acceleration is discontinuous +at the junction between the two circular profiles at the basis of the structure and thus the system of +governing equations becomes non-smooth. +Piecewise-smooth dynamical systems are common in the mechanics of solids and structures. In +fact, elastoplasticity and contact with friction are based on piece-wise linear equations of the rate type; +bi-linear elasticity defines solids with different tensile and compressive elastic moduli; structures im- +pacting against unilateral constraints involve two sets of equations of motion. These systems are known +to exhibit peculiar forms of instability, such as for instance stick and slip motion for frictional contact +[6], or blowing-up vibrations for non-associative elastoplasticity [7, 8]. In particular, every elastoplastic +constitutive equation is always piecewise linear in the rate response and nonassociative flow rules lead +to a lack of symmetry similar in essence to that induced by follower loads in structural elements. There- +fore, the structures designed in the present paper are governed by equations sharing strong similarity +with elastoplasticity, so that our results lead to important conjectures in that field. More in detail, stabil- +ity analysis in elastoplasticity is performed on the so-called ‘comparison solids’ [9], which are the exact +counterpart of the component structures introduced here. Stability of the comparison solids is usually +assumed to imply stability of the true piecewise linear behaviour, but our structural examples demon- +strate that this may be false, an implication that would completely revolutionize the stability theory for +nonassociative elastoplasticity. +Accordingly to its interest in mechanics, instability of piecewise-smooth dynamical systems has re- +cently attracted a growing interest. The first proof that a compound system generated as the fusion +between two stable systems can be unstable is due to Carmona et al. [2]. For non-smooth dynamical +systems, they have provided a sufficient condition for instability, based on the detection of a so-called +‘invariant cone’. This condition has been further developed in various directions [10–13]. All these +works provide a mathematical framework and open new directions for research in bifurcation theory. +However, applications of the mathematical setting to mechanics are scarce and so far limited to simpli- +fied systems characterized by Coulomb friction [14–16]. Therefore, the objective of the present article is +to develop the analysis of invariant cone to demonstrate instability of the structure shown in Fig. 1 on +the right, obtained as the fusion of two stable structures. +Although a general concept, the invariant cone can practically be applied only to the linearized equa- +tions of motion governing a mechanical model, so that structures which are proven to be unstable on the +basis of a linearized analytical treatment are also numerically investigated in this article, to provide a +complete picture of their mechanical behaviour. In this way it is proven that in all cases in which the lin- +earized equations display instability, the latter persists also when the fully nonlinear piecewise-smooth +problem is examined. +It has to be highlighted that the mathematical background so far developed only consists in suffi- +cient conditions for instability, so that when these conditions are not fulfilled, nothing can be concluded +concerning stability. This situation is reflected in the results presented in the present article, where only +examples of piecewise-smooth unstable structures are obtained, while the behaviour of the same struc- +tures at different loads is usually unknown (though open to numerical investigation), as the sufficient +condition fails. +The paper is organized as follows. After the introduction of the class of the addressed elastic struc- +tures (Section 2) , the differential equations governing their dynamics are formulated in Section 3, where +the concept of invariant cone is provided as a sufficient condition for instability. The properties of invari- +ant cones are demonstrated in Section 4, where it is shown, under broad hypotheses, that an unstable +cone is always attractive. An algorithm to detect invariant cones for two d.o.f. mechanical systems is +presented in Section 5 and numerical results on the linearized analysis are developed demonstrating the +instability, which is also finally confirmed through numerical solutions where the nonlinear behaviour +3 + +is fully kept into account. +2 +Elastic structure on a curved constraint with a jump in curvature +2.1 +Nonlinear dynamics +A two d.o.f. elastic structure is considered, Fig. 2, composed of a rigid bar, of mass density ρ and length +l, which is loaded at one end (point L) with a follower force, positive when tensile, of constant modulus +and parallel to the bar. At the other end (point P ), the bar is connected to an elastic hinge of rotational +stiffness k2, which is constrained to move, without friction, along a smooth profile γ. The elastic hinge +at the lower end of the rigid bar is linked to a fixed point S (singled out by the coordinates xs and ys), +with a longitudinal linear spring of stiffness k1. +The rigid smooth profile, along which point P is constrained to move, plays a fundamental role in +the mechanics of the structure shown in Fig. 2. The formulation presented below is general enough to +include profiles with discontinuous curvature, provided that higher-order derivatives are understood +in the generalized sense. +The profile may be described by parametric equations (x(ξ), y(ξ)) in the plane Oxy defined by the +two unit vectors e1 and e2, so that the tangent, the unit tangent, and the principal normal at the generic +point P (ξ) of the profile are +P ′ = x′(ξ)e1 + y′(ξ)e2, +t = P ′ +|P ′|, +n = t′ +|t′|, +(1) +in which a dash ( )′ denotes differentiation with respect to the parameter ξ. The signed curvature of the +profile is defined as +κ = α′ +|P ′|, +where α is the angle between P ′ and the x-axis, +α(ξ) = arctan y′(ξ) +x′(ξ), +(2) +defined in such a way that t and the unit vector obtained through an anticlockwise rotation of t by π/2 +may be represented as +t = cos α(ξ) e1 + sin α(ξ) e2, +m = − sin α(ξ) e1 + cos α(ξ) e2. +The derivative of equation (2) with respect to ξ leads to +α′ = x′y′′ − x′′y′ +x′2 + y′2 += m · t′. +4 + +k1 +k2 +ξ(t) +φ(t) +α(ξ) +α(ξ) +t(ξ) +F +L +L +S +O +P +e2 +e1 +γ +(x(ξ), y(ξ)) +l, ρ +m(ξ) +F +el +eφ +Figure 2: A 2 d.o.f. elastic structure made up of a rigid bar constrained to move with an elastic hinge on a curved +profile and subject to a tensile follower force (remaining parallel to the bar). +Note that the parameter ξ may be identified with the arc length of the profile and in such case +|P ′| = 1 so that α′ coincides with the signed curvature κ. +The deformation of the structure is described by two generalized coordinates: the arc length ξ of the +curve describing the profile and the angle φ between the rigid bar and the y-axis, positive if clockwise, +which are assumed to be continuous functions of time, namely, ξ = ξ(t) and φ = φ(t). +Two unit vectors el and eφ are defined, attached to the rigid bar and aligned parallel and transverse +to it respectively, as +el = sin φ e1 + cos φ e2, +eφ = cos φ e1 − sin φ e2. +(3) +Differentiation of equations (3) yields (denoting with a superimposed dot the derivative with respect to +time) +˙el = ˙φ eφ, +˙eφ = − ˙φ el, +and therefore the positions of the end points P and L and of the generic point R of the bar (at distance +r from P ) can be written as +P = x(ξ)e1 + y(ξ)e2 + O, +L = lel + P , +R = r el + P , +where it can be noted that R(l) = L. The velocities of points P , L and R are +˙P = ˙ξP ′, +˙L = l ˙φeφ + ˙P , +˙R = r ˙φeφ + ˙P , +where P ′ provides the tangent to the rigid profile at P , equation (1)1. The accelerations can be calcu- +lated as +¨P = ¨ξP ′ + ˙ξ2P ′′, +¨L = l ¨φeφ − l ˙φ2el + ¨P , +¨R = r ¨φeφ − r ˙φ2el + ¨P . +The follower force has a constant modulus F and remains always parallel to the rigid bar, +F = F el. +The longitudinal spring with stiffness k1 produces an elastic force proportional to the vector P − S +F s = −k1(P − S), +while the rotational spring of stiffness k2 applies a moment (positive when anticlockwise) to the end P +of the rigid bar, which is given by +M = −k2(φ + α), +5 + +where α has been defined in formula (2). +The equations of motion governing the dynamics of the mechanical system under analysis can be +found using the principle of virtual work, that can be written as +F · δL − k1(P − S) · δP − k2(φ + α)(δφ + δα) − ρ +� l +0 +¨R · δR dr = 0. +(4) +The external work due to the follower force is +F · δL = Fel · δP , +while +� l +0 +¨R · δR dr = l2 +2 +�2l +3 +¨φ + ¨P · eφ +� +δφ + l +� l +2 +¨φeφ − l +2 +˙φ2el + ¨P +� +· δP , +so that equation (4) can be rewritten as +�� +F + ρl2 +2 +˙φ2 +� +el − ρl2 +2 +¨φeφ − k1(P − S) − ρl ¨P +� +· P ′δξ − k2(φ + α)α′δξ +− +� +k2(φ + α) + ρl2 +2 +�2l +3 +¨φ + ¨P · eφ +�� +δφ = 0, +which, invoking the arbitrariness of δξ and δφ, can be split into the two equations governing the dy- +namics of the structure +� +F + ρl2 +2 +˙φ2 +� +(x′ sin φ + y′ cos φ) − ρl2 +2 +¨φ (x′ cos φ − y′ sin φ) − k1 [x′(x − xS) + y′(y − yS)] +− ρl +� +¨ξ +� +x′2 + y′2� ++ ˙ξ2 (x′x′′ + y′y′′) +� +− k2(φ + α)α′ = 0, +k2(φ + α) + ρl3 +3 +¨φ + ρl2 +2 +� +¨ξ (x′ cos φ − y′ sin φ) + ˙ξ2 (x′′ cos φ − y′′ sin φ) +� += 0. +(5) +The nonlinear system (5) can be solved for ξ and φ, so that it can be equivalently written as +¨q(t) = g(q(t), ˙q(t)), +(6) +where q(t) = [ξ(t), φ(t)]T is a vector collecting the Lagrangian coordinates. +Alternatively to the above, the Hamiltonian formulation can be used, so that the system (6) becomes +a first-order differential nonlinear system +˙y(t) = f(y(t)), +where the phase vector y(t) = [q(t), ˙q(t)]T = [ξ, φ, ˙ξ, ˙φ]T contains the vector of Lagrangian generalized +coordinates and its first derivative in time, so defining a 4-dimensional phase space. +2.2 +Linearized dynamics +The nonlinear differential system (5) can be linearized near ξ = φ = 0 as +ρl2 +2 +¨φx′(0) + [k2α′ − Fx′]ξ=0 φ + ρl¨ξ +� +x′2 + y′2� +ξ=0 ++ +� +k1 +� +x′′(x − xS) + x′2 + y′′(y − yS) + y′2� +− Fy′′ + k2 +� +α′2 + αα′′�� +ξ=0 ξ ++ [k1 (x′(x − xS) + y′(y − yS)) − Fy′ + k2α′α]ξ=0 = 0, +ρl3 +3 +¨φ + k2φ + ρl2 +2 x′(0)¨ξ + k2α′(0)ξ + k2α(0) = 0. +(7) +6 + +Furthermore, with the introduction of the vector collecting the Lagrangian generalized coordinates, +q = [ξ, φ]T , equations (7) can be written in matrix form as +M ¨q(t) + Kq(t) = f(t), +(8) +where M is the mass matrix, K the stiffness matrix and f the vector of generalized forces, respectively +M = ρl +� +��� +x′2 + y′2 +l +2x′ +l +2x′ +l2 +3 +� +��� +ξ=0 +, +K = +� +� +k1 +� +x′′(x − xS) + x′2 + y′′(y − yS) + y′2� +− Fy′′ + k2 +� +α′2 + αα′′� +k2α′ − Fx′ +k2α′ +k2 +� +� +ξ=0 +, +and +f = [Fy′ − k1 (x′(x − xS) + y′(y − yS)) − k2α′α, +−k2α]T +ξ=0 . +The trivial solution ξ = φ = 0 is an equilibrium configuration only when f = 0, which implies +y′(0) = 0, +x(0) = xS, +(9) +so that the tangent to the profile has to be horizontal at ξ = 0, and the fixed point S of the linear spring +must be aligned vertically with the point of the curve at ξ = 0. +2.3 +Piecewise-smooth structure: doubly circular profile +All equations obtained in the previous Sections 2.1 and 2.2 can be applied to a profile with discontinuous +curvature, and hold for both branches of the profile. However, the discontinuity has to be made explicit. +The introduced structure can be particularised through the implementation of a specific curve for the +constraint, given as a parametric function of the arc length ξ. +Circular curves will be addressed with positive and negative curvatures, see Fig. 3, so that the coor- +dinates of the point P along the profile singled out by the arc length ξ are +x(ξ) = R± sin ξ +R± +, +y(ξ) = ±R± +� +1 − cos ξ +R± +� +, +(10) +where R± > 0 is the radius of curvature and where the ‘+’ sign (the ‘−’ sign) applies for positive (for +negative) curvature. For a circular curve described by the parametric representation (10), the nonlinear +governing equations are obtained from (5) by substituting +x′(ξ) = cos ξ +R± +, +y′(ξ) = ± sin ξ +R± +, +x′′(ξ) = − 1 +R± +sin ξ +R± +, +y′′(ξ) = ± 1 +R± +cos ξ +R± +α(ξ) = ± ξ +R± +, +α′(ξ) = ± 1 +R± +. +(11) +7 + +k1 +k2 +ξ(t) +φ(t) +F +F +G +G +L +L +O +P +e2 +e1 +R+ +l, ρ +k1 +k2 +ξ(t) +φ(t) +F +F +G +G +L +L +O +P +e2 +e1 +R− +l, ρ +Figure 3: Two elastic structures of the type shown in Fig. 2 with circular sliding profiles, having positive (on the +left) and negative (on the right) curvatures. +Moreover, conditions (9) for a trivial equilibrium solution are fulfilled provided that xS = 0. Refer- +ring to the linearized equations (8), the mass matrix remains the same for both positive and negative +curvatures, while the stiffness matrix is different in the two cases, namely, +M = ρl +� +� +1 +l/2 +l/2 +l2/3 +� +� , +K± = +� +��� +k1 + k2 +R2 +± +∓ k1ys +R± +∓ F +R± +± k2 +R± +− F +± k2 +R± +k2 +� +��� . +(12) +A third elastic structure shown in Fig. 4 is now considered, described by two degrees of freedom +and obtained as the ‘fusion’ of the two previously described subsystems with positive and negative cur- +vatures. Specifically, the profile on the left (colored blue in the figure) is a circular path with negative +curvature, while the profile on the right (colored red in the figure) is a circular path with positive cur- +vature. The tangent to the profile at the junction is horizontal and at this point the longitudinal spring +is unloaded, so that the vertical configuration of the rigid rod is the trivial equilibrium configuration. +8 + +k1 +k2 +ξ(t) +φ(t) +α(ξ) +α(ξ) +τ(ξ) +F +H +F +G +G +L +L +S +O +P +e2 +e1 +R+ +R− +γ +{x(ξ), y(ξ)} +l, ρ +Figure 4: An elastic structure is obtained as the ‘fusion’ of the two elastic structures shown in Fig. 3. The profile, +on which one end of the structure is forced to slide without friction, is composed of two circular paths, +having negative and positive curvatures on the left and the right, respectively. +This new structure is characterized by a smooth sliding profile, which however evidences a jump in +the curvature at ξ = 0, so that the dynamics is characterized by piecewise smooth differential equations. +Specifically, the equations of motion are obtained through a substitution of equation (11) into equation +(5) and considering the equations referred to the + system (− system) for ξ > 0 (ξ < 0). +Correspondingly, the small amplitude vibrations of the system are described by the following piece- +wise linear equations of motion +� +M ¨q(t) + K−q(t) = 0, +ξ < 0, +M ¨q(t) + K+q(t) = 0, +ξ > 0. +(13) +Although composed of two linear differential problems, the piecewise system (13) is globally nonlinear, +due to switching between two different sets of equations of motion. +Using the notation so far introduced, it will be shown that, due to the presence of the follower (non- +conservative) load, the stability properties of this structure are non-trivial and, in particular, an unstable +structure may result from the union of two structures which are stable when considered alone. +3 +Dynamics and instability for smooth and non-smooth structures +The analysis of a piecewise-smooth dynamical system described in the Lagrangian formalism by the +equation of motions (13) is not trivial due to its nonlinear nature. In particular, as for stability analysis, +the classical Lyapunov theorem on linear analysis cannot be applied, because the Jacobian matrix cal- +culated at equilibrium is not unique, so all the standard criteria based on the nature of the eigenvalues +of the Jacobian matrix are impracticable. Moreover, the complexity of the analysis increases due to the +fact that the time intervals in which the solution is associated to a specific subdomain (ξ > 0 or ξ < 0) +are a priori unknown, since they depend on the initial conditions applied to the structure. +The aim of the present article is the definition of some general criteria that allow the design of +a mechanical structure with given stability properties. In particular, the unusual unstable behaviour +related to the coupling of two stable systems is investigated. In this Section, some extensions of the +classical stability theory are introduced to deal with piecewise linear dynamical system, with the aim of +linking the stability properties of each single smooth system to those of the entire non-smooth structure. +9 + +3.1 +Linearized behaviour for a smooth structure +Before analyzing the non-smooth mechanical system of Fig. 4, the stability of the smooth subsystems +reported in Fig. 3 is analyzed. The treatment applies to both smooth mechanical systems with positive +or negative curvature, so that for simplicity the symbols ‘±’ will be dropped in the following. The lin- +earized dynamics is governed by the following system of two linear second-order differential equations +M ¨q + Kq = 0, +(14) +to be complemented by initial conditions on position and velocity, q(0) = q0 and ˙q(0) = ˙q0, respectively. +The solution to equation (14) can be expressed using exponential functions as +q(t) = ψ(j)eλjt, +leading to the eigenvalue problem +� +λ2 +jM + K +� +ψ(j) = 0, +(15) +which provides two values for λ2 +j. The eigenvalues λj are related to the natural frequencies ωj of the +mechanical system through λj = iωj, where i = √−1 is the imaginary unit. +The generalized coordinates q(t) that satisfy equation (14) can now be written as the linear combi- +nation of four exponential terms +q(t) = +2 +� +j=1 +ψ(j) � +Ajeλjt + Bje−λjt� +, +where Aj and Bj are four arbitrary constants that can be obtained from the initial conditions q0 and ˙q0. +The response of the dynamical system to initial conditions near the trivial equilibrium configuration is +now determined. +The linear equation (14) can be rewritten in the Hamiltonian form, namely, as a system of four linear +first-order differential equations +˙y(t) = A y(t), +(16) +where +A = +� +� +0 +I +−M −1K +0 +� +� = +� +� +0 +I +Γ +0 +� +� +is a 4 × 4 matrix, I is the 2 × 2 identity matrix, and the phase vector y(t) = [q(t), ˙q(t)]T = [ξ, φ, ˙ξ, ˙φ]T +contains the vector of Lagrangian generalized coordinates and its first derivative in time, so defining a +4-dimensional phase space. The differential equation (16) can be solved as for the Lagrangian formulation +using an exponential ansatz y(t) = v(j)eλjt, leading to the eigenvalue problem +A v(j) = λjv(j), +whose eigenvalues λj are the same appearing in equation (15), while the eigenvectors v(j) are related to +the eigenvectors ψ(j) through +v(1,2) =[ψ(1) +1 , ψ(1) +2 , ±λ1ψ(1) +1 , ±λ1ψ(1) +2 ]T, +v(3,4) =[ψ(2) +1 , ψ(2) +2 , ±λ2ψ(2) +1 , ±λ2ψ(2) +2 ]T. +The solution of the initial value problem with assigned initial conditions y(0) = y0 is unique and can +be related to y0 through the so-called fundamental solution matrix, which is the matrix exponential eAt +defined in such a way that +y(t) = eAt y0, +(17) +which can also be written in extenso for the case under analysis as +� +��� +ξ(t) +φ(t) +˙ξ(t) +˙φ(t) +� +��� = eAt +� +��� +ξ(0) +φ(0) +˙ξ(0) +˙φ(0) +� +��� . +10 + +The stability analysis of the equilibrium configuration for a given smooth mechanical system can be +performed using the Lyapunov theorem. In particular, this analysis is based on the nature of the eigen- +values λj of the matrix A, +λj = ± +� +I1 ± +� +I2 +1 − 4I2 +2 +, +(18) +where +I1 = tr Γ +and +I2 = det Γ +(19) +are the first and second invariants of the matrix Γ = −M −1K. +The Lyapunov theorem [3, 17] states that the equilibrium configuration of a nonlinear dynamical +system is stable when the real parts of all eigenvalues of the Jacobian matrix are negative, whereas it is +unstable when the real part of at least one eigenvalue is positive. The case in which the real part of one +or more eigenvalues is zero and all the others have negative real part is not covered by the Lyapunov +theorem and is referred to as critical case, in the context of mechanics [3], or as marginally stable, in the +context of abstract dynamical systems [17]. Due to the symmetry of the eigenvalues with respect to the +imaginary axis given by the structure of the formula (18), only three cases can be distinguished, see +Fig. 5: +• (Marginal) stability: the two squared eigenvalues λ2 +j are real and negative, so that the correspond- +ing λj ∈ iR are two purely imaginary conjugate pairs, say, λ1,2 = ±iω1 and λ3,4 = ±iω2. To be +more specific, this is the critical case mentioned above. Note that the sub-structures originating +the piecewise-smooth structure analyzed in this article are stable in the sense considered here. +• Divergence instability: at least one of λ2 +j is real and positive, that produces λ1,2 = ±ω1. +• Flutter instability: λ2 +j are complex conjugate pairs, so that λ1,2,3,4 = ±(α ± iβ), α ̸= 0. The +behaviour is unstable and the presence of a non-null imaginary part produces oscillations in the +solution. +I1 +I2 +I2 = I2 +1 +4 +complex eigenvalues +with not vanishing real +part(flutter, instability) +two purely imaginary +and two real eigenvalues +(saddle-node, instability) +purely imaginary +eigenvalues +(marginal stability) +real eigenvalues +(divergence, instability) +Figure 5: Representation in the plane I1 − I2 of the stability domains for a dynamical system characterized by 2 +degrees of freedom +The aim of the present article is to show that two smooth elastic structures which are stable taken +separately may lead to an unstable structure when combined together. Therefore, our interest is in the +11 + +case when both smooth subsystems are stable, for which λ1,2 = ±iω1 and λ3,4 = ±iω2. Assuming +ω1 ̸= ω2 1 , matrix A is diagonalizable as A = UJU −1 and the matrix exponential fulfils the relation +eA = UeJU −1, +where +J = +� +��� +iω1 +0 +0 +0 +0 +−iω1 +0 +0 +0 +0 +iω2 +0 +0 +0 +0 +−iω2 +� +��� , +U = +� +����� +ψ(1) +1 +ψ(1) +1 +ψ(2) +1 +ψ(2) +1 +ψ(1) +2 +ψ(1) +2 +ψ(2) +2 +ψ(2) +2 +iω1ψ(1) +1 +−iω1ψ(1) +1 +iω2ψ(2) +1 +−iω2ψ(2) +1 +iω1ψ(1) +2 +−iω1ψ(1) +2 +iω2ψ(2) +2 +−iω2ψ(2) +2 +� +����� +, +and the vectors ψ(1) and ψ(2) are solutions of equation (15), while the inverse of U is given by +U −1 = +1 +2(ψ(1) +1 ψ(2) +2 +− ψ(1) +2 ψ(2) +1 ) +� +����� +ψ(2) +2 +−ψ(2) +1 +−iψ(2) +2 /ω1 +iψ(2) +1 /ω1 +ψ(2) +2 +−ψ(2) +1 +iψ(2) +2 /ω1 +−iψ(2) +1 /ω1 +−ψ(1) +2 +ψ(1) +1 +iψ(1) +2 /ω2 +−iψ(1) +1 /ω2 +−ψ(1) +2 +ψ(1) +1 +−iψ(1) +2 /ω2 +iψ(1) +1 /ω2 +� +����� +. +Therefore, the matrix exponential eJt becomes +eJt = +� +��� +cos ω1t + i sin ω1t +0 +0 +0 +0 +cos ω1t − i sin ω1t +0 +0 +0 +0 +cos ω2t + i sin ω2t +0 +0 +0 +0 +cos ω2t − i sin ω2t +� +��� +and finally, the matrix exponential eAt for the four-dimensional smooth mechanical system considered +here is +eAt = +� +�������� +a1 cos ω1t−a2 cos ω2t +a1−a2 +a1a2(cos ω2t−cos ω1t) +a1−a2 +a1ω2 sin ω1t−a2ω1 sin ω2t +a1ω1ω2−a2ω1ω2 +a1a2(ω1 sin ω2t−ω2 sin ω1t) +a1ω1ω2−a2ω1ω2 +cos ω1t−cos ω2t +a1−a2 +a1 cos ω2t−a2 cos ω1t +a1−a2 +ω2 sin ω1t−ω1 sin ω2t +a1ω1ω2−a2ω1ω2 +a1ω1 sin ω2t−a2ω2 sin ω1t +a1ω1ω2−a2ω1ω2 +a2ω2 sin ω2t−a1ω1 sin ω1t +a1−a2 +a1a2(ω1 sin ω1t−ω2 sin ω2t) +a1−a2 +a1 cos ω1t−a2 cos ω2t +a1−a2 +a1a2(cos ω2t−cos ω1t) +a1−a2 +ω2 sin ω2t−ω1 sin ω1t +a1−a2 +a2ω1 sin ω1t−a1ω2 sin ω2t +a1−a2 +cos ω1t−cos ω2t +a1−a2 +a1 cos ω2t−a2 cos ω1t +a1−a2 +� +�������� +, +(20) +where +a1 = ψ(1) +1 /ψ(1) +2 += −ω2 +1 + Γ22 +Γ21 +, +a2 = ψ(2) +1 /ψ(2) +2 += −ω2 +2 + Γ22 +Γ21 +. +It will be instrumental later to calculate the inverse of the matrix given by equation (20). Because of the +property +AB = BA =⇒ eAeB = eA+B +the inverse of the exponential matrix can be obtained as +(eAt)−1 = e−At, +corresponding to a change in the sign of t in equation (20). +As described above, for a smooth mechanical system, the stability can simply be judged through +the calculation of the eigenvalues λ2 +j of the jacobian matrix. However, this is not enough for piecewise +smooth structures, for which a complete understanding of the dynamics of the system is required, by +a direct solution of the equations of motion. This analysis will be performed in the next paragraph +through the computation of the solution of the piecewise linear system (13). +1The coalescence ω1 = ω2 would denote a grazing of an unstable boundary, an occurrence which is excluded here. +12 + +3.2 +Linearized behaviour for a piecewise-smooth structure +When a piecewise smooth dynamical system is considered, equation (13) can be rewritten in the Hamil- +tonian form as +˙y(t) = +� +A−y(t), +y1 < 0, +A+y(t), +y1 > 0, +(21) +where y1 = ξ. A dynamical system with discontinuous right-hand side such as that expressed by +equation (21) is referred to as ‘Filippov system’ [18]. +Note that, although the accelerations ˙y3 = ¨ξ and ˙y4 = ¨φ suffer a jump across the discontinuity in the +curvature of the profile, y1 = 0, the velocities ˙y1 = y3 = ˙ξ and ˙y2 = y4 = ˙φ remain continuous, so that +e1 · A−y(t) = e1 · A+y(t), and e2 · A−y(t) = e2 · A+y(t), when y1 = 0. +Equation (21) shows that the piecewise smooth structure under consideration defines a 4-dimensional +phase space, which can be divided into two subdomains V± within which the system can be considered +smooth. The manifold separating the two subdomains, called switching manifold, is the hyperplane +Σ = {y ∈ R4 : y1 = 0}. The negative part of equations (21) applies to the subdomain defined by +V− = +� +y ∈ R4 : y1 < 0 +� +, while the positive part applies to V+ = +� +y ∈ R4 : y1 > 0 +� +. Note that the +origin of the phase space (ξ, φ) = (y1, y2) = (0, 0) represents the trivial equilibrium configuration of the +structure and belongs to the switching manifold Σ. +The discontinuous system (21) does not define the time derivative ˙y(t) when the configuration of +the system y(t) is on the switching boundary Σ. To overcome this difficulty, Filippov has developed a +technique, known as Filippov’s convex method, that extends the discontinuous system (21) to a differential +inclusion of the form +˙y(t) ∈ +� +� +� +� +� +A−y(t), +y1 < 0, +co{A−y(t), A+y(t)}, +y1 = 0, +A+y(t), +y1 > 0, +(22) +where the closed convex hull co of the two right-hand sides f − and f + is defined by +co{f −, f +} = {(1 − η)f − + ηf +, ∀η ∈ [0, 1]}. +A solution in the Filippov sense of the discontinuous system (21) is a solution of the differential inclusion +(22). In this sense, the mechanical system admits among the possible solutions, the so-called sliding +modes, namely, motions in which the system remains on the switching manifold Σ. +More precisely, there are three possible ways in which the mechanical system behave around the +switching boundary Σ, namely: +• Transverse intersection: both vector fields A+y(t) and A−y(t) point on the same side of Σ +� +e1 · A+y(t) +�� +e1 · A−y(t) +� +> 0, +(23) +so that a solution, that evolves in one subdomain and at some instant of time hits Σ, will cross it +transversely and proceed in the other subdomain. In this case the solution is locally unique. +• Attractive sliding modes: both vector fields A+y(t) and A−y(t) point to Σ +e1 · A+y(t) < 0 +and +e1 · A−y(t)] > 0, +hence a solution that hits the switching boundary Σ will not leave it and will therefore move along +Σ. Also in this case the solution is locally unique. +• Repulsive sliding modes: both vector fields A+y(t) and A−y(t) point in the opposite direction to +Σ +e1 · A+y(t) > 0 +and +e1 · A−y(t)] < 0, +which implies that a solution emanating from Σ can remain in Σ or leave it by entering either +subdomain. Consequently, the solution in this case is not unique. +13 + +For the two d.o.f. elastic structure under consideration, the condition of transverse intersection, +equation (23), is satisfied almost everywhere, that is for any configuration belonging to Σ and such that +y3 = ˙ξ ̸= 0, since the velocity ˙ξ is continuous across the switching manifold Σ. Sliding modes are only +possible in a zero-measure subset of Σ, namely the set {y ∈ R4 : y · e1 = y · e3 = 0}, and will not be +investigated further. +Assuming that transverse intersection always prevail at the intersections of the orbits with the +switching manifold Σ, a solution that evolves in one subdomain and hits Σ necessarily crosses the +hyperplane and enters the other subdomain. The intersection point between the orbit and the hyper- +plane Σ can then be used as initial condition for the subsequent evolution in the subdomain in which +the orbit is entering. Therefore, the solution of the piecewise-linear system (21) can be obtained as the +composition of exponential matrices as follows +y(t) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +eA−(t−t0)y0, +t0 ≤ t < t1, +eA+(t−t1)eA−(t1−t0)y0, +t1 ≤ t < t2, +· · · +· · · +eA−(t−tk−1)eA+(tk−1−tk−2) · · · eA+(t2−t1)eA−(t1−t0)y0, +tk−1 ≤ t < tk, +eA+(t−tk)eA−(tk−tk−1) · · · eA+(t2−t1)eA−(t1−t0)y0, +tk ≤ t < tk+1, +· · · +· · · +(24) +where, without loss of generality, it is assumed that the initial condition belongs to the negative subdo- +main, y(0) = y0 ∈ V−, and {t1, t2, · · · , tk−1, tk, · · · } is the sequence of intersection times at which the or- +bit crosses the switching boundary and changes subdomain. However, it should be pointed out that the +piecewise solution (24) is not completely determined, because the intersection times {t1, t2, · · · , tk−1, tk, · · · } +are a priori unknown and depend on the initial condition y0. To completely define the solution (24), the +intersection times have to be determined by tracing the evolution of the orbit from the initial condition +y0 and numerically detecting the roots of the crossing condition ξ = 0. +A classical expedient to simplify the description of a dynamical system is the introduction of a +discrete map, known as Poincaré map. A Poincaré map transforms a n-dimensional continuous-time +system into a (n−1)-dimensional discrete-time system, with the introduction of a hyperplane embedded +in the n-dimensional phase space, called Poincaré section. +For the two d.o.f. elastic system under consideration, described by the 4-dimensional non-smooth +dynamical system (21), a natural and convenient choice for the 3-dimensional Poincaré section is the +switching manifold Σ. In this case, the Poincaré map P : Σ → Σ links points on Σ through the orbits +defined by the solution (24). +More precisely, let us consider a point x ∈ Σ and let us assume, without loss of generality, that the +orbit starting from the initial condition x enters the negative subspace V−. The time evolution in the +negative subsystem is described by the first expression in (24) +y(t) = eA−(t−t0)x, +until the orbit reaches Σ at point ξ = y(t1) = exp (A−∆t−) x in a given time interval, namely ∆t− = +t1 − t0 measured from the initial time t0. Then the orbit crosses the switching manifold and enters in +the positive subsystem V+, following the orbit described by +y(t) = eA+(t−t1)eA−∆t−x = eA+(t−t1)ξ, +so that the point ξ can be interpreted as a new initial condition for the second part of the orbit, evolving +within the positive subdomain V+. The orbit remains inside V+ until it hits Σ for a second time at point +η = y(t2) = exp (A+∆t+) ξ in a time interval ∆t+ = t2 − t1, measured from t = t1. +The just described sequence defines two Poincaré half-maps +ξ = P −(x) = eA−∆t−(x)x, +η = P +(ξ) = eA+∆t+(ξ)ξ, +(25) +and that the complete Poincaré map P : Σ → Σ may be obtained through the composition of the two +half-maps, P = P + ◦ P −, such that +η = P (x) = P +(P −(x)) = eA+∆t+(ξ(x))eA−∆t−(x)x. +(26) +14 + +It should be noted that the time intervals ∆t−(x) and ∆t+(ξ) are non linear functions of the initial +conditions x and ξ, respectively, and can be defined as +∆t−(x) = inf +� +∆t > 0 : e1 · eA−∆tx = 0 +� +, +∆t+(ξ) = inf +� +∆t > 0 : e1 · eA+∆tξ = 0 +� +, +(27) +where e1 is the normal to the switching manifold Σ. These definitions simply represent the fact that +the points ξ and η must be on the switching manifold and that they identify the first two intersections +between the considered trajectory and the hyperplane Σ. +The use of the Poincaré map will be crucial in the next section for the analysis of stability of the +piecewise smooth elastic structure. +e1 +O +Σ +V+ +V− +x +ξ = P −(x) +η = P +(ξ) +Figure 6: A pictorial view (in which a 4-dimensional space is reduced to a 3D sketch) of an unstable cone with +the Poincaré half-maps P − and P +, separated by the hyperplane Σ. The three points x, ξ, η identify a +solution belonging to the invariant cone. +3.3 +Invariant cones: instability of piecewise-linear mechanical systems +The stability analysis of the piecewise-linear system (21) cannot be pursued using standard methods for +smooth system, such as the analysis of eigenvalues at the equilibrium point. In this section a tool will +be developed, based on the existence of special invariant sets, called invariant cones. +An invariant set of an autonomous dynamical system, such that described by equation (16) for +smooth structures or by equation (21) for piecewise smooth structures, is a subset S of the phase space +such that y(t0) ∈ S implies y(t) ∈ S, for all times t > t0, [19–21]. This subset, for an n-dimensional +dynamical system, is assumed to be an (n − 1)-dimensional invariant manifold. +A special type of invariant set, called invariant cone, see [2, 12, 22, 23], is defined by the condition +that a vector n belonging to a Poincaré section Σ exists, for which the Poincaré map P describing the +15 + +evolution of the dynamical system fulfills the condition +P (n) = µ n, +where µ ∈ R+ is a positive real number and n ∈ Σ. +In the specific case of the piecewise-linear system (21), the Poincaré map is given by the composition +of two half-maps (25), hence an invariant cone exists for the elastic system under investigation if and +only if there exist a scalar µ > 0 and a vector x ∈ Σ such that +η = eA+∆t+(ξ(x))eA−∆t−(x)x = µ x. +(28) +Equation (28) has the structure of an eigenvalue problem: the multiplier µ can be interpreted as a gen- +eralized eigenvalue and the vector x as a generalized eigenvector of the matrix eA+∆t+(ξ(x))eA−∆t−(x), +which defines the Poincaré map. However, the eigenvalue problem is nonlinear, because the matrix +itself depends on the eigenvector, hence the solution of (28) in terms of µ and x is not trivial and the +usual procedures for linear eigenvalue problems cannot be adopted. +A solution of eq. (28), provided it exists, identifies an invariant cone and is given by the list +{∆t−, ∆t+, x, µ}, +comprising the time intervals ∆t− and ∆t+, expended by the mechanical system in passing through +the subdomains V− and V+, respectively, an eigenvector x belonging to the invariant cone (and also to +the Poincaré section), and finally the corresponding eigenvalue µ. +Before proceeding, it is instrumental to establish some fundamental properties of invariant cones. +First of all, it is noted that the assumption that the system enters first in the negative subdomain V−, +used in eq. (28), is not restrictive, as the same invariant cone may be identified by assuming instead that +the system enters first the positive subdomain V+ and, thus, by solving the eigenvalue problem +eA−∆t−eA+∆t+ξ = γ ξ, +(29) +where ∆t− and ∆t+ are the same as in eq. (28). It is easy to check that ξ = eA−∆t−x solves the problem +(29) with γ = µ. This means that the solution {∆t+, ∆t−, ξ, µ} of eq. (29) identifies the same invariant +cone as the solution {∆t−, ∆t+, x, µ} of eq. (28). To simplify notations, here and in the sequel, the three +points x, ξ, η identify a solution belonging to the invariant cone. +Next, it is noted that the time intervals ∆t−(x) and ∆t+(ξ) are homogeneous functions of degree +zero, +∆t−(αx) = ∆t−(x), +∆t+(αξ) = ∆t+(ξ), +∀α > 0, +(30) +as can be easily checked from the definitions (27). These conditions are a direct consequence of the lin- +earity of the dynamical system within each individual subdomain, V− and V+, and define the structure +of the invariant cone. In fact, when the point η is assumed as an initial condition for the motion of the +system after the two initial half-maps, ξ = P −(x) and η = P +(ξ), the new intersection time intervals +become +∆t−(η) = ∆t−(µx) = ∆t−(x), +∆t+(µξ) = ∆t+(ξ), +hence the time intervals ∆t± are constants and do not change in the application of further half-maps. +The properties (30) immediately imply that both the Poincaré half-maps P −(x) and P +(ξ), defined +according to equation (25) as well as the complete Poincaré map P (x), defined by equation (26), are +homogeneous functions of degree one +P −(αx) = αP −(x), +P +(αξ) = αP +(ξ), +P (αx) = αP (x), +∀α > 0. +The latter equations have a clear geometrical interpretation, in that the Poincaré half-maps P −(x) and +P +(ξ), plus the Poincaré map P (x), transform straight half-lines belonging to the switching boundary +Σ and intersecting the origin into straight half-lines also belonging to Σ and intersecting the origin. +The last observation is crucial for the stability analysis of dynamical system when an invariant cone +is present. In fact, when an initial condition y0 belongs to the Poincaré section and to the invariant cone, +16 + +the evolution of the system can be described by the recursive applications of Poincaré maps such that +y(∆t− + ∆t+) = eA+∆t+eA−∆t−y0 = µ y0, +y(2∆t− + 2∆t+) = +� +eA+∆t+eA−∆t−�2 +y0 = µ2 y0, +y(3∆t− + 3∆t+) = +� +eA+∆t+eA−∆t−�3 +y0 = µ3 y0, +· · · +y(k∆t− + k∆t+) = +� +eA+∆t+eA−∆t−�k +y0 = µk y0, +· · · +(31) +defining a discrete exponential relation. +Accordingly, the behaviour of the mechanical system can easily be understood, when an invariant +cone is present. This is because the orbits may spiral in or out along the cone, depending on the value +of the eigenvalue µ, and may evolve either towards the vertex at the equilibrium point or away from it. +In particular, the following conclusions can be drawn: +• If µ > 1, a family of ‘spiraling out’ trajectories, belonging to the invariant cone, exists, moving +away for t > 0 from the vertex of the cone, which represents the equilibrium configuration of the +system. When an initial condition is selected, belonging to the invariant cone, the motion of the +structure evolves in time diverging from the fixed point, hence the equilibrium configuration at +the origin is unstable. +• If µ < 1, a family of ‘spiraling in’ trajectories belonging to the invariant cone exists, moving +towards the vertex of the cone. Although this corresponds to a stable behaviour, other trajectories +different from those on the invariant cone may exist being divergent, so that stability of the fixed +point cannot be guaranteed. +• If µ = 1, a family of periodic trajectories, belonging to the invariant cone, exists. In this case, +nothing can be concluded about the stability of the equilibrium configuration. +Summarising the above statements, for the dynamical system represented by equation (21), the exis- +tence of an invariant cone, eq. (28), with µ > 1 is a sufficient condition for instability. Vice-versa, assuming +the existence of an invariant cone, fulfillment of condition (28) with µ ≤ 1 is a necessary condition for +stability. +It is important to emphasise that these conditions hold true, regardless of the stability behaviour +of each single subsystem. Therefore, an equilibrium configuration of a piecewise-linear mechanical +system, which results stable when analysed separately for each constituting subsystem, can become +unstable when the composed structure is considered. +A proposition is now proven which establishes a general property of invariant cones for autonomous +non-dissipative mechanical systems. Although this proposition is proven for the 2 d.o.f. mechanical +system under investigation, it can be easily extended to a system with any number of degrees of free- +dom. +A proposition on reciprocal eigenvalues. +If the autonomous non-dissipative mechanical system (21) +admits an invariant cone, that is a solution {∆t−, ∆t+, x, µ} of the eigenvalue problem (28) with µ ̸= 1, +then another invariant cone {∆t−, ∆t+, Jξ, 1/µ} exists, associated to the eigenvalue 1/µ, reciprocal of +µ, where +ξ = eA−∆t−x, +J = +� +��� +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +−1 +� +��� . +Consequently, +when the problem (28) admits a real eigenvalue µ ̸= 1, it admits also the eigenvalue 1/µ. Therefore, +when a solution of equation (28) is found with µ ̸= 1, the trivial equilibrium configuration is certainly +unstable. +17 + +The above proposition can be proven by preliminarily noting that the application of matrix J to a +state vector y = [ξ, φ, ˙ξ, ˙φ]T has the effect of changing the sign of the velocities +J +� +��� +ξ +φ +˙ξ +˙φ +� +��� = +� +��� +ξ +φ +− ˙ξ +− ˙φ +� +��� . +Note also that J = J−1, e−A±t± = JeA±t±J and that vector Jξ enters in the negative subdomain, as +the vector ξ, without the minus signs, enters in the positive subdomain, being a solution of the half-map +(25)1 by assumption. +The above statement follows from the two properties +Jξ = e−A−∆t−Jx, +Jη = e−A+∆t+Jξ, +(32) +equations that can be directly checked considering the form of matrix eAt, equation (20). The above +properties have a clear mechanical meaning, as they correspond to an inversion of the motion: the time +is inverted by changing the sign to the variable t and, correspondingly, the velocities in the vectors x, ξ +and η are inverted through multiplication by matrix J. +A multiplication of equation (32)1 by eA+∆t+eA−∆t− and subsequent use of x = η/µ, as well as +consideration of equation (32)2 leads to +eA+∆t+eA−∆t−Jξ = eA+∆t+Jx = 1 +µeA+∆t+Jη = 1 +µJξ, +which proves the proposition. +3.4 +Mechanical energy for the piecewise smooth system +The unusual unstable behaviour of the piecewise smooth structure composed of two stable substruc- +tures can be qualitatively explained from a mechanical point of view by analyzing the mechanical en- +ergy characterizing the non-smooth system. +The stiffness matrix K±, defined in equation (12), collects the effect of the elastic springs, k1 and k2, +and of the follower force F and can be decomposed into the sum of a symmetric and an unsymmetric +components, K± = � +K± + G±, +� +K +± = +� +���� +k1 +� +1 ∓ ys +R± +� ++ k2 +R2 +± +± k2 +R± +± k2 +R± +k2 +� +���� , +G± = F +� +�� +∓ 1 +R± +−1 +0 +0 +� +�� , +where � +K +± is a symmetric matrix collecting only the spring stiffnesses and, possibly, other conservative +loads applied to the structure, while G± contains only the nonconservative forces. +A scalar product of equation (14) by ˙q and a factorization of the time derivative yield +d +dt +�1 +2 ˙q · M ˙q + 1 +2q · � +Kq +� += − ˙q · Gq. +(33) +The mechanical energy H is defined as the sum of the kinetic and the total potential energy +H = 1 +2 ˙q · M ˙q +� +�� +� +Kinetic energy ++ +1 +2q · � +Kq +� +�� +� +Total potential energy +, +(34) +where the total potential energy is the sum of the elastic energy and the potential energy of conservative +external loads, when present. +18 + +Equation (34) can be rewritten in matrix notation as a function of the state vector y(t) as +H(t) = 1 +2y(t) · Dy(t), +(35) +where +D = +� +� +� +K +0 +0 +M +� +� , +is a symmetric matrix. An integration of equation (33) yields +H(tf) − H(t0) = +− +� tf +t0 +˙q · Gq dt +� +�� +� +Work done by the nonconservative loads +, +therefore, a change in the mechanical energy equals a corresponding work done on the structure by the +nonconservative loads. +When in a smooth system only the symmetric part � +K of the stiffness matrix is present, the mechan- +ical energy remains constant. On the contrary, when a nonconservative contribution G is present, the +mechanical energy H(t) varies in time. +The mechanical energy for a solution near a stable equilibrium configuration of a smooth system can +be studied by substituting the solution (17) with the matrix exponential (20) into equation (35), hence +H = 1 +2y0 · (eAt)T DeAty0. +(36) +When a solution is stable, the mechanical energy (36) results as a sum of trigonometric functions, which +in general is not a periodic function, but it is bounded, so that in a stable system the mechanical energy +cannot indefinitely increase. Formally, this can be shown by considering that the mechanical energy +(36) can be bounded as +y0 · (eAt)T DeAty0 ≤ ||(eAt)T DeAt|| y2 +0 ≤ ||eAt||2 ||D|| y2 +0, +where, in the case of stability, matrix eAt is given by equation (20), so that its norm is bounded in time +for every value of t ∈ [0, ∞). +On the contrary, in case of a piecewise linear system described by equation (21), the mechanical +energy is not necessarily bounded, although the subsystems forming the non smooth structure are both +stable. +In fact, the mechanical energy at time t = 0, equation (36), for initial conditions y0 is +H0 = H(0) = 1 +2y0 · Dy0, +where D can be identified with either D− or D+, because both matrices provide the same result, being +the energy continuous at the switching manifold ξ = 0. +During the motion, the variation of mechanical energy with time can be computed by substituting +the piecewise solution (24) into eq. (35), thus obtaining +H(t) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 +2eA−(t−t0)y0 · D−eA−(t−t0)y0, +t0 ≤ t < t1, +1 +2eA+(t−t1)eA−(t1−t0)y0 · D+eA+(t−t1)eA−(t1−t0)y0, +t1 ≤ t < t2, +· · · +· · · +1 +2eA−(t−tk−1) · · · eA−(t1−t0)y0 · D−eA−(t−tk−1) · · · eA−(t1−t0)y0, +tk−1 ≤ t < tk, +1 +2eA+(t−tk) · · · eA−(t1−t0)y0 · D+eA+(t−tk) · · · eA−(t1−t0)y0, +tk ≤ t < tk+1, +· · · +· · · +19 + +Assuming that an invariant cone exists, eq. (31), the mechanical energy at the intersection time k∆t− + +k∆t+ (where k is a positive integer) can be computed as +H(k∆t− + k∆t+) = 1 +2µky0 · Dµky0 = µ2kH0. +Therefore, when µ > 1 the mechanical energy suffers an unbounded exponential growth in time, revealing an +unstable behaviour of the system. This exponential growth is accompanied by oscillations, a situation +similar to flutter instability in smooth systems loaded by follower forces, so that the instability be- +haviour under investigation can be interpreted as a condition of flutter, but for non-smooth mechanical +systems. +The evolution in time (made dimensionless through division by a reference time T) of the mechani- +cal energy H is reported in Fig. 7 for the non-smooth structure reported in Fig. 4, characterized by the +values of parameters listed in (49), together with the two ‘component’ smooth structures shown in Fig. +3 (responses reported dashed). +Instability of the non-smooth system corresponds to a blow up of the mechanical energy (note the +vivid representation in the inset, showing exponential growth), while the two stable substructures evi- +dence a bounded evolution. +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +5.5 +6 +6.5 +7 +7.5 +0.04 +0.045 +0.05 +0.055 +0.06 +energy of the negative subsystem +neglecting the switching conditions +energy of the positive subsystem +neglecting the switching conditions +τ +H +100 +200 +500 +1,000 +τ +H +Figure 7: Evolution in time of the mechanical energy for the non-smooth structure [shown in Fig. 4 and charac- +terized by the parameters in the list (49)]. The behaviours of the two ‘component’ smooth mechanical +structures (shown in Fig. 3) are also reported with dotted lines. The energy in the latter case remains +bounded, denoting stability. The exponential growth of the mechanical energy corresponding to a flutter +instability is visible in the main graph, but may be vividly observed in the inset, referred to a longer time +interval. +4 +Attractivity of the cone: the effects of perturbations +It has been demonstrated in the previous sections that the existence of an invariant cone with eigenvalue +µ ̸= 1 in the phase space of a non-smooth dynamical system of the type (21) is a sufficient condition for +instability. Now the effect of perturbations in the initial conditions has to be analyzed. In particular, the +question arises whether a motion generated from an initial condition slightly outside an unstable cone +20 + +will still display unstable unbounded growth. When the latter behaviour is found, the cone is called +attractive, so that it will be asymptotically approached by a trajectory originated from initial conditions +sufficiently close to it. +The answer to the above question is provided in the following, showing that for the structure under +consideration, the unstable cone (associated to µ > 1) is always attractive. In other words, the instability +detected with µ > 1 can be considered ‘genuine’. +A proposition on the attractivity of the cone. +A cone with eigenvector x = y(0) = y0 and its as- +sociated eigenvalue µ are assumed to exist, solution of equation (28), together with the corresponding +vectors ξ = y(∆t−) = eA−∆t−y0 and η = y(∆t− + ∆t+) = eA+∆t+eA−∆t−y0, defining the state of the +system at the first crossing (from V− to V+) and the second crossing (from V+ to V−) of the switching +manifold Σ. These vectors describe a solution y(t) belonging to the invariant cone that is considered as +a reference solution and that is perturbed in order to study the attractivity of the cone. The reference +solution y(t) crosses the switching manifold Σ separating the negative and the positive subspaces the +first time at the instant t− = ∆t− and a second time at the instant t+ = ∆t− + ∆t+. +V− +V+ +Σ +e1 +A−y(t−) +A+y(t−) +y0 +y(t) +˜y0 +˜y(t) +δy0 +δy(t) +δy(t− + δt−) +y(t−) +˜y(t− + δt−) +y(t− + δt−) +Figure 8: Sketch of the effect of a perturbation δy0 of a given piecewise solution (developing from ˜y0) for a non- +smooth dynamical system. The perturbed motion, starting at ˜y0 + δy0, is assumed to develop in a close +neighborhood of the unperturbed motion, starting at ˜y0 +A generic small perturbation δy0 in the initial conditions is assumed, with the restriction that ˜y0 = +y0 + δy0 belongs to the negative subspace V−, as sketched in Fig. 8. The perturbed initial condition +˜y0 = y0 + δy0 gives rise to a perturbed solution ˜y(t) = y(t) + δy(t) which crosses for the first time the +switching manifold Σ at a different instant of time t− + δt−. Therefore, equation (17) can be applied to +obtain +˜y(t− + δt−) = eA−(t−+δt−)(y0 + δy0), +where, due to the smallness of δy0, it is assumed that δt− is also sufficiently small to justify the following +Taylor series approximation +˜y(t− + δt−) = eA−t−y0 + eA−t−δy0 + δt−A−eA−t−y0. +21 + +The reference solution at time t− + δt− belongs to the positive subspace V+ and can be calculated with +an analogous Taylor expansion (assuming that δt− is sufficiently small) as +y(t− + δt−) = eA+δt−eA−t−y0 = eA−t−y0 + δt−A+eA−t−y0, +where A+eA−t−y0 = A+y(t−) = ˙y(t−) is the orbital velocity of the reference solution entering at the +time instant t− into the positive subspace. +The evolution of the perturbation after the crossing of the switching manifold is described by the +quantity δy(t− + δt−) = ˜y(t− + δt−) − y(t− + δt−), that can be computed as +δy(t− + δt−) = eA−t−δy0 + δt−(A− − A+)eA−t−y0. +(37) +The small time interval δt− can be calculated noting that the vector ˜y(t− + δt−) − y(t−) belongs to the +switching manifold and is orthogonal to the vector e1, hence +�˜y(t− + δt−) − y(t−) +� +· e1 = +� +eA−t−δy0 + δt−A−eA−t−y0 +� +· e1 = 0, +an equation that can be solved in δt−, leading to +δt− = − +1 +˙ξ(t−) +e1 · eA−t−δy0, +(38) +where ˙ξ(t−) = e1 · A−y(t−). Note also that ˙ξ(t−) = e1 · A−y(t−) = e1 · A+y(t−), because the velocities +˙ξ and ˙φ remain continuous crossing the switching manifold. +Substituting eq. (38) into eq. (37), the evolution of the perturbation after the crossing can be written +as +δy(t− + δt−) = S−eA−t−δy0, +where the saltation matrix S− is defined as +S− = I + +1 +˙ξ(t−) +� +A+y(t−) − A−y(t−) +� +⊗ e1. +The saltation matrix defines the perturbation δy(t− + δt−) after the crossing of the switching manifold +Σ, for a given perturbation δy(t−) = eA−t−δy0 defined just before the crossing. +Due to the structure of the saltation matrix and because det(I + a ⊗ b) = 1 + a · b, it follows that +det(S−) = 1. +A similar calculation can be performed when the unperturbed solution crosses the switching mani- +fold from the positive to the negative subspace at time t+ = ∆t− + ∆t+ and the perturbed solution at +time t+ + δt+, so that a new saltation matrix can be defined as +S+ = I + +1 +˙ξ(t+) +� +A−y(t+) − A+y(t+) +� +⊗ e1, +in which ˙ξ(t+) = e1 · A+y(t+) and y(t+) = eA+∆t+eA−∆t−y0. +Therefore, the initial perturbation δy0 evolves in time in a way that after two crossings of the switch- +ing manifold, the difference between the reference and the perturbed solutions is governed by the rela- +tion +δy(t+ + δt+) = S+eA+∆t+S−eA−∆t−δy0, +where the matrix +ΦT = S+eA+∆t+S−eA−∆t− +is referred to as the monodromy matrix in the context of stability of periodic solutions. Consequently, +the attractivity of the invariant cone is related to the four eigenvalues of the monodromy matrix, the +so-called Floquet multipliers. +The monodromy matrix for the 2 d.o.f. mechanical system under consideration possesses the fol- +lowing properties: +22 + +• The determinant is equal to the unit, +det +� +S+eA+∆t+S−eA−∆t−� += 1, +because det eAt = etr At = 1 and det S− = det S+ = 1; +• Two eigenvectors and two eigenvalues coincide with those of the eigenvalue problem (28) defin- +ing the invariant cone +� +S+eA+∆t+S−eA−∆t−� +y0 = µ y0, +� +S+eA+∆t+S−eA−∆t−� +Jy(t−) = 1 +µJy(t−), +two identities following directly from S−y(t−) = S+y(t−) = y(t−) and S−Jy0 = S+Jy0 = +Jy0 (the saltation matrices S− and S+ leave unchanged any vector belonging to the switching +manifold Σ). +• A third eigenvector is equal to A−y0, with corresponding eigenvalue µ, +� +S+eA+∆t+S−eA−∆t−� +A−y0 = µ A−y0, +which follow from the commutativity, eA±∆t±A± = A±eA±∆t±, and from the two identities +S−A−y(t−) = A+y(t−) +and +S+A+y(t+) = A−y(t+). +It follows from all the above that the monodromy matrix S+eA+∆t+S−eA−∆t− possesses the four +eigenvalues {µ, µ, 1/µ, 1/µ}. The first two eigenvalues {µ, µ} are associated with initial perturbations +δy0 along the directions y0 and A−y0, i.e. belonging to the invariant cone. The other two {1/µ, 1/µ} +are associated to perturbations outside the invariant cone. +In conclusion, +since a generic perturbation can always be decomposed along the eigenvectors of the monodromy ma- +trix S+eA+∆t+S−eA−∆t−, the unstable cone associated to the eigenvalue µ > 1 is always attractive. +Therefore, when an eigenvalue µ ̸= 1 is found as the solution of problem (28), the structure admits +a stable (non attractive) cone, associated to µ < 1, and an unstable (attractive) cone, associated to +µ > 1. +Note that in the case µ = 1 the motion is periodic and the cone is not attractive, hence conclu- +sions about instability of the mechanical system cannot be reached. This case is therefore not further +considered. +5 +Numerical examples +5.1 +Numerical algorithm for the identification of invariant cones for piecewise lin- +ear systems +Assume that all the mechanical parameters of the system, together with the applied follower force, are +given. A numerical procedure is proposed in this section for the identification of possible invariant +cones, i.e. solutions of the generalized nonlinear eigenvalue problem (28), recalled here for convenience +eA+∆t+eA−∆t−x = µ x, +where the initial condition vector x belongs to the switching manifold Σ, so that it has the following +form +x = [0, x2, x3, x4]T . +23 + +Note that the unknowns of the problem are the eigenvalue µ, the eigenvector x and the two time in- +tervals ∆t− and ∆t+. Note also that the modulus of the vector x is arbitrary, given the structure of the +equation. Thus eq. (28) can be scaled as follows +eA+∆t+eA−∆t− +� +��� +0 +x2 +x3 +1 +� +��� = µ +� +��� +0 +x2 +x3 +1 +� +��� , +(39) +from which it is clear that there are five scalar unknowns to be determined ∆t−,∆t+, x2, x3 and µ. +However, the system (39) provides only four scalar equations, and thus it has to be complemented by +an additional scalar equation, which is provided by the condition that also the intermediate point ξ +belongs to the switching manifold, +ξ1 = [eA−∆t−x]1 = 0. +(40) +The system of equations (39) and (40) is nonlinear, so that an algorithm is proposed below to partially +decouple the system and reduce it to two equations for the unknowns ∆t− and ∆t+. +The starting point x and the final point η of the Poincaré map (26) can be expressed in terms of the +intermediate point ξ using the Poincaré half maps (25) as follows +x = e−A−∆t−ξ, +η = eA+∆t+ξ. +(41) +The condition that both x and η belong to the switching manifold provides two equations, +x1 = +� +e−A−∆t−ξ +� +1 = 0, +η1 = +� +eA+∆t+ξ +� +1 = 0, +(42) +that are linear in ξ2, ξ3, and ξ4, and hence can be solved for ξ2 and ξ3 as +ξ2 = ξ4 h(∆t−, ∆t+), +ξ3 = ξ4 k(∆t−, ∆t+), +(43) +where the coefficients h(∆t−, ∆t+) and k(∆t−, ∆t+) are the following functions of ∆t− and ∆t+ +h(∆t−, ∆t+) = +� +eA+∆t+� +14 +� +e−A−∆t−� +13 − +� +eA+∆t+� +13 +� +e−A−∆t−� +14 +� +eA+∆t+� +13 +� +e−A−∆t−� +12 − +� +eA+∆t+� +12 +� +e−A−∆t−� +13 +, +k(∆t−, ∆t+) = +� +eA+∆t+� +14 +� +e−A−∆t−� +12 − +� +eA+∆t+� +12 +� +e−A−∆t−� +14 +� +eA+∆t+� +12 +� +e−A−∆t−� +13 − +� +eA+∆t+� +13 +� +e−A−∆t−� +12 +. +Turning back the attention to equations (41), these can be rewritten, using equations (43), as +� +� +x2 +x3 +x4 +� +� = ξ4 +� +e−A−∆t− +� +� +h(∆t−, ∆t+) +k(∆t−, ∆t+) +1 +� +� , +� +� +η2 +η3 +η4 +� +� = ξ4 � +eA+∆t+ +� +� +h(∆t−, ∆t+) +k(∆t−, ∆t+) +1 +� +� , +(44) +where +� +e−A−∆t− and � +eA+∆t+ are the submatrices obtained from the original matrices through elimina- +tion of the first row and column. +Now, to be solutions of the eigenvalue problem (28), the two vectors given by equation (44) have to +be parallel, namely, +η2 +x2 += η3 +x3 += η4 +x4 += µ, +(45) +24 + +providing two equations for the unknowns ∆t− and ∆t+, as follows +� +eA+∆t+� +22 h(∆t−, ∆t+) + +� +eA+∆t+� +23 k(∆t−, ∆t+) + +� +eA+∆t+� +24 +� +e−A−∆t−� +22 h(∆t−, ∆t+) + +� +e−A−∆t−� +23 k(∆t−, ∆t+) + +� +e−A−∆t−� +24 += +� +eA+∆t+� +32 h(∆t−, ∆t+) + +� +eA+∆t+� +33 k(∆t−, ∆t+) + +� +eA+∆t+� +34 +� +e−A−∆t−� +32 h(∆t−, ∆t+) + +� +e−A−∆t−� +33 k(∆t−, ∆t+) + +� +e−A−∆t−� +34 += +� +eA+∆t+� +42 h(∆t−, ∆t+) + +� +eA+∆t+� +43 k(∆t−, ∆t+) + +� +eA+∆t+� +44 +� +e−A−∆t−� +42 h(∆t−, ∆t+) + +� +e−A−∆t−� +43 k(∆t−, ∆t+) + +� +e−A−∆t−� +44 +. +(46) +The system (46) is nonlinear and must be solved numerically. Once the time intervals ∆t− and ∆t+ +are known, the eigenvector x can be computed from (44)1, whereas the eigenvalue µ is obtained from +equation (45). +The calculation of the time intervals ∆t− and ∆t+ is far from trivial, as the determining equations in- +volve transcendental trigonometric functions, so that there are infinite values of ∆t− and ∆t+ satisfying +them. Among these infinite values, only those corresponding to motions crossing the switching mani- +fold the first time at ∆t− and the second at ∆t− + ∆t+ have to be retained, while the other disregarded, +because they refer to orbits that cross the switching manifold multiple times, but erroneously remain in +the same subdomain. This situation, sketched in Fig. 9, has been solved through: (i) an estimation of +the maximum time intervals ∆t± +max = max{3π/(2ω± +1 ), 3π/(2ω± +2 )} within which the first intersections of +the orbit with the switching manifold Σ occur, so that the solution is searched for in the bounded time +domain [0, ∆t− +max] × [0, ∆t+ +max] (details are reported in [24]); (ii) a systematic elimination of the solutions +lacking mechanical meaning. In fact, an obtained solution is meaningful if and only if the following +conditions are met: +• First crossings. The time instants ∆t− and ∆t− +∆t+ correspond respectively to the first and sec- +ond intersection time of an orbit on the invariant cone with the switching manifold Σ. Therefore, +given the initial vector x (obtained from the above algorithm), the conditions have to be checked: +(i.) that the value ∆t− is the smallest solution of equation (42)1, and (ii.) that the value ∆t+ is the +smallest solution of equation (42)2. +• Switching conditions. Any solution representing an invariant cone must fulfill +x3 = ˙ξ(0) < 0, +ξ3 = ˙ξ(∆t−) > 0, +η3 = ˙ξ(∆t− + ∆t+) < 0, +(47) +denoting the fact that the orbit on the invariant cone must initially cross the switching manifold +and enter in the negative subdomain (at t = 0), while at t = ∆t− has to enter into the positive one, +and finally at ∆t− + ∆t+ the negative subdomain has to be entered again. However, a solution +to the problem (46) yields a vector x defined except for the sign, which can always be adjusted, +so that instead of conditions (47) for a solution to be meaningful, the time intervals ∆t− and ∆t+ +have to satisfy +˙ξ(0) ˙ξ(∆t− + ∆t+) > 0, +˙ξ(0) ˙ξ(∆t−) < 0, +meaning that the assumption made on the first entered subdomain (the negative) is arbitrary and +the opposite assumption could be equally made. +25 + +x +ξ +η +Σ +V+ +V− +x +ξ +η +x +ξ +η +wrong solution: y(t−) is not the first intersection of +the blue orbit with the switching manifold +wrong solution: y(t− + t+) is not the first intersec- +tion of the red orbit with the switching manifold +Figure 9: Sketch of a motion starting at point x and initially developing inside the negative branch of the mechan- +ical system (blue solid line). At point ξ the motion crosses the switching manifold and further develops +inside the positive branch (red solid line). However, from a purely mathematical point of view, the con- +tinuation with wrong equations referred to the negative branch (blue dashed line) are still solutions of +equations (46) and their orbits cross the switching manifold at several subsequent points. All the latter +points have to be disregarded, because they represent the solution of a smooth system, different from that +under analysis. Something analogous happens with the solution continuation of the red line (represented +dashed). Therefore, there are infinite solutions on the switching manifold, and the selection of the correct +points is the harder problem to be solved in finding instability cones. +5.2 +A non-smooth structure with an unstable invariant cone +5.2.1 +Existence of an unstable invariant cone +It is instrumental now to introduce the following non-dimensional parameters +ζ± = R± +l +k = k1l2 +k2 +γ = Fl +k2 +σ = ys +l , +(48) +so that equations (13) governing the dynamics of the piecewise-smooth structure become +Θ +� +�� +1 +1 +2 +1 +2 +1 +3 +� +�� +� +� +˜ξ,ττ(τ) +˜φ,ττ(τ) +� +� + +� +��� +k +ζ± +(ζ± ∓ σ) + 1 ∓ γζ± +ζ2 +± +−γζ± ± 1 +ζ± +± 1 +ζ± +1 +� +��� +� +� +˜ξ(τ) +˜φ(τ) +� +� = 0 +where the non-dimensional time τ = t/T and the non-dimensional Lagrangian coordinates ˜ξ(τ) = +ξ(t)/l, ˜φ(τ) = φ(t) are introduced, together with the dimensionless mass density +Θ = +ρl3 +T 2k2 +. +26 + +Moreover, the non-dimensional parameter singling out the ratio between the two radii of the two con- +straints is defined as +χ = ζ− +ζ+ . +The presence of an unstable invariant cone has been numerically detected for a broad range of val- +ues of the above parameters. As a paradigmatic example, the following values for the parameters are +considered in this section +ζ+ = R+ +l += 0.6, +k = k1l2 +k2 += 0.3, +γ = Fl +k2 += 0.06, +σ = ys +l = 0, +Θ = +ρl3 +T 2k2 += 1, +χ = ζ− +ζ+ = 6 +(49) +This reference numerical example contains all the most relevant features of the new kind of unstable +structural behaviour disclosed in the present article. In the sequel all quantities are dimensionless, +according to the normalization (48), and a superimposed dot stands for the derivative with respect to +the dimensionless time τ. However the dimensionless Lagrangian coordinates are denoted ξ and φ +(without tilde) to ease the notation. +For the above geometry and loading, a piecewise invariant cone described by the initial condition +x = y0 = +� +ξ(0), φ(0), ˙ξ(0), ˙φ(0) +� += [0, −0.00838564, −0.372424, 0.928025] +is present, with a multiplier µ = 1.079995. The intersection time intervals, calculated using the algo- +rithm presented in the previous section for the detection of the invariant cone (and later confirmed by +the numerical simulation of the mechanical system), are ∆τ − = 0.637108 and ∆τ + = 2.981694. +The critical loads for flutter and divergence instability for the smooth substructures that compose +the non-smooth mechanical structure can be analytically determined as +γ+ +flu,div = +−2ζ+(3ζ+ − 2)(ζ+(ζ+(k + 3) − kσ − 3) + 1) ± 2 +√ +3 +� +ζ5 ++(3ζ+ − 2)2k(ζ+ − σ) +(2 − 3ζ+)2ζ2 ++ +, +γ− +flu,div = +−2ζ−(3ζ− + 2)(ζ−(ζ−(k + 3) + kσ + 3) + 1) ± 2 +√ +3 +� +ζ5 +−(3ζ− + 2)2k(ζ− + σ) +ζ2 +−(3ζ− + 2)2 +, +(50) +where γ+ +flu,div (γ− +flu,div) refers to the structure with positive (negative) curvature, while the critical loads +for flutter correspond to the minimum absolute values. Therefore, a substitution of the values (49) into +(50) leads to the conclusion that the substructures are both stable when considered separately for an +assumed tensile load γ = 0.06, because their critical loads are much higher in absolute value (one is +negative and therefore compressive): +γ− +flu = −1.83477, +and +γ+ +flu = 0.774567. +The stability of each smooth subsystem can be observed in the phase portraits reported in Figs. 10 +and 11, for the system with negative and positive curvature, respectively. The portraits refer to the +linearized solution, so that the orbits evolve remaining confined within the neighbourhood of the origin, +representing the equilibrium point. +27 + +−0.4 −0.2 +0 +0.2 +0.4 +−0.5 +0 +0.5 +ξ +˙ξ +−0.5 +0 +0.5 +−1 +−0.5 +0 +0.5 +1 +φ +˙φ +−0.4 −0.2 +0 +0.2 +0.4 +−0.5 +0 +0.5 +ξ +φ +Figure 10: Phase portraits, showing stable response, for one (with positive curvature, see the inset on the left) of +the two smooth systems forming the piecewise linear structure analyzed in Figs. 12 and 13 +−0.2 +0 +0.2 +−0.5 +0 +0.5 +ξ +˙ξ +−0.5 +0 +0.5 +−1 +−0.5 +0 +0.5 +1 +φ +˙φ +−0.2 +0 +0.2 +−0.5 +0 +0.5 +ξ +φ +Figure 11: Phase portraits, showing stable response, for one (with negative curvature, see the inset on the left) of +the two smooth systems forming the piecewise linear structure analyzed in Figs. 12 and 13 +Although the two subsystems forming the piecewise linear structure are stable when considered +separately, the combination of them is unstable, as can be appreciated in Figs. 12 and 13, reporting a +solution of the linearized equations of motion belonging to the unstable invariant cone. The evolution +in time of the Lagrangian coordinates ξ and φ shows an exponential increase of the amplitude of mo- +tion, analogous to the behaviour of a smooth mechanical system when flutter instability occurs. The +phase portraits show an orbit laying on the invariant cone, so that each phase portrait is a projection +of this 4-dimensional invariant manifold onto a 2-dimensional plane. The orbit starting close to the +origin (representing the trivial equilibrium configuration) evolves following a spiralling out motion, +corresponding to an unstable behaviour. +28 + +0 +50 +100 +150 +200 +250 +0 +20 +40 +60 +τ +ξ +0 +50 +100 +150 +200 +250 +−100 +−50 +0 +τ +φ +Figure 12: Evolution of the Lagrangian parameters in (dimensionless) time, for a non-smooth elastic structure +(shown in the inset on the right) composed of two stable substructures. Exponential blow-up demon- +stratesinstability. +−0.5 +0 +0.5 +1 +1.5 +2 +2.5 +−2 +0 +2 +ξ +˙ξ +−5 +−4 +−3 +−2 +−1 +0 +1 +−4 +−2 +0 +2 +4 +6 +φ +˙φ +−0.5 +0 +0.5 +1 +1.5 +2 +2.5 +−4 +−2 +0 +ξ +φ +Figure 13: Phase portraits for the non-smooth mechanical system analyzed in Fig. 12 (and shown on the lower +part, right), evidencing instability although the two component substructures are stable. +29 + +5.2.2 +Instability of the structure in the nonlinear range +All the theoretical and numerical results obtained in the previous sections are based on the linearization +of the equations of motion describing the system and, in particular, on the piecewise linear response +resulting from the combination of the two linearized responses for each subsystem forming the non- +smooth structure. For a smooth dynamical system, the linearization of the equations of motion near +an equilibrium configuration is a classical strategy for determining whether or not the considered con- +figuration is stable. When the equilibrium of the linearized system is not marginally stable (i.e. the real +part of the eigenvalues of the Jacobian matrix vanishes), the Lyapunov theorem assures that the results +obtained for the linearized case can be extended to the original nonlinear one. +A structure with a piecewise-linear behaviour cannot be further linearized, so that the Lyapunov +theorem cannot be applied. An extension of this theorem to the nonlinear case of piecewise smooth +dynamical system has been provided, see [12], under regularity assumptions which are satisfied for the +structures under examination. +Following this extension, the instability of the structures in a fully nonlinear range is expected, and +indeed a direct integration of the nonlinear equations of motion for the previously analyzed structures +confirms the presence of an unstable behaviour. +0 +100 +200 +300 +400 +500 +600 +0 +0.1 +0.2 +0.3 +τ +ξ +0 +100 +200 +300 +400 +500 +600 +−0.6 +−0.4 +−0.2 +0 +0.2 +τ +φ +0 +1 +2 +·10−3 +−2 +0 +2 +·10−3 +ξ +˙ξ +−4 +−2 +0 +·10−3 +−5 +0 +5 +·10−3 +φ +˙φ +0 +1 +2 +·10−3 +−4 +−2 +0 +·10−3 +ξ +φ +Figure 14: Phase portrait and evolution diagrams for the non-smooth structure (its linearized analysis is reported +in Figs. 12 and 13 nonlinear case +In particular, the numerical solution of the nonlinear dynamics describing the reference structure is +depicted in Fig. 14, for the same initial conditions used in the piecewise linear analysis, namely, vector +y0, is selected now with a sufficiently small modulus to start from a neighbourhood of the equilibrium +configuration, thus y0 has been scaled as +y0 = +� +ξ(0), φ(0), ˙ξ(0), ˙φ(0) +� += fs × [0, −0.00838564, −0.372424, 0.928025] , +with the scaling factor fs = 0.001. +30 + +The evolution of the Lagrangian generalized coordinates presents an exponential growth for small +values of τ that can be associated to the presence of a quasi-invariant cone, as defined in [12]. The orbits +do not reach a limit cycle as for the Hopf bifurcation in smooth structures, but evidence blowing-up +oscillations which reach a peak and then decrease in amplitude until near the initial amplitude. This +motion repeats itself several times (in a way similar to beats) and the peak values are found to be almost +constant and independent of the modulus of the initial condition y0 (the cases fs = 10−4 and fs = 10−5 +have also been tested and for all cases a peak value of approximately ξ = 0.3 and φ = −0.6 have been +found, Fig. 14 a-b). These features of the structural dynamics denote a complex unstable behaviour. +5.3 +A non-smooth structure evidencing flutter in both tension and compression +5.3.1 +Existence of an unstable invariant cone in both tension and compression +The reference solution considered in the previous section is just an example of instability related to the +non-smoothness of the structure. Considering other combinations of parameters leading to instability, +the topological structure of the solution in the phase space has been found to remain similar, so that an +orbit is found which spirals out from the origin. In certain cases, the evolution of the solution can be +more or less irregular, as can be seen in the example reported in Fig. 15, where the wide difference in +the values of ∆t− and ∆t+ leads to an orbit that remains for a longer time in the negative subsystem. +It is interesting to note that non-smooth structures presenting this kind of instability can be found +both for tensile and compressive follower forces. For instance, consider the following set of design +parameters +ζ+ = 0.5, +k = 0.1, +σ = 0, +Θ = 1, +χ = 2, +together with the two cases of tensile or compressive follower force +γA = Fl +k2 += −1.5, +γB = Fl +k2 += 0.75. +(51) +Note that at loads (51) the substructures are both stable, as their critical loads for flutter are +γ− +flu = −2.62091, +γ+ +flu = 1.10455. +In the case of compressive load γA = −1.5, the initial condition defining the invariant cone is +xA = yA +0 = +� +ξ(0), φ(0), ˙ξ(0), ˙φ(0) +� += [0, −0.00594364, −0.608652, 0.793415] , +and the eigenvalue is equal to µ = 2.481844, a value higher than the reference structure, in which the +growth rate of the phase vector after each period ∆t− + ∆t+ was only 7.99%. The intersection time +intervals can be calculated to be ∆t− = 9.797295 and ∆t+ = 1.595396. +In the case of tensile load γB = 0.75, the initial condition defining the invariant cone is +xB = yB +0 = +� +ξ(0), φ(0), ˙ξ(0), ˙φ(0) +� += [0, 0.086944, −0.360442, 0.928721] +and the eigenvalue is equal to µ = 2.486877, very close to the value for the compressive load, so that +the growth rate of the solution on the invariant cone after each cycle is almost the same for both cases. +The intersection time intervals can be calculated to be ∆t− = 0.311784 and ∆t+ = 4.132277. +31 + +0 +20 +40 +60 +80 +−100 +0 +100 +τ +ξ +0 +20 +40 +60 +80 +−200 +−100 +0 +100 +τ +φ +−20 +−10 +0 +10 +−20 +0 +20 +ξ +˙ξ +−30 +−20 +−10 +0 +−40 +−20 +0 +20 +40 +φ +˙φ +−20 +−10 +0 +10 +−30 +−20 +−10 +0 +ξ +φ +Figure 15: Phase portrait and evolution diagrams for a non-smooth elastic structure displaying instability in com- +pression (although the component substructures are stable). +32 + +0 +20 +40 +60 +80 +0 +2 +4 +6 +·106 +τ +ξ +0 +20 +40 +60 +80 +−1 +−0.5 +0 +·107 +τ +φ +0 +100 +200 +−200 +−100 +0 +100 +ξ +˙ξ +−600 +−400 +−200 +0 +0 +500 +φ +˙φ +0 +100 +200 +−600 +−400 +−200 +0 +ξ +φ +Figure 16: Phase portrait and evolution diagrams for a non-smooth elastic structure displaying instability in ten- +sion (although the component substructures are stable). +5.3.2 +Instability of the nonlinear structure +A nonlinear analysis has been performed for the structure linearly analyzed in Figs. 15 and 16, evidenc- +ing unstable behaviour both in tension and compression, to confirm the instability detected from the +linearized analysis. The initial condition has been scaled as +yA +0 = +� +ξ(0), φ(0), ˙ξ(0), ˙φ(0) +� += fs × [0, −0.00594364, −0.608652, 0.793415] , +yB +0 = +� +ξ(0), φ(0), ˙ξ(0), ˙φ(0) +� += fs × [0, 0.086944, −0.360442, 0.928721] , +for tensile and compressive loads, respectively, with the scaling factor set equal to fs = 10−6. +The behaviour of these cases, shown in Fig. 17 and Fig. 18, is qualitatively different from that of the +nonlinear reference structure reported in Sec. 5.2.2, because ‘beats’ are not present. However, the orbits +are more irregular, showing an erratic behaviour. However, the instability of the system is evident for +both tensile and compressive loads, since the orbits evolve along the invariant cone spiraling away from +the equilibrium configuration. +33 + +0 +100 +200 +300 +400 +500 +−0.4 +−0.2 +0 +0.2 +τ +ξ +0 +100 +200 +300 +400 +500 +−0.6 +−0.4 +−0.2 +0 +τ +φ +−0.1 −0.05 +0 +0.05 +−0.1 +0 +0.1 +ξ +˙ξ +−0.1 +0 +−0.2 +0 +0.2 +φ +˙φ +−0.1 −0.05 +0 +0.05 +−0.1 +0 +ξ +φ +Figure 17: Fully nonlinear behaviour of the structure linearly analyzed in Fig. 15. The phase portrait and evolution +diagrams for the case confirm instability in tension, resulting in a highly irregular motion. +34 + +0 +20 +40 +60 +80 +0 +0.2 +0.4 +0.6 +0.8 +τ +ξ +0 +20 +40 +60 +80 +−1.5 +−1 +−0.5 +0 +τ +φ +0 +0.2 +0.4 +0.6 +0.8 +−0.5 +0 +0.5 +1 +ξ +˙ξ +−1.5 +−1 +−0.5 +0 +−1 +0 +1 +φ +˙φ +0 +0.2 +0.4 +0.6 +0.8 +−1.5 +−1 +−0.5 +0 +ξ +φ +Figure 18: Fully nonlinear behaviour of the structure linearly analyzed in Fig. 16. The phase portrait and evolution +diagrams for the case confirm instability in tension, resulting in a blowing up motion. +6 +Conclusions +A class of elastic structures has been disclosed, exposed to a new kind of instability, which, although +already elaborated from a mathematical point of view, was never directly related to elastic structural +systems. The instability results from the combination of both non-conservative follower load and non- +smoothness of the equations governing the dynamics of the mechanical system. From this point of view, +the proposed model is a discrete and simplified prototype of nonassociative elastoplasticity or frictional +sliding, as it shares with these theories both the lack of self-adjointness of the governing differential +operator and the piecewise linearity. +We provide the first example of the use of the invariant cone theory as an instability criterion for +elastic structures, permitting the design of an unstable structure as a fusion of two stable structures. +The instability is fully explained and motivated from a mechanical perspective and is shown to be +similar to the flutter instability occurring in smooth systems under follower loads. +From the point of view of applications, keeping into account that nonconservative follower forces +are fully feasible [25], we introduce a new design paradigm to avoid previously unknown structural +instabilities or to design extremely deformable structures to be employed as sensors, or for energy +harvesting, or as building blocks for archtected materials. +7 +Acknowledgments +The authors acknowledge financial support from ERC-ADG-2021-101052956-BEYOND. +35 + +8 +Appendix +8.1 +Flutter and divergence critical loads for smooth constraints +For a smooth profile with continuous curvature, see Fig. 2, the flutter and divergence critical loads +can be computed by solving the linearized equations (7), as described in Sec. 3.1. Consequently, any +profile can be approximated by its osculating circle, since only the local curvature enters the linearized +equations. +Therefore, assuming a circular profile defined by the dimensionless signed curvature κ = ±l/R± = +±1/ζ±, where ζ± = R±/l > 0 is the normalized radius of curvature, the first and second invariants I1 +and I2 of the matrix Γ = −M −1K, see eq. (19), are given by +I1 = 2 +Θ +� +γ(2κ − 3) − 2(3 + k + κ2 − 3κ) + 2kκσ +� +, +I2 = 12k +Θ2 (1 − κσ), +in terms of the dimensionless quantities defined in (48). The flutter and divergence critical loads can +then be computed from the equation +I2 +1 − 4I2 = 0, +of the parabola defining the critical condition in the I1−I2 plane, see Fig. 5. The solution of this equation +provides the critical loads +γflu,div = +2 +2κ − 3 +� +3 + k − 3κ + κ2 − kκσ ± +� +3k(1 − κσ) +� +, +where it is understood that the critical load for flutter (divergence) corresponds to the minimum (max- +imum) absolute value. Since the term inside the braces is always positive, for any k > 0, κ ∈ R and +σ < 1/κ, it follows that the follower and divergence loads are compressive for κ < 3/2 and tensile for +κ > 3/2, see Fig. 19. +36 + +−10 +−5 +0 +5 +10 +−40 +−20 +0 +20 +40 +κ = 3/2 +k = 2.5 +κ +γ +−10 +−5 +0 +5 +10 +−40 +−20 +0 +20 +40 +κ = 3/2 +k = 5 +κ +γ +−10 +−5 +0 +5 +10 +−40 +−20 +0 +20 +40 +κ = 3/2 +k = 25 +κ +γ +−10 +−5 +0 +5 +10 +−40 +−20 +0 +20 +40 +κ = 3/2 +k = 50 +κ +γ +Flutter critical load +Divergence critical load +Flutter instability +Divergence instability +Figure 19: Flutter and divergence instabilities for smooth circular constraints: the flutter (blue curve) and diver- +gence (yellow curve) critical loads are reported as a function of the dimensionless signed curvature κ +and four values of the dimensionless stiffness k = {2.5, 5, 25, 50}. +37 + +References +[1] +M. Branicky. “Multiple Lyapunov functions and other analysis tools for switched and hybrid +systems”. 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DOI: 10.1016/j. +jmps.2018.03.024. +39 + diff --git a/XtE4T4oBgHgl3EQfNgyO/content/tmp_files/load_file.txt b/XtE4T4oBgHgl3EQfNgyO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4c233b36874929ac865e91595ec03b647d5445b3 --- /dev/null +++ b/XtE4T4oBgHgl3EQfNgyO/content/tmp_files/load_file.txt @@ -0,0 +1,1112 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf,len=1111 +page_content='Fusion of two stable elastic structures resulting in an unstable system MARCO ROSSI1, ANDREA PICCOLROAZ1, AND DAVIDE BIGONI*1 1DICAM, University of Trento, via Mesiano 77, Trento, Italy Abstract It is shown that a compound elastic structure, which displays a dynamic instability, may be de- signed as the union (or ‘fusion’) of two structures which are stable when separately analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The compound elastic structure has two degrees of freedom and is made up of a rigid rod connected with two springs to a smooth support, which evidences a jump in the curvature at the equilibrium configura- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Instability is proven in a linearized context and is related to the application of a non-conservative load of the follower type, so that the instability disappears under dead loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In the fully nonlinear range, the instability is also confirmed through numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The obtained results may be useful in the design of new mechanical sensors, or devices for energy harvesting, or architected mate- rials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In addition, our findings have conceptual implications on piecewise-linear theories of mechanics such as for instance plasticity or frictional contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Keywords: Flutter instability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Tensile instability 1 Introduction Consider two elastic structures loaded within their stable range and imagine that these represent two parts of a third elastic structure obtained through the ‘fusion’ of the initial structures, becoming the ‘constituents’ of the new structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Stability of the compound structure is expected when the latter is subject to the same load at which the two component structures are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' This intuitive belief is true for dead loading and smooth systems, but it is shown in the present article to be false for follower loads and non-smooth mechanical behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' From a purely mathematical point of view, the singularity of a non-smooth behaviour obtained as combination of two stable dynamical systems may be expected, as was advocated through a purely mathematical example by Branicky [1] and Carmona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' [2] for abstract systems of non-smooth differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' However, an elastic structure exhibiting this peculiar kind of instability has never been discovered so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The purpose of the present article is to fill this gap through the invention and the theoretical and numerical analysis of a structure designed to demonstrate instability of a simple two d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' non-smooth mechanical system that is composed of two stable smooth subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' This finding brings a mathematical result into the realm of mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The elastic structures considered here are of the type shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 1, consisting of a rigid rod with an end sliding on a smooth profile, while the other end is subject to a tangentially follower load, remaining parallel to the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The deformed configuration of the structure is defined by two degrees of freedom, specifically the arc length distance characterizing the roller position along the profile and the angle of rotation of the bar with respect to the vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The structure is stiffened by a longitudinal spring connecting the roller to a fixed point and a rotational spring interposed betweeen the rod and the roller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Corresponding author: e-mail: bigoni@ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='unitn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' phone: +39 0461 282507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='04957v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='class-ph] 12 Jan 2023 0 20 40 60 80 0 1 2 τ ξ 0 20 40 60 80 0 1 2 τ ξ 0 20 40 60 80 0 1 2 τ ξ Figure 1: Two stable smooth subsystems with positive and negative curvature of a sliding constraint (upper part: left and center) and the fusion of these two structures, namely, a compound non-smooth structure dis- playing instability (upper part: right), although the two ‘components’ are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The trajectories of the end of the structures is also reported for vibrational motion, together with the corresponding arc-length ξ vs time τ behaviours, showing sinusoidal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' stable, lower part: left and center) and unstable (lower part: right) oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The tensile force acting at the free end of the rods is tangentially follower and the same for all the three structures, laying well below the critical load for instability in the case of the two smooth ‘component systems’, both displaying motions confined in a neighborhood of the trivial equilib- rium configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Differently, a flutter-like instability is observed for the composite structure (upper part: right), as evidenced by the unstable and exponentially growing oscillations of the loaded end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The first two structures (from left) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 1 are characterized by a circular profile (with posi- tive and negative curvature, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' These structures are described by smooth dynamical systems and suffer flutter and divergence instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Assuming that the magnitude of the follower force is well below the critical values, the vertical trivial equilibrium configuration is stable, so that a small perturba- tion in the initial conditions generates a motion which remains confined within a small neighborhood of the fixed point, as shown in the lower part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 1 (obtained through numerical integration of the nonlinear equations of motion and representing the arc-length distance ξ traveled by the roller as a function of the elapsed time τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The third elastic structure shown on the right in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 1 is also described by two degrees of freedom and is obtained as the ‘fusion’ of the two structures sketched on the left and center of the same figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' This new structure is characterized by a smooth sliding profile, which evidences a jump in the curvature so that the dynamics is characterized by piecewise smooth differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The tangent to the profile at the junction is horizontal and at this point the longitudinal spring is unloaded, so that the vertical configuration of the rigid rod is the trivial equilibrium configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' If this structure is subject to a load smaller than the critical loads of the two ‘generating structures’, it might be expected that the structure would be stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' This becomes true when the load is conservative, but now the load is follower, so that: it is shown in this article that the non-smooth structure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 1 right), obtained as the fusion of two smooth structures (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 1 left and center), may be dynamically unstable at a load well inside the stability domains of both the generating structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Indeed, at a load well below both critical loads of the smooth constituent structures, the compound 2 non-smooth structure exhibits the exponentially growing oscillation illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 1 (lower part, obtained through numerical integration of the nonlinear equations of motion), resembling the flutter instability, occurring in smooth systems, for instance the celebrated Ziegler double pendulum [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Note that the structures shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 1 are similar to those investigated in [4] and [5], but now the load is non-conservative and follower, so that the instability landscape results completely changed and, in particular, the possibility arises of finding instabilities unrelated to the instabilities of the component structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The above-stated result, referred to the structure shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 1 on the right, follows from the com- bination of two features, namely, the presence of (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=') a non-conservative follower load and (ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=') a jump in curvature in the sliding constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The latter feature implies that the acceleration is discontinuous at the junction between the two circular profiles at the basis of the structure and thus the system of governing equations becomes non-smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Piecewise-smooth dynamical systems are common in the mechanics of solids and structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In fact, elastoplasticity and contact with friction are based on piece-wise linear equations of the rate type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' bi-linear elasticity defines solids with different tensile and compressive elastic moduli;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' structures im- pacting against unilateral constraints involve two sets of equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' These systems are known to exhibit peculiar forms of instability, such as for instance stick and slip motion for frictional contact [6], or blowing-up vibrations for non-associative elastoplasticity [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In particular, every elastoplastic constitutive equation is always piecewise linear in the rate response and nonassociative flow rules lead to a lack of symmetry similar in essence to that induced by follower loads in structural elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' There- fore, the structures designed in the present paper are governed by equations sharing strong similarity with elastoplasticity, so that our results lead to important conjectures in that field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' More in detail, stabil- ity analysis in elastoplasticity is performed on the so-called ‘comparison solids’ [9], which are the exact counterpart of the component structures introduced here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Stability of the comparison solids is usually assumed to imply stability of the true piecewise linear behaviour, but our structural examples demon- strate that this may be false, an implication that would completely revolutionize the stability theory for nonassociative elastoplasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Accordingly to its interest in mechanics, instability of piecewise-smooth dynamical systems has re- cently attracted a growing interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The first proof that a compound system generated as the fusion between two stable systems can be unstable is due to Carmona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' For non-smooth dynamical systems, they have provided a sufficient condition for instability, based on the detection of a so-called ‘invariant cone’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' This condition has been further developed in various directions [10–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' All these works provide a mathematical framework and open new directions for research in bifurcation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' However, applications of the mathematical setting to mechanics are scarce and so far limited to simpli- fied systems characterized by Coulomb friction [14–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Therefore, the objective of the present article is to develop the analysis of invariant cone to demonstrate instability of the structure shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 1 on the right, obtained as the fusion of two stable structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Although a general concept, the invariant cone can practically be applied only to the linearized equa- tions of motion governing a mechanical model, so that structures which are proven to be unstable on the basis of a linearized analytical treatment are also numerically investigated in this article, to provide a complete picture of their mechanical behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In this way it is proven that in all cases in which the lin- earized equations display instability, the latter persists also when the fully nonlinear piecewise-smooth problem is examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' It has to be highlighted that the mathematical background so far developed only consists in suffi- cient conditions for instability, so that when these conditions are not fulfilled, nothing can be concluded concerning stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' This situation is reflected in the results presented in the present article, where only examples of piecewise-smooth unstable structures are obtained, while the behaviour of the same struc- tures at different loads is usually unknown (though open to numerical investigation), as the sufficient condition fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' After the introduction of the class of the addressed elastic struc- tures (Section 2) , the differential equations governing their dynamics are formulated in Section 3, where the concept of invariant cone is provided as a sufficient condition for instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The properties of invari- ant cones are demonstrated in Section 4, where it is shown, under broad hypotheses, that an unstable cone is always attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' An algorithm to detect invariant cones for two d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' mechanical systems is presented in Section 5 and numerical results on the linearized analysis are developed demonstrating the instability, which is also finally confirmed through numerical solutions where the nonlinear behaviour 3 is fully kept into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 2 Elastic structure on a curved constraint with a jump in curvature 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='1 Nonlinear dynamics A two d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' elastic structure is considered, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 2, composed of a rigid bar, of mass density ρ and length l, which is loaded at one end (point L) with a follower force, positive when tensile, of constant modulus and parallel to the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' At the other end (point P ), the bar is connected to an elastic hinge of rotational stiffness k2, which is constrained to move, without friction, along a smooth profile γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The elastic hinge at the lower end of the rigid bar is linked to a fixed point S (singled out by the coordinates xs and ys), with a longitudinal linear spring of stiffness k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The rigid smooth profile, along which point P is constrained to move, plays a fundamental role in the mechanics of the structure shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The formulation presented below is general enough to include profiles with discontinuous curvature, provided that higher-order derivatives are understood in the generalized sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The profile may be described by parametric equations (x(ξ), y(ξ)) in the plane Oxy defined by the two unit vectors e1 and e2, so that the tangent, the unit tangent, and the principal normal at the generic point P (ξ) of the profile are P ′ = x′(ξ)e1 + y′(ξ)e2, t = P ′ |P ′|, n = t′ |t′|, (1) in which a dash ( )′ denotes differentiation with respect to the parameter ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The signed curvature of the profile is defined as κ = α′ |P ′|, where α is the angle between P ′ and the x-axis, α(ξ) = arctan y′(ξ) x′(ξ), (2) defined in such a way that t and the unit vector obtained through an anticlockwise rotation of t by π/2 may be represented as t = cos α(ξ) e1 + sin α(ξ) e2, m = − sin α(ξ) e1 + cos α(ξ) e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The derivative of equation (2) with respect to ξ leads to α′ = x′y′′ − x′′y′ x′2 + y′2 = m · t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 4 k1 k2 ξ(t) φ(t) α(ξ) α(ξ) t(ξ) F L L S O P e2 e1 γ (x(ξ), y(ξ)) l, ρ m(ξ) F el eφ Figure 2: A 2 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' elastic structure made up of a rigid bar constrained to move with an elastic hinge on a curved profile and subject to a tensile follower force (remaining parallel to the bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Note that the parameter ξ may be identified with the arc length of the profile and in such case |P ′| = 1 so that α′ coincides with the signed curvature κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The deformation of the structure is described by two generalized coordinates: the arc length ξ of the curve describing the profile and the angle φ between the rigid bar and the y-axis, positive if clockwise, which are assumed to be continuous functions of time, namely, ξ = ξ(t) and φ = φ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Two unit vectors el and eφ are defined, attached to the rigid bar and aligned parallel and transverse to it respectively, as el = sin φ e1 + cos φ e2, eφ = cos φ e1 − sin φ e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (3) Differentiation of equations (3) yields (denoting with a superimposed dot the derivative with respect to time) ˙el = ˙φ eφ, ˙eφ = − ˙φ el, and therefore the positions of the end points P and L and of the generic point R of the bar (at distance r from P ) can be written as P = x(ξ)e1 + y(ξ)e2 + O, L = lel + P , R = r el + P , where it can be noted that R(l) = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The velocities of points P , L and R are ˙P = ˙ξP ′, ˙L = l ˙φeφ + ˙P , ˙R = r ˙φeφ + ˙P , where P ′ provides the tangent to the rigid profile at P , equation (1)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The accelerations can be calcu- lated as ¨P = ¨ξP ′ + ˙ξ2P ′′, ¨L = l ¨φeφ − l ˙φ2el + ¨P , ¨R = r ¨φeφ − r ˙φ2el + ¨P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The follower force has a constant modulus F and remains always parallel to the rigid bar, F = F el.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The longitudinal spring with stiffness k1 produces an elastic force proportional to the vector P − S F s = −k1(P − S), while the rotational spring of stiffness k2 applies a moment (positive when anticlockwise) to the end P of the rigid bar, which is given by M = −k2(φ + α), 5 where α has been defined in formula (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The equations of motion governing the dynamics of the mechanical system under analysis can be found using the principle of virtual work, that can be written as F · δL − k1(P − S) · δP − k2(φ + α)(δφ + δα) − ρ � l 0 ¨R · δR dr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (4) The external work due to the follower force is F · δL = Fel · δP ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' while � l 0 ¨R · δR dr = l2 2 �2l 3 ¨φ + ¨P · eφ � δφ + l � l 2 ¨φeφ − l 2 ˙φ2el + ¨P � δP ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' so that equation (4) can be rewritten as �� F + ρl2 2 ˙φ2 � el − ρl2 2 ¨φeφ − k1(P − S) − ρl ¨P � P ′δξ − k2(φ + α)α′δξ − � k2(φ + α) + ρl2 2 �2l 3 ¨φ + ¨P · eφ �� δφ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' which,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' invoking the arbitrariness of δξ and δφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' can be split into the two equations governing the dy- namics of the structure � F + ρl2 2 ˙φ2 � (x′ sin φ + y′ cos φ) − ρl2 2 ¨φ (x′ cos φ − y′ sin φ) − k1 [x′(x − xS) + y′(y − yS)] − ρl � ¨ξ � x′2 + y′2� + ˙ξ2 (x′x′′ + y′y′′) � − k2(φ + α)α′ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' k2(φ + α) + ρl3 3 ¨φ + ρl2 2 � ¨ξ (x′ cos φ − y′ sin φ) + ˙ξ2 (x′′ cos φ − y′′ sin φ) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (5) The nonlinear system (5) can be solved for ξ and φ, so that it can be equivalently written as ¨q(t) = g(q(t), ˙q(t)), (6) where q(t) = [ξ(t), φ(t)]T is a vector collecting the Lagrangian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Alternatively to the above, the Hamiltonian formulation can be used, so that the system (6) becomes a first-order differential nonlinear system ˙y(t) = f(y(t)), where the phase vector y(t) = [q(t), ˙q(t)]T = [ξ, φ, ˙ξ, ˙φ]T contains the vector of Lagrangian generalized coordinates and its first derivative in time, so defining a 4-dimensional phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 Linearized dynamics The nonlinear differential system (5) can be linearized near ξ = φ = 0 as ρl2 2 ¨φx′(0) + [k2α′ − Fx′]ξ=0 φ + ρl¨ξ � x′2 + y′2� ξ=0 + � k1 � x′′(x − xS) + x′2 + y′′(y − yS) + y′2� − Fy′′ + k2 � α′2 + αα′′�� ξ=0 ξ + [k1 (x′(x − xS) + y′(y − yS)) − Fy′ + k2α′α]ξ=0 = 0, ρl3 3 ¨φ + k2φ + ρl2 2 x′(0)¨ξ + k2α′(0)ξ + k2α(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (7) 6 Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' with the introduction of the vector collecting the Lagrangian generalized coordinates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' q = [ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' φ]T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' equations (7) can be written in matrix form as M ¨q(t) + Kq(t) = f(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (8) where M is the mass matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' K the stiffness matrix and f the vector of generalized forces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' respectively M = ρl � ��� x′2 + y′2 l 2x′ l 2x′ l2 3 � ��� ξ=0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' K = � � k1 � x′′(x − xS) + x′2 + y′′(y − yS) + y′2� − Fy′′ + k2 � α′2 + αα′′� k2α′ − Fx′ k2α′ k2 � � ξ=0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' and f = [Fy′ − k1 (x′(x − xS) + y′(y − yS)) − k2α′α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' −k2α]T ξ=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The trivial solution ξ = φ = 0 is an equilibrium configuration only when f = 0, which implies y′(0) = 0, x(0) = xS, (9) so that the tangent to the profile has to be horizontal at ξ = 0, and the fixed point S of the linear spring must be aligned vertically with the point of the curve at ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='3 Piecewise-smooth structure: doubly circular profile All equations obtained in the previous Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 can be applied to a profile with discontinuous curvature, and hold for both branches of the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' However, the discontinuity has to be made explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The introduced structure can be particularised through the implementation of a specific curve for the constraint, given as a parametric function of the arc length ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Circular curves will be addressed with positive and negative curvatures, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 3, so that the coor- dinates of the point P along the profile singled out by the arc length ξ are x(ξ) = R± sin ξ R± , y(ξ) = ±R± � 1 − cos ξ R± � , (10) where R± > 0 is the radius of curvature and where the ‘+’ sign (the ‘−’ sign) applies for positive (for negative) curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' For a circular curve described by the parametric representation (10), the nonlinear governing equations are obtained from (5) by substituting x′(ξ) = cos ξ R± , y′(ξ) = ± sin ξ R± , x′′(ξ) = − 1 R± sin ξ R± , y′′(ξ) = ± 1 R± cos ξ R± α(ξ) = ± ξ R± , α′(ξ) = ± 1 R± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (11) 7 k1 k2 ξ(t) φ(t) F F G G L L O P e2 e1 R+ l, ρ k1 k2 ξ(t) φ(t) F F G G L L O P e2 e1 R− l, ρ Figure 3: Two elastic structures of the type shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 2 with circular sliding profiles, having positive (on the left) and negative (on the right) curvatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Moreover, conditions (9) for a trivial equilibrium solution are fulfilled provided that xS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Refer- ring to the linearized equations (8), the mass matrix remains the same for both positive and negative curvatures, while the stiffness matrix is different in the two cases, namely, M = ρl � � 1 l/2 l/2 l2/3 � � , K± = � ��� k1 + k2 R2 ± ∓ k1ys R± ∓ F R± ± k2 R± − F ± k2 R± k2 � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (12) A third elastic structure shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 4 is now considered, described by two degrees of freedom and obtained as the ‘fusion’ of the two previously described subsystems with positive and negative cur- vatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Specifically, the profile on the left (colored blue in the figure) is a circular path with negative curvature, while the profile on the right (colored red in the figure) is a circular path with positive cur- vature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The tangent to the profile at the junction is horizontal and at this point the longitudinal spring is unloaded, so that the vertical configuration of the rigid rod is the trivial equilibrium configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 8 k1 k2 ξ(t) φ(t) α(ξ) α(ξ) τ(ξ) F H F G G L L S O P e2 e1 R+ R− γ {x(ξ), y(ξ)} l, ρ Figure 4: An elastic structure is obtained as the ‘fusion’ of the two elastic structures shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The profile, on which one end of the structure is forced to slide without friction, is composed of two circular paths, having negative and positive curvatures on the left and the right, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' This new structure is characterized by a smooth sliding profile, which however evidences a jump in the curvature at ξ = 0, so that the dynamics is characterized by piecewise smooth differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Specifically, the equations of motion are obtained through a substitution of equation (11) into equation (5) and considering the equations referred to the + system (− system) for ξ > 0 (ξ < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Correspondingly, the small amplitude vibrations of the system are described by the following piece- wise linear equations of motion � M ¨q(t) + K−q(t) = 0, ξ < 0, M ¨q(t) + K+q(t) = 0, ξ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (13) Although composed of two linear differential problems, the piecewise system (13) is globally nonlinear, due to switching between two different sets of equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Using the notation so far introduced, it will be shown that, due to the presence of the follower (non- conservative) load, the stability properties of this structure are non-trivial and, in particular, an unstable structure may result from the union of two structures which are stable when considered alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 3 Dynamics and instability for smooth and non-smooth structures The analysis of a piecewise-smooth dynamical system described in the Lagrangian formalism by the equation of motions (13) is not trivial due to its nonlinear nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In particular, as for stability analysis, the classical Lyapunov theorem on linear analysis cannot be applied, because the Jacobian matrix cal- culated at equilibrium is not unique, so all the standard criteria based on the nature of the eigenvalues of the Jacobian matrix are impracticable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Moreover, the complexity of the analysis increases due to the fact that the time intervals in which the solution is associated to a specific subdomain (ξ > 0 or ξ < 0) are a priori unknown, since they depend on the initial conditions applied to the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The aim of the present article is the definition of some general criteria that allow the design of a mechanical structure with given stability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In particular, the unusual unstable behaviour related to the coupling of two stable systems is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In this Section, some extensions of the classical stability theory are introduced to deal with piecewise linear dynamical system, with the aim of linking the stability properties of each single smooth system to those of the entire non-smooth structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='1 Linearized behaviour for a smooth structure Before analyzing the non-smooth mechanical system of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 4, the stability of the smooth subsystems reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 3 is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The treatment applies to both smooth mechanical systems with positive or negative curvature, so that for simplicity the symbols ‘±’ will be dropped in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The lin- earized dynamics is governed by the following system of two linear second-order differential equations M ¨q + Kq = 0, (14) to be complemented by initial conditions on position and velocity, q(0) = q0 and ˙q(0) = ˙q0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The solution to equation (14) can be expressed using exponential functions as q(t) = ψ(j)eλjt, leading to the eigenvalue problem � λ2 jM + K � ψ(j) = 0, (15) which provides two values for λ2 j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The eigenvalues λj are related to the natural frequencies ωj of the mechanical system through λj = iωj, where i = √−1 is the imaginary unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The generalized coordinates q(t) that satisfy equation (14) can now be written as the linear combi- nation of four exponential terms q(t) = 2 � j=1 ψ(j) � Ajeλjt + Bje−λjt� , where Aj and Bj are four arbitrary constants that can be obtained from the initial conditions q0 and ˙q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The response of the dynamical system to initial conditions near the trivial equilibrium configuration is now determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The linear equation (14) can be rewritten in the Hamiltonian form, namely, as a system of four linear first-order differential equations ˙y(t) = A y(t), (16) where A = � � 0 I −M −1K 0 � � = � � 0 I Γ 0 � � is a 4 × 4 matrix, I is the 2 × 2 identity matrix, and the phase vector y(t) = [q(t), ˙q(t)]T = [ξ, φ, ˙ξ, ˙φ]T contains the vector of Lagrangian generalized coordinates and its first derivative in time, so defining a 4-dimensional phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The differential equation (16) can be solved as for the Lagrangian formulation using an exponential ansatz y(t) = v(j)eλjt, leading to the eigenvalue problem A v(j) = λjv(j), whose eigenvalues λj are the same appearing in equation (15), while the eigenvectors v(j) are related to the eigenvectors ψ(j) through v(1,2) =[ψ(1) 1 , ψ(1) 2 , ±λ1ψ(1) 1 , ±λ1ψ(1) 2 ]T, v(3,4) =[ψ(2) 1 , ψ(2) 2 , ±λ2ψ(2) 1 , ±λ2ψ(2) 2 ]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The solution of the initial value problem with assigned initial conditions y(0) = y0 is unique and can be related to y0 through the so-called fundamental solution matrix, which is the matrix exponential eAt defined in such a way that y(t) = eAt y0, (17) which can also be written in extenso for the case under analysis as � ��� ξ(t) φ(t) ˙ξ(t) ˙φ(t) � ��� = eAt � ��� ξ(0) φ(0) ˙ξ(0) ˙φ(0) � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 10 The stability analysis of the equilibrium configuration for a given smooth mechanical system can be performed using the Lyapunov theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In particular, this analysis is based on the nature of the eigen- values λj of the matrix A, λj = ± � I1 ± � I2 1 − 4I2 2 , (18) where I1 = tr Γ and I2 = det Γ (19) are the first and second invariants of the matrix Γ = −M −1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The Lyapunov theorem [3, 17] states that the equilibrium configuration of a nonlinear dynamical system is stable when the real parts of all eigenvalues of the Jacobian matrix are negative, whereas it is unstable when the real part of at least one eigenvalue is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The case in which the real part of one or more eigenvalues is zero and all the others have negative real part is not covered by the Lyapunov theorem and is referred to as critical case, in the context of mechanics [3], or as marginally stable, in the context of abstract dynamical systems [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Due to the symmetry of the eigenvalues with respect to the imaginary axis given by the structure of the formula (18), only three cases can be distinguished, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 5: (Marginal) stability: the two squared eigenvalues λ2 j are real and negative, so that the correspond- ing λj ∈ iR are two purely imaginary conjugate pairs, say, λ1,2 = ±iω1 and λ3,4 = ±iω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' To be more specific, this is the critical case mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Note that the sub-structures originating the piecewise-smooth structure analyzed in this article are stable in the sense considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Divergence instability: at least one of λ2 j is real and positive, that produces λ1,2 = ±ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Flutter instability: λ2 j are complex conjugate pairs, so that λ1,2,3,4 = ±(α ± iβ), α ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The behaviour is unstable and the presence of a non-null imaginary part produces oscillations in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' I1 I2 I2 = I2 1 4 complex eigenvalues with not vanishing real part(flutter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' instability) two purely imaginary and two real eigenvalues (saddle-node,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' instability) purely imaginary eigenvalues (marginal stability) real eigenvalues (divergence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' instability) Figure 5: Representation in the plane I1 − I2 of the stability domains for a dynamical system characterized by 2 degrees of freedom The aim of the present article is to show that two smooth elastic structures which are stable taken separately may lead to an unstable structure when combined together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Therefore, our interest is in the 11 case when both smooth subsystems are stable, for which λ1,2 = ±iω1 and λ3,4 = ±iω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Assuming ω1 ̸= ω2 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' matrix A is diagonalizable as A = UJU −1 and the matrix exponential fulfils the relation eA = UeJU −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' where J = � ��� iω1 0 0 0 0 −iω1 0 0 0 0 iω2 0 0 0 0 −iω2 � ��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' U = � ����� ψ(1) 1 ψ(1) 1 ψ(2) 1 ψ(2) 1 ψ(1) 2 ψ(1) 2 ψ(2) 2 ψ(2) 2 iω1ψ(1) 1 −iω1ψ(1) 1 iω2ψ(2) 1 −iω2ψ(2) 1 iω1ψ(1) 2 −iω1ψ(1) 2 iω2ψ(2) 2 −iω2ψ(2) 2 � ����� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' and the vectors ψ(1) and ψ(2) are solutions of equation (15),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' while the inverse of U is given by U −1 = 1 2(ψ(1) 1 ψ(2) 2 − ψ(1) 2 ψ(2) 1 ) � ����� ψ(2) 2 −ψ(2) 1 −iψ(2) 2 /ω1 iψ(2) 1 /ω1 ψ(2) 2 −ψ(2) 1 iψ(2) 2 /ω1 −iψ(2) 1 /ω1 −ψ(1) 2 ψ(1) 1 iψ(1) 2 /ω2 −iψ(1) 1 /ω2 −ψ(1) 2 ψ(1) 1 −iψ(1) 2 /ω2 iψ(1) 1 /ω2 � ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' the matrix exponential eJt becomes eJt = � ��� cos ω1t + i sin ω1t 0 0 0 0 cos ω1t − i sin ω1t 0 0 0 0 cos ω2t + i sin ω2t 0 0 0 0 cos ω2t − i sin ω2t � ��� and finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' the matrix exponential eAt for the four-dimensional smooth mechanical system considered ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='here is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='eAt = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='�������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1 cos ω1t−a2 cos ω2t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1−a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1a2(cos ω2t−cos ω1t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1−a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1ω2 sin ω1t−a2ω1 sin ω2t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1ω1ω2−a2ω1ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1a2(ω1 sin ω2t−ω2 sin ω1t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1ω1ω2−a2ω1ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='cos ω1t−cos ω2t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1−a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1 cos ω2t−a2 cos ω1t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1−a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='ω2 sin ω1t−ω1 sin ω2t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1ω1ω2−a2ω1ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1ω1 sin ω2t−a2ω2 sin ω1t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1ω1ω2−a2ω1ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a2ω2 sin ω2t−a1ω1 sin ω1t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1−a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1a2(ω1 sin ω1t−ω2 sin ω2t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1−a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1 cos ω1t−a2 cos ω2t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1−a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1a2(cos ω2t−cos ω1t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1−a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='ω2 sin ω2t−ω1 sin ω1t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1−a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a2ω1 sin ω1t−a1ω2 sin ω2t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1−a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='cos ω1t−cos ω2t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1−a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1 cos ω2t−a2 cos ω1t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='a1−a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='�������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (20) where a1 = ψ(1) 1 /ψ(1) 2 = −ω2 1 + Γ22 Γ21 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' a2 = ψ(2) 1 /ψ(2) 2 = −ω2 2 + Γ22 Γ21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' It will be instrumental later to calculate the inverse of the matrix given by equation (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Because of the property AB = BA =⇒ eAeB = eA+B the inverse of the exponential matrix can be obtained as (eAt)−1 = e−At, corresponding to a change in the sign of t in equation (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' As described above, for a smooth mechanical system, the stability can simply be judged through the calculation of the eigenvalues λ2 j of the jacobian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' However, this is not enough for piecewise smooth structures, for which a complete understanding of the dynamics of the system is required, by a direct solution of the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' This analysis will be performed in the next paragraph through the computation of the solution of the piecewise linear system (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 1The coalescence ω1 = ω2 would denote a grazing of an unstable boundary, an occurrence which is excluded here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 Linearized behaviour for a piecewise-smooth structure When a piecewise smooth dynamical system is considered, equation (13) can be rewritten in the Hamil- tonian form as ˙y(t) = � A−y(t), y1 < 0, A+y(t), y1 > 0, (21) where y1 = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' A dynamical system with discontinuous right-hand side such as that expressed by equation (21) is referred to as ‘Filippov system’ [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Note that, although the accelerations ˙y3 = ¨ξ and ˙y4 = ¨φ suffer a jump across the discontinuity in the curvature of the profile, y1 = 0, the velocities ˙y1 = y3 = ˙ξ and ˙y2 = y4 = ˙φ remain continuous, so that e1 · A−y(t) = e1 · A+y(t), and e2 · A−y(t) = e2 · A+y(t), when y1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Equation (21) shows that the piecewise smooth structure under consideration defines a 4-dimensional phase space, which can be divided into two subdomains V± within which the system can be considered smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The manifold separating the two subdomains, called switching manifold, is the hyperplane Σ = {y ∈ R4 : y1 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The negative part of equations (21) applies to the subdomain defined by V− = � y ∈ R4 : y1 < 0 � , while the positive part applies to V+ = � y ∈ R4 : y1 > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Note that the origin of the phase space (ξ, φ) = (y1, y2) = (0, 0) represents the trivial equilibrium configuration of the structure and belongs to the switching manifold Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The discontinuous system (21) does not define the time derivative ˙y(t) when the configuration of the system y(t) is on the switching boundary Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' To overcome this difficulty, Filippov has developed a technique, known as Filippov’s convex method, that extends the discontinuous system (21) to a differential inclusion of the form ˙y(t) ∈ � � � � � A−y(t), y1 < 0, co{A−y(t), A+y(t)}, y1 = 0, A+y(t), y1 > 0, (22) where the closed convex hull co of the two right-hand sides f − and f + is defined by co{f −, f +} = {(1 − η)f − + ηf +, ∀η ∈ [0, 1]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' A solution in the Filippov sense of the discontinuous system (21) is a solution of the differential inclusion (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In this sense, the mechanical system admits among the possible solutions, the so-called sliding modes, namely, motions in which the system remains on the switching manifold Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' More precisely, there are three possible ways in which the mechanical system behave around the switching boundary Σ, namely: Transverse intersection: both vector fields A+y(t) and A−y(t) point on the same side of Σ � e1 · A+y(t) �� e1 · A−y(t) � > 0, (23) so that a solution, that evolves in one subdomain and at some instant of time hits Σ, will cross it transversely and proceed in the other subdomain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In this case the solution is locally unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Attractive sliding modes: both vector fields A+y(t) and A−y(t) point to Σ e1 · A+y(t) < 0 and e1 · A−y(t)] > 0, hence a solution that hits the switching boundary Σ will not leave it and will therefore move along Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Also in this case the solution is locally unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Repulsive sliding modes: both vector fields A+y(t) and A−y(t) point in the opposite direction to Σ e1 · A+y(t) > 0 and e1 · A−y(t)] < 0, which implies that a solution emanating from Σ can remain in Σ or leave it by entering either subdomain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Consequently, the solution in this case is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 13 For the two d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' elastic structure under consideration, the condition of transverse intersection, equation (23), is satisfied almost everywhere, that is for any configuration belonging to Σ and such that y3 = ˙ξ ̸= 0, since the velocity ˙ξ is continuous across the switching manifold Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Sliding modes are only possible in a zero-measure subset of Σ, namely the set {y ∈ R4 : y · e1 = y · e3 = 0}, and will not be investigated further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Assuming that transverse intersection always prevail at the intersections of the orbits with the switching manifold Σ, a solution that evolves in one subdomain and hits Σ necessarily crosses the hyperplane and enters the other subdomain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The intersection point between the orbit and the hyper- plane Σ can then be used as initial condition for the subsequent evolution in the subdomain in which the orbit is entering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' the solution of the piecewise-linear system (21) can be obtained as the composition of exponential matrices as follows y(t) = � � � � � � � � � � � � � � � � � eA−(t−t0)y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' t0 ≤ t < t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' eA+(t−t1)eA−(t1−t0)y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' t1 ≤ t < t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' · · · · eA−(t−tk−1)eA+(tk−1−tk−2) · · · eA+(t2−t1)eA−(t1−t0)y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' tk−1 ≤ t < tk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' eA+(t−tk)eA−(tk−tk−1) · · · eA+(t2−t1)eA−(t1−t0)y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' tk ≤ t < tk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' · · · · (24) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' without loss of generality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' it is assumed that the initial condition belongs to the negative subdo- main,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' y(0) = y0 ∈ V−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' and {t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' tk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' tk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' · · · } is the sequence of intersection times at which the or- bit crosses the switching boundary and changes subdomain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' However, it should be pointed out that the piecewise solution (24) is not completely determined, because the intersection times {t1, t2, · · · , tk−1, tk, · · · } are a priori unknown and depend on the initial condition y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' To completely define the solution (24), the intersection times have to be determined by tracing the evolution of the orbit from the initial condition y0 and numerically detecting the roots of the crossing condition ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' A classical expedient to simplify the description of a dynamical system is the introduction of a discrete map, known as Poincaré map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' A Poincaré map transforms a n-dimensional continuous-time system into a (n−1)-dimensional discrete-time system, with the introduction of a hyperplane embedded in the n-dimensional phase space, called Poincaré section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' For the two d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' elastic system under consideration, described by the 4-dimensional non-smooth dynamical system (21), a natural and convenient choice for the 3-dimensional Poincaré section is the switching manifold Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In this case, the Poincaré map P : Σ → Σ links points on Σ through the orbits defined by the solution (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' More precisely, let us consider a point x ∈ Σ and let us assume, without loss of generality, that the orbit starting from the initial condition x enters the negative subspace V−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The time evolution in the negative subsystem is described by the first expression in (24) y(t) = eA−(t−t0)x, until the orbit reaches Σ at point ξ = y(t1) = exp (A−∆t−) x in a given time interval, namely ∆t− = t1 − t0 measured from the initial time t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Then the orbit crosses the switching manifold and enters in the positive subsystem V+, following the orbit described by y(t) = eA+(t−t1)eA−∆t−x = eA+(t−t1)ξ, so that the point ξ can be interpreted as a new initial condition for the second part of the orbit, evolving within the positive subdomain V+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The orbit remains inside V+ until it hits Σ for a second time at point η = y(t2) = exp (A+∆t+) ξ in a time interval ∆t+ = t2 − t1, measured from t = t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The just described sequence defines two Poincaré half-maps ξ = P −(x) = eA−∆t−(x)x, η = P +(ξ) = eA+∆t+(ξ)ξ, (25) and that the complete Poincaré map P : Σ → Σ may be obtained through the composition of the two half-maps, P = P + ◦ P −, such that η = P (x) = P +(P −(x)) = eA+∆t+(ξ(x))eA−∆t−(x)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (26) 14 It should be noted that the time intervals ∆t−(x) and ∆t+(ξ) are non linear functions of the initial conditions x and ξ, respectively, and can be defined as ∆t−(x) = inf � ∆t > 0 : e1 · eA−∆tx = 0 � , ∆t+(ξ) = inf � ∆t > 0 : e1 · eA+∆tξ = 0 � , (27) where e1 is the normal to the switching manifold Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' These definitions simply represent the fact that the points ξ and η must be on the switching manifold and that they identify the first two intersections between the considered trajectory and the hyperplane Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The use of the Poincaré map will be crucial in the next section for the analysis of stability of the piecewise smooth elastic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' e1 O Σ V+ V− x ξ = P −(x) η = P +(ξ) Figure 6: A pictorial view (in which a 4-dimensional space is reduced to a 3D sketch) of an unstable cone with the Poincaré half-maps P − and P +, separated by the hyperplane Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The three points x, ξ, η identify a solution belonging to the invariant cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='3 Invariant cones: instability of piecewise-linear mechanical systems The stability analysis of the piecewise-linear system (21) cannot be pursued using standard methods for smooth system, such as the analysis of eigenvalues at the equilibrium point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In this section a tool will be developed, based on the existence of special invariant sets, called invariant cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' An invariant set of an autonomous dynamical system, such that described by equation (16) for smooth structures or by equation (21) for piecewise smooth structures, is a subset S of the phase space such that y(t0) ∈ S implies y(t) ∈ S, for all times t > t0, [19–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' This subset, for an n-dimensional dynamical system, is assumed to be an (n − 1)-dimensional invariant manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' A special type of invariant set, called invariant cone, see [2, 12, 22, 23], is defined by the condition that a vector n belonging to a Poincaré section Σ exists, for which the Poincaré map P describing the 15 evolution of the dynamical system fulfills the condition P (n) = µ n, where µ ∈ R+ is a positive real number and n ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In the specific case of the piecewise-linear system (21), the Poincaré map is given by the composition of two half-maps (25), hence an invariant cone exists for the elastic system under investigation if and only if there exist a scalar µ > 0 and a vector x ∈ Σ such that η = eA+∆t+(ξ(x))eA−∆t−(x)x = µ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (28) Equation (28) has the structure of an eigenvalue problem: the multiplier µ can be interpreted as a gen- eralized eigenvalue and the vector x as a generalized eigenvector of the matrix eA+∆t+(ξ(x))eA−∆t−(x), which defines the Poincaré map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' However, the eigenvalue problem is nonlinear, because the matrix itself depends on the eigenvector, hence the solution of (28) in terms of µ and x is not trivial and the usual procedures for linear eigenvalue problems cannot be adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' A solution of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (28), provided it exists, identifies an invariant cone and is given by the list {∆t−, ∆t+, x, µ}, comprising the time intervals ∆t− and ∆t+, expended by the mechanical system in passing through the subdomains V− and V+, respectively, an eigenvector x belonging to the invariant cone (and also to the Poincaré section), and finally the corresponding eigenvalue µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Before proceeding, it is instrumental to establish some fundamental properties of invariant cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' First of all, it is noted that the assumption that the system enters first in the negative subdomain V−, used in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (28), is not restrictive, as the same invariant cone may be identified by assuming instead that the system enters first the positive subdomain V+ and, thus, by solving the eigenvalue problem eA−∆t−eA+∆t+ξ = γ ξ, (29) where ∆t− and ∆t+ are the same as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' It is easy to check that ξ = eA−∆t−x solves the problem (29) with γ = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' This means that the solution {∆t+, ∆t−, ξ, µ} of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (29) identifies the same invariant cone as the solution {∆t−, ∆t+, x, µ} of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' To simplify notations, here and in the sequel, the three points x, ξ, η identify a solution belonging to the invariant cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Next, it is noted that the time intervals ∆t−(x) and ∆t+(ξ) are homogeneous functions of degree zero, ∆t−(αx) = ∆t−(x), ∆t+(αξ) = ∆t+(ξ), ∀α > 0, (30) as can be easily checked from the definitions (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' These conditions are a direct consequence of the lin- earity of the dynamical system within each individual subdomain, V− and V+, and define the structure of the invariant cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In fact, when the point η is assumed as an initial condition for the motion of the system after the two initial half-maps, ξ = P −(x) and η = P +(ξ), the new intersection time intervals become ∆t−(η) = ∆t−(µx) = ∆t−(x), ∆t+(µξ) = ∆t+(ξ), hence the time intervals ∆t± are constants and do not change in the application of further half-maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The properties (30) immediately imply that both the Poincaré half-maps P −(x) and P +(ξ), defined according to equation (25) as well as the complete Poincaré map P (x), defined by equation (26), are homogeneous functions of degree one P −(αx) = αP −(x), P +(αξ) = αP +(ξ), P (αx) = αP (x), ∀α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The latter equations have a clear geometrical interpretation, in that the Poincaré half-maps P −(x) and P +(ξ), plus the Poincaré map P (x), transform straight half-lines belonging to the switching boundary Σ and intersecting the origin into straight half-lines also belonging to Σ and intersecting the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The last observation is crucial for the stability analysis of dynamical system when an invariant cone is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In fact, when an initial condition y0 belongs to the Poincaré section and to the invariant cone, 16 the evolution of the system can be described by the recursive applications of Poincaré maps such that y(∆t− + ∆t+) = eA+∆t+eA−∆t−y0 = µ y0, y(2∆t− + 2∆t+) = � eA+∆t+eA−∆t−�2 y0 = µ2 y0, y(3∆t− + 3∆t+) = � eA+∆t+eA−∆t−�3 y0 = µ3 y0, · · y(k∆t− + k∆t+) = � eA+∆t+eA−∆t−�k y0 = µk y0, · · (31) defining a discrete exponential relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Accordingly, the behaviour of the mechanical system can easily be understood, when an invariant cone is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' This is because the orbits may spiral in or out along the cone, depending on the value of the eigenvalue µ, and may evolve either towards the vertex at the equilibrium point or away from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In particular, the following conclusions can be drawn: If µ > 1, a family of ‘spiraling out’ trajectories, belonging to the invariant cone, exists, moving away for t > 0 from the vertex of the cone, which represents the equilibrium configuration of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' When an initial condition is selected, belonging to the invariant cone, the motion of the structure evolves in time diverging from the fixed point, hence the equilibrium configuration at the origin is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' If µ < 1, a family of ‘spiraling in’ trajectories belonging to the invariant cone exists, moving towards the vertex of the cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Although this corresponds to a stable behaviour, other trajectories different from those on the invariant cone may exist being divergent, so that stability of the fixed point cannot be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' If µ = 1, a family of periodic trajectories, belonging to the invariant cone, exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In this case, nothing can be concluded about the stability of the equilibrium configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Summarising the above statements, for the dynamical system represented by equation (21), the exis- tence of an invariant cone, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (28), with µ > 1 is a sufficient condition for instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Vice-versa, assuming the existence of an invariant cone, fulfillment of condition (28) with µ ≤ 1 is a necessary condition for stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' It is important to emphasise that these conditions hold true, regardless of the stability behaviour of each single subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Therefore, an equilibrium configuration of a piecewise-linear mechanical system, which results stable when analysed separately for each constituting subsystem, can become unstable when the composed structure is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' A proposition is now proven which establishes a general property of invariant cones for autonomous non-dissipative mechanical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Although this proposition is proven for the 2 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' mechanical system under investigation, it can be easily extended to a system with any number of degrees of free- dom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' A proposition on reciprocal eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' If the autonomous non-dissipative mechanical system (21) admits an invariant cone, that is a solution {∆t−, ∆t+, x, µ} of the eigenvalue problem (28) with µ ̸= 1, then another invariant cone {∆t−, ∆t+, Jξ, 1/µ} exists, associated to the eigenvalue 1/µ, reciprocal of µ, where ξ = eA−∆t−x, J = � ��� 1 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 −1 � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Consequently, when the problem (28) admits a real eigenvalue µ ̸= 1, it admits also the eigenvalue 1/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Therefore, when a solution of equation (28) is found with µ ̸= 1, the trivial equilibrium configuration is certainly unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 17 The above proposition can be proven by preliminarily noting that the application of matrix J to a state vector y = [ξ, φ, ˙ξ, ˙φ]T has the effect of changing the sign of the velocities J � ��� ξ φ ˙ξ ˙φ � ��� = � ��� ξ φ − ˙ξ − ˙φ � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Note also that J = J−1, e−A±t± = JeA±t±J and that vector Jξ enters in the negative subdomain, as the vector ξ, without the minus signs, enters in the positive subdomain, being a solution of the half-map (25)1 by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The above statement follows from the two properties Jξ = e−A−∆t−Jx, Jη = e−A+∆t+Jξ, (32) equations that can be directly checked considering the form of matrix eAt, equation (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The above properties have a clear mechanical meaning, as they correspond to an inversion of the motion: the time is inverted by changing the sign to the variable t and, correspondingly, the velocities in the vectors x, ξ and η are inverted through multiplication by matrix J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' A multiplication of equation (32)1 by eA+∆t+eA−∆t− and subsequent use of x = η/µ, as well as consideration of equation (32)2 leads to eA+∆t+eA−∆t−Jξ = eA+∆t+Jx = 1 µeA+∆t+Jη = 1 µJξ, which proves the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='4 Mechanical energy for the piecewise smooth system The unusual unstable behaviour of the piecewise smooth structure composed of two stable substruc- tures can be qualitatively explained from a mechanical point of view by analyzing the mechanical en- ergy characterizing the non-smooth system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The stiffness matrix K±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' defined in equation (12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' collects the effect of the elastic springs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' k1 and k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' and of the follower force F and can be decomposed into the sum of a symmetric and an unsymmetric components,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' K± = � K± + G±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' � K ± = � ���� k1 � 1 ∓ ys R± � + k2 R2 ± ± k2 R± ± k2 R± k2 � ���� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' G± = F � �� ∓ 1 R± −1 0 0 � �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' where � K ± is a symmetric matrix collecting only the spring stiffnesses and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' possibly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' other conservative loads applied to the structure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' while G± contains only the nonconservative forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' A scalar product of equation (14) by ˙q and a factorization of the time derivative yield d dt �1 2 ˙q · M ˙q + 1 2q · � Kq � = − ˙q · Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (33) The mechanical energy H is defined as the sum of the kinetic and the total potential energy H = 1 2 ˙q · M ˙q � �� � Kinetic energy + 1 2q · � Kq � �� � Total potential energy , (34) where the total potential energy is the sum of the elastic energy and the potential energy of conservative external loads, when present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 18 Equation (34) can be rewritten in matrix notation as a function of the state vector y(t) as H(t) = 1 2y(t) · Dy(t), (35) where D = � � � K 0 0 M � � , is a symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' An integration of equation (33) yields H(tf) − H(t0) = − � tf t0 ˙q · Gq dt � �� � Work done by the nonconservative loads , therefore, a change in the mechanical energy equals a corresponding work done on the structure by the nonconservative loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' When in a smooth system only the symmetric part � K of the stiffness matrix is present, the mechan- ical energy remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' On the contrary, when a nonconservative contribution G is present, the mechanical energy H(t) varies in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The mechanical energy for a solution near a stable equilibrium configuration of a smooth system can be studied by substituting the solution (17) with the matrix exponential (20) into equation (35), hence H = 1 2y0 · (eAt)T DeAty0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (36) When a solution is stable, the mechanical energy (36) results as a sum of trigonometric functions, which in general is not a periodic function, but it is bounded, so that in a stable system the mechanical energy cannot indefinitely increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Formally, this can be shown by considering that the mechanical energy (36) can be bounded as y0 · (eAt)T DeAty0 ≤ ||(eAt)T DeAt|| y2 0 ≤ ||eAt||2 ||D|| y2 0, where, in the case of stability, matrix eAt is given by equation (20), so that its norm is bounded in time for every value of t ∈ [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' On the contrary, in case of a piecewise linear system described by equation (21), the mechanical energy is not necessarily bounded, although the subsystems forming the non smooth structure are both stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In fact, the mechanical energy at time t = 0, equation (36), for initial conditions y0 is H0 = H(0) = 1 2y0 · Dy0, where D can be identified with either D− or D+, because both matrices provide the same result, being the energy continuous at the switching manifold ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' During the motion, the variation of mechanical energy with time can be computed by substituting the piecewise solution (24) into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (35), thus obtaining H(t) = � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 1 2eA−(t−t0)y0 · D−eA−(t−t0)y0, t0 ≤ t < t1, 1 2eA+(t−t1)eA−(t1−t0)y0 · D+eA+(t−t1)eA−(t1−t0)y0, t1 ≤ t < t2, · · · · 1 2eA−(t−tk−1) · · · eA−(t1−t0)y0 · D−eA−(t−tk−1) · · · eA−(t1−t0)y0, tk−1 ≤ t < tk, 1 2eA+(t−tk) · · · eA−(t1−t0)y0 · D+eA+(t−tk) · · · eA−(t1−t0)y0, tk ≤ t < tk+1, · · · · 19 Assuming that an invariant cone exists, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (31), the mechanical energy at the intersection time k∆t− + k∆t+ (where k is a positive integer) can be computed as H(k∆t− + k∆t+) = 1 2µky0 · Dµky0 = µ2kH0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Therefore, when µ > 1 the mechanical energy suffers an unbounded exponential growth in time, revealing an unstable behaviour of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' This exponential growth is accompanied by oscillations, a situation similar to flutter instability in smooth systems loaded by follower forces, so that the instability be- haviour under investigation can be interpreted as a condition of flutter, but for non-smooth mechanical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The evolution in time (made dimensionless through division by a reference time T) of the mechani- cal energy H is reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 7 for the non-smooth structure reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 4, characterized by the values of parameters listed in (49), together with the two ‘component’ smooth structures shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 3 (responses reported dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Instability of the non-smooth system corresponds to a blow up of the mechanical energy (note the vivid representation in the inset, showing exponential growth), while the two stable substructures evi- dence a bounded evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='06 energy of the negative subsystem neglecting the switching conditions energy of the positive subsystem neglecting the switching conditions τ H 100 200 500 1,000 τ H Figure 7: Evolution in time of the mechanical energy for the non-smooth structure [shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 4 and charac- terized by the parameters in the list (49)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The behaviours of the two ‘component’ smooth mechanical structures (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 3) are also reported with dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The energy in the latter case remains bounded, denoting stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The exponential growth of the mechanical energy corresponding to a flutter instability is visible in the main graph, but may be vividly observed in the inset, referred to a longer time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 4 Attractivity of the cone: the effects of perturbations It has been demonstrated in the previous sections that the existence of an invariant cone with eigenvalue µ ̸= 1 in the phase space of a non-smooth dynamical system of the type (21) is a sufficient condition for instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Now the effect of perturbations in the initial conditions has to be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In particular, the question arises whether a motion generated from an initial condition slightly outside an unstable cone 20 will still display unstable unbounded growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' When the latter behaviour is found, the cone is called attractive, so that it will be asymptotically approached by a trajectory originated from initial conditions sufficiently close to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The answer to the above question is provided in the following, showing that for the structure under consideration, the unstable cone (associated to µ > 1) is always attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In other words, the instability detected with µ > 1 can be considered ‘genuine’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' A proposition on the attractivity of the cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' A cone with eigenvector x = y(0) = y0 and its as- sociated eigenvalue µ are assumed to exist, solution of equation (28), together with the corresponding vectors ξ = y(∆t−) = eA−∆t−y0 and η = y(∆t− + ∆t+) = eA+∆t+eA−∆t−y0, defining the state of the system at the first crossing (from V− to V+) and the second crossing (from V+ to V−) of the switching manifold Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' These vectors describe a solution y(t) belonging to the invariant cone that is considered as a reference solution and that is perturbed in order to study the attractivity of the cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The reference solution y(t) crosses the switching manifold Σ separating the negative and the positive subspaces the first time at the instant t− = ∆t− and a second time at the instant t+ = ∆t− + ∆t+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' V− V+ Σ e1 A−y(t−) A+y(t−) y0 y(t) ˜y0 ˜y(t) δy0 δy(t) δy(t− + δt−) y(t−) ˜y(t− + δt−) y(t− + δt−) Figure 8: Sketch of the effect of a perturbation δy0 of a given piecewise solution (developing from ˜y0) for a non- smooth dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The perturbed motion, starting at ˜y0 + δy0, is assumed to develop in a close neighborhood of the unperturbed motion, starting at ˜y0 A generic small perturbation δy0 in the initial conditions is assumed, with the restriction that ˜y0 = y0 + δy0 belongs to the negative subspace V−, as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The perturbed initial condition ˜y0 = y0 + δy0 gives rise to a perturbed solution ˜y(t) = y(t) + δy(t) which crosses for the first time the switching manifold Σ at a different instant of time t− + δt−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Therefore, equation (17) can be applied to obtain ˜y(t− + δt−) = eA−(t−+δt−)(y0 + δy0), where, due to the smallness of δy0, it is assumed that δt− is also sufficiently small to justify the following Taylor series approximation ˜y(t− + δt−) = eA−t−y0 + eA−t−δy0 + δt−A−eA−t−y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 21 The reference solution at time t− + δt− belongs to the positive subspace V+ and can be calculated with an analogous Taylor expansion (assuming that δt− is sufficiently small) as y(t− + δt−) = eA+δt−eA−t−y0 = eA−t−y0 + δt−A+eA−t−y0, where A+eA−t−y0 = A+y(t−) = ˙y(t−) is the orbital velocity of the reference solution entering at the time instant t− into the positive subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The evolution of the perturbation after the crossing of the switching manifold is described by the quantity δy(t− + δt−) = ˜y(t− + δt−) − y(t− + δt−), that can be computed as δy(t− + δt−) = eA−t−δy0 + δt−(A− − A+)eA−t−y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (37) The small time interval δt− can be calculated noting that the vector ˜y(t− + δt−) − y(t−) belongs to the switching manifold and is orthogonal to the vector e1, hence �˜y(t− + δt−) − y(t−) � e1 = � eA−t−δy0 + δt−A−eA−t−y0 � e1 = 0, an equation that can be solved in δt−, leading to δt− = − 1 ˙ξ(t−) e1 · eA−t−δy0, (38) where ˙ξ(t−) = e1 · A−y(t−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Note also that ˙ξ(t−) = e1 · A−y(t−) = e1 · A+y(t−), because the velocities ˙ξ and ˙φ remain continuous crossing the switching manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Substituting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (38) into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (37), the evolution of the perturbation after the crossing can be written as δy(t− + δt−) = S−eA−t−δy0, where the saltation matrix S− is defined as S− = I + 1 ˙ξ(t−) � A+y(t−) − A−y(t−) � ⊗ e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The saltation matrix defines the perturbation δy(t− + δt−) after the crossing of the switching manifold Σ, for a given perturbation δy(t−) = eA−t−δy0 defined just before the crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Due to the structure of the saltation matrix and because det(I + a ⊗ b) = 1 + a · b, it follows that det(S−) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' A similar calculation can be performed when the unperturbed solution crosses the switching mani- fold from the positive to the negative subspace at time t+ = ∆t− + ∆t+ and the perturbed solution at time t+ + δt+, so that a new saltation matrix can be defined as S+ = I + 1 ˙ξ(t+) � A−y(t+) − A+y(t+) � ⊗ e1, in which ˙ξ(t+) = e1 · A+y(t+) and y(t+) = eA+∆t+eA−∆t−y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Therefore, the initial perturbation δy0 evolves in time in a way that after two crossings of the switch- ing manifold, the difference between the reference and the perturbed solutions is governed by the rela- tion δy(t+ + δt+) = S+eA+∆t+S−eA−∆t−δy0, where the matrix ΦT = S+eA+∆t+S−eA−∆t− is referred to as the monodromy matrix in the context of stability of periodic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Consequently, the attractivity of the invariant cone is related to the four eigenvalues of the monodromy matrix, the so-called Floquet multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The monodromy matrix for the 2 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' mechanical system under consideration possesses the fol- lowing properties: 22 The determinant is equal to the unit, det � S+eA+∆t+S−eA−∆t−� = 1, because det eAt = etr At = 1 and det S− = det S+ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Two eigenvectors and two eigenvalues coincide with those of the eigenvalue problem (28) defin- ing the invariant cone � S+eA+∆t+S−eA−∆t−� y0 = µ y0, � S+eA+∆t+S−eA−∆t−� Jy(t−) = 1 µJy(t−), two identities following directly from S−y(t−) = S+y(t−) = y(t−) and S−Jy0 = S+Jy0 = Jy0 (the saltation matrices S− and S+ leave unchanged any vector belonging to the switching manifold Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' A third eigenvector is equal to A−y0, with corresponding eigenvalue µ, � S+eA+∆t+S−eA−∆t−� A−y0 = µ A−y0, which follow from the commutativity, eA±∆t±A± = A±eA±∆t±, and from the two identities S−A−y(t−) = A+y(t−) and S+A+y(t+) = A−y(t+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' It follows from all the above that the monodromy matrix S+eA+∆t+S−eA−∆t− possesses the four eigenvalues {µ, µ, 1/µ, 1/µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The first two eigenvalues {µ, µ} are associated with initial perturbations δy0 along the directions y0 and A−y0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' belonging to the invariant cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The other two {1/µ, 1/µ} are associated to perturbations outside the invariant cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In conclusion, since a generic perturbation can always be decomposed along the eigenvectors of the monodromy ma- trix S+eA+∆t+S−eA−∆t−, the unstable cone associated to the eigenvalue µ > 1 is always attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Therefore, when an eigenvalue µ ̸= 1 is found as the solution of problem (28), the structure admits a stable (non attractive) cone, associated to µ < 1, and an unstable (attractive) cone, associated to µ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Note that in the case µ = 1 the motion is periodic and the cone is not attractive, hence conclu- sions about instability of the mechanical system cannot be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' This case is therefore not further considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 5 Numerical examples 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='1 Numerical algorithm for the identification of invariant cones for piecewise lin- ear systems Assume that all the mechanical parameters of the system, together with the applied follower force, are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' A numerical procedure is proposed in this section for the identification of possible invariant cones, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' solutions of the generalized nonlinear eigenvalue problem (28), recalled here for convenience eA+∆t+eA−∆t−x = µ x, where the initial condition vector x belongs to the switching manifold Σ, so that it has the following form x = [0, x2, x3, x4]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 23 Note that the unknowns of the problem are the eigenvalue µ, the eigenvector x and the two time in- tervals ∆t− and ∆t+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Note also that the modulus of the vector x is arbitrary, given the structure of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Thus eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (28) can be scaled as follows eA+∆t+eA−∆t− � ��� 0 x2 x3 1 � ��� = µ � ��� 0 x2 x3 1 � ��� , (39) from which it is clear that there are five scalar unknowns to be determined ∆t−,∆t+, x2, x3 and µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' However, the system (39) provides only four scalar equations, and thus it has to be complemented by an additional scalar equation, which is provided by the condition that also the intermediate point ξ belongs to the switching manifold, ξ1 = [eA−∆t−x]1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (40) The system of equations (39) and (40) is nonlinear, so that an algorithm is proposed below to partially decouple the system and reduce it to two equations for the unknowns ∆t− and ∆t+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The starting point x and the final point η of the Poincaré map (26) can be expressed in terms of the intermediate point ξ using the Poincaré half maps (25) as follows x = e−A−∆t−ξ, η = eA+∆t+ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (41) The condition that both x and η belong to the switching manifold provides two equations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' x1 = � e−A−∆t−ξ � 1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' η1 = � eA+∆t+ξ � 1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (42) that are linear in ξ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ξ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' and ξ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' and hence can be solved for ξ2 and ξ3 as ξ2 = ξ4 h(∆t−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ∆t+),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ξ3 = ξ4 k(∆t−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ∆t+),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (43) where the coefficients h(∆t−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ∆t+) and k(∆t−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ∆t+) are the following functions of ∆t− and ∆t+ h(∆t−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ∆t+) = � eA+∆t+� 14 � e−A−∆t−� 13 − � eA+∆t+� 13 � e−A−∆t−� 14 � eA+∆t+� 13 � e−A−∆t−� 12 − � eA+∆t+� 12 � e−A−∆t−� 13 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' k(∆t−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ∆t+) = � eA+∆t+� 14 � e−A−∆t−� 12 − � eA+∆t+� 12 � e−A−∆t−� 14 � eA+∆t+� 12 � e−A−∆t−� 13 − � eA+∆t+� 13 � e−A−∆t−� 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Turning back the attention to equations (41), these can be rewritten, using equations (43), as � � x2 x3 x4 � � = ξ4 � e−A−∆t− � � h(∆t−, ∆t+) k(∆t−, ∆t+) 1 � � , � � η2 η3 η4 � � = ξ4 � eA+∆t+ � � h(∆t−, ∆t+) k(∆t−, ∆t+) 1 � � , (44) where � e−A−∆t− and � eA+∆t+ are the submatrices obtained from the original matrices through elimina- tion of the first row and column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' to be solutions of the eigenvalue problem (28),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' the two vectors given by equation (44) have to be parallel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' η2 x2 = η3 x3 = η4 x4 = µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (45) 24 providing two equations for the unknowns ∆t− and ∆t+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' as follows � eA+∆t+� 22 h(∆t−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ∆t+) + � eA+∆t+� 23 k(∆t−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ∆t+) + � eA+∆t+� 24 � e−A−∆t−� 22 h(∆t−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ∆t+) + � e−A−∆t−� 23 k(∆t−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ∆t+) + � e−A−∆t−� 24 = � eA+∆t+� 32 h(∆t−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ∆t+) + � eA+∆t+� 33 k(∆t−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ∆t+) + � eA+∆t+� 34 � e−A−∆t−� 32 h(∆t−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ∆t+) + � e−A−∆t−� 33 k(∆t−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ∆t+) + � e−A−∆t−� 34 = � eA+∆t+� 42 h(∆t−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ∆t+) + � eA+∆t+� 43 k(∆t−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ∆t+) + � eA+∆t+� 44 � e−A−∆t−� 42 h(∆t−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ∆t+) + � e−A−∆t−� 43 k(∆t−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ∆t+) + � e−A−∆t−� 44 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (46) The system (46) is nonlinear and must be solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Once the time intervals ∆t− and ∆t+ are known, the eigenvector x can be computed from (44)1, whereas the eigenvalue µ is obtained from equation (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The calculation of the time intervals ∆t− and ∆t+ is far from trivial, as the determining equations in- volve transcendental trigonometric functions, so that there are infinite values of ∆t− and ∆t+ satisfying them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Among these infinite values, only those corresponding to motions crossing the switching mani- fold the first time at ∆t− and the second at ∆t− + ∆t+ have to be retained, while the other disregarded, because they refer to orbits that cross the switching manifold multiple times, but erroneously remain in the same subdomain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' This situation, sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 9, has been solved through: (i) an estimation of the maximum time intervals ∆t± max = max{3π/(2ω± 1 ), 3π/(2ω± 2 )} within which the first intersections of the orbit with the switching manifold Σ occur, so that the solution is searched for in the bounded time domain [0, ∆t− max] × [0, ∆t+ max] (details are reported in [24]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (ii) a systematic elimination of the solutions lacking mechanical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In fact, an obtained solution is meaningful if and only if the following conditions are met: First crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The time instants ∆t− and ∆t− +∆t+ correspond respectively to the first and sec- ond intersection time of an orbit on the invariant cone with the switching manifold Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Therefore, given the initial vector x (obtained from the above algorithm), the conditions have to be checked: (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=') that the value ∆t− is the smallest solution of equation (42)1, and (ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=') that the value ∆t+ is the smallest solution of equation (42)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Switching conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Any solution representing an invariant cone must fulfill x3 = ˙ξ(0) < 0, ξ3 = ˙ξ(∆t−) > 0, η3 = ˙ξ(∆t− + ∆t+) < 0, (47) denoting the fact that the orbit on the invariant cone must initially cross the switching manifold and enter in the negative subdomain (at t = 0), while at t = ∆t− has to enter into the positive one, and finally at ∆t− + ∆t+ the negative subdomain has to be entered again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' However, a solution to the problem (46) yields a vector x defined except for the sign, which can always be adjusted, so that instead of conditions (47) for a solution to be meaningful, the time intervals ∆t− and ∆t+ have to satisfy ˙ξ(0) ˙ξ(∆t− + ∆t+) > 0, ˙ξ(0) ˙ξ(∆t−) < 0, meaning that the assumption made on the first entered subdomain (the negative) is arbitrary and the opposite assumption could be equally made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 25 x ξ η Σ V+ V− x ξ η x ξ η wrong solution: y(t−) is not the first intersection of the blue orbit with the switching manifold wrong solution: y(t− + t+) is not the first intersec- tion of the red orbit with the switching manifold Figure 9: Sketch of a motion starting at point x and initially developing inside the negative branch of the mechan- ical system (blue solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' At point ξ the motion crosses the switching manifold and further develops inside the positive branch (red solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' However, from a purely mathematical point of view, the con- tinuation with wrong equations referred to the negative branch (blue dashed line) are still solutions of equations (46) and their orbits cross the switching manifold at several subsequent points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' All the latter points have to be disregarded, because they represent the solution of a smooth system, different from that under analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Something analogous happens with the solution continuation of the red line (represented dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Therefore, there are infinite solutions on the switching manifold, and the selection of the correct points is the harder problem to be solved in finding instability cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 A non-smooth structure with an unstable invariant cone 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='1 Existence of an unstable invariant cone It is instrumental now to introduce the following non-dimensional parameters ζ± = R± l k = k1l2 k2 γ = Fl k2 σ = ys l ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (48) so that equations (13) governing the dynamics of the piecewise-smooth structure become Θ � �� 1 1 2 1 2 1 3 � �� � � ˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='ττ(τ) ˜φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='ττ(τ) � � + � ��� k ζ± (ζ± ∓ σ) + 1 ∓ γζ± ζ2 ± −γζ± ± 1 ζ± ± 1 ζ± 1 � ��� � � ˜ξ(τ) ˜φ(τ) � � = 0 where the non-dimensional time τ = t/T and the non-dimensional Lagrangian coordinates ˜ξ(τ) = ξ(t)/l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' ˜φ(τ) = φ(t) are introduced,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' together with the dimensionless mass density Θ = ρl3 T 2k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 26 Moreover, the non-dimensional parameter singling out the ratio between the two radii of the two con- straints is defined as χ = ζ− ζ+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The presence of an unstable invariant cone has been numerically detected for a broad range of val- ues of the above parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' As a paradigmatic example, the following values for the parameters are considered in this section ζ+ = R+ l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='6, k = k1l2 k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='3, γ = Fl k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='06, σ = ys l = 0, Θ = ρl3 T 2k2 = 1, χ = ζ− ζ+ = 6 (49) This reference numerical example contains all the most relevant features of the new kind of unstable structural behaviour disclosed in the present article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In the sequel all quantities are dimensionless, according to the normalization (48), and a superimposed dot stands for the derivative with respect to the dimensionless time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' However the dimensionless Lagrangian coordinates are denoted ξ and φ (without tilde) to ease the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' For the above geometry and loading, a piecewise invariant cone described by the initial condition x = y0 = � ξ(0), φ(0), ˙ξ(0), ˙φ(0) � = [0, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='00838564, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='372424, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='928025] is present, with a multiplier µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='079995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The intersection time intervals, calculated using the algo- rithm presented in the previous section for the detection of the invariant cone (and later confirmed by the numerical simulation of the mechanical system), are ∆τ − = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='637108 and ∆τ + = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='981694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The critical loads for flutter and divergence instability for the smooth substructures that compose the non-smooth mechanical structure can be analytically determined as γ+ flu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='div = −2ζ+(3ζ+ − 2)(ζ+(ζ+(k + 3) − kσ − 3) + 1) ± 2 √ 3 � ζ5 +(3ζ+ − 2)2k(ζ+ − σ) (2 − 3ζ+)2ζ2 + ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' γ− flu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='div = −2ζ−(3ζ− + 2)(ζ−(ζ−(k + 3) + kσ + 3) + 1) ± 2 √ 3 � ζ5 −(3ζ− + 2)2k(ζ− + σ) ζ2 −(3ζ− + 2)2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (50) where γ+ flu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='div (γ− flu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='div) refers to the structure with positive (negative) curvature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' while the critical loads for flutter correspond to the minimum absolute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Therefore, a substitution of the values (49) into (50) leads to the conclusion that the substructures are both stable when considered separately for an assumed tensile load γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='06, because their critical loads are much higher in absolute value (one is negative and therefore compressive): γ− flu = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='83477, and γ+ flu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='774567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The stability of each smooth subsystem can be observed in the phase portraits reported in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 10 and 11, for the system with negative and positive curvature, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The portraits refer to the linearized solution, so that the orbits evolve remaining confined within the neighbourhood of the origin, representing the equilibrium point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 27 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 ξ ˙ξ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 1 φ ˙φ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 ξ φ Figure 10: Phase portraits, showing stable response, for one (with positive curvature, see the inset on the left) of the two smooth systems forming the piecewise linear structure analyzed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 12 and 13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 ξ ˙ξ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 1 φ ˙φ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 ξ φ Figure 11: Phase portraits, showing stable response, for one (with negative curvature, see the inset on the left) of the two smooth systems forming the piecewise linear structure analyzed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 12 and 13 Although the two subsystems forming the piecewise linear structure are stable when considered separately, the combination of them is unstable, as can be appreciated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 12 and 13, reporting a solution of the linearized equations of motion belonging to the unstable invariant cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The evolution in time of the Lagrangian coordinates ξ and φ shows an exponential increase of the amplitude of mo- tion, analogous to the behaviour of a smooth mechanical system when flutter instability occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The phase portraits show an orbit laying on the invariant cone, so that each phase portrait is a projection of this 4-dimensional invariant manifold onto a 2-dimensional plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The orbit starting close to the origin (representing the trivial equilibrium configuration) evolves following a spiralling out motion, corresponding to an unstable behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 28 0 50 100 150 200 250 0 20 40 60 τ ξ 0 50 100 150 200 250 −100 −50 0 τ φ Figure 12: Evolution of the Lagrangian parameters in (dimensionless) time, for a non-smooth elastic structure (shown in the inset on the right) composed of two stable substructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Exponential blow-up demon- stratesinstability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 −2 0 2 ξ ˙ξ −5 −4 −3 −2 −1 0 1 −4 −2 0 2 4 6 φ ˙φ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 −4 −2 0 ξ φ Figure 13: Phase portraits for the non-smooth mechanical system analyzed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 12 (and shown on the lower part, right), evidencing instability although the two component substructures are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 Instability of the structure in the nonlinear range All the theoretical and numerical results obtained in the previous sections are based on the linearization of the equations of motion describing the system and, in particular, on the piecewise linear response resulting from the combination of the two linearized responses for each subsystem forming the non- smooth structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' For a smooth dynamical system, the linearization of the equations of motion near an equilibrium configuration is a classical strategy for determining whether or not the considered con- figuration is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' When the equilibrium of the linearized system is not marginally stable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' the real part of the eigenvalues of the Jacobian matrix vanishes), the Lyapunov theorem assures that the results obtained for the linearized case can be extended to the original nonlinear one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' A structure with a piecewise-linear behaviour cannot be further linearized, so that the Lyapunov theorem cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' An extension of this theorem to the nonlinear case of piecewise smooth dynamical system has been provided, see [12], under regularity assumptions which are satisfied for the structures under examination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Following this extension, the instability of the structures in a fully nonlinear range is expected, and indeed a direct integration of the nonlinear equations of motion for the previously analyzed structures confirms the presence of an unstable behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 0 100 200 300 400 500 600 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='3 τ ξ 0 100 200 300 400 500 600 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 τ φ 0 1 2 10−3 −2 0 2 10−3 ξ ˙ξ −4 −2 0 10−3 −5 0 5 10−3 φ ˙φ 0 1 2 10−3 −4 −2 0 10−3 ξ φ Figure 14: Phase portrait and evolution diagrams for the non-smooth structure (its linearized analysis is reported in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 12 and 13 nonlinear case In particular, the numerical solution of the nonlinear dynamics describing the reference structure is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 14, for the same initial conditions used in the piecewise linear analysis, namely, vector y0, is selected now with a sufficiently small modulus to start from a neighbourhood of the equilibrium configuration, thus y0 has been scaled as y0 = � ξ(0), φ(0), ˙ξ(0), ˙φ(0) � = fs × [0, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='00838564, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='372424, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='928025] , with the scaling factor fs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 30 The evolution of the Lagrangian generalized coordinates presents an exponential growth for small values of τ that can be associated to the presence of a quasi-invariant cone, as defined in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The orbits do not reach a limit cycle as for the Hopf bifurcation in smooth structures, but evidence blowing-up oscillations which reach a peak and then decrease in amplitude until near the initial amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' This motion repeats itself several times (in a way similar to beats) and the peak values are found to be almost constant and independent of the modulus of the initial condition y0 (the cases fs = 10−4 and fs = 10−5 have also been tested and for all cases a peak value of approximately ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='3 and φ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='6 have been found, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 14 a-b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' These features of the structural dynamics denote a complex unstable behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='3 A non-smooth structure evidencing flutter in both tension and compression 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='1 Existence of an unstable invariant cone in both tension and compression The reference solution considered in the previous section is just an example of instability related to the non-smoothness of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Considering other combinations of parameters leading to instability, the topological structure of the solution in the phase space has been found to remain similar, so that an orbit is found which spirals out from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In certain cases, the evolution of the solution can be more or less irregular, as can be seen in the example reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 15, where the wide difference in the values of ∆t− and ∆t+ leads to an orbit that remains for a longer time in the negative subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' It is interesting to note that non-smooth structures presenting this kind of instability can be found both for tensile and compressive follower forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' For instance, consider the following set of design parameters ζ+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5, k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='1, σ = 0, Θ = 1, χ = 2, together with the two cases of tensile or compressive follower force γA = Fl k2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5, γB = Fl k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (51) Note that at loads (51) the substructures are both stable, as their critical loads for flutter are γ− flu = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='62091, γ+ flu = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='10455.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In the case of compressive load γA = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5, the initial condition defining the invariant cone is xA = yA 0 = � ξ(0), φ(0), ˙ξ(0), ˙φ(0) � = [0, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='00594364, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='608652, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='793415] , and the eigenvalue is equal to µ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='481844, a value higher than the reference structure, in which the growth rate of the phase vector after each period ∆t− + ∆t+ was only 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The intersection time intervals can be calculated to be ∆t− = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='797295 and ∆t+ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='595396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In the case of tensile load γB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='75, the initial condition defining the invariant cone is xB = yB 0 = � ξ(0), φ(0), ˙ξ(0), ˙φ(0) � = [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='086944, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='360442, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='928721] and the eigenvalue is equal to µ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='486877, very close to the value for the compressive load, so that the growth rate of the solution on the invariant cone after each cycle is almost the same for both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The intersection time intervals can be calculated to be ∆t− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='311784 and ∆t+ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='132277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 31 0 20 40 60 80 −100 0 100 τ ξ 0 20 40 60 80 −200 −100 0 100 τ φ −20 −10 0 10 −20 0 20 ξ ˙ξ −30 −20 −10 0 −40 −20 0 20 40 φ ˙φ −20 −10 0 10 −30 −20 −10 0 ξ φ Figure 15: Phase portrait and evolution diagrams for a non-smooth elastic structure displaying instability in com- pression (although the component substructures are stable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 32 0 20 40 60 80 0 2 4 6 106 τ ξ 0 20 40 60 80 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 0 107 τ φ 0 100 200 −200 −100 0 100 ξ ˙ξ −600 −400 −200 0 0 500 φ ˙φ 0 100 200 −600 −400 −200 0 ξ φ Figure 16: Phase portrait and evolution diagrams for a non-smooth elastic structure displaying instability in ten- sion (although the component substructures are stable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 Instability of the nonlinear structure A nonlinear analysis has been performed for the structure linearly analyzed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 15 and 16, evidenc- ing unstable behaviour both in tension and compression, to confirm the instability detected from the linearized analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The initial condition has been scaled as yA 0 = � ξ(0), φ(0), ˙ξ(0), ˙φ(0) � = fs × [0, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='00594364, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='608652, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='793415] , yB 0 = � ξ(0), φ(0), ˙ξ(0), ˙φ(0) � = fs × [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='086944, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='360442, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='928721] , for tensile and compressive loads, respectively, with the scaling factor set equal to fs = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The behaviour of these cases, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 17 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 18, is qualitatively different from that of the nonlinear reference structure reported in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2, because ‘beats’ are not present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' However, the orbits are more irregular, showing an erratic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' However, the instability of the system is evident for both tensile and compressive loads, since the orbits evolve along the invariant cone spiraling away from the equilibrium configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 33 0 100 200 300 400 500 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 τ ξ 0 100 200 300 400 500 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 0 τ φ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='1 ξ ˙ξ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='1 0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 φ ˙φ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='1 0 ξ φ Figure 17: Fully nonlinear behaviour of the structure linearly analyzed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The phase portrait and evolution diagrams for the case confirm instability in tension, resulting in a highly irregular motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 34 0 20 40 60 80 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='8 τ ξ 0 20 40 60 80 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 0 τ φ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 1 ξ ˙ξ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 0 −1 0 1 φ ˙φ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 0 ξ φ Figure 18: Fully nonlinear behaviour of the structure linearly analyzed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The phase portrait and evolution diagrams for the case confirm instability in tension, resulting in a blowing up motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 6 Conclusions A class of elastic structures has been disclosed, exposed to a new kind of instability, which, although already elaborated from a mathematical point of view, was never directly related to elastic structural systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The instability results from the combination of both non-conservative follower load and non- smoothness of the equations governing the dynamics of the mechanical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' From this point of view, the proposed model is a discrete and simplified prototype of nonassociative elastoplasticity or frictional sliding, as it shares with these theories both the lack of self-adjointness of the governing differential operator and the piecewise linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' We provide the first example of the use of the invariant cone theory as an instability criterion for elastic structures, permitting the design of an unstable structure as a fusion of two stable structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The instability is fully explained and motivated from a mechanical perspective and is shown to be similar to the flutter instability occurring in smooth systems under follower loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' From the point of view of applications, keeping into account that nonconservative follower forces are fully feasible [25], we introduce a new design paradigm to avoid previously unknown structural instabilities or to design extremely deformable structures to be employed as sensors, or for energy harvesting, or as building blocks for archtected materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 7 Acknowledgments The authors acknowledge financial support from ERC-ADG-2021-101052956-BEYOND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 35 8 Appendix 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='1 Flutter and divergence critical loads for smooth constraints For a smooth profile with continuous curvature, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 2, the flutter and divergence critical loads can be computed by solving the linearized equations (7), as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Consequently, any profile can be approximated by its osculating circle, since only the local curvature enters the linearized equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Therefore, assuming a circular profile defined by the dimensionless signed curvature κ = ±l/R± = ±1/ζ±, where ζ± = R±/l > 0 is the normalized radius of curvature, the first and second invariants I1 and I2 of the matrix Γ = −M −1K, see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' (19), are given by I1 = 2 Θ � γ(2κ − 3) − 2(3 + k + κ2 − 3κ) + 2kκσ � , I2 = 12k Θ2 (1 − κσ), in terms of the dimensionless quantities defined in (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The flutter and divergence critical loads can then be computed from the equation I2 1 − 4I2 = 0, of the parabola defining the critical condition in the I1−I2 plane, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' The solution of this equation provides the critical loads γflu,div = 2 2κ − 3 � 3 + k − 3κ + κ2 − kκσ ± � 3k(1 − κσ) � , where it is understood that the critical load for flutter (divergence) corresponds to the minimum (max- imum) absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Since the term inside the braces is always positive, for any k > 0, κ ∈ R and σ < 1/κ, it follows that the follower and divergence loads are compressive for κ < 3/2 and tensile for κ > 3/2, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 36 −10 −5 0 5 10 −40 −20 0 20 40 κ = 3/2 k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='−10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='−5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='−40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='−20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='κ = 3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='k = 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='−10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='−5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='−40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='−20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='κ = 3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='k = 25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='−10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='−5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='−40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='−20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='κ = 3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='k = 50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='Flutter critical load ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='Divergence critical load ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='Flutter instability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='Divergence instability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='Figure 19: Flutter and divergence instabilities for smooth circular constraints: the flutter (blue curve) and diver- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='gence (yellow curve) critical loads are reported as a function of the dimensionless signed curvature κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='and four values of the dimensionless stiffness k = {2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='5, 5, 25, 50}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 37 References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' Branicky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' “Multiple Lyapunov functions and other analysis tools for switched and hybrid systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In: IEEE Transactions on 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' “The deformation of an elastic rod with a clamp sliding along a smooth and curved profile”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' In: International Journal of Solids and Structures 69-70 (Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' 491–497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE4T4oBgHgl3EQfNgyO/content/2301.04957v1.pdf'} +page_content='1016/j.' 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+1,2964 @@ + +1 +Aerial Image Object Detection With Vision Transformer Detector (ViTDet) + +Liya Wang, Alex Tien + +The MITRE Corporation, McLean, VA, 22102, United States + +ABSTRACT + +The past few years have seen an increased interest in aerial +image object detection due to its critical value to large-scale +geoscientific research like environmental studies, urban +planning, and intelligence monitoring. However, the task is +very challenging due to the bird’s-eye view perspective, +complex backgrounds, large and various image sizes, +different appearances of objects, and the scarcity of well- +annotated datasets. Recent advances in computer vision have +shown promise tackling the challenge. Specifically, Vision +Transformer Detector (ViTDet) was proposed to extract +multi-scale features for object detection. The empirical study +shows that ViTDet’s simple design achieves good +performance on natural scene images and can be easily +embedded into any detector architecture. To date, ViTDet’s +potential benefit to challenging aerial image object detection +has not been explored. Therefore, in our study, 25 +experiments were carried out to evaluate the effectiveness of +ViTDet for aerial image object detection on three well-known +datasets: Airbus Aircraft, RarePlanes, and Dataset of Object +DeTection in Aerial images (DOTA). Our results show that +ViTDet can consistently outperform its convolutional neural +network counterparts on horizontal bounding box (HBB) +object detection by a large margin (up to 17% on average +precision) and that it achieves the competitive performance +for oriented bounding box (OBB) object detection. Our +results also establish a baseline for future research. + +Index Terms— Aerial image, Object detection, +Computer vision, ViTDet, HBB, OBB + +1. INTRODUCTION +Aerial image object detection has been a vibrant research +topic for its essential role in large-scale geoscientific research +like environmental science, ecology, agricultural studies, +wildfire monitoring, urban planning, intelligence monitoring, +and emergency rescue. However, the task is very challenging +due +to +the +bird’s-eye +view +perspective, +complex +backgrounds, large and various image sizes, various +appearances of objects, and the scarcity of well-annotated +datasets [1]. In addition, the objects in aerial images are often +arbitrarily oriented. For that reason, instead of using common +horizontal bounding boxes (HBBs) (Fig. 1a), oriented +bounding boxes (OBBs) (Fig. 1b) have been alternatively +used to avoid mismatching between bounding boxes and +corresponding objects [1]. +In the past few years, deep learning techniques have +dominated the object detection domain for their effective +feature learning capability. Fig. 2 shows the milestones of +deep learning algorithms in object detection since 2014. The +green and orange rectangles in Fig. 2 highlight one-stage and +two-stage methods for HBB object detection, respectively. +Two-stage methods have two separate processes, region +proposal and detection proposal, while in one-stage methods +these two processes are combined. In general, two-stage +methods have better performance than one-stage methods at +the expense of computational workload. The purple rectangle +in Fig. 2 shows the algorithms designed specifically for OBB +object detection; the yellow rectangle calls out the important +deep learning frameworks for building object detection +algorithms, where ResNet [2], Vision Transformer (ViT) [3], +and Swin-T [4] are commonly used backbones for feature +extraction; in particular, Vision Transformer Detector +(ViTDet) [5] was a newly proposed backbone for object +detection; feature pyramid network (FPN) [6] is often used as +neck after backbone for feature fusion. +Feature learning is always essential in any computer vision +(CV) machine learning methods. Following the advent of +ViTs [3] in 2021, an exciting self-supervised learning (SSL) +method, Masked Autoencoder (MAE) [7], was proposed to +learn effective visual representation features. MAE adopts +Masked Image Modeling (MIM) technique and tries to infer +masked image patches from unmasked ones. To date, MAE +has attracted unprecedented attention because of its superior +performance over its supervised learning and contrastive +learning counterparts. The encouraging success of MAE has +inspired a wide range of applications in the areas of video, +audio, medical images, earth observation, multimodal, point +cloud, 3D mesh data, reinforcement learning, and graphs (see +Table 1 for the summary). It is noticeable that several efforts +have been devoted to object detection, including ViTDet [5], +a new backbone designed especially for object detection with +support of MAE pretrained ViT. +ViTDet was particularly designed to enhance the +effectiveness of ViT backbone on object detection problems. +Although MAE pretrained ViTs are effective for image +classification tasks, they are less effective for object +detection, which usually requires multi-scale features. ViT is +a plain, non-hierarchical architecture that maintains a single- +scale feature map throughout, which indicates that ViT +åbackbone is not sufficient for object detection tasks, + + +2 +especially when dealing with multi-scale objects and high- +resolution images [8]. To deal with the deficiency, ViTDet +was invented to extract multi-scale features for object +detection. ViTDet has demonstrated its superior performance +on natural scene image (e.g., COCO [9]) object detection [5], +and its simple design can also make it embeddable in any +detector architecture. +To the authors’ knowledge, no research has ever adopted +ViTDet and examined its performance for aerial image object +detection. Therefore, this research aims to evaluate and gain +insight on the potential benefits of ViTDet for both aerial +image HBB and OBB object detection. The remainder of the +paper is organized as follows: Section II describes related +work. Section III discusses the aerial image datasets used for +performance evaluation. Section IV gives the details of the +implementation platforms for carrying out the experiments. +The results are presented in Section V. Section VI is the +conclusion. + + +(a). HBB +(b). OBB +Fig. 1 Illustration of different bounding box types. + +Fig. 2 Milestones of deep learning algorithms in object detection. + + + + + + + +ReDet +Oriented +Oriented RCNN +Rol Transformer +Gliding Vertex +R'Det +Rotated RetinaNet +Rotated RepPoints +Rotated ATSS +S?A-Net +Rotated FasterRCNN +Rotated FCOS +Beyond Boundina-Box +RCNN +Fast RCNN +Mask RCNN +Two-stage Horizontal +SPPNet +Faster RCNN +Cascade RCNN +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021 +2022 +YOLOv4 +YOLOVT +YOLO +RetinaNet +YOLOv3 +YOLOR +YOLOv2 +YOLOv5 +DETR +SDD +One-stage +ViT +ResNet +FPN +ViTDet +Swin-T +3 +Table 1 Summary of latest MIM research +Domain +Sub-Domain +Research Papers +Vision +Image +BEiT v1 [10], v2 [11], MAE [7], SimMIM [12], ADIOS [13], AMT [14], AttMask [15], Beyond-Masking [16], +BootMAE [17], CAE [18], CAN [19], ConvMAE [20], Contrastive MAE [21], ContrastMask [22], dBOT +[23], DMAE [24], Denoising MAE [25], GreenMAE [26], iBOT [27], LoMaR [28], LS-MAE [29], +MaskAlign [30], MaskDistill [31], MaskFeat [32], MaskTune [33], MetaMask [34], MFM [35], MILAN +[36], MixMask [37], MixMIM [38], MRA [39], MSN [40], MST [41], MultiMAE [42], MVP [43], RC- +MAE [44], SDMAE [45], SemMAE [46], SdAE [47], SupMAE [48], U-MAE [49], UM-MAE [50] +Video +AdaMAE [51], Bevt [52], MAM2 [53], MAR [54], MaskViT [55], M3Video [56], MCVD [57], MotionMAE +[58], OmniMAE [59], Spatial-Temporal [60], SSVH [61], VideoMAE [62], Vimpac [63], VRL [64] +Medical Image +DAMA [65], GCMAE [66], SD-MAE [67], SMIT [68] +Satellite Image +SatMAE [69] +Image Classification +MUST [70] +Object Detection +imTED [71], Mask DINO [72], ObjMAE [73], PACMAC [74], ViTDet [5] +Segmentation +kMaX-DeepLab [75], Mask-CLIP [76], MaskDistill [77], Mask Transfiner [78], MOVE [79], NameMask [80] +Image Generation +DiffEdit [81], MAGE [82], MaskGIT [83], Divide-and-Revise [84] +Face Recognition +FaceMAE [85], FFR-Net [86], MFR [87] +Text Recognition +MaskOCR [88] +Multimodal +Vision-Language +Data2vec [89], M3AE [90], MAMO [91], MaskCLIP [92], Masked V+L [93], M3AE [94], MLIM [95], ViCHA +[96], VL-BEiT [97], VLC [98], VIOLET [99], VLMAE [100] +Audio-Language +CAV-MAE [101] +Others +Audio +Audio-MAE [102], Group-MAE [103], MAE-AST [104], MSM [105], M2D [106] +Anomaly Detection +MAEDAY [107], SSMCTB [108], ST-MAE [109] +Graph +MGAP [110], GMAE [111], GMAE-AS [112], GraphMAE [113], HGMAE [114], MGAE [115], MaskGAE +[116] +Point Cloud +Point-Bert [117], Point-MAE [118], Point-M2AE [119], Mask-Point [120], Masked [121], Voxel-MAE [122] +Skeleton +SimMC [123] +Depth Estimation +Depth Refinement [124], FM-Net [125] +Reinforcement +Learning +MLR [126], Motor Control, Visual Control [127] +3D Mesh Data +MeshMAE [128] +Adversarial Attack +MAD [129] +Miscellaneous +D-MAE [130], MAEEG [131], MGD [132], Extra-MAE , MADE [133], MaskDP [134], i2i [135], Lifetime +Prediction [136], MET [137], MIL [138], Robot Training [139], Time Series [140] +Survey +MIM Survey [141] +Theory +CL vs MIM [142], Contextual Representation Learning[143] , Data Scaling [144], EVA [145], i-MAE [146], +Revealing MIM [147], Understanding MAE [148], Understanding MIM [149], Understanding DR [150], +Architecture +Deeper vs Wider [151], Masked BNN [152], ViT Back to CNN [153], ConvNeXt V2 [154] + + + +4 +2. RELATED RESEARCH +2.1 Backbones +In CV deep learning methods, backbones are commonly used +to extract discriminative object feature representation, and +they have been a driving force for rapid object detection +performance improvement [155]. Popular backbones for +object detection are ResNet [2], ResNeXt [156], and Swin-T +[4] because of their deep hierarchical architectures, which can +produce the needed multi-scale features. Backbone +pretraining is usually carried out on ImageNet-1k [157] with +either supervised learning or SSL methods like contrastive +learning or MAE, which will be presented next. +2.2 MAE +MAE is an asymmetric autoencoder that uses ViTs in both its +encoder and decoder, and the size of decoder is smaller than +the encoder, as illustrated in Fig. 3. It directly infers masked +patches from the unmasked ones with a simple loss of mean +squared error (MSE). To save computation, the encoder +works on only the unmasked patches; in contrast, the decoder +works on both masked and unmasked patches trying to +predict the original images. The masking ratio can be set up +to 75%, which is considerably higher than that in BERT +(typically 15%) [158] or earlier MIM methods (20% to 50%) +[10], [159]. MAE’s ablation study also points out that a high +masking ratio is good for fine-tuning and linear probing [7]. +With those meticulous designs, MAE is three times (or more) +faster than Bidirectional Encoder representation from Image +Transformers (BEiT) [10] while achieving superior +performance [7]. + +Fig. 3 MAE architecture [7]. +2.3 ViTDet +ViTDet was designed to extract multi-scale features for +object detection with minimal adaptation to MAE pretrained +ViT. Fig. 4 shows the architecture of ViTDet building a +simple feature pyramid from only the last feature map of a +plain ViT backbone, and it adopts non-overlapping window +attention for efficient feature extraction. To propagate +information, ViTDet uses a small number of cross-window +blocks, which can be implemented with global attention or +convolutions. The adaptation takes place only during fine- +tuning; therefore, they do not affect the upstream pretraining. +The empirical study shows that ViTDet’s simple design +achieves good results on natural scene image object detection +[5], which further proves that the general-purpose pretrained +ViT from MAE can serve object detection as well. ViTDet’s +simple design makes it easily plug into any detector +architecture. Investigating ViTDet’s effectiveness for +challenging aerial image object detection is the focus of this +study. + +Fig. 4 ViTDet backbone architecture. ViTDet builds a +simple pyramid from only the last, large-stride (16) +feature map of a plain backbone [5]. +2.4 Object Detection +Object detection is one of the most fundamental yet +challenging CV tasks. The task is to identify and localize all +the objects in an image. Each object will have a label, and its +location is commonly defined by an HBB (𝑥, 𝑦, 𝑤, ℎ), where +𝑥 and 𝑦 are center coordinates of the box, and 𝑤 and ℎ are +width and height of the box (illustrated in Fig. 5). + +Fig. 5 Illustration of HBB object detection. +However, in aerial images, the objects could be arbitrarily +oriented. The methods relying on HBBs often introduce +mismatches between the Regions of Interest (RoI) and +objects, which further deteriorate the final object +classification confidence and localization accuracy [1]. For +example, in Fig. 6, a RoI (top) may contain several instances, +leading to difficulties for the subsequent classification and +location task [1]. For this reason, research has proposed OBB +annotations (𝑥, 𝑦, 𝑤, ℎ, 𝜃) (see Fig. 7 for illustration), where + +encoded +masked +25% +....... +.atent ++ +encoder +decoder +(MiT) +input +target1/16 +1/32 +neck/head: +1/16 ++ +1/16 +backbone: +1/16 +1/8 +1/16 +114dog/1.00 +catj0.99 +5 +𝑥 and 𝑦 are center coordinates of the box and 𝑤, ℎ, and 𝜃 are +the width, height, and angle of an OBB. It should be noted +that 𝑤 and ℎ of the OBBs are measured in different rotating +coordinate systems for each object. OBBs make more +accurate orientation information, especially when detecting +aerial objects with a large aspect ratio, arbitrary orientation, +and dense distribution. Furthermore, OBBs can also have +more accurate RoIs and allow for better discriminative +feature extraction for object detection. Deep learning +algorithms such as oriented Region-based Convolutional +Neural Network (RCNN) [160], RoI Transformer [1], and +Rotation-equivalent Detector (ReDet) [161] have been +proposed particularly for OBB detection. They usually adopt +numerous rotated anchors with different angles, scales, and +aspect ratios for better regression, resulting in significant +computation burden. + +Fig. 6 HBB (top) vs OBB (bottom) illustration in an +image with many densely packed objects [1]. + +Fig. 7 OBB definition, where x and y are center +coordinates of the box and w, h, and θ are the width, +height, and angle of an OBB [162]. +2.5 Object Detection Algorithms +As mentioned in Fig. 2, there are several types of object +detection methods. Famous detection challenges have shown +that two-stage methods achieve better performance than one- +stage methods if heavy computation workload is not a +concern. In our work, we focus on two-stage methods for +their better performance. The two-stage object detection +methods usually consist of the three steps proposed in RCNN +[162] (illustrated in Fig. 8). The first step is region proposal, +which generates a series of candidate region proposals (about +2,000) that may contain objects. The second step is feature +extraction for the proposed regions. Following that, the third +step is classification, where the candidate regions are +distinguished as object classes or background and furtherly +fine-tuned for the coordinates of the bounding boxes. As +research advances, various types of algorithms have been +proposed to hone the components for better performances. +Next, the most fundamental deep learning method, Faster +RCNN [163], and the algorithms explored in this research +will be presented. + +Fig. 8 Object detection system overview [163]. +2.6 Faster RCNN +Faster RCNN [163] is the first end-to-end object detection +method fully using deep learning techniques, which is more +efficient than its predecessors, RCNN [162] and Fast RCNN +[164]. Faster RCNN proposes a novel idea called region +proposal network (RPN), which fully utilizes convolutional +layers extracted features to generate proposals. Compared +with conventional region proposal generation algorithms like +Selective Search [165], which is an offline algorithm and +makes it impossible to train whole algorithm from end to end, +RPN is much more efficient. After RPN, Faster RCNN then +uses the RoI pooling layer to extract a fixed-length feature +vector from each region proposal. Fig. 9 depicts the +architecture of Faster RCNN and the sequential relationship +among backbone (convolutional layers), RPN, and the RoI +pooling layer. Based on Faster RCNN, several variants have +been proposed to improve the performance of object +detection. Next, we present the relevant methods tested in our +study. +2.7 Mask RCNN +Mask RCNN [166] is an extension of Faster RCNN. Besides +Faster RCNN’s two outputs for each candidate object—a +class label and a bounding box—a third type of output, object +mask, is proposed. Fig. 10 illustrates the architecture of Mask +RCNN. The backbone of Mask RCNN is for feature +extraction, and it can be traditional ResNet [2], Swin-T [4], + +y +0E[-90°,-0"] +(x, y.w, h, 0)warped region +aeroplane? no. +Mperson?yes. +tvmonitor?no. +1. Input +2. Extract region +3.Compute +4. Classify +image +proposals (~2k) +CNNfeatures +regions +6 +or newly proposed ViTDet [5]. RPN is the same as the one in +Faster RCNN. The novel element of Mask RCNN is the +RoIAlign layer, which can preserve the pixel-level spatial +correspondence and address the shortfalls of Fast/Faster +RCNN. The mask head is a Fully Convolutional Network +(FCN) [167] on top of a feature map. Mask RCNN is still +simple to train and generalizes well, but it introduces a small +computation overhead to Faster RCNN. + +Fig. 9 Faster RCNN architecture [164]. + + +Fig. 10 Mask RCNN architecture [169]. +2.8 Cascade RCNN +Cascade RCNN [168] adopts a new trick for better +performance, classifying with multistage classifiers. The +trick works in such a way that early stages can discard many +easy negative samples; therefore, later stages can focus on +handling more difficult examples. Fig. 11 illustrates the +architecture of Cascade RCNN, where “I” is input image, +“conv” is the convolutions backbone, “pool” is for the region- +wise feature extraction, “H” represents various network head, +“B” is the bounding box, “C” is classification, and “B0” is +proposals in all architectures. An object detection architecture +like Faster RCNN can be deemed as a cascade (i.e., the RPN +removing large amounts of background and the following +detector head classifying the remaining regions). Therefore, +Cascade RCNN extends the idea to multiple stages in the +classification layer to enhance the performance. When mask +head is also included in the output, the algorithm is called +Cascade Mask RCNN, which is used for HBB object +detection in our study. + +Fig. 11 Cascade RCNN architecture [170]. +2.9 RoI Transformer +RoI Transformer [1] was designed specifically for OBB +object detection. In the past, rotated anchors have been used +to tackle the OBB object detection problem. The design +always multiplies the number of anchors, which considerably +increases the computation burden. Hence, RoI Transformer +was tried for reducing the computation burden. Fig. 12 +illustrates the architecture of RoI Transformer. In specific, it +first adopts a Rotated Region of Interest (RRoI) learner to +transform a Horizontal Region of Interest (HRoI). Based on +the RRoIs, it then uses a Rotated Position Sensitive RoI Align +(RPS-RoI-Align) module to extract rotation-invariant +features, which are then used for enhancing subsequent +classification and regression performance. RoI Transformer +is a light-weighted component and can be easily plugged into +any detector framework for OBB object detection. +2.10 Rotation-Equivalent Detector (ReDet) +ReDet [161] was also proposed to solve OBB aerial image +object detection problems. It introduces rotation-equivariant +networks into the detector to extract rotation-equivariant +features, which can accurately predict the orientation and +result in a huge reduction in model size. Fig. 13 illustrates the +working mechanism of ReDet. Fig. 13a shows the overall +architecture of ReDet, which first uses the rotation- +equivariant backbone to extract rotation-equivariant features, +followed by an RPN and RoI Transformer (RT) [1] to +generate RRoIs. After that, a novel Rotation-Invariant RoI +Align (RiRoI Align) is used to produce rotation-invariant +features for RoI-wise classification and bounding box +regression. Fig. 13b shows rotation-equivariant feature maps. +Under the cyclic group 𝐶!, the rotation-equivariant feature +maps with the size (𝐾, 𝑁, 𝐻, 𝑊) have 𝑁 orientation channels, + +classifier +Rol pooling +proposals +RegionProposalNetwork +featuremaps +conv layersRaw image ++ +Backbone +RPN +RolAlign +Object detection +Mask generation +head +head +Class +Box +MaskBO +C1 +B1 +C2 +B2 +C3 +B3 +H1 +H2 +H3 +pool +1 +conv +7 +and each orientation channel corresponds to an element in 𝐶!. +Fig. 13c illustrates RiRoI Align. The proposed RiRoI Align +consists of two parts: spatial alignment and orientation +alignment. For an RRoI, spatial alignment warps the RRoI +from the spatial dimension, while orientation alignment +circularly switches orientation channels and interpolates +features to produce completely rotation-invariant features. +ReDet has achieved state-of-the-art performance [169]; +therefore, it was selected in our study to test ViTDet for OBB +object detection. Next, we give more details about aerial +image datasets to test ViTDet backbone. + + +Fig. 12 Architecture of RoI Transformer [1]. + + +Fig. 13 ReDet architecture [161]. + +RRolLearner +Classification +Decoder +10channels +Rotated Position Sensitive Rol Align +RolTransformer +10chammels +490chanmnels +RegressionRPN+RT +classification +feature maps : +backbon +RiRol Align +Rol feature +(a) Overall architecture +c() +c2 + spatial alignment +orientation alignment +FC layer +(2) +RRol +switch +channels +interpolate +2r +(c) Rotation-invariant Rol Align +8 +3. DATASETS +To evaluate the newly proposed backbone of MAE pretrained +ViTDet for aerial image object detection, we conducted 25 +experiments across three distinct datasets of aerial images: 1) +Airbus Aircraft Detection [170], 2) RarePlanes [171], and 3) +Dataset of Object DeTection in Aerial images (DOTA) [172]. +The smallest dataset is Airbus Aircraft Detection, with 103 +images, and the largest dataset is RarePlanes, with about +68,000 images. These two both use HBB annotations. DOTA +is the most complicated dataset with OBB annotations. Table +2 gives the details of the three datasets. A short introduction +about each dataset will be provided next. +3.1 Airbus Aircraft Detection +The Airbus Aircraft Detection [173] dataset is collected from +Airbus’ Pleiades twin satellites for earth observation, which +collect pictures of airports worldwide on a regular basis. This +dataset contains 103 images with 0.5 m resolution (see Fig. +14 for an example). Each image is stored as a JPEG file of +size 2,560 x 2,560 pixels (i.e., 1,280 meters x 1,280 meters). +Some airports could appear multiple times at different +acquisition dates. Some images may include fog or cloud +because of weather. The annotations are provided in the form +of closed GeoJSON polygons. A CSV file named +annotations.csv provides all annotations—one annotation per +line with the corresponding object ID; filename as image ID; +annotation box; and class label, mainly Aircraft (3,316 +instances) or Truncated_Aircraft (109 instances) when an +aircraft is located at the border of an image. The minimum +and maximum number of aircraft in an image are 5 and 92, +respectively. +3.2 RarePlanes +RarePlanes [171] is an open-source dataset that includes both +real and synthetically generated satellite images. The +RarePlanes dataset is specifically designed to automatically +detect aircraft and their attributes in satellite images (see Fig. +15 for examples). To date, RarePlanes is the largest openly +available high-resolution dataset created to test the value of +synthetic data from an overhead perspective. The real images +are collected from 253 Maxar WorldView-3 satellite scenes, +spanning 112 locations and 2,142 km2 with 14,700 hand- +annotated aircraft. The accompanying synthetic dataset is +generated via AI.Reverie’s simulation platform and has about +60,000 synthetic satellite images covering a total area of +9,331 km2 with about 630,000 aircraft annotations. +Both the real and synthetically generated aircraft have been +given 10 fine-grained attributes—aircraft length, wingspan, +wing shape, wing position, wingspan class, propulsion, +number of engines, number of vertical stabilizers, presence of +canards, and aircraft role—which are derived from the +previous nine attributes. Seven role classes have been +defined; Table 3 summarizes aircraft role count for real +dataset, in which the first column lists seven “aircraft role” +classes. As demonstrated in Table 3, the most common +aircraft role is Small Civil Transport/Utility, and the least +common one is Military Bomber. More detail on role +definitions can be found in the “RarePlanes User Guide” at +https://www.cosmiqworks.org/rareplanes-public-user- +guide/. We conducted two types of object detection tasks— +aircraft and aircraft role—on both sub-datasets to evaluate +MAE pretrained ViTDet backbone’s performance. +3.3 DOTA +DOTA [174] is the largest aerial image dataset for OBB +object detection (see Fig. 16 for some examples), and it is +deemed as the most challenging dataset in the earth +observation community for its various image sizes and +densely packed objects. It has released three different +versions. DOTA-v1.0 contains 2,806 aerial images, with the +size ranging from 800 × 800 to 4,000 × 4,000 and containing +188,282 instances. DOTA-v1.0 has 15 common categories: +Plane (PL), Baseball diamond (BD), Bridge (BR), Ground +track field (GTF), Small vehicle (SV), Large vehicle (LV), +Ship (SH), Tennis court (TC), Basketball court (BC), Storage +tank (ST), Soccer-ball field (SBF), Roundabout (RA), Harbor +(HA), Swimming pool (SP), and Helicopter (HC). The +second version DOTA-v1.5 was released for 2019 Detecting +Objects in Aerial Images (DOAI) Challenge. Compared with +v1.0, it has an extra category, Container crane, and more +extremely small instances (less than 10 pixels), resulting in +402,089 instances. The third version, DOTA-v2.0, collects +more aerial images from Google Earth and GF-2 Satellite. +DOTA-v2.0 has 18 categories, 11,268 images, and 1,793,658 +instances. Compared with DOTA-v1.5, it further adds the +new categories of Airport and Helipad. Our study focused on +DOTA-v1.0 due to abundant baseline benchmarks available +for evaluating ViTDet’s performance. +Table 2 Tested aerial image datasets +Datasets +Subsets +Tasks +# Object Types +# Images +# Instances +Image Width +Annotation +Year Available +Airbus Aircraft Detection +- +Aircraft +2 +103 +3,425 +2,560 +HBB +2021 +RarePlanes +Real +Aircraft +1 +8,527 +14,700 +512 +HBB +2020 +Synthetic +Aircraft +1 +60,000 +629,551 +1,920 +Real +Aircraft role +7 +8,527 +14,700 +512 +Synthetic +Aircraft role +7 +60,000 +629,551 +1,920 +DOTA +v 1.0 +Objects +15 +2,806 +188,282 +800-4,000 +OBB +2018 + + +9 + +Fig. 14 Airbus Aircraft Detection image example. +Table 3 Real dataset role count + + +汉Aircraft role +Count +Small Civil Transport/Utility +8002 +Medium Civil Transport/Utility +5132 +Large Civil Transport/Utility +1098 +Military Transport/Utility/AWAC +283 +Military Fighter/lnterceptor/Attack +171 +Military Trainer +15 +Military Bomber +6 +10 + +Fig. 15 Examples of the real and synthetic datasets present in RarePlanes [173]. + +Fig. 16 Examples of annotated images in DOTA [176]. + +X10000 +Helicoprer +Bridor +Shir +romnd track field +mall-vehicle +Harbor +Baseball dic +Storage tan +11 +4. IMPLEMENTATION PLATFORMS +To evaluate the new backbone ViTDet in the aforementioned +algorithms, we chose two well-known platforms in the CV +field: Detectron2 [175] and MMRotate [169]. Detectron2 is +the official implementation site for ViTDet and is used for +HBB object detection. MMRotate is selected because it has +most of state-of-the-art algorithms for OBB object detection, +which Detectron2 lacks. +4.1 Detectron2 +Detectron2 [175] is an open-source research platform +developed by Facebook AI Research [175]. The platform is +implemented in PyTorch. It provides many state-of-the-art +detection and segmentation algorithms, including FPNs, +numerous variants of the pioneering Mask RCNN model +family, and the latest MAE pretrained ViTDet backbone. +Therefore, we used Detectron2 to implement aerial image +HBB object detection with its provided pretrained models. +4.2 MMRotate +OpenMMLab [176] is another open-source platform to +provide powerful CV packages like Detectron2. For general +HBB object detection, MMDetection in OpenMMLab is the +go-to package and forms the basis for MMRotate [169], +which is specially designed for OBB object detection. +According to Table 4 provided by Zhou et al. [169], +MMRotate provides 18 OBB algorithms and four famous +datasets. In addition, its modular design with multiple choices +of orientation angle representations, backbones, necks, and +detection heads makes it very easy and flexible to set up a +new model. For example, it can support multiple angle +representations. Popular OpenCV definition, long edge 90° +definition, and long edge 135° are all supported in +MMRotate. MMRotate also provides baseline benchmarks +for comparison. Therefore, we selected MMRotate for +customization of RoI Transformer and ReDet, where ViTDet +will be used as the backbone. Note that at the time of this +research, ViTDet has not officially been implemented in +MMRotate. We used a non-official version of ViTDet from +[177] for OBB object detection. + +5. RESULTS +This section presents the experiment results of aerial image +object detection using the MAE pretrained ViTDet backbone. +5.1 Experimental Setup +To make a comprehensive evaluation, we conducted 25 +experiments on the selected three datasets: 1) Airbus Aircraft +Detection [170], 2) RarePlanes [171], and 3) DOTA [172]. +The experiments tested three types of backbones (i.e., ResNet +[2], Swin Transformer [4], and ViTDet [5]) in four object +detection algorithms (i.e., Mask RCNN [166], Cascade Mask +RCNN [168], RoI Transformer [1], and ReDet [161]). For the +Airbus Aircraft and RarePlanes datasets, we tested Mask +RCNN and Cascade Mask RCNN algorithms on the +Detectron2 [175] platform. For the DOTA dataset, we tested +RoI Transformer and ReDet on the MMRotate [169] +platform. The MAE pretrained ViTs were downloaded from +https://github.com/facebookresearch/mae [178]. All the +experiments were carried out on four A100 GPUs with +160GB memory. More specific implementation details for +each dataset will be presented in the corresponding sections. +5.2 Evaluation Metrics +Average precision (AP) is a commonly used metric to +evaluate object detection algorithms, and it is derived from +precision and recall. There are several variants of AP. +Different platforms may adopt different versions of AP. In +details, Detectron2 uses COCO-defined AP metrics (see +Table 5 for the detailed list), which mainly focus on the +accuracy of the bounding box. COCO-defined AP is averaged +across all classes and 10 Intersection Over Union (IOU) +values ranging from 0.5 to 0.95 in steps of 0.05 [155]. By +contrast, in the MMRotate platform, AP is calculated +separately for each class, and mean AP (mAP) is calculated +by averaging AP over all classes. To have a fair comparison, +we calculated the default evaluation metrics defined by the +two platforms. +Table 4 Open source rotated object detection benchmarks [169] + + + + + +Benchmark +AerialDet +JDet +OBBDet +AlphaRotate +MMRotate +DL library +PyTorch +Jittor +PyTorch +TensorFlow +PyTorch +Inference +PyTorch +Jittor +PyTorch +TensorFlow +PyTorch +engine +onnx runtime +Windows +Windows +OS +Linux +Linux +Windows +Linux +Linux +Linux +Algorithm +5 +8 +9 +16 +18 +Dataset +1 +4 +5 +11 +4 +Doc. +- +- +Easy install +- +- +- +Maintain +V +12 +Table 5 COCO-defined AP evaluation metrics [154], used in default by Detectron2 +5.3 Airbus Aircraft Object Detection Results +To detect aircraft in this small dataset, we have taken the +following three steps: +Step 1. Data preparation +o +Split dataset (103 images) into training (92 images) and +testing (11 images) subsets. +o +Convert he data into COCO format for easy use of +ViTDet in Detectron2 packages. +Step 2. Experiment setup +o +Downloaded COCO pretrained models of Mask RCNN +and Cascade Mask RCNN from the website +https://github.com/facebookresearch/detectron2/tree/ma +in/projects/ViTDet. +o +Set up the configuration files for model training. +Step 3. Model fine-tuning +Table 6 shows the experiments conducted and the +performance evaluation results. The tested backbones are as +follows: ResNeXt-101 [156] is a convolutional neural +network (CNN) backbone with 101 layers and is pretrained +in a supervised manner. ViTDet, ViT-L is ViTDet backbone +built with a large version of ViT that has 24 layers and 1024- +dimension output. ViTDet, ViT-H is ViTDet backbone built +with a huge version of ViT that has 32 layers and 1280- +dimension output. The column FT-epoch is the epochs for +fine-tuning. +Yellow color highlights the best metrics in Table 6Table 6. +As expected, Cascade Mask RCNN performs better than +Mask RCNN; larger backbones achieve better performance. +Cascade Mask RCNN with backbone of ViTDet, ViT-H +achieves the best performance in all evaluation metrics except +for AP75, an evaluation metric when IOU equals 0.75. Most +importantly, ViTDet outperforms ResNeXt-101 in most of +evaluation metrics, and ResNeXt-101 is deemed as one of top +CNN backbones. According to Table 6, ViTDet performs +much better (20-50% improvement) than ResNeXt-101 on +APs, which measures AP for small object detection. For APl, +a metric to measure AP for large object detection, ViTDet +also beats ResNeXt-101 by a large margin of 16-20%. For +AP, 6-10% improvement has been achieved by ViTDet. In +short, the new backbone ViTDet greatly improves object +detection performance on this small dataset. +Fig. 17 shows object detection results on a testing image. +There are about 90 aircraft in this testing image; all but three +are detected and one is falsely labeled. Therefore, ViTDet +backbone does a good job for this testing image. +Table 6. Airbus Aircraft object detection results comparison. Note APs, APm, and APl represent COCO-defined APsmall, +APmedium, and APlarge listed in Table 5, respectively. + + + + +AveragePrecision(AP): +AP + AP at IoU=.50:.05:.95 (primary challenge metric) +APIoU=.50 +APatIoU=.50(PASCALVOCmetric) +APIoU=.75 +APatIoU=.75 +(strictmetric) +AP Across Scales: +Apsmall + AP for small objects:area< 322 +Apmedium + AP for medium objects: 322 < area < 962 +Aplarge + AP for large objects:area >96? +Average Recall +(AR): +ARmax=1 + AR given l detection per image +ARmax=10 + ARgiven lodetections per image +ARmax=100 +AR given loodetectionsper image +AR Across Scales: +ARsmall + AR for small objects: area < 32? +ARmedium + AR for medium objects: 322< area < 962 +ARlarge + AR for large objects: area > 96?Method +Backbone +Pre-train +FT-epoch +Learning rate +AP +AP50 +AP75 +APs +APm +API +Mask RCNN +ResNeXt-101 +IN1K, sup +1000 +0.00010 +48.36 +72.91 +64.21 +0.00 +47.76 +52.56 +Mask RCNN +ViTDet, ViT-L +IN1K, MAE +1000 +0.00010 +54.80 +79.55 +62.93 +20.00 +50.56 +69.49 +CascadeMaskRCNN +ViTDet, ViT-L +IN1K, MAE +1000 +0.00010 +57.08 +80.38 +75.63 +50.00 +53.11 +68.64 +Cascade Mask RCNN +ViTDet, ViT-H +IN1K, MAE +750 +0.00001 +59.75 +83.62 +67.50 +50.00 +55.25 +73.21 +13 + +Fig. 17 Example of detection results on the Airbus Aircraft dataset with ViTDet, ViT-L backbone. + +500 +1000 +1500 +2000 +2500 +500 +1000 +1500 +2000 +2500 +14 +5.4 RarePlanes Object Detection Results +The experiment steps for RarePlanes are the same as the ones +used on the Airbus Aircraft dataset, except for dataset split +because RarePlanes already provides training and testing sub- +datasets. Table 7 lists the information of the provided training +and testing sub-datasets. We ran experiments for two types of +object detection tasks: aircraft and aircraft role. Next, the +detailed results on four experiments will be presented. +Table 7 Training and testing datasets of RarePlanes + +Real +Synthetic +Training +5,815 +45,000 +Testing +2,710 +5,000 +Total +8,525 +50,000 +5.4.1. Aircraft Object Detection Results for the Real Image +Dataset +Table 8 shows aircraft object detection results for the real +dataset. The best metrics across algorithms are highlighted in +yellow. Like the previous findings, Cascade Mask RCNN still +outperforms Mask RCNN, and ViTDet still beats CNN +backbone in all evaluation metrics. For small object +detection, ViTDet can beat the CNN counterpart by 7-11% +on APs, which implies ViTDet backbone can better detect +small objects. Fig. 18 shows an example of object detection +results on a testing image. The two aircraft are tested with +high confidence value (>=98%). +5.4.2. Aircraft Object Detection Results for the Synthetic +Image Dataset +Table 9 presents aircraft object detection results for the +synthetic dataset. As with the previous testing results, +Cascade Mask RCNN still consistently outperforms Mask +RCNN; ViTDet still beats CNN backbone in all evaluation +metrics. For this dataset, the performance improvement of +small object detection is not so large as in the two previously +tested datasets. Fig. 19 shows an example of aircraft object +detection on a testing image. In this case, all aircraft are +identified. However, several non-aircraft objects are mis- +labeled as aircraft. +5.4.3. Aircraft Role Object Detection Results for the Real +Image Dataset +Table 10 shows aircraft role object detection results for the +real dataset. As with the above three experiment cases, +Cascade Mask RCNN still outperforms Mask RCNN, except +on AP50. For AP, ViTDet backbone still beats CNN +backbone with large margins of improvement (14-17%). Fig. +20 shows an example of aircraft role object detection on a +testing image, where the aircraft are labeled by their role +names of “large civil transportation utility.” +5.4.4. Aircraft Role Object Detection Results for the +Synthetic Image Dataset +Table 11 presents aircraft role object detection results for the +synthetic dataset. Similarly, Cascade Mask RCNN still +performs better than Mask RCNN. On AP, ViTDet backbone +still beats CNN backbone with large margins (12-16%). Fig. +21 shows an example of aircraft role object detection on a +testing image, and roles are identified according to their sizes. +As in Fig. 19, all aircraft objects are correctly identified with +their roles. However, several non-aircraft objects are wrongly +labeled as aircraft role. +As seen in the above four experiments for the RarePlanes +dataset, obviously ViTDet backbone performs much better +than the CNN counterpart. For AP, the improvement ranges +from 5% to 17%. The above experiments focused on HBB +object detection performed with Detectron2. Next, we will +present OBB object detection with MMRotate. + +Table 8 RarePlanes real dataset aircraft object detection results comparison + + +Method +Backbone +Pre-train +Task +FT-epoch +Learning rate +AP +AP50 +AP75 +APs +APm +API +Mask RCNN +ResNeXt-101 +1K,sup +aircarft +1000 +0.0001 +69.17 +96.33 +85.78 +58.66 +68.57 +83.86 +Mask RCNN +ViTDet,ViT-L +1K,MAE +aircarft +1000 +0.0001 +74.72 +98.29 +88.62 +65.55 +73.22 +84.78 +CascadeMaskRCNN +ViTDet,ViT-L +1K,MAE +aircarft +1000 +0.0001 +77.45 +97.60 +89.97 +70.02 +75.81 +87.17 +15 + +Fig. 18 Example of aircraft object detection results on the RarePlanes real testing dataset. +Table 9 RarePlanes synthetic dataset aircraft object detection results comparison + + +Fig. 19 Example of aircraft object detection results on the RarePlanes synthetic testing dataset. + +100 +ircraft100% +200 +raft98% +300 +400 +500 +0 +100 +200 +300 +400 +500Method +Backbone +Pre-train +Task +FT-epoch +Learningrate +AP +AP50 +AP75 +APs +APm +API +MaskRCNN +ResNeXt-101 +1K,sup +aircarft +1000 +0.0001 +69.43 +91.89 +84.66 +39.91 +67.13 +81.02 +Mask RCNN +ViTDet,ViT-L +1K,MAE +aircarft +1000 +0.0001 +74.80 +96.69 +84.85 +40.13 +69.32 +88.42 +Cascade Mask RCNN +ViTDet,ViT-L +1K,MAE +aircarft +1000 +0.0001 +78.06 +96.70 +87.17 +43.33 +73.06 +91.58200 +400 +aircraft 999% +600 +craft 100% +800 +1000 +0 +250 +500 +1000 +1250 +1500 +1750 +16 +Table 10 RarePlanes real dataset aircraft role object detection result comparison + + +Fig. 20 Example of aircraft role object detection results on the RarePlanes real dataset. +Table 11 RarePlanes synthetic dataset aircraft role object detection results comparison + + + +Method +Backbone +Pre-train +Task +FT-epoch +Learningrate +AP +AP50 +AP75 +APs +APm +API +Mask RCNN +ResNeXt-101 +1K,sup +aircarft role +1000 +0.0001 +56.96 +82.58 +71.35 +36.28 +44.23 +60.45 +Mask RCNN +ViTDet,ViT-L +1K,MAE +aircarft role +1000 +0.0001 +71.00 +93.41 +84.29 +45.51 +56.29 +81.03 +CascadeMaskRCNN +ViTDet,ViT-L +1K,MAE +aircarft role +1000 +0.0001 +73.94 +92.38 +85.74 +52.88 +59.26 +83.040 +100 +LargeCivilTransport/ilityi20%% +200 +Transdort/Utility.11%/ +300 +400 +500 +0 +100 +200 +300 +400 +500Method +Backbone +Pre-train +Task +FT-epoch +Learningrate +AP +AP50 +AP75 +APs +APm +API +Mask RCNN +ResNeXt-101 +1K,sup +aircarft role +1000 +0.0001 +52.66 +71.31 +66.11 +14.23 +39.32 +40.78 +Mask RCNN +ViTDet,ViT-L +1K,MAE +aircarft role +1000 +0.0001 +64.62 +90.93 +70.56 +23.46 +56.17 +54.02 +CascadeMaskRCNN +ViTDet,ViT-L +1K,MAE +aircarft role +1000 +0.0001 +68.67 +91.78 +75.32 +27.56 +60.88 +57.27 +17 + +Fig. 21 Example of aircraft role object detection results for RarePlanes synthetic dataset. +5.5 DOTA-v1.0 Object Detection Results +Training models for DOTA-v1.0 is more complicated than +the previous HBB object detection experiments. We followed +five steps to carry out OBB experiments: +Step 1. Data preprocessing +For a fair comparison, we followed the same data +preprocessing steps laid out in Ding et al. [1] and Han et al. +[161]. Specifically, we first combined both training and +validation sub-datasets to train models, and the testing dataset +was used for final evaluation. Note that the testing dataset +does not provide data labels in the downloaded folders, and +the DOTA web evaluation server [172] must be used for the +final results. DOTA’s image size ranges from 800 x 800 to +4,000 x 4,000; therefore, we also followed the image splitting +practice and cropped the original images into 1,024 × 1,024 +patches with a stride of 824. Just as importantly, we also +carried out data augmentation to get a variant of DOTA-v1.0 +for training, in which we also adopted standard random +rotation (RR) and multi-scale (MS) transformation at three +scales {0.5, 1.0, and 1.5} for a fair comparison. After all the +necessary steps, we have two transformed datasets for the +model training; one is only with splitting, and the other is with +splitting and data augmentation. +Step 2. Pretrained models downloading +From the RoI Transformer and ReDet model zoo, we +downloaded the pretrained models with the ResNet and Swin- +T backbones for comparing. For RoI Transformer, models +were +downloaded +from +the +following +webpage: +https://github.com/open- +mmlab/mmrotate/blob/main/configs/roi_trans/README.md +. For ReDet, models were downloaded from the DOTA-v.10 +table on the following webpage: https://github.com/open- +mmlab/mmrotate/blob/main/configs/redet/README.md. +Step 3. Configuration files customization +We set up configuration files for ViTDet backbone in the +selected algorithms: RoI Transformer [1] and ReDet [161]. +We used the default configuration files provided in the +MMRotate platform as exemplar and created corresponding +ones for ViTDet backbone. In details, the angle +representation was set to 1e90, the learning rate was 0.0001, +AdamW optimizer was used, and the number of training +epochs was 12. + +TrarsponUtigy 62% +y58 +200 +MediumCivilTransport/Utility69 +400 +Medium CivilTransport/Utility 696 +Large Civil Transport/Utility 93966 +600 +edunCMlTransgortUnityBa +800 +LUnlty T965H +1000 +um CMi TransportUslity Gre +0 +250 +500 +750 +1000 +1250 +1500 +1750 +18 +Step 4. Fine-tuning models for ViTDet backbone +We fine-tuned four models with ViTDet backbone in RoI +Transformer and ReDet on two preprocessed datasets in Step +1. +Step 5. Evaluation on the testing dataset +When all 9 models were ready, we evaluated them on the +testing datasets. We then submitted the predicated object +detection results to the DOTA-v1.0 official evaluation server, +which in turn gave us AP for each class and mAP for all +classes shown in Table 12. +Table 12 presents the detailed nine experiment results. The +three backbones are as follows: R50 stands for ResNet-50; +Swin-T represents Swin Transformer tiny version; and +ViTDet, ViT-B is ViTDet backbone built with a base version +of ViT that has 12 layers and 768-dimension output. With +consideration of OBB algorithms’ heavy computation +burden, we did not evaluate ViTDet, ViT-L and ViTDet, ViT- +H. The column “aug.” shows whether data augmentation was +used. As demonstrated in Table 12, given the same +backbones, RetDet performs slightly better than RoI +Transformer. Without data augmentation, ViTDet, ViT-B +backbone is slightly worse than the other two backbones. +However, with data augmentation, ViTDet, ViT-B achieves +the best performance on mAP (80.89%), which is very +comparable to the best published benchmark of 80.90% in +Zhou et al. [169]. That benchmark was achieved with a +combination of RoI Transformer, Swin-T backbone, +Kullback-Leibler Divergence (KLD) trick [179], and data +augmentation. In comparison, ViTDet can much more easily +achieve comparable best performance without the need of the +complicated KLD trick, in which the rotated bounding box +must be converted into a 2-D Gaussian distribution and then +KLD between the Gaussian distributions are calculated as the +regression loss. Moreover, for helicopter detection (HC), +ViTDet, ViT-B performs the best, improving about 23% at +large. Overall, for a complicated dataset like DOTA, data +augmentation still plays a bigger role than backbones. +Fig. 22 shows an example of detection results. In a compacted +parking lot like the one pictured, most of the vehicles are +detected with high confidence values. In short, for OBB +object detection, ViTDet, ViT-B still achieves comparable +performance with other backbones; the computation burden +of ViTDet is heavier than R50 and Swin-T backbones. +Therefore, more research may be needed to improve +ViTDet’s performance for OBB detection. +Table 12 Accuracy comparison of rotated object detection on DOTA-v1.0 + + +method +backbone +pre-train +aug. +PL +BD +BR +GTF +sV +LV +HS +TC +BC +1s +SBF +RA +HA +SP +HC +mAP +Rol Trans +R50 +1K,sup +- +88.97 +82.14 +54.59 +76.28 +79.29 +77.94 +87.94 +90.88 +87.19 +85.62 +62.21 +62.63 +74.62 +72.43 +59.23 +76.13 +Rol Trans +Swin-T +1K,sup +89.08 +83.60 +54.84 +72.10 +79.02 +84.45 +87.97 +90.90 +87.14 +86.64 +64.65 +66.50 +76.65 +72.30 +66.90 +77.52 +Rol Trans +ViTDet,ViT-B +1K,MAE +89.42 +81.13 +52.99 +72.25 +78.35 +84.26 +88.16 +90.89 +84.58 +86.63 +53.09 +66.37 +75.37 +72.63 +59.20 +75.69 +Rol Trans +R50 +1K,sup +MS+RR +88.76 +84.47 +59.20 +78.65 +79.65 +85.50 +88.26 +90.90 +87.05 +88.21 +69.73 +68.77 +78.90 +81.48 +71.36 +80.06 +Rol Trans +ViTDet,ViT-B +1K,MAE +MS+RR +88.87 +84.51 +60.31 +75.48 +81.04 +86.00 +88.47 +90.84 +84.63 +87.62 +62.81 +72.07 +78.86 +82.47 +75.94 +79.99 +ReDet +R50 +1K,sup +89.20 +83.79 +52.23 +73.31 +78.06 +82.48 +88.23 +90.86 +87.26 +85.97 +65.64 +62.87 +75.90 +70.02 +66.79 +76.84 +ReDet +R50 +1K,sup +MS+RR +89.19 +85.75 +62.13 +81.20 +78.98 +86.01 +88.67 +90.90 +89.20 +88.23 +69.81 +66.54 +79.13 +78.72 +71.19 +80.38 +ReDet +ViTDet,ViT-B +1K,MAE +- +89.24 +80.37 +53.84 +71.37 +78.21 +84.19 +88.12 +90.90 +85.51 +86.28 +52.20 +65.94 +75.91 +70.93 +62.63 +75.71 +ReDet +ViTDet,ViT-B +1K,MAE +MS+RR +87.75 +85.22 +61.37 +81.12 +80.62 +85.82 +88.37 +90.88 +85.93 +87.79 +63.31 +73.15 +78.96 +80.13 +82.88 +80.89 +19 + +Fig. 22 Example of object detection results on DOTA-v1.0 with ViTDet backbone. + +e-veh +20 +6. CONCLUSION +This study has explored the newly proposed MAE pretrained +ViTDet backbone for challenging aerial image object +detection problems. We carried out 25 experiments on three +well-known +aerial +image +datasets: +Airbus +Aircraft, +RarePlanes, and DOTA. Our experiments demonstrated that +ViTDet backbone consistently beats its CNN counterparts in +HBB object detection (up to 17% improvement on AP) and +that it achieves on-par performance for OBB object detection. +Our results also provided a baseline for future research. +ACKNOWLEDGMENTS +The authors sincerely thank Dr. Kris Rosfjord and Dr. Heath +Farris for their generous support of this project. We would +also like to thank Mike Robinson, Bill Bateman, Lixia Song, +Erik Vargo, and Paul A. Diffenderfer of The MITRE +Corporation for their valuable discussions, insights, and +encouragement. +NOTICE +This work was sponsored by MITRE’s independent research +and development program. 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Available: +http://arxiv.org/abs/2106.01883 + + diff --git a/aNFLT4oBgHgl3EQfWy86/content/tmp_files/load_file.txt b/aNFLT4oBgHgl3EQfWy86/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..179c3a4f6624c84fbddb72343cbe5674a504c173 --- /dev/null +++ b/aNFLT4oBgHgl3EQfWy86/content/tmp_files/load_file.txt @@ -0,0 +1,2706 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf,len=2705 +page_content='1 Aerial Image Object Detection With Vision Transformer Detector (ViTDet) Liya Wang, Alex Tien The MITRE Corporation, McLean, VA, 22102, United States ABSTRACT The past few years have seen an increased interest in aerial image object detection due to its critical value to large-scale geoscientific research like environmental studies, urban planning, and intelligence monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' However, the task is very challenging due to the bird’s-eye view perspective, complex backgrounds, large and various image sizes, different appearances of objects, and the scarcity of well- annotated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Recent advances in computer vision have shown promise tackling the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Specifically, Vision Transformer Detector (ViTDet) was proposed to extract multi-scale features for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The empirical study shows that ViTDet’s simple design achieves good performance on natural scene images and can be easily embedded into any detector architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' To date, ViTDet’s potential benefit to challenging aerial image object detection has not been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Therefore, in our study, 25 experiments were carried out to evaluate the effectiveness of ViTDet for aerial image object detection on three well-known datasets: Airbus Aircraft, RarePlanes, and Dataset of Object DeTection in Aerial images (DOTA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Our results show that ViTDet can consistently outperform its convolutional neural network counterparts on horizontal bounding box (HBB) object detection by a large margin (up to 17% on average precision) and that it achieves the competitive performance for oriented bounding box (OBB) object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Our results also establish a baseline for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Index Terms— Aerial image, Object detection, Computer vision, ViTDet, HBB, OBB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' INTRODUCTION Aerial image object detection has been a vibrant research topic for its essential role in large-scale geoscientific research like environmental science, ecology, agricultural studies, wildfire monitoring, urban planning, intelligence monitoring, and emergency rescue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' However, the task is very challenging due to the bird’s-eye view perspective, complex backgrounds, large and various image sizes, various appearances of objects, and the scarcity of well-annotated datasets [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' In addition, the objects in aerial images are often arbitrarily oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' For that reason, instead of using common horizontal bounding boxes (HBBs) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 1a), oriented bounding boxes (OBBs) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 1b) have been alternatively used to avoid mismatching between bounding boxes and corresponding objects [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' In the past few years, deep learning techniques have dominated the object detection domain for their effective feature learning capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 2 shows the milestones of deep learning algorithms in object detection since 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The green and orange rectangles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 2 highlight one-stage and two-stage methods for HBB object detection, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Two-stage methods have two separate processes, region proposal and detection proposal, while in one-stage methods these two processes are combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' In general, two-stage methods have better performance than one-stage methods at the expense of computational workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The purple rectangle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 2 shows the algorithms designed specifically for OBB object detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' the yellow rectangle calls out the important deep learning frameworks for building object detection algorithms, where ResNet [2], Vision Transformer (ViT) [3], and Swin-T [4] are commonly used backbones for feature extraction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' in particular, Vision Transformer Detector (ViTDet) [5] was a newly proposed backbone for object detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' feature pyramid network (FPN) [6] is often used as neck after backbone for feature fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Feature learning is always essential in any computer vision (CV) machine learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Following the advent of ViTs [3] in 2021, an exciting self-supervised learning (SSL) method, Masked Autoencoder (MAE) [7], was proposed to learn effective visual representation features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MAE adopts Masked Image Modeling (MIM) technique and tries to infer masked image patches from unmasked ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' To date, MAE has attracted unprecedented attention because of its superior performance over its supervised learning and contrastive learning counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The encouraging success of MAE has inspired a wide range of applications in the areas of video, audio, medical images, earth observation, multimodal, point cloud, 3D mesh data, reinforcement learning, and graphs (see Table 1 for the summary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' It is noticeable that several efforts have been devoted to object detection, including ViTDet [5], a new backbone designed especially for object detection with support of MAE pretrained ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' ViTDet was particularly designed to enhance the effectiveness of ViT backbone on object detection problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Although MAE pretrained ViTs are effective for image classification tasks, they are less effective for object detection, which usually requires multi-scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' ViT is a plain, non-hierarchical architecture that maintains a single- scale feature map throughout, which indicates that ViT åbackbone is not sufficient for object detection tasks, 2 especially when dealing with multi-scale objects and high- resolution images [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' To deal with the deficiency, ViTDet was invented to extract multi-scale features for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' ViTDet has demonstrated its superior performance on natural scene image (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=', COCO [9]) object detection [5], and its simple design can also make it embeddable in any detector architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' To the authors’ knowledge, no research has ever adopted ViTDet and examined its performance for aerial image object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Therefore, this research aims to evaluate and gain insight on the potential benefits of ViTDet for both aerial image HBB and OBB object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The remainder of the paper is organized as follows: Section II describes related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Section III discusses the aerial image datasets used for performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Section IV gives the details of the implementation platforms for carrying out the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The results are presented in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Section VI is the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' HBB (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' OBB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 1 Illustration of different bounding box types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 2 Milestones of deep learning algorithms in object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=" ReDet Oriented Oriented RCNN Rol Transformer Gliding Vertex R'Det Rotated RetinaNet Rotated RepPoints Rotated ATSS S?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='A-Net Rotated FasterRCNN Rotated FCOS Beyond Boundina-Box RCNN Fast RCNN Mask RCNN Two-stage Horizontal SPPNet Faster RCNN Cascade RCNN 2014 2015 2016 2017 2018 2019 2020 2021 2022 YOLOv4 YOLOVT YOLO RetinaNet YOLOv3 YOLOR YOLOv2 YOLOv5 DETR SDD One-stage ViT ResNet FPN ViTDet Swin-T 3 Table 1 Summary of latest MIM research Domain Sub-Domain Research Papers Vision Image BEiT v1 [10],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' v2 [11],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MAE [7],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' SimMIM [12],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' ADIOS [13],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' AMT [14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' AttMask [15],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Beyond-Masking [16],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' BootMAE [17],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' CAE [18],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' CAN [19],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' ConvMAE [20],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Contrastive MAE [21],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' ContrastMask [22],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' dBOT [23],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' DMAE [24],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Denoising MAE [25],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' GreenMAE [26],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' iBOT [27],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' LoMaR [28],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' LS-MAE [29],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MaskAlign [30],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MaskDistill [31],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MaskFeat [32],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MaskTune [33],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MetaMask [34],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MFM [35],' metadata={'source': 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[42],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MVP [43],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' RC- MAE [44],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' SDMAE [45],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' SemMAE [46],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' SdAE [47],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' SupMAE [48],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' U-MAE [49],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' UM-MAE [50] Video AdaMAE [51],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Bevt [52],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MAM2 [53],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MAR [54],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MaskViT [55],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' M3Video [56],' metadata={'source': 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+page_content=' Vimpac [63],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' VRL [64] Medical Image DAMA [65],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' GCMAE [66],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' SD-MAE [67],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' SMIT [68] Satellite Image SatMAE [69] Image Classification MUST [70] Object Detection imTED [71],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Mask DINO [72],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' ViCHA [96],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' VL-BEiT [97],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' VLC [98],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' VIOLET [99],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' VLMAE [100] Audio-Language CAV-MAE [101] Others Audio Audio-MAE [102],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Group-MAE [103],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MAE-AST [104],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MSM [105],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' M2D [106] Anomaly Detection MAEDAY [107],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' SSMCTB [108],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' ST-MAE [109] Graph MGAP [110],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' GMAE [111],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' GMAE-AS [112],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' GraphMAE [113],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' HGMAE [114],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MGAE [115],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MaskGAE [116] Point Cloud Point-Bert [117],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Point-MAE [118],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Point-M2AE [119],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Mask-Point [120],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Masked [121],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Voxel-MAE [122] Skeleton SimMC [123] Depth Estimation Depth Refinement [124],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' FM-Net [125] Reinforcement Learning MLR [126],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Motor Control,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Visual Control [127] 3D Mesh Data MeshMAE [128] Adversarial Attack MAD [129] Miscellaneous D-MAE [130],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MAEEG [131],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MGD [132],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Extra-MAE ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MADE [133],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MaskDP [134],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' i2i [135],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Lifetime Prediction [136],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MET [137],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MIL [138],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Robot Training [139],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Time Series [140] Survey MIM Survey [141] Theory CL vs MIM [142],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Contextual Representation Learning[143] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Data Scaling [144],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' EVA [145],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' i-MAE [146],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Revealing MIM [147],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Understanding MAE [148],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Understanding MIM [149],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Understanding DR [150],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Architecture Deeper vs Wider [151],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Masked BNN [152],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' ViT Back to CNN [153],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' ConvNeXt V2 [154] 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' RELATED RESEARCH 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='1 Backbones In CV deep learning methods, backbones are commonly used to extract discriminative object feature representation, and they have been a driving force for rapid object detection performance improvement [155].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Popular backbones for object detection are ResNet [2], ResNeXt [156], and Swin-T [4] because of their deep hierarchical architectures, which can produce the needed multi-scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Backbone pretraining is usually carried out on ImageNet-1k [157] with either supervised learning or SSL methods like contrastive learning or MAE, which will be presented next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='2 MAE MAE is an asymmetric autoencoder that uses ViTs in both its encoder and decoder, and the size of decoder is smaller than the encoder, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' It directly infers masked patches from the unmasked ones with a simple loss of mean squared error (MSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' To save computation, the encoder works on only the unmasked patches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' in contrast, the decoder works on both masked and unmasked patches trying to predict the original images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The masking ratio can be set up to 75%, which is considerably higher than that in BERT (typically 15%) [158] or earlier MIM methods (20% to 50%) [10], [159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MAE’s ablation study also points out that a high masking ratio is good for fine-tuning and linear probing [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' With those meticulous designs, MAE is three times (or more) faster than Bidirectional Encoder representation from Image Transformers (BEiT) [10] while achieving superior performance [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 3 MAE architecture [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='3 ViTDet ViTDet was designed to extract multi-scale features for object detection with minimal adaptation to MAE pretrained ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 4 shows the architecture of ViTDet building a simple feature pyramid from only the last feature map of a plain ViT backbone, and it adopts non-overlapping window attention for efficient feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' To propagate information, ViTDet uses a small number of cross-window blocks, which can be implemented with global attention or convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The adaptation takes place only during fine- tuning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' therefore, they do not affect the upstream pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The empirical study shows that ViTDet’s simple design achieves good results on natural scene image object detection [5], which further proves that the general-purpose pretrained ViT from MAE can serve object detection as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' ViTDet’s simple design makes it easily plug into any detector architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Investigating ViTDet’s effectiveness for challenging aerial image object detection is the focus of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 4 ViTDet backbone architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' ViTDet builds a simple pyramid from only the last, large-stride (16) feature map of a plain backbone [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='4 Object Detection Object detection is one of the most fundamental yet challenging CV tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The task is to identify and localize all the objects in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Each object will have a label, and its location is commonly defined by an HBB (𝑥, 𝑦, 𝑤, ℎ), where 𝑥 and 𝑦 are center coordinates of the box, and 𝑤 and ℎ are width and height of the box (illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 5 Illustration of HBB object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' However, in aerial images, the objects could be arbitrarily oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The methods relying on HBBs often introduce mismatches between the Regions of Interest (RoI) and objects, which further deteriorate the final object classification confidence and localization accuracy [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 6, a RoI (top) may contain several instances, leading to difficulties for the subsequent classification and location task [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' For this reason, research has proposed OBB annotations (𝑥, 𝑦, 𝑤, ℎ, 𝜃) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 7 for illustration), where encoded masked 25% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='atent + encoder decoder (MiT) input target1/16 1/32 neck/head: 1/16 + 1/16 backbone: 1/16 1/8 1/16 114dog/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='00 catj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='99 5 𝑥 and 𝑦 are center coordinates of the box and 𝑤, ℎ, and 𝜃 are the width, height, and angle of an OBB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' It should be noted that 𝑤 and ℎ of the OBBs are measured in different rotating coordinate systems for each object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' OBBs make more accurate orientation information, especially when detecting aerial objects with a large aspect ratio, arbitrary orientation, and dense distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Furthermore, OBBs can also have more accurate RoIs and allow for better discriminative feature extraction for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Deep learning algorithms such as oriented Region-based Convolutional Neural Network (RCNN) [160], RoI Transformer [1], and Rotation-equivalent Detector (ReDet) [161] have been proposed particularly for OBB detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' They usually adopt numerous rotated anchors with different angles, scales, and aspect ratios for better regression, resulting in significant computation burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 6 HBB (top) vs OBB (bottom) illustration in an image with many densely packed objects [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 7 OBB definition, where x and y are center coordinates of the box and w, h, and θ are the width, height, and angle of an OBB [162].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='5 Object Detection Algorithms As mentioned in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 2, there are several types of object detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Famous detection challenges have shown that two-stage methods achieve better performance than one- stage methods if heavy computation workload is not a concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' In our work, we focus on two-stage methods for their better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The two-stage object detection methods usually consist of the three steps proposed in RCNN [162] (illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The first step is region proposal, which generates a series of candidate region proposals (about 2,000) that may contain objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The second step is feature extraction for the proposed regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Following that, the third step is classification, where the candidate regions are distinguished as object classes or background and furtherly fine-tuned for the coordinates of the bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' As research advances, various types of algorithms have been proposed to hone the components for better performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Next, the most fundamental deep learning method, Faster RCNN [163], and the algorithms explored in this research will be presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 8 Object detection system overview [163].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='6 Faster RCNN Faster RCNN [163] is the first end-to-end object detection method fully using deep learning techniques, which is more efficient than its predecessors, RCNN [162] and Fast RCNN [164].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Faster RCNN proposes a novel idea called region proposal network (RPN), which fully utilizes convolutional layers extracted features to generate proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Compared with conventional region proposal generation algorithms like Selective Search [165], which is an offline algorithm and makes it impossible to train whole algorithm from end to end, RPN is much more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' After RPN, Faster RCNN then uses the RoI pooling layer to extract a fixed-length feature vector from each region proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 9 depicts the architecture of Faster RCNN and the sequential relationship among backbone (convolutional layers), RPN, and the RoI pooling layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Based on Faster RCNN, several variants have been proposed to improve the performance of object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Next, we present the relevant methods tested in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='7 Mask RCNN Mask RCNN [166] is an extension of Faster RCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Besides Faster RCNN’s two outputs for each candidate object—a class label and a bounding box—a third type of output, object mask, is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 10 illustrates the architecture of Mask RCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The backbone of Mask RCNN is for feature extraction, and it can be traditional ResNet [2], Swin-T [4], y 0E[-90°,-0"] (x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='w, h, 0)warped region aeroplane?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Mperson?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' tvmonitor?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Input 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Extract region 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='Compute 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Classify image proposals (~2k) CNNfeatures regions 6 or newly proposed ViTDet [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' RPN is the same as the one in Faster RCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The novel element of Mask RCNN is the RoIAlign layer, which can preserve the pixel-level spatial correspondence and address the shortfalls of Fast/Faster RCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The mask head is a Fully Convolutional Network (FCN) [167] on top of a feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Mask RCNN is still simple to train and generalizes well, but it introduces a small computation overhead to Faster RCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 9 Faster RCNN architecture [164].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 10 Mask RCNN architecture [169].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='8 Cascade RCNN Cascade RCNN [168] adopts a new trick for better performance, classifying with multistage classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The trick works in such a way that early stages can discard many easy negative samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' therefore, later stages can focus on handling more difficult examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 11 illustrates the architecture of Cascade RCNN, where “I” is input image, “conv” is the convolutions backbone, “pool” is for the region- wise feature extraction, “H” represents various network head, “B” is the bounding box, “C” is classification, and “B0” is proposals in all architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' An object detection architecture like Faster RCNN can be deemed as a cascade (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=', the RPN removing large amounts of background and the following detector head classifying the remaining regions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Therefore, Cascade RCNN extends the idea to multiple stages in the classification layer to enhance the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' When mask head is also included in the output, the algorithm is called Cascade Mask RCNN, which is used for HBB object detection in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 11 Cascade RCNN architecture [170].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='9 RoI Transformer RoI Transformer [1] was designed specifically for OBB object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' In the past, rotated anchors have been used to tackle the OBB object detection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The design always multiplies the number of anchors, which considerably increases the computation burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Hence, RoI Transformer was tried for reducing the computation burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 12 illustrates the architecture of RoI Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' In specific, it first adopts a Rotated Region of Interest (RRoI) learner to transform a Horizontal Region of Interest (HRoI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Based on the RRoIs, it then uses a Rotated Position Sensitive RoI Align (RPS-RoI-Align) module to extract rotation-invariant features, which are then used for enhancing subsequent classification and regression performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' RoI Transformer is a light-weighted component and can be easily plugged into any detector framework for OBB object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='10 Rotation-Equivalent Detector (ReDet) ReDet [161] was also proposed to solve OBB aerial image object detection problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' It introduces rotation-equivariant networks into the detector to extract rotation-equivariant features, which can accurately predict the orientation and result in a huge reduction in model size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 13 illustrates the working mechanism of ReDet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 13a shows the overall architecture of ReDet, which first uses the rotation- equivariant backbone to extract rotation-equivariant features, followed by an RPN and RoI Transformer (RT) [1] to generate RRoIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' After that, a novel Rotation-Invariant RoI Align (RiRoI Align) is used to produce rotation-invariant features for RoI-wise classification and bounding box regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 13b shows rotation-equivariant feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Under the cyclic group 𝐶!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=', the rotation-equivariant feature maps with the size (𝐾, 𝑁, 𝐻, 𝑊) have 𝑁 orientation channels, classifier Rol pooling proposals RegionProposalNetwork featuremaps conv layersRaw image + Backbone RPN RolAlign Object detection Mask generation head head Class Box MaskBO C1 B1 C2 B2 C3 B3 H1 H2 H3 pool 1 conv 7 and each orientation channel corresponds to an element in 𝐶!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='. Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 13c illustrates RiRoI Align.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The proposed RiRoI Align consists of two parts: spatial alignment and orientation alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' For an RRoI, spatial alignment warps the RRoI from the spatial dimension, while orientation alignment circularly switches orientation channels and interpolates features to produce completely rotation-invariant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' ReDet has achieved state-of-the-art performance [169];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' therefore, it was selected in our study to test ViTDet for OBB object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Next, we give more details about aerial image datasets to test ViTDet backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 12 Architecture of RoI Transformer [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 13 ReDet architecture [161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' RRolLearner Classification Decoder 10channels Rotated Position Sensitive Rol Align RolTransformer 10chammels 490chanmnels RegressionRPN+RT classification feature maps : backbon RiRol Align Rol feature (a) Overall architecture c() c2 spatial alignment orientation alignment FC layer (2) RRol switch channels interpolate 2r (c) Rotation-invariant Rol Align 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' DATASETS To evaluate the newly proposed backbone of MAE pretrained ViTDet for aerial image object detection, we conducted 25 experiments across three distinct datasets of aerial images: 1) Airbus Aircraft Detection [170], 2) RarePlanes [171], and 3) Dataset of Object DeTection in Aerial images (DOTA) [172].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The smallest dataset is Airbus Aircraft Detection, with 103 images, and the largest dataset is RarePlanes, with about 68,000 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' These two both use HBB annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' DOTA is the most complicated dataset with OBB annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Table 2 gives the details of the three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' A short introduction about each dataset will be provided next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='1 Airbus Aircraft Detection The Airbus Aircraft Detection [173] dataset is collected from Airbus’ Pleiades twin satellites for earth observation, which collect pictures of airports worldwide on a regular basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' This dataset contains 103 images with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='5 m resolution (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 14 for an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Each image is stored as a JPEG file of size 2,560 x 2,560 pixels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=', 1,280 meters x 1,280 meters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Some airports could appear multiple times at different acquisition dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Some images may include fog or cloud because of weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The annotations are provided in the form of closed GeoJSON polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' A CSV file named annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='csv provides all annotations—one annotation per line with the corresponding object ID;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' filename as image ID;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' annotation box;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' and class label, mainly Aircraft (3,316 instances) or Truncated_Aircraft (109 instances) when an aircraft is located at the border of an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The minimum and maximum number of aircraft in an image are 5 and 92, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='2 RarePlanes RarePlanes [171] is an open-source dataset that includes both real and synthetically generated satellite images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The RarePlanes dataset is specifically designed to automatically detect aircraft and their attributes in satellite images (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 15 for examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' To date, RarePlanes is the largest openly available high-resolution dataset created to test the value of synthetic data from an overhead perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The real images are collected from 253 Maxar WorldView-3 satellite scenes, spanning 112 locations and 2,142 km2 with 14,700 hand- annotated aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The accompanying synthetic dataset is generated via AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='Reverie’s simulation platform and has about 60,000 synthetic satellite images covering a total area of 9,331 km2 with about 630,000 aircraft annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Both the real and synthetically generated aircraft have been given 10 fine-grained attributes—aircraft length, wingspan, wing shape, wing position, wingspan class, propulsion, number of engines, number of vertical stabilizers, presence of canards, and aircraft role—which are derived from the previous nine attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Seven role classes have been defined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Table 3 summarizes aircraft role count for real dataset, in which the first column lists seven “aircraft role” classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' As demonstrated in Table 3, the most common aircraft role is Small Civil Transport/Utility, and the least common one is Military Bomber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' More detail on role definitions can be found in the “RarePlanes User Guide” at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='cosmiqworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='org/rareplanes-public-user- guide/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' We conducted two types of object detection tasks— aircraft and aircraft role—on both sub-datasets to evaluate MAE pretrained ViTDet backbone’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='3 DOTA DOTA [174] is the largest aerial image dataset for OBB object detection (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 16 for some examples), and it is deemed as the most challenging dataset in the earth observation community for its various image sizes and densely packed objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' It has released three different versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' DOTA-v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0 contains 2,806 aerial images, with the size ranging from 800 × 800 to 4,000 × 4,000 and containing 188,282 instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' DOTA-v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0 has 15 common categories: Plane (PL), Baseball diamond (BD), Bridge (BR), Ground track field (GTF), Small vehicle (SV), Large vehicle (LV), Ship (SH), Tennis court (TC), Basketball court (BC), Storage tank (ST), Soccer-ball field (SBF), Roundabout (RA), Harbor (HA), Swimming pool (SP), and Helicopter (HC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The second version DOTA-v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='5 was released for 2019 Detecting Objects in Aerial Images (DOAI) Challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Compared with v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0, it has an extra category, Container crane, and more extremely small instances (less than 10 pixels), resulting in 402,089 instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The third version, DOTA-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0, collects more aerial images from Google Earth and GF-2 Satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' DOTA-v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0 has 18 categories, 11,268 images, and 1,793,658 instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Compared with DOTA-v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='5, it further adds the new categories of Airport and Helipad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Our study focused on DOTA-v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0 due to abundant baseline benchmarks available for evaluating ViTDet’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Table 2 Tested aerial image datasets Datasets Subsets Tasks # Object Types # Images # Instances Image Width Annotation Year Available Airbus Aircraft Detection Aircraft 2 103 3,425 2,560 HBB 2021 RarePlanes Real Aircraft 1 8,527 14,700 512 HBB 2020 Synthetic Aircraft 1 60,000 629,551 1,920 Real Aircraft role 7 8,527 14,700 512 Synthetic Aircraft role 7 60,000 629,551 1,920 DOTA v 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0 Objects 15 2,806 188,282 800-4,000 OBB 2018 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 14 Airbus Aircraft Detection image example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Table 3 Real dataset role count 汉Aircraft role Count Small Civil Transport/Utility 8002 Medium Civil Transport/Utility 5132 Large Civil Transport/Utility 1098 Military Transport/Utility/AWAC 283 Military Fighter/lnterceptor/Attack 171 Military Trainer 15 Military Bomber 6 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 15 Examples of the real and synthetic datasets present in RarePlanes [173].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 16 Examples of annotated images in DOTA [176].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' X10000 Helicoprer Bridor Shir romnd track field mall-vehicle Harbor Baseball dic Storage tan 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' IMPLEMENTATION PLATFORMS To evaluate the new backbone ViTDet in the aforementioned algorithms, we chose two well-known platforms in the CV field: Detectron2 [175] and MMRotate [169].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Detectron2 is the official implementation site for ViTDet and is used for HBB object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MMRotate is selected because it has most of state-of-the-art algorithms for OBB object detection, which Detectron2 lacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='1 Detectron2 Detectron2 [175] is an open-source research platform developed by Facebook AI Research [175].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The platform is implemented in PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' It provides many state-of-the-art detection and segmentation algorithms, including FPNs, numerous variants of the pioneering Mask RCNN model family, and the latest MAE pretrained ViTDet backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Therefore, we used Detectron2 to implement aerial image HBB object detection with its provided pretrained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='2 MMRotate OpenMMLab [176] is another open-source platform to provide powerful CV packages like Detectron2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' For general HBB object detection, MMDetection in OpenMMLab is the go-to package and forms the basis for MMRotate [169], which is specially designed for OBB object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' According to Table 4 provided by Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' [169], MMRotate provides 18 OBB algorithms and four famous datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' In addition, its modular design with multiple choices of orientation angle representations, backbones, necks, and detection heads makes it very easy and flexible to set up a new model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' For example, it can support multiple angle representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Popular OpenCV definition, long edge 90° definition, and long edge 135° are all supported in MMRotate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' MMRotate also provides baseline benchmarks for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Therefore, we selected MMRotate for customization of RoI Transformer and ReDet, where ViTDet will be used as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Note that at the time of this research, ViTDet has not officially been implemented in MMRotate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' We used a non-official version of ViTDet from [177] for OBB object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' RESULTS This section presents the experiment results of aerial image object detection using the MAE pretrained ViTDet backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='1 Experimental Setup To make a comprehensive evaluation, we conducted 25 experiments on the selected three datasets: 1) Airbus Aircraft Detection [170], 2) RarePlanes [171], and 3) DOTA [172].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The experiments tested three types of backbones (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=', ResNet [2], Swin Transformer [4], and ViTDet [5]) in four object detection algorithms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=', Mask RCNN [166], Cascade Mask RCNN [168], RoI Transformer [1], and ReDet [161]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' For the Airbus Aircraft and RarePlanes datasets, we tested Mask RCNN and Cascade Mask RCNN algorithms on the Detectron2 [175] platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' For the DOTA dataset, we tested RoI Transformer and ReDet on the MMRotate [169] platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The MAE pretrained ViTs were downloaded from https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='com/facebookresearch/mae [178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' All the experiments were carried out on four A100 GPUs with 160GB memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' More specific implementation details for each dataset will be presented in the corresponding sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='2 Evaluation Metrics Average precision (AP) is a commonly used metric to evaluate object detection algorithms, and it is derived from precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' There are several variants of AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Different platforms may adopt different versions of AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' In details, Detectron2 uses COCO-defined AP metrics (see Table 5 for the detailed list), which mainly focus on the accuracy of the bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' COCO-defined AP is averaged across all classes and 10 Intersection Over Union (IOU) values ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='5 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='95 in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='05 [155].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' By contrast, in the MMRotate platform, AP is calculated separately for each class, and mean AP (mAP) is calculated by averaging AP over all classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' To have a fair comparison, we calculated the default evaluation metrics defined by the two platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Table 4 Open source rotated object detection benchmarks [169] Benchmark AerialDet JDet OBBDet AlphaRotate MMRotate DL library PyTorch Jittor PyTorch TensorFlow PyTorch Inference PyTorch Jittor PyTorch TensorFlow PyTorch engine onnx runtime Windows Windows OS Linux Linux Windows Linux Linux Linux Algorithm 5 8 9 16 18 Dataset 1 4 5 11 4 Doc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Easy install Maintain V 12 Table 5 COCO-defined AP evaluation metrics [154], used in default by Detectron2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='3 Airbus Aircraft Object Detection Results To detect aircraft in this small dataset, we have taken the following three steps: Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Data preparation o Split dataset (103 images) into training (92 images) and testing (11 images) subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' o Convert he data into COCO format for easy use of ViTDet in Detectron2 packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Experiment setup o Downloaded COCO pretrained models of Mask RCNN and Cascade Mask RCNN from the website https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='com/facebookresearch/detectron2/tree/ma in/projects/ViTDet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' o Set up the configuration files for model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Model fine-tuning Table 6 shows the experiments conducted and the performance evaluation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The tested backbones are as follows: ResNeXt-101 [156] is a convolutional neural network (CNN) backbone with 101 layers and is pretrained in a supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' ViTDet, ViT-L is ViTDet backbone built with a large version of ViT that has 24 layers and 1024- dimension output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' ViTDet, ViT-H is ViTDet backbone built with a huge version of ViT that has 32 layers and 1280- dimension output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The column FT-epoch is the epochs for fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Yellow color highlights the best metrics in Table 6Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' As expected, Cascade Mask RCNN performs better than Mask RCNN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' larger backbones achieve better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Cascade Mask RCNN with backbone of ViTDet, ViT-H achieves the best performance in all evaluation metrics except for AP75, an evaluation metric when IOU equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Most importantly, ViTDet outperforms ResNeXt-101 in most of evaluation metrics, and ResNeXt-101 is deemed as one of top CNN backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' According to Table 6, ViTDet performs much better (20-50% improvement) than ResNeXt-101 on APs, which measures AP for small object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' For APl, a metric to measure AP for large object detection, ViTDet also beats ResNeXt-101 by a large margin of 16-20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' For AP, 6-10% improvement has been achieved by ViTDet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' In short, the new backbone ViTDet greatly improves object detection performance on this small dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 17 shows object detection results on a testing image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' There are about 90 aircraft in this testing image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' all but three are detected and one is falsely labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Therefore, ViTDet backbone does a good job for this testing image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Airbus Aircraft object detection results comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Note APs, APm, and APl represent COCO-defined APsmall, APmedium, and APlarge listed in Table 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' AveragePrecision(AP): AP AP at IoU=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='50:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='05:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='95 (primary challenge metric) APIoU=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='50 APatIoU=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='50(PASCALVOCmetric) APIoU=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='75 APatIoU=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='75 (strictmetric) AP Across Scales: Apsmall AP for small objects:area< 322 Apmedium AP for medium objects: 322 < area < 962 Aplarge AP for large objects:area >96?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Average Recall (AR): ARmax=1 AR given l detection per image ARmax=10 ARgiven lodetections per image ARmax=100 AR given loodetectionsper image AR Across Scales: ARsmall AR for small objects: area < 32?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' ARmedium AR for medium objects: 322< area < 962 ARlarge AR for large objects: area > 96?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='Method Backbone Pre-train FT-epoch Learning rate AP AP50 AP75 APs APm API Mask RCNN ResNeXt-101 IN1K, sup 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='00010 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='36 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='91 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='00 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='76 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='56 Mask RCNN ViTDet, ViT-L IN1K, MAE 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='00010 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='80 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='55 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='93 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='00 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='56 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='49 CascadeMaskRCNN ViTDet, ViT-L IN1K, MAE 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='00010 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='08 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='38 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='63 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='00 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='11 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='64 Cascade Mask RCNN ViTDet, ViT-H IN1K, MAE 750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='00001 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='75 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='62 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='50 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='00 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='25 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='21 13 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 17 Example of detection results on the Airbus Aircraft dataset with ViTDet, ViT-L backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 500 1000 1500 2000 2500 500 1000 1500 2000 2500 14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='4 RarePlanes Object Detection Results The experiment steps for RarePlanes are the same as the ones used on the Airbus Aircraft dataset, except for dataset split because RarePlanes already provides training and testing sub- datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Table 7 lists the information of the provided training and testing sub-datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' We ran experiments for two types of object detection tasks: aircraft and aircraft role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Next, the detailed results on four experiments will be presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Table 7 Training and testing datasets of RarePlanes Real Synthetic Training 5,815 45,000 Testing 2,710 5,000 Total 8,525 50,000 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Aircraft Object Detection Results for the Real Image Dataset Table 8 shows aircraft object detection results for the real dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The best metrics across algorithms are highlighted in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Like the previous findings, Cascade Mask RCNN still outperforms Mask RCNN, and ViTDet still beats CNN backbone in all evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' For small object detection, ViTDet can beat the CNN counterpart by 7-11% on APs, which implies ViTDet backbone can better detect small objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 18 shows an example of object detection results on a testing image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The two aircraft are tested with high confidence value (>=98%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Aircraft Object Detection Results for the Synthetic Image Dataset Table 9 presents aircraft object detection results for the synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' As with the previous testing results, Cascade Mask RCNN still consistently outperforms Mask RCNN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' ViTDet still beats CNN backbone in all evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' For this dataset, the performance improvement of small object detection is not so large as in the two previously tested datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 19 shows an example of aircraft object detection on a testing image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' In this case, all aircraft are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' However, several non-aircraft objects are mis- labeled as aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Aircraft Role Object Detection Results for the Real Image Dataset Table 10 shows aircraft role object detection results for the real dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' As with the above three experiment cases, Cascade Mask RCNN still outperforms Mask RCNN, except on AP50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' For AP, ViTDet backbone still beats CNN backbone with large margins of improvement (14-17%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 20 shows an example of aircraft role object detection on a testing image, where the aircraft are labeled by their role names of “large civil transportation utility.” 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Aircraft Role Object Detection Results for the Synthetic Image Dataset Table 11 presents aircraft role object detection results for the synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Similarly, Cascade Mask RCNN still performs better than Mask RCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' On AP, ViTDet backbone still beats CNN backbone with large margins (12-16%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 21 shows an example of aircraft role object detection on a testing image, and roles are identified according to their sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 19, all aircraft objects are correctly identified with their roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' However, several non-aircraft objects are wrongly labeled as aircraft role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' As seen in the above four experiments for the RarePlanes dataset, obviously ViTDet backbone performs much better than the CNN counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' For AP, the improvement ranges from 5% to 17%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The above experiments focused on HBB object detection performed with Detectron2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Next, we will present OBB object detection with MMRotate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Table 8 RarePlanes real dataset aircraft object detection results comparison Method Backbone Pre-train Task FT-epoch Learning rate AP AP50 AP75 APs APm API Mask RCNN ResNeXt-101 1K,sup aircarft 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0001 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='17 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='33 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='78 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='66 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='57 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='86 Mask RCNN ViTDet,ViT-L 1K,MAE aircarft 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0001 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='72 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='29 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='62 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='55 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='22 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='78 CascadeMaskRCNN ViTDet,ViT-L 1K,MAE aircarft 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0001 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='45 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='60 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='97 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='02 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='81 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='17 15 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 18 Example of aircraft object detection results on the RarePlanes real testing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Table 9 RarePlanes synthetic dataset aircraft object detection results comparison Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 19 Example of aircraft object detection results on the RarePlanes synthetic testing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 100 ircraft100% 200 raft98% 300 400 500 0 100 200 300 400 500Method Backbone Pre-train Task FT-epoch Learningrate AP AP50 AP75 APs APm API MaskRCNN ResNeXt-101 1K,sup aircarft 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0001 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='43 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='89 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='66 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='91 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='13 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='02 Mask RCNN ViTDet,ViT-L 1K,MAE aircarft 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0001 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='80 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='69 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='85 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='13 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='32 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='42 Cascade Mask RCNN ViTDet,ViT-L 1K,MAE aircarft 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0001 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='06 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='70 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='17 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='33 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='06 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='58200 400 aircraft 999% 600 craft 100% 800 1000 0 250 500 1000 1250 1500 1750 16 Table 10 RarePlanes real dataset aircraft role object detection result comparison Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 20 Example of aircraft role object detection results on the RarePlanes real dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Table 11 RarePlanes synthetic dataset aircraft role object detection results comparison Method Backbone Pre-train Task FT-epoch Learningrate AP AP50 AP75 APs APm API Mask RCNN ResNeXt-101 1K,sup aircarft role 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0001 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='96 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='58 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='35 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='28 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='23 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='45 Mask RCNN ViTDet,ViT-L 1K,MAE aircarft role 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0001 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='00 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='41 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='29 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='51 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='29 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='03 CascadeMaskRCNN ViTDet,ViT-L 1K,MAE aircarft role 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0001 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='94 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='38 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='74 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='88 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='26 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='040 100 LargeCivilTransport/ilityi20%% 200 Transdort/Utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='11%/ 300 400 500 0 100 200 300 400 500Method Backbone Pre-train Task FT-epoch Learningrate AP AP50 AP75 APs APm API Mask RCNN ResNeXt-101 1K,sup aircarft role 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0001 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='66 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='31 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='11 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='23 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='32 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='78 Mask RCNN ViTDet,ViT-L 1K,MAE aircarft role 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0001 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='62 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='93 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='56 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='46 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='17 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='02 CascadeMaskRCNN ViTDet,ViT-L 1K,MAE aircarft role 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0001 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='67 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='78 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='32 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='56 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='88 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='27 17 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 21 Example of aircraft role object detection results for RarePlanes synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='5 DOTA-v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0 Object Detection Results Training models for DOTA-v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0 is more complicated than the previous HBB object detection experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' We followed five steps to carry out OBB experiments: Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Data preprocessing For a fair comparison, we followed the same data preprocessing steps laid out in Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' [1] and Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' [161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Specifically, we first combined both training and validation sub-datasets to train models, and the testing dataset was used for final evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Note that the testing dataset does not provide data labels in the downloaded folders, and the DOTA web evaluation server [172] must be used for the final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' DOTA’s image size ranges from 800 x 800 to 4,000 x 4,000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' therefore, we also followed the image splitting practice and cropped the original images into 1,024 × 1,024 patches with a stride of 824.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Just as importantly, we also carried out data augmentation to get a variant of DOTA-v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0 for training, in which we also adopted standard random rotation (RR) and multi-scale (MS) transformation at three scales {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='5} for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' After all the necessary steps, we have two transformed datasets for the model training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' one is only with splitting, and the other is with splitting and data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Pretrained models downloading From the RoI Transformer and ReDet model zoo, we downloaded the pretrained models with the ResNet and Swin- T backbones for comparing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' For RoI Transformer, models were downloaded from the following webpage: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='com/open- mmlab/mmrotate/blob/main/configs/roi_trans/README.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='md .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' For ReDet, models were downloaded from the DOTA-v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='10 table on the following webpage: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='com/open- mmlab/mmrotate/blob/main/configs/redet/README.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Configuration files customization We set up configuration files for ViTDet backbone in the selected algorithms: RoI Transformer [1] and ReDet [161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' We used the default configuration files provided in the MMRotate platform as exemplar and created corresponding ones for ViTDet backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' In details, the angle representation was set to 1e90, the learning rate was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0001, AdamW optimizer was used, and the number of training epochs was 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' TrarsponUtigy 62% y58 200 MediumCivilTransport/Utility69 400 Medium CivilTransport/Utility 696 Large Civil Transport/Utility 93966 600 edunCMlTransgortUnityBa 800 LUnlty T965H 1000 um CMi TransportUslity Gre 0 250 500 750 1000 1250 1500 1750 18 Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fine-tuning models for ViTDet backbone We fine-tuned four models with ViTDet backbone in RoI Transformer and ReDet on two preprocessed datasets in Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Evaluation on the testing dataset When all 9 models were ready, we evaluated them on the testing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' We then submitted the predicated object detection results to the DOTA-v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0 official evaluation server, which in turn gave us AP for each class and mAP for all classes shown in Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Table 12 presents the detailed nine experiment results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The three backbones are as follows: R50 stands for ResNet-50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Swin-T represents Swin Transformer tiny version;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' and ViTDet, ViT-B is ViTDet backbone built with a base version of ViT that has 12 layers and 768-dimension output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' With consideration of OBB algorithms’ heavy computation burden, we did not evaluate ViTDet, ViT-L and ViTDet, ViT- H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The column “aug.” shows whether data augmentation was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' As demonstrated in Table 12, given the same backbones, RetDet performs slightly better than RoI Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Without data augmentation, ViTDet, ViT-B backbone is slightly worse than the other two backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' However, with data augmentation, ViTDet, ViT-B achieves the best performance on mAP (80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='89%), which is very comparable to the best published benchmark of 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='90% in Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' [169].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' That benchmark was achieved with a combination of RoI Transformer, Swin-T backbone, Kullback-Leibler Divergence (KLD) trick [179], and data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' In comparison, ViTDet can much more easily achieve comparable best performance without the need of the complicated KLD trick, in which the rotated bounding box must be converted into a 2-D Gaussian distribution and then KLD between the Gaussian distributions are calculated as the regression loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Moreover, for helicopter detection (HC), ViTDet, ViT-B performs the best, improving about 23% at large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Overall, for a complicated dataset like DOTA, data augmentation still plays a bigger role than backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 22 shows an example of detection results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' In a compacted parking lot like the one pictured, most of the vehicles are detected with high confidence values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' In short, for OBB object detection, ViTDet, ViT-B still achieves comparable performance with other backbones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' the computation burden of ViTDet is heavier than R50 and Swin-T backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Therefore, more research may be needed to improve ViTDet’s performance for OBB detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Table 12 Accuracy comparison of rotated object detection on DOTA-v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0 method backbone pre-train aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' PL BD BR GTF sV LV HS TC BC 1s SBF RA HA SP HC mAP Rol Trans R50 1K,sup 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='97 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='14 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='59 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='28 79.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='94 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='91 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='93 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='63 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='71 ReDet ViTDet,ViT-B 1K,MAE MS+RR 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='75 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='22 61.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 22 Example of object detection results on DOTA-v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='0 with ViTDet backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' e-veh 20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' CONCLUSION This study has explored the newly proposed MAE pretrained ViTDet backbone for challenging aerial image object detection problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' We carried out 25 experiments on three well-known aerial image datasets: Airbus Aircraft, RarePlanes, and DOTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Our experiments demonstrated that ViTDet backbone consistently beats its CNN counterparts in HBB object detection (up to 17% improvement on AP) and that it achieves on-par performance for OBB object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Our results also provided a baseline for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors sincerely thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Kris Rosfjord and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Heath Farris for their generous support of this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' We would also like to thank Mike Robinson, Bill Bateman, Lixia Song, Erik Vargo, and Paul A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Diffenderfer of The MITRE Corporation for their valuable discussions, insights, and encouragement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' NOTICE This work was sponsored by MITRE’s independent research and development program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' The contents of this document reflect the views of the authors and do not necessarily reflect the views of the Federal Aviation Administration (FAA) or the Department of Transportation (DOT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Neither the FAA nor the DOT makes any warranty or guarantee, expressed or implied, concerning the content or accuracy of these views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' ã 2023 THE MITRE CORPORATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' ALL RIGHTS RESERVED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' APPROVED FOR PUBLIC RELEASE, DISTRIBUTION UNLIMITED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' PRS CASE 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' 04, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content=' Available: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='org/abs/2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} +page_content='01883' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFLT4oBgHgl3EQfWy86/content/2301.12058v1.pdf'} diff --git a/bNE4T4oBgHgl3EQfoQ2m/content/tmp_files/2301.05183v1.pdf.txt b/bNE4T4oBgHgl3EQfoQ2m/content/tmp_files/2301.05183v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..361330a15cd867e6e96d3ef7426412341e8aa60d --- /dev/null +++ b/bNE4T4oBgHgl3EQfoQ2m/content/tmp_files/2301.05183v1.pdf.txt @@ -0,0 +1,2205 @@ +DarkSPHERE: Exploring light dark matter with a spherical proportional counter +electroformed underground at the Boulby Underground Laboratory +L. Balogh,1 C. Beaufort,2 M. Chapellier,3 E. C. Corcoran,4 J.-M. Coquillat,3 A. Dastgheibi-Fard,2 Y. Deng,5 +D. Durnford,5 C. Garrah,5 G. Gerbier,3 I. Giomataris,6 G. Giroux,3 P. Gorel,7 M. Gros,6 P. Gros,3 +O. Guillaudin,2 E. W. Hoppe,8 I. Katsioulas,9 F. Kelly,4 P. Knights,9, ∗ P. Lautridou,10 I. Manthos,9 +R.D. Martin,3 J. Matthews,9 J.-F. Muraz,2 T. Neep,9 K. Nikolopoulos,9 P. O’Brien,5 M.-C. Piro,5 +N. Rowe,3 D. Santos,2 G. Savvidis,3 I. Savvidis,11 F. Vazquez de Sola Fernandez,10 and R. Ward9 +(NEWS-G Collaboration) +E. Banks,12 L. Hamaide,13 C. McCabe,13 K. Mimasu,13 and S. Paling12 +1Department of Mechanical and Materials Engineering, +Queen’s University, Kingston, Ontario K7L 3N6, Canada +2LPSC, Universit´e Grenoble-Alpes, CNRS-IN2P3, Grenoble, 38026, France +3Department of Physics, Engineering Physics & Astronomy, +Queen’s University, Kingston, Ontario, K7L 3N6, Canada +4Chemistry & Chemical Engineering Department, +Royal Military College of Canada, Kingston, Ontario K7K 7B4, Canada +5Department of Physics, University of Alberta, Edmonton, T6G 2E1, Canada +6IRFU, CEA, Universit´e Paris-Saclay, F-91191 Gif-sur-Yvette, France +7SNOLAB, Lively, Ontario, P3Y 1N2, Canada +8Pacific Northwest National Laboratory, Richland, Washington 99354, USA +9School of Physics and Astronomy, University of Birmingham, Birmingham, B15 2TT, United Kingdom +10SUBATECH, IMT-Atlantique/CNRS-IN2P3/Nantes University, Nantes, 44307, France +11Aristotle University of Thessaloniki, Thessaloniki, 54124 Greece +12STFC Boulby Underground Laboratory, Boulby Mine, Redcar-and-Cleveland, TS13 4UZ, UK +13Department of Physics, King’s College London, Strand, London, WC2R 2LS, United Kingdom +We present the conceptual design and the physics potential of DarkSPHERE, a proposed 3 m +in diameter spherical proportional counter electroformed underground at the Boulby Underground +Laboratory. This effort builds on the R&D performed and experience acquired by the NEWS-G +Collaboration. DarkSPHERE is primarily designed to search for nuclear recoils from light dark +matter in the 0.05–10 GeV mass range. Electroforming the spherical shell and the implementation +of a shield based on pure water ensures a background level below 0.01 dru. These, combined with +the proposed helium-isobutane gas mixture, will provide sensitivity to the spin-independent nucleon +cross-section of 2 × 10−41 (2 × 10−43) cm2 for a dark matter mass of 0.1 (1) GeV. The use of a +hydrogen-rich gas mixture with a natural abundance of 13C provides sensitivity to spin-dependent +nucleon cross-sections more than two orders of magnitude below existing constraints for dark matter +lighter than 1 GeV. The characteristics of the detector also make it suitable for searches of other +dark matter signatures, including scattering of MeV-scale dark matter with electrons, and super- +heavy dark matter with masses around the Planck scale that leave extended ionisation tracks in the +detector. +Keywords: Particle dark matter, dark matter detection, spherical proportional counter, electroformation +I. +INTRODUCTION +The nature of Dark Matter (DM) is one of the most +pressing questions in physics, as reflected by the num- +ber of ongoing and planned activities, spanning orders +of magnitude in scale and complexity [1, 2]. The 10 – +1000 GeV mass region has been under intense experi- +mental scrutiny as the preferred range for Weakly Inter- +acting Massive Particles (WIMPs) [3]. However, the lack +of conclusive evidence to-date, including from searches +at colliders [4, 5], demands the broadening of the DM +search strategy. Searches for lighter DM candidates be- +low the Lee-Weinberg bound of about 2 GeV [6] are +coming increasingly into focus, supported by a growing +number of theory paradigms, e.g., asymmetric dark mat- +ter [7, 8], hidden sectors [9–11], and scenarios with a +modified early-universe cosmology [12–15]. +The large-scale liquid noble gas detectors that pro- +vide the most stringent constraints on WIMP inter- +actions with an atomic nucleus (see e.g., [16–18]) are +not optimised for the light DM region, below approx- +imately 5 GeV, due to poor kinematic matching be- +tween the DM and target nucleus. +There are ongoing +attempts to recover sensitivity through processes such +as the Migdal effect [19–22], which is itself under in- +tense investigation [23, 24]. Several collaborations includ- +ing CRESST [25], DAMIC [26], EDELWEISS [27] and +SuperCDMS [28] have demonstrated sensitivity to light +arXiv:2301.05183v1 [hep-ex] 12 Jan 2023 + +2 +Fig. 1. +Schematic and principle of operation of the spher- +ical proportional counter. +Particle interactions within the +gas produce ionisation electrons, which drift towards the an- +ode at the centre of the sphere. +The anode shown is of +the simplest form, comprising a single-anode read-out, while +DarkSPHERE will operate with a multi-anode ACHINOS +read-out sensor. The copper used for the grounded cathode +and grounded rod will be electroformed underground to min- +imise the background rate. +DM candidates by utilising smaller, cryogenic solid-state +detectors with sub-keV detection thresholds. However, +scaling to larger exposures while maintaining low energy +thresholds and low background rates is challenging. +The New Experiments With Spheres-Gas (NEWS-G) +Collaboration searches for light DM using spherical pro- +portional counters [29, 30]. In its simplest form, a spher- +ical proportional counter consists of a grounded, spheri- +cal, metallic vessel filled with an appropriate gas mixture +and a spherical anode of O(1 mm) in radius at the cen- +tre, as depicted in Figure 1. The anode is supported by +a grounded metallic rod, which also shields the wire used +to apply a positive voltage to the anode and read out the +signal. The electric field varies approximately as 1/r2, +dividing the gas region into a drift and an avalanche +volume. +Particle interactions in the gas may result in +the ionisation of electrons, which subsequently drift to +the anode. Within approximately 100 µm from the an- +ode, the electric field becomes sufficiently intense for an +avalanche to occur, providing signal amplification. +Spherical proportional counters exhibit several key fea- +tures that make them ideal for performing light DM +searches [31]. Firstly, they have a small detector capaci- +tance by virtue of the spherical shape, and together with +the ability to operate in high gas gains, allow for the de- +tection of nuclear recoils with sub-keV energy [32, 33]. +A crucial advantage is that the small detector capaci- +tance is independent of the outer diameter (∅) of the +detector, thus the detector can be scaled to a larger size +without impacting the energy threshold. Secondly, the +simplicity of the design, while also greatly easing detec- +tor operation, enables construction from a small num- +ber of radio-pure components. This allows for low back- +ground rates to be achieved. Thirdly, the ability to op- +erate the detector with different gas mixtures and pres- +sures offers two significant benefits: a) the use of different +atoms or molecules in the gas mixture allows kinematic +DM candidate-target matching; and b) changes to the +gas mixture and pressure provide additional handles to +disentangle potential signals from unknown instrumen- +tal backgrounds. +Finally, analysis of the pulse shape +provides a powerful handle for background rejection and +fiducialisation [34]. +The first DM results with a spherical proportional +counter were produced using SEDINE, a ∅60 cm detec- +tor operating at the Laboratoire Souterrain de Modane +(LSM), France [35]. +At the time, SEDINE provided +the best sensitivity to the spin-independent DM-nucleon +scattering cross-section at 0.5 GeV [34]. The data from +SEDINE was also used to perform a search for Kaluza- +Klein axions produced in the Sun, and a 90% confidence +level upper limit of gaγγ < 8.99 × 10−13 GeV−1 was set +on the axion-photon coupling [36]. +S140, a ∅140 cm +spherical proportional counter made of 99.99% pure cop- +per is currently operating in SNOLAB, Canada [37]. +First preliminary results provide the strongest constraint +on spin-dependent WIMP-proton cross-section in the 0.2- +2 GeV DM mass range [38]. +S140’s active volume is +internally shielded with a 500 µm thick layer of ultra- +radiopure copper that has been deposited on the in- +ner surface by adapting a low-background electroforming +method to hemispheres. This procedure was undertaken +at LSM [39, 40]. Despite using 99.99% pure copper and +the electroformed internal shield, the dominant remain- +ing background in S140 is the radioactive contamina- +tion and cosmogenic activation of the copper. Neverthe- +less, this can be mitigated by fully electroforming future +detectors directly in the underground laboratory where +they will be operated. This is the objective of Electro- +formed Cuprum Manufacturing Experiment (ECuME), +a ∅140 cm spherical proportional counter that will be +fully electroformed underground in SNOLAB. By fully +electroforming the intact detector, additional radioactive +contamination brought by machining and welding pro- +cesses is avoided, and by conducting this underground, +cosmogenic activation is minimised. +ECuME will be +operated with a neon–methane gas mixture (Ne:CH4, +90%:10%) at 2 bar. The electroformed ECuME detector +will be installed in the shielding currently used for S140 +upon conclusion of its physics exploitation. +To further reduce background rates, the detector + +Drift Region +e +0 +Grounded Cathode +400 +core years). +In the following subsections we present the details of +the analysis and computational strategies used to gen- +erate the LCR data products. +3.1. Analysis Technique & Tools +The characterization of LAT sources is typically per- +formed using a maximum likelihood analysis (Abdo +et al. 2009), in which the parameters of a model de- +scribing the point sources and diffuse isotropic gamma- +ray emission in a given region of the sky are jointly opti- +mized to best describe the observed photon distribution. +The light curves of the LCR are obtained by performing +an unbinned likelihood analysis, in which the full spatial +and spectral information of each photon is used in the +maximum likelihood optimization. + +Number of sources +102 +TTTT +101 +100 +NLS10-8 +口 +BLL +Not in LCR +FSRQ +Other classes +in LCR +BCU +10-9 +口 +10-10 +10-11 +TTTT +12 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +Power-law index (r)Fractional variability +10. +10-1 +BLL +口 +Not in LCR +FSRQ +Other classes +in LCR +BCU +101 +102 +103 +104 +105 +Variability indexThe Fermi-LAT Light Curve Repository +5 +Figure 2. +A model map, with 0.1 degree resolution, for +a single weekly time bin of the region surrounding FSRQ +4C +28.07. This source contains 15 other variable sources +within a 12◦ radius, highlighting the need to model all vari- +able sources within the ROI. +The LCR analysis is performed with the standard LAT +Fermitools3 (version 1.0.5) using the P8R3 SOURCE V2 +instrument response functions on P8R3 SOURCE class +(Atwood et al. 2013; Bruel et al. 2018) photons selected +over the energy range covering 100 MeV–100 GeV. Note +that the energy dispersion is neglected in the unbinned +analysis mode. This is not expected to impact the qual- +ity of the analysis. However, repeating the analysis with +a binned likelihood approach would return somewhat +(mostly non significant) results. +For each source and +time bin, photons are selected from a circular region of +interest (ROI) of radius 12◦ centered on the location of +the target source. The ROIs are analyzed separately. By +contrast, for the standard FGL catalogs, fluxes and spec- +tral parameters of multiple sources are extracted from +the same ROI. The size of the ROI is conservatively cho- +sen to be as large as the 95% containment radius of the +LAT energy-dependent point-spread function (PSF) at +100 MeV. Additional data selection cuts are imposed to +exclude photons associated with regions and periods of +known solar flares and gamma-ray bursts. A zenith an- +gle limit of 90◦ to strongly reduce contamination from +gamma rays produced through interactions of cosmic +rays with Earth’s atmosphere (Earth limb). +Following the 4FGL-DR2 catalog, the LCR sources +can have one of three different spectral types: +3 https://fermi.gsfc.nasa.gov/ssc/data/analysis/software/ +• Power-law (PL): +dN/dE = N0(E/E0)−Γ +(1) +• log-parabola (LP): +dN/dE = N0(E/Eb)−(α+β log(E/Eb)) +(2) +• and subexponentially cutoff power-law (PLEC): +dN/dE = N0(E/E0)−γ1e−aEγ2 +(3) +A majority of LCR sources are best described by a PL +or LP spectral shape. The PLEC spectral shape best +represents 11 additional sources, comprising the 9 pul- +sars in the LCR sample and the two brightest FSRQs +(CTA 102 and 3C 454.3). +The normalization of the source spectrum in the +model is left free to vary, while the spectral shape pa- +rameters are initially fixed to their 4FGL-DR2 cata- +log values. +The model also includes all gamma-ray +sources in the 4FGL-DR2 catalog within a radius of +30◦ from the ROI center. +The normalization of each +variable source in the ROI is also left free to vary +in the model, with spectral shape parameters fixed +to their catalog values. +In addition to the point +sources, Galactic and isotropic background components +are included in the model. +The Galactic component, +gll iem v07.fits4, is a spatial and spectral template +that accounts for interstellar diffuse gamma-ray emis- +sion from the Milky Way. +The isotropic component, +iso P8R3 SOURCE V3 v1, provides a spectral template to +model the remaining isotropic events, including contri- +butions from the residual charged-particle background +and the isotropic celestial gamma-ray emission. +The +normalizations of both the Galactic and isotropic com- +ponents are allowed to vary during the fit. The free pa- +rameters of the model are varied to maximize the likeli- +hood of observing the data given the model. An iterative +fitting strategy, which varies the required fit tolerance5 +over three steps (1, 10−4 and 10−8), is employed to min- +imize the number of time bins in which the likelihood fit +does not successfully converge. Once fit convergence is +achieved with the tightest tolerance, a second round of +4 https://fermi.gsfc.nasa.gov/ssc/data/access/lat/ +BackgroundModels.html +5 The fit tolerance is the relative convergence tolerance that is +specified in the gtlike routine and passed to the optimization +algorithm used to maximize the log likelihood function. The fit +tolerance can be loosened to help achieve fit convergence. On the +other hand, fits with tighter fit tolerances exhibit overall lower +fractional errors on the resulting flux estimations. Therefore an +iterative approach has been adopted for the LCR. + +101 ++40° +4FGLJ023 +4FGLJ0252.9+3834 +18.9+3643 +Counts (0.1-100 GeV) +4FGL +4FGL +J0221.1+3556 ++32° +4FGLJ0324.8+3412 +4FGL +0159 +4FGLJ0253.5+3216 +4FGL +Dec (2000) +3042 +100 +4FGL ++2848 +4FGL ++24° +4FGLJ0245.4+2408 +4FGLJ0258.1+2030 ++16° +4FGL10224.9+1843 ++8° +10-1 +56° +48° +40° +32° +24° +RA (J2000)6 +The Fermi-LAT Collaboration +Figure 3. An example 3-day light curve as can be found in the LCR. It spans more than 11 years of LAT data and refers to +the source at the center of Figure 2, FSRQ 4C +28.07. This light curve is obtained by specifying the analysis option that the +spectral index of the source is free to vary. The data gap in 2018 is due to the temporary shut down of the instrument because +of a solar panel anomaly. +fitting is performed in which a spectral shape parameter +of the target source is allowed to vary, namely, photon +index (Γ) for PL, α for LP, and γ1 for the PLEC model. +All other parameters remain fixed, i.e., β for LP, and a +and γ2 for the PLEC model. +Figure 2 shows a model map for a single time bin +of the region surrounding FSRQ 4C +28.07 (4FGL +J0237.8+2848), which contains 15 other variable sources +within a 12◦ radius. Figure 3 shows the resulting 3-day +light curve spanning more than 11 years of LAT data +for 4FGL J0237.8+2848. +A likelihood ratio test (Neyman & Pearson 1928) is +used to quantify the significance of the target source +above the background; specifically the test statistic +(Mattox et al. 1996): +TS = −2 log(L0/L). +(4) +The TS compares the maximum value of the likelihood +function L0 evaluated for the parameter values that +maximize the likelihood under a background-only null +hypothesis (i.e., a model that does not include a tar- +get source), with L, the likelihood function evaluated at +the best-fit model parameters when including the target +source. +In the null hypothesis, TS is distributed ap- +proximately as χ2 (Wilks 1938), and the analysis rejects +the null hypothesis when the test statistic is greater than +TS ≥ 4, which is roughly equivalent to a 2 σ rejection cri- +terion for a single degree of freedom6. Using this thresh- +old value of TS as the detection criterion, the LCR em- +6 The null hypothesis is tested against the presence of one source +at a known position (at the center of the ROI) +ploys a Bayesian profile likelihood method7 to calculate +the 95% confidence level upper limits for any interval +that yields a TS ≤ 4, and also extracting a flux estima- +tion of the target source for any interval with a TS ≥ 1. +The resulting best-fit values (or upper limits) for photon +flux (cm−2 s−1), energy flux8 (GeV cm−2 s−1), and asso- +ciated spectral shape are saved to the LCR database, for +both the fixed and free photon index analyses. This pro- +cedure ultimately allows for a user-selectable detection +threshold and spectral fit method, as the flux estimates +and upper limits between 1 ≤ TS ≤ 4 are both recorded +in the LCR database, as well as the results from both +the fixed and free spectral analyses. +In Fig. 4 we compare, for the case of the bright quasar +Ton 599, the best-fit flux values found using a fixed spec- +tral index vs. leaving the spectral index free to vary. The +colors in the plot refer to the three cadences analysed. +The results are generally stable, with greater uncertain- +ties for lower fluxes for higher cadences due to lower +statistics in the ROI in each time bin. +3.2. Computational Strategy +For analyses of a relatively small number (a few thou- +sands) of photons, an unbinned likelihood analysis can +be performed rapidly (a few minutes), but as the num- +ber of events increases, the time to perform the analysis +can become prohibitive. +This limitation becomes in- +creasingly burdensome when the need arises to perform +7 A description of the Bayesian profile likelihood method employed +by the LCR can be found at https://fermi.gsfc.nasa.gov/ssc/ +data/p7rep/analysis/scitools/python tutorial.html. +8 The energy flux is obtained from the photon differential flux dN +dE +as +� +E dN +dE dE + +2.00e-6 +三 +1.50e-6 +1.00e-6 +5.00e-7 +0.00e+0 +2009 +2010 +2011 +2012 +2013 +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021 +Date (UTC)The Fermi-LAT Light Curve Repository +7 +Figure 4. Comparison of the fluxes [ph/cm2/s] in time bins +obtained with the fitting pipeline keeping the spectral index +fixed versus letting the spectral index be free. Empty circles +indicate results from analyses that did not converge, i.e., +MINUIT Return Code q ̸= 0. +MINUIT (James 1994) is the +optimizer adopted in the likelihood analysis. The plot shows +the fluxes measured for the quasar Ton 599. In this case, the +fit for the highest-flux point (with TS∼ 4) converged when +the spectral index was set free, but did not converge when +the spectral index was fixed. +a source characterization over a large number of time +bins. A binned likelihood analysis could alleviate this +issue. However, information is lost when binning data. +The LCR tackles the computational overhead by paral- +lelizing the process of performing a full unbinned likeli- +hood analysis. In order to produce a high cadence light +curve over the entire lifetime of the mission in a reason- +able amount of time, the LCR distributes the analyses of +the light curve bins to separate nodes in a computer clus- +ter hosted at the SLAC National Accelerator Laboratory +and utilizing the IBM Spectrum LSF workload manage- +ment platform. The parallelization allows for thousands +of time bins to be analyzed simultaneously, with the net +effect of drastically reducing the time to generate light +curves over the entire duration of the mission. +3.3. Exposure analysis +The Fermi spacecraft has been executing a sky- +scanning strategy for the LAT for the great majority of +the mission, generally reaching an almost uniform full- +sky coverage daily, in fact, every two orbits (i.e., ap- +proximately every three hours). During the mission the +scanning strategy has been interrupted occasionally for +targeted observations. The longest-duration such obser- +vation was a modified observing strategy executed from +2013 December to 2015 July that favored the Galactic +center region. In March 2018, the seizing of the drive +motor for one of the solar panels forced the temporary +shut down of the LAT and a redefinition of the survey +mode9. This caused the exposure to be limited to a part +of the sky for some period of time. The effects of these +periods of uneven exposure can be seen in the 3-day light +curves. Therefore we quantify and discuss these effects +in this section. +The number of photons in a given ROI is modulated +by the exposure of the observation, which is given by the +product of the LAT livetime (during the brief readout +time of a photon or cosmic-ray interaction the LAT, the +instrument is ‘dead’ to triggering on other interactions) +and the energy and angle-dependent effective area of +the LAT. In general, analyses integrating over long ob- +servation times can have small fractional exposure vari- +ations across an ROI, but shorter-timescale analysis can +be significantly impacted by low exposure for a particu- +lar region of the sky. In fact, since the beginning of the +post-anomaly modification of the sky-scanning strategy +in February 2019, a number of the 3-day and 7-day ca- +dence time bins have no exposure in particular portions +of the sky. This is due to the constraints on the direc- +tion of the zenith of spacecraft with respect to the Sun10. +Fig. 5 shows the range of values of the LAT exposure in +the sky for each time bin of 3-day cadence. The maps +in Fig. 6 illustrate the positions of all the sources ana- +lyzed in the LCR, overlaid on the counts map (top) and +an exposure map (bottom) for an typical 3-day cadence +time bin with a zero minimum exposure: some sources +fall in the region of the sky with zero exposure, which +naturally translates into zero events to analyze in the +ROI. +In these cases, or in those with minimum exposure +orders of magnitude lower than the maximum (shaded +pink regions in the maps in Fig. 6), the pipeline typically +returns an upper limit on the source flux. However, in +some cases it could still find a solution that maximizes +the likelihood, but warns of a poor fit quality or an un- +reasonably low error value. For this reason, we recom- +mend LCR users use caution when using the 3-day and +7-day cadence light curve for these low-exposure inter- +vals, or exclude the affected time bins entirely from their +analyses. +To illustrate the effect of low exposure on the photon +counts, we generated binned all-sky counts and related +exposure maps in the same time bins used for the LCR +data analysis. We consider events in the 0.1–100 GeV +9 https://fermi.gsfc.nasa.gov/ssc/observations/types/ +post anomaly/ +10 Note that the ‘gaps’ in the exposure for different time bins are +not always in the same portion of the sky + +4FGL J1159.5+2914 +LCR, daily +LCR, daily, q±0 +LCR, weekly +LCR, weekly, q± 0 +LCR, monthly8 +The Fermi-LAT Collaboration +Figure 5. The gray band marks the range of LAT on-sky exposures for each time bin of 3-day cadence. The pink bands, in +order, mark time ranges of the original rocking angle mission profile(†), the Galactic center monitoring, and the time gap between +the solar panel anomaly and the beginning of the new survey mode. Note that some time bins have a minimum exposure value +of zero: this means that some part of the sky was not observed during that time interval. The exposure maps are computed at +the central energy in the 0.1–100 GeV band. +(†) Details on the Observatory sky-survey profiles can be found at https://fermi.gsfc.nasa.gov/ssc/observations/types/allsky/. +energy range, and the maps are produced in HEALPix11 +format with the tool gtexpcube212, using the same anal- +ysis setup and IRFs as for the automated likelihood anal- +ysis. The HEALPix format allows us to easily extract +the average exposure in every ROI, and the pixel res- +olution (HEALPix order 6) matches the resolution of +the pre-computed livetime cubes provided for LAT data +analyses. +For each source considered in the LCR and each time +bin of 3-day and 7-day cadences, we extract the average +exposure and the total photon counts in the ROI cen- +tered on the source between 0.1 and 100 GeV. In Fig. 7 +we illustrate the statistics of sources with a given num- +ber of time bins in the light curve that had no photons +in the ROI or had fewer than 20 photons (considered as +an arbitrary threshold for low-statistics analysis). Hun- +dreds of sources have up to 30 time bins with zero events +in the light curve and/or less than 20 events, while tens +of sources have fewer than 20 photons in more than 50 +time bins (note that the total number of time bins in +11 http://healpix.jpl.nasa.gov/ (G´orski et al. 2005) +12 The Fermitools, as part of the maximum likelihood calculation, +automatically account for the exposure, as describet on the of- +ficial mission web page at https://fermi.gsfc.nasa.gov/ssc/data/ +analysis/documentation/Cicerone/Cicerone Data Exploration/ +livetime and exposure.html. +The module gtexpmap is used to +compute the exposure when performing unbinned analyses, as +for the in the LCR analysis pipeline. In this section, we compute +the exposure through the gtexpcube2 module, which is used +when performing binned likelihood analyses. We stress that this +is for purely illustrative purposes; the exposure maps computed +in this section were not used in the LCR likelihood analyses. +a 3-day cadence light curve is more than 1680 as of +now). +The monthly cadence, due to the longer inte- +gration time, should not be affected by this issue; how- +ever, we still recommend that users check the exposure +of each bin used in their analyses. The exposures within +the 12◦ radius ROIs for each source for the 3-day and +7-day cadences are provided in the LCR downloadable +data. +4. USAGE CAVEATS +In this section we provide a number of caveats to be +mindful of when using the LCR data for scientific re- +search. This list is also available and will be updated +periodically on the LCR data portal. +1. The LCR provides fit results from likelihood anal- +yses that both did and did not converge. +How- +ever, it is important that the end user is aware +that results from analyses that did not converge +should be considered suspect and not be used for +higher-level analyses (e.g., multi-frequency cross- +correlation, power spectral density, or studies of +variability). The convergence status for a partic- +ular time bin is recorded in the MINUIT Return +Code parameter and non-convergent analyses are +hidden by default, but are optionally accessible to +the user. A review of the current version of the +LCR fit results for the first fourteen years of mis- +sion data shows that the analyses for 0.7% of time +bins, for all sources, did not converge when the +spectral indices were held fixed, and that ∼35% +did not converge when the spectral indices were +free to vary. + +109. +108 +[cm +107 +OLD ROCKING +106. +GC POINTING +SOLAR PANEL ANOMALY +3-day exposure range in the sky +3 +4 +5 +Time [MET] +1e8The Fermi-LAT Light Curve Repository +9 +Figure 6. Top: Counts map for a 3-day time bin with 0 min- +imum exposure; Bottom: Exposure map for the same time +bin. The white pixels mark the positions of LCR sources. +The light blue and magenta circles in the maps highlight the +cases of two ROIs lying in a region of the sky with zero and +non-flat exposures, respectively. +2. From its definition, equation 4, the TS is a man- +ifestly positive quantity. +However, negative TS +values can sometimes be obtained when the pa- +rameters reach the limits of their allowed intervals +without having maximized the likelihood profile. +Fit results obtained from intervals that resulted in +negative TS values should be considered suspect +and not used in higher-level analyses. +The cur- +rent version of the LCR data products for the first +fourteen years of the mission for all LCR sources +have only a few bins with negative TS results, per +cadence, for both the fixed and free fits. +3. While time intervals containing gamma-ray bursts +(GRBs) and solar flares have been removed from +the LAT data prior to the likelihood analyses, pos- +sible contamination by the proximity of the qui- +escent Sun has not been accounted for, nor have +those time ranges been excluded. The angular sep- +aration of the Sun from the target source is pro- +vided for each time bin. A total of 175 GRBs and +266 solar flare time intervals were excluded from +the data prior to performing the maximum like- +Figure 7. In light blue, we show the distribution of LCR +sources by number of time bins with zero photons within +the ROI used for their analysis. In magenta, we show the +distribution of LCR sources according the number of time +bins with fewer than 20 photons within the ROI used for +their analysis. Darker shades of the lines refer to the 3-day +cadence, while lighter shades refer to the weekly cadence. +The low-exposure time bins represent the 0.1% of the total +time bins. +lihood analysis for the LCR first fourteen years +of light curve data. This affects less than about +0.01% of the time bins for all the LCR sources per +cadence. +4. Because the LCR analyses are made available in +real-time (new analysis results are generally made +available within 24 hours of being processed), the +results are not validated by the LAT Collaboration +prior to release. Users are encouraged to perform +sanity checks by examining the ratio of flux to flux +uncertainty vs. +the square root of the TS, and +the distributions of fit results, e.g., flux, flux un- +certainties, spectral indices mean values (photon +index Γ for PL, α for LP, and γ1 for the PLEC +model) and their uncertainties. Some examples of +these are shown in Fig. 8 for the specific case of +the FSRQ 4C +28.07 (4FGL J0237.8+2848). The +flux over the flux uncertainty ratio is expected to +be approximately proportional to the square root +of the TS. Any outliers should be either further +investigated or removed before using the data for +higher-level analyses, as should any extreme out- +liers from the data distributions. +5. The free-spectral-index light curves provided by +the LCR were produced using a model of the per- +tinent region of the sky for which only the spectral +index of the target source is set free, and those of + +Exp0sure map △T=491788801-492048001 MET +0 +9.21022e+083-day Counts < 20 +3-day Counts = 0 +Weekly Counts < 20 +Weekly Counts = 0 +Number of sources +102 +101 +0 +5 +10 +15 +20 +25 +30 +35 +Number of binsCounts map △T=491788801-492048001 MET +0 +111510 +The Fermi-LAT Collaboration +Figure 8. Example of validation plots for the FSRQ 4C +28.07. The top panels show the case for which the spectral index +(α for this source) is fixed. In all the plots we are considering the 0.1–100 GeV energy range. The bottom panels show the +case for which the spectral index is free to vary. The middle panels show the distributions of flux statistical uncertainties. Flux +distributions are shown in the right panels. In this example, distributions are good except for the outliers highlighted within +green squares, which should be either further investigated or removed. In the histograms, dashed lines represent results from +all time bins. Solid lines represent bins that did not converge. Notice that outliers are not necessarily the results from analyses +that did not converge. +all the other sources were fixed to the 4FGL-DR2 +catalog values. Therefore, contamination induced +by possible changes in the spectral indices of the +sources surrounding the target are not taken into +account. For instance, sources undergoing bright +flares have been seen to also experience dramatic +changes in their spectral indices, including changes +in the curvature, at the same time (e.g., harder- +when-brighter behavior). Therefore, bright, vari- +able sources in the ROI can induce this type of con- +tamination and must be considered when a target +source is in close proximity to any bright variable +sources. +6. Some erroneously high flux values during periods +of zero or low exposure, often associated to a very +small (or zero) error, have been found for several +sources (see Section 3.3). We recommend check- +ing the exposure value indicated for each time bin +and source ROI in the provided tables. If the light +curve has fluxes with very small fractional uncer- +tainties, rather than flux upper limit, in the time +bins with zero or low exposure, the reported fluxes +and uncertainties should be considered unreliable +and those time bins should be excluded from con- +sideration. Time bins with zero error on the flux +estimations are automatically hidden but are op- +tionally available to the user. +7. A bug in the make4FGLxml.py tool that the +LCR uses to generate the ROI models was re- +cently identified, in which the reported exten- +sion of the extended sources with RadialGauss +spatial profile is the 68% C.L. value instead of +the sigma value used in the XML model. Thir- +teen 4FGL-DR2 sources have this spatial profile, +namely, Crab IC, IC 443, Monoceros, HESS J1303- +631, FHES J1501.0−6310, FHES J1626.9−2431, +FHES J1723.5−0501, HESS J1825−137, W 41, +Cygnus Cocoon, +FHES J2129.9+5833, +FHES +J2208.4+6443 and FHES J2304.0+5406. None of +these sources is deemed variable in the 4FGL-DR2 +catalog. However, they are present in the ROIs of +101 LCR sources, which might result in a system- +atic bias in the target flux values across the whole +light curve. Correcting this issue will require re- +processing the data for these sources, which is al- +ready underway. Meanwhile users should be mind- +ful of this issue when considering sources in the +vicinities of these extended sources. +8. Particular care should be taken when using the +light curve for the synchrotron component of the +Crab. The Crab has different components treated +as separate entries in the 4FGL-DR2 catalog: the +extended emission, the inverse Compton emission +from the PWN and the synchrotron emission from +the pulsar. The LCR analysis generates only the +light curve for the synchrotron (variable) emission, + +4FGL J0237.8+2848 +LCR, daily +LCR, monthly +LCR, weekly103 +bin +102 +b +102 +occurrence +nce +occurrer +101 +101 +100 +100 +-12 +-10 +-8 +7 +5 +-4 +log10(oF) +log10(F)uig +102 +102 +occurrence +occurrence +101 +101 +100 +100 +-10 +-9 +-7 +-8 +.4 +-2 +8 +log10(oF) +log10(F)The Fermi-LAT Light Curve Repository +11 +while keeping the parameters of the other two com- +ponents frozen in the fit. This could result in some +contamination of the synchrotron component light +curve deriving from any unaccounted-for variabil- +ity of the other Crab components. On the other +hand, the LCR approach overrides the apparent +(not real) variability reported for the pulsar in +4FGL-DR2 (and all other versions of the 4FGL +catalog). +5. PROSPECTS AND CONCLUSIONS +The development of the Fermi-LAT LCR was moti- +vated by the need for a coherent collection of light curves +of variable gamma-ray sources observed by the Fermi- +LAT in support of the time-domain and multi-messenger +communities. By continuously reporting the flux evolu- +tion and transition to high-flux states for many variable +sources, the LCR is a valuable resource for triggering ob- +servations of other observatories. Furthermore, the LCR +can be used to validate the study of variable activity in +neighboring faint sources, helping to identify potential +contamination from flaring activity of a bright source. +In this manuscript we described the automated analysis +pipeline which will continuously update the repository +with new data as soon as new observations by the LAT +become available. We invite the community to use the +LCR data products, and report any issues or suggestions +to the LCR contacts at the Fermi Science Support Cen- +ter. A brief guide for navigating the LCR to navigate +the web site is provided in Appendix. +In the future, the LCR source list and the resulting +light curve data will be updated with every new Fermi- +LAT source catalog, with the next LCR version planned +to be released alongside the 5FGL catalog. Due to the +computational expense of re-analyzing the full-mission +light curves for the entire LCR sample, the LCR sample +will not be updated for each catalog sub-release (e.g., the +newly available 4FGL-DR3 (Abdollahi et al. 2022b) and +the upcoming 4FGL-DR4). Currently, the LCR does not +provide permanent identifiers that allow distinguishing +between from different versions due to data reprocess- +ing. This will be implemented in a future version of the +repository. +In conclusion, in this era of large surveys, the Fermi- +LAT is the only high-energy gamma-ray observatory +to continuously monitor variable sources, providing the +all-sky coverage to identify gamma-ray counterparts to +transient events at other wavelengths. We expect that +the LCR will greatly enhance the usefulness of LAT +data to the time-domain, multi-messenger and multi- +wavelength communities. +ACKNOWLEDGMENTS +MN and JV acknowledge that the material is based +upon work supported by NASA under award number +80GSFC21M0002. +DK and MN acknowledge support +to this work from NASA Fermi GI Program under +grant number 80NSSC23K0242. +AB is supported by +the NASA Postdoctoral Program at NASA Goddard +Space Flight Center, administered by Oak Ridge As- +sociated Universities. +INFN and ASI personnel per- +formed in part under ASI-INFN Agreements No. 2021- +43-HH.0. The Fermi LAT Collaboration acknowledges +generous ongoing support from a number of agencies +and institutes that have supported both the develop- +ment and the operation of the LAT as well as scientific +data analysis. These include the National Aeronautics +and Space Administration and the Department of En- +ergy in the United States, the Commissariat `a l’Energie +Atomique and the Centre National de la Recherche Sci- +entifique / Institut National de Physique Nucl´eaire et de +Physique des Particules in France, the Agenzia Spaziale +Italiana and the Istituto Nazionale di Fisica Nucleare in +Italy, the Ministry of Education, Culture, Sports, Sci- +ence and Technology (MEXT), High Energy Accelerator +Research Organization (KEK) and Japan Aerospace Ex- +ploration Agency (JAXA) in Japan, and the K. A. Wal- +lenberg Foundation, the Swedish Research Council and +the Swedish National Space Board in Sweden. +Additional support for science analysis during the op- +erations phase is gratefully acknowledged from the Is- +tituto Nazionale di Astrofisica in Italy and the Cen- +tre National d’´Etudes Spatiales in France. This work +performed in part under DOE Contract DE-AC02- +76SF00515. +REFERENCES +Aartsen, M. G., Abraham, K., Ackermann, M., et al. 2017, +ApJ, 835, 45, doi: 10.3847/1538-4357/835/1/45 +Aartsen, M. G., et al. 2018a, Science, 361, eaat1378, +doi: 10.1126/science.aat1378 +—. 2018b, Science, 361, 147, doi: 10.1126/science.aat2890 +Abbasi, R., Ackermann, M., Adams, J., et al. 2022, ApJ, +928, 50, doi: 10.3847/1538-4357/ac4d29 +Abdo, A. 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A., Ackermann, M., et al. 2009, +The Astrophysical Journal, 697, 1071, +doi: 10.1088/0004-637x/697/2/1071 +Ballet, J., Burnett, T. H., Digel, S. W., & Lott, B. 2020, +arXiv, doi: 10.48550/ARXIV.2005.11208 +Bruel, P., Burnett, T. H., Digel, S. W., et al. 2018, arXiv +e-prints, arXiv:1810.11394. +https://arxiv.org/abs/1810.11394 +G´orski, K. M., Hivon, E., Banday, A. J., et al. 2005, ApJ, +622, 759, doi: 10.1086/427976 +Hayashida, M., Madejski, G. M., Nalewajko, K., et al. 2012, +ApJ, 754, 114, doi: 10.1088/0004-637X/754/2/114 +Hayashida, M., Nalewajko, K., Madejski, G. M., et al. 2015, +ApJ, 807, 79, doi: 10.1088/0004-637X/807/1/79 +James, F. 1994 +Kocevski, D., et al. 2021, The Astronomer’s Telegram. +https://www.astronomerstelegram.org/?read=15110 +Liodakis, I., Romani, R. W., Filippenko, A. V., Kocevski, +D., & Zheng, W. 2019, ApJ, 880, 32, +doi: 10.3847/1538-4357/ab26b7 +Mattox, J. R., Bertsch, D. L., Chiang, J., et al. 1996, ApJ, +461, 396, doi: 10.1086/177068 +Neyman, J., & Pearson, E. S. 1928, Biometrika, 20A, 175 +Wilks, S. S. 1938, Ann. Math. Statist., 9, 60, +doi: 10.1214/aoms/1177732360 + +The Fermi-LAT Light Curve Repository +13 +APPENDIX +A. QUICK USER GUIDE +In this appendix we provide a comprehensive list of the main features of the LCR website13. +• The main page of the website features an interactive Catalog Map plotting the positions of all 4FGL-DR2 +sources. Optionally, additional data may be overlaid on the map , e.g., real time Sun or Moon position, positions +of IceCube neutrino alerts, and GRB error circles as reported in the Second LAT GRB catalog (Ajello et al. +2019). The 1525 LCR sources are highlighted in dark gray, while by default the non-variable 4FGL-DR2 sources +are marked in light gray. Hovering over any source displays a tooltip box showing its name and key characteristics +as well as linking to its 4FGL light curve and spectrum, related FAVA entry, and LCR light curve if applicable14. +Below the map, a table is shown listing the 4FGL sources and important parameters. Clicking the name of a +source included in the LCR opens a separate page dedicated to that source. +• A Map Options menu provides numerous options related to the display of the Catalog Map. Options are provided +to change the coordinate system and celestial projection. The marker label and color can be changed, as can the +meaning of its size to indicate the variability index, average significance, or time-resolved significance in 3-day +bins. +• A Catalog Search toolbox allows the user to search for a specific source by name or Right Ascension and +Declination. The results of the search are highlighted in the Catalog Map. Clicking on the linked name of +the target source opens a dedicated page for the source in a new tab. +• A Data Overlays toolbox allows the user to visualize a number of additional catalog overlays in the sky map. +These catalogs include: the Fermi-LAT Gamma-ray Burst Catalog (2FLGC; Ajello et al. 2019), the IceCube +Neutrino Alerts15, and the FAVA Flare Catalog (2FAV; Abdollahi et al. 2017). Activating any of the catalogs +will also add the related table under the map. +• The dedicated page for each source shows an interactive light curve displaying the detections and upper limits for +that source. Options are provided to show the 3 day, 7 day (1 week), or 30 day (1 month) cadence light curve. By +default only the significant flux points (with TS≥4) are shown, while upper limits are shown for less-significant +time bins. However the user can choose to change the minimum detection significance through the drop-down +menu, by selecting among the available options (TSmin=4,3,2,1). The Spectral Fitting option allows the user to +choose to visualize either the best-fit values obtained with the fixed spectral index fit or the ones resulting from +the fit with spectral index free to vary. A table listing the main characteristics of the selected source is provided. +Additional information about the fit convergence, fit tolerance and detection ratio are reported for diagnostics +purposes below the light curve plot. +• A Data Download toolbox on each dedicated source page provides all data for that source for download. The +data are provided in CSV and JSON formats16. All data points are provided, potentially including unconstrained +and possibly TS< 0 data points, or from analyses that did not converge. Guidelines for cleaning the data before +use in analysis are given in Sec. 4. +• Finally, the LCR contains a Usage Notes page which reviews the data analysis and modeling details, fitting +strategy, and caveats for usage (see Sec. 4). +13 https://fermi.gsfc.nasa.gov/ssc/data/access/lat/ +LightCurveRepository/ +14 This option can be disabled by unchecking the Source info option +from the Map Options menu. +15 https://gcn.gsfc.nasa.gov/amon.html +16 A description of the file formats can be found at https:// +fermi.gsfc.nasa.gov/ssc/data/access/lat/LightCurveRepository/ +table description.html + diff --git a/iNAzT4oBgHgl3EQfo_3Q/content/tmp_files/load_file.txt b/iNAzT4oBgHgl3EQfo_3Q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ca826e13617dc4a750da9ea8e8e62d0871de63cb --- /dev/null +++ b/iNAzT4oBgHgl3EQfo_3Q/content/tmp_files/load_file.txt @@ -0,0 +1,940 @@ 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Bruel,17 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Burns,18 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Caputo,16 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Caraveo,19 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Cibrario,13, 14 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Ciprini,20, 21 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Cristarella Orestano,10, 22 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Cutini,22 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' D’Ammando,23 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' De Gaetano,12, 11 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Digel,24 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Di Lalla,24 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Di Venere,11, 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Dom´ınguez,25 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Ferrara,16, 26, 27 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Fiori,3 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Fukazawa,28 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Fusco,11, 12 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Gammaldi,29 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Gargano,12 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Garrappa ,30 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Gasbarra,20, 31 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Gasparrini,20, 21 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Giglietto,11, 12 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Giordano,11, 12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Giroletti,23 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Green,32 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Grenier,33 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Guiriec,34, 16 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Gustafsson,35 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Horan,17 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Hou,36, 37 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' J´ohannesson,38, 39 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Kerr,40 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Kocevski ,41 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Kuss,9 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Latronico,13 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Li,42, 43 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Liodakis,44 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Longo,45, 46 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Loparco,11, 12 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Lorusso,11, 12 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Lott,47 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Lovellette,48 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Lubrano,22 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Maldera,13 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Manfreda,3 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Mart´ı-Devesa,49 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Mazziotta,12 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='Mereu,10, 22 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Michelson,24 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Mizuno,50 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Monzani,24, 51 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Morselli,20 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Moskalenko,24 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Negro ,52, 16, 53 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Omodei,24 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Orlando,54, 24 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Ormes,55 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Panzarini,11, 12 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Perkins,16 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Persic,46, 56 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Pesce-Rollins,9 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Pillera,11, 12 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Porter,24 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Principe,45, 46, 23 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Rain`o,11, 12 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Rando,6, 5, 7 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Rani,57, 16, 58 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Razzano,3 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Razzaque,59, 34 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Reimer,49 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Reimer,49 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' S´anchez-Conde,60, 29 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Saz Parkinson,61 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Serini,12 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Sgr`o,9 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Siskind,62 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Spandre,9 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Spinelli,11, 12 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Suson,63 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Tajima,64, 24 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Thompson,16 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Torres,65, 66 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Valverde ,67, 16 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Wadiasingh,16 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Wagner,68, 69 and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Wood70 1IRAP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Universit´e de Toulouse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' UPS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' CNES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' F-31028 Toulouse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' France 2Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Clemson University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Kinard Lab of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Clemson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' SC 29634-0978,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' USA 3Universit`a di Pisa and Istituto Nazionale di Fisica Nucleare,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Sezione di Pisa I-56127 Pisa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Italy 4Universit´e Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Universit´e Paris Cit´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' CEA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' AIM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' F-91191 Gif-sur-Yvette Cedex,' metadata={'source': 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Higashi-Hiroshima,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Hiroshima 739-8526,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Japan mnegro1@umbc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='edu janeth@umbc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='edu daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='kocevski@nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gov aryeh.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Universidad Aut´onoma de Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 28049 Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Spain 30Ruhr University Bochum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Faculty of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Astronomical Institute (AIRUB),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 44780 Bochum,' metadata={'source': 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University Northwest, Hammond, IN 46323, USA 64Solar-Terrestrial Environment Laboratory, Nagoya University, Nagoya 464-8601, Japan 65Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Magrans s/n, E-08193 Barcelona, Spain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' and Institut d’Estudis Espacials de Catalunya (IEEC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' E-08034 Barcelona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Spain 66Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' E-08010 Barcelona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Spain 67Department of Physics and Center for Space Sciences and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' University of Maryland Baltimore County,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Baltimore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' MD 21250,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' USA 68Institute for Theoretical Physics and Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Universit¨at W¨urzburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' D-97074 W¨urzburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Germany 69Kavli Institute for Particle Astrophysics and Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Stanford University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Stanford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' CA 94305,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' USA 70Praxis Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=', Alexandria, VA 22303, resident at Naval Research Laboratory, Washington, DC 20375, USA ABSTRACT The Fermi Large Area Telescope (LAT) light curve repository (LCR) is a publicly available, continu- ally updated library of gamma-ray light curves of variable Fermi-LAT sources generated over multiple timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The Fermi-LAT LCR aims to provide publication-quality light curves binned on timescales of 3 days, 7 days, and 30 days for 1525 sources deemed variable in the source catalog of the first 10 years of Fermi-LAT observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The repository consists of light curves generated through full likelihood analyses that model the sources and the surrounding region, providing fluxes and photon The Fermi-LAT Light Curve Repository 3 indices for each time bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The LCR is intended as a resource for the time-domain and multi-messenger communities by allowing users to quickly search LAT data to identify correlated variability and flaring emission episodes from gamma-ray sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' We describe the sample selection and analysis employed by the LCR and provide an overview of the associated data access portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Keywords: Fermi-LAT, gamma-rays, time domain, active galaxies 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' INTRODUCTION Study of variability of astronomical objects has led to many discoveries in modern astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Identified as one of the central themes of the 2020 Decadal Review, the in- creasing realization of time-domain and multi-messenger astronomy is now opening an entirely new window on the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' A high duty cycle and long-term monitoring of the gamma-ray sky has made the Fermi Large Area Tele- scope (LAT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Atwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2009) a pivotal tool in the study of time-domain and multi-messenger astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' More than a decade of LAT observations has provided the identification and regular monitoring of thousands of transient, variable, and steady-state sources (Abdollahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The long-term monitoring of the gamma- ray sky by the LAT played a crucial role in the first as- sociation of a high-energy neutrino detected by the Ice- Cube neutrino observatory and the flaring blazar TXS 0506+056 (Aartsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2018a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The TXS 0506+056 association provided the first tan- talizing clue to the origin of the high-energy cosmic neu- trino flux detected by the IceCube neutrino observa- tory (Abbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Though blazars have long been suggested as a possible source of extragalactic high-energy neutrinos, constraints on neutrino emission from bright gamma-ray blazars have largely disfavored steady-state active galactic nuclei (AGN) as the pri- mary source of the observed neutrino flux (Aartsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The TXS 0506+056 association, however, has shown that at least some of the astrophysical neutrinos detected by IceCube could be attributed to high fluence AGN that undergo periods of intense flaring activity on top of a much lower quiescent gamma-ray emission that may be undetectable to the LAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The long-term monitoring provided by the LAT has also been indispensable to multi-wavelength campaigns that aim to study long-term correlated variability in AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Observations of radio and optical flares have been used to examine the location of the gamma-ray emitting region, as well as the particle acceleration and radiation processes in the relativistic jets in these sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' For example, the long-term correlated variability between optical and gamma-ray flares has allowed for the de- tection of a systematic lag in the optical-to-gamma-ray flares in 3C 279 (Hayashida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2012, 2015), the evi- dence of quasi-periodic variations in the BL Lac object PG 1553+113 (Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2015), and constraints on the rate of orphan optical flares with no gamma-ray counterpart (Liodakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Such studies of cor- related multi-wavelength variability are a crucial diag- nostic of the physics of relativistic jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Despite the importance of long-term variability stud- ies of LAT sources, very few existing resources enable the community to easily access the whole-mission data to perform correlative analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Resources like the Fermi All-Sky Variability Analysis (FAVA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Abdollahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2017) and the 3FGL (Acero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2015) monthly aper- ture photometry light curves1 allow users to quickly ex- amine a source for relative flux increases, but do not provide flux calibrated characterization of a source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Do- ing so requires a full likelihood analysis of the region that takes into account the flux variations of all nearby sources, and generating a high cadence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=', daily) light curve using a full likelihood treatment over the entire duration of the mission can be very time consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' To address this need, we developed the Fermi-LAT Light Curve Repository (LCR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Kocevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2021)2, consisting of a public database of light curves for vari- able Fermi-LAT sources on a variety of timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The repository provides publication-quality light curves on timescales of 3 days, 7 days (weekly), and 30 days (monthly) for 1525 sources deemed variable in the 4FGL-DR2 catalog (Ballet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The repository consists of light curves generated through full likelihood analyses of the sources and surrounding regions, provid- ing flux and spectral index measurements for each time bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Hosted at NASA’s Fermi Science Support Center (FSSC), the LCR provides users with on-demand access to this light curve data, which is continually updated as new data becomes available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The repository is a new resource to the time-domain and multi-messenger com- munities for associating and monitoring LAT sources, in particular facilitating identification of time intervals with high fluence or flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 1 https://fermi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gov/ssc/data/access/lat/4yr catalog/ ap lcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='php 2 https://fermi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gov/ssc/data/access/lat/ LightCurveRepository/ 4 The Fermi-LAT Collaboration Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Top: Histogram of the source types included in the LCR population, defined as having a variability index above 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='67, corresponds to <1% chance of steady emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Center and bottom: The population of sources included in the LCR (empty markers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In grey we report the 4FGL-DR2 population not included in the LCR as reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The energy range of integration is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='1−100 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2 we describe the gamma-ray sources included in the LCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 3 is devoted to the description and discussion of the automated data analysis employed by the reposi- tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' We discuss analysis caveats and data usage best practices in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' We briefly summarize and conclude in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' We provide a Quick Guide in Appendix A with the aim to help familiarize users with the LCR data portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' THE SOURCE SAMPLE Motivated by the science described above, the LCR focuses on the sources in the 4FGL-DR2 catalog that have variability indices greater than 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='67, where the variability index can be thought of as a proxy for the av- erage fractional variability δF/F, with δF measured on timescales of 1 year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' As defined in the 4FGL-DR2 cat- alog, which is based on 10 years of survey data, sources with such a variability index over 10 years are estimated to have a less than 1% chance of being steady.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The re- sulting sample consists of 1525 sources, or roughly 26% of the 4FGL-DR2 catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The vast majority of these sources is blazars, further classified as flat spectrum ra- dio quasars (FSRQ), BL Lacs (BLL), and blazar candi- dates of unknown type (BCU), making up roughly 38%, 31%, and 24% of the repository sample, respectively, or 77%, 36% and 26% of their respective 4FGL-DR2 class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' This is consistent with the fact that the LAT class with the largest number of variable sources are FSRQs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' We also count 9 pulsars (PSR) and several sources of other kinds, as illustrated in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In the middle and bottom panels of the same figure, we show plots of the spectral shape parameters (power-law index and total energy flux) and variability parameters (vari- ability index and fractional variability from DR2) for the sources included in the LCR (colored points) overplotted on the full 4FGL population (grey points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The former shows that most of the sources not included are fainter with harder spectra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=', mainly BL Lac objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' This is because, as can be seen in the bottom panel of the same figure, BL Lac objects are intrinsically less variable than FSRQs in the Fermi-LAT energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' AUTOMATED DATA ANALYSIS Generating 3-day, 7-day, and 30-day light curves for each of these sources for over 13 years of data requires the analysis of over 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='7 million individual time bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The LCR analysis pipeline runs on a cluster hosted at SLAC National Accelerator Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Generating the light curves over the entire mission to date for the in- cluded sources requires approximately 3 months (>400 core years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In the following subsections we present the details of the analysis and computational strategies used to gen- erate the LCR data products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Analysis Technique & Tools The characterization of LAT sources is typically per- formed using a maximum likelihood analysis (Abdo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2009), in which the parameters of a model de- scribing the point sources and diffuse isotropic gamma- ray emission in a given region of the sky are jointly opti- mized to best describe the observed photon distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The light curves of the LCR are obtained by performing an unbinned likelihood analysis, in which the full spatial and spectral information of each photon is used in the maximum likelihood optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Number of sources 102 TTTT 101 100 NLS10-8 口 BLL Not in LCR FSRQ Other classes in LCR BCU 10-9 口 10-10 10-11 TTTT 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='5 Power-law index (r)Fractional variability 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 10-1 BLL 口 Not in LCR FSRQ Other classes in LCR BCU 101 102 103 104 105 Variability indexThe Fermi-LAT Light Curve Repository 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' A model map, with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='1 degree resolution, for a single weekly time bin of the region surrounding FSRQ 4C +28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' This source contains 15 other variable sources within a 12◦ radius, highlighting the need to model all vari- able sources within the ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The LCR analysis is performed with the standard LAT Fermitools3 (version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='5) using the P8R3 SOURCE V2 instrument response functions on P8R3 SOURCE class (Atwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Bruel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2018) photons selected over the energy range covering 100 MeV–100 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Note that the energy dispersion is neglected in the unbinned analysis mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' This is not expected to impact the qual- ity of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' However, repeating the analysis with a binned likelihood approach would return somewhat (mostly non significant) results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' For each source and time bin, photons are selected from a circular region of interest (ROI) of radius 12◦ centered on the location of the target source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The ROIs are analyzed separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' By contrast, for the standard FGL catalogs, fluxes and spec- tral parameters of multiple sources are extracted from the same ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The size of the ROI is conservatively cho- sen to be as large as the 95% containment radius of the LAT energy-dependent point-spread function (PSF) at 100 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Additional data selection cuts are imposed to exclude photons associated with regions and periods of known solar flares and gamma-ray bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' A zenith an- gle limit of 90◦ to strongly reduce contamination from gamma rays produced through interactions of cosmic rays with Earth’s atmosphere (Earth limb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Following the 4FGL-DR2 catalog, the LCR sources can have one of three different spectral types: 3 https://fermi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gov/ssc/data/analysis/software/ Power-law (PL): dN/dE = N0(E/E0)−Γ (1) log-parabola (LP): dN/dE = N0(E/Eb)−(α+β log(E/Eb)) (2) and subexponentially cutoff power-law (PLEC): dN/dE = N0(E/E0)−γ1e−aEγ2 (3) A majority of LCR sources are best described by a PL or LP spectral shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The PLEC spectral shape best represents 11 additional sources, comprising the 9 pul- sars in the LCR sample and the two brightest FSRQs (CTA 102 and 3C 454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The normalization of the source spectrum in the model is left free to vary, while the spectral shape pa- rameters are initially fixed to their 4FGL-DR2 cata- log values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The model also includes all gamma-ray sources in the 4FGL-DR2 catalog within a radius of 30◦ from the ROI center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The normalization of each variable source in the ROI is also left free to vary in the model, with spectral shape parameters fixed to their catalog values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In addition to the point sources, Galactic and isotropic background components are included in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The Galactic component, gll iem v07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='fits4, is a spatial and spectral template that accounts for interstellar diffuse gamma-ray emis- sion from the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The isotropic component, iso P8R3 SOURCE V3 v1, provides a spectral template to model the remaining isotropic events, including contri- butions from the residual charged-particle background and the isotropic celestial gamma-ray emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The normalizations of both the Galactic and isotropic com- ponents are allowed to vary during the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The free pa- rameters of the model are varied to maximize the likeli- hood of observing the data given the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' An iterative fitting strategy, which varies the required fit tolerance5 over three steps (1, 10−4 and 10−8), is employed to min- imize the number of time bins in which the likelihood fit does not successfully converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Once fit convergence is achieved with the tightest tolerance, a second round of 4 https://fermi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gov/ssc/data/access/lat/ BackgroundModels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='html 5 The fit tolerance is the relative convergence tolerance that is specified in the gtlike routine and passed to the optimization algorithm used to maximize the log likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The fit tolerance can be loosened to help achieve fit convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' On the other hand, fits with tighter fit tolerances exhibit overall lower fractional errors on the resulting flux estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Therefore an iterative approach has been adopted for the LCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 101 +40° 4FGLJ023 4FGLJ0252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='9+3834 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='9+3643 Counts (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='1-100 GeV) 4FGL 4FGL J0221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='1+3556 +32° 4FGLJ0324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='8+3412 4FGL 0159 4FGLJ0253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='5+3216 4FGL Dec (2000) 3042 100 4FGL +2848 4FGL +24° 4FGLJ0245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='4+2408 4FGLJ0258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='1+2030 +16° 4FGL10224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='9+1843 +8° 10-1 56° 48° 40° 32° 24° RA (J2000)6 The Fermi-LAT Collaboration Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' An example 3-day light curve as can be found in the LCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' It spans more than 11 years of LAT data and refers to the source at the center of Figure 2, FSRQ 4C +28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' This light curve is obtained by specifying the analysis option that the spectral index of the source is free to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The data gap in 2018 is due to the temporary shut down of the instrument because of a solar panel anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' fitting is performed in which a spectral shape parameter of the target source is allowed to vary, namely, photon index (Γ) for PL, α for LP, and γ1 for the PLEC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' All other parameters remain fixed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=', β for LP, and a and γ2 for the PLEC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Figure 2 shows a model map for a single time bin of the region surrounding FSRQ 4C +28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='07 (4FGL J0237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='8+2848), which contains 15 other variable sources within a 12◦ radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Figure 3 shows the resulting 3-day light curve spanning more than 11 years of LAT data for 4FGL J0237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='8+2848.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' A likelihood ratio test (Neyman & Pearson 1928) is used to quantify the significance of the target source above the background;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' specifically the test statistic (Mattox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 1996): TS = −2 log(L0/L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' (4) The TS compares the maximum value of the likelihood function L0 evaluated for the parameter values that maximize the likelihood under a background-only null hypothesis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=', a model that does not include a tar- get source), with L, the likelihood function evaluated at the best-fit model parameters when including the target source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In the null hypothesis, TS is distributed ap- proximately as χ2 (Wilks 1938), and the analysis rejects the null hypothesis when the test statistic is greater than TS ≥ 4, which is roughly equivalent to a 2 σ rejection cri- terion for a single degree of freedom6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Using this thresh- old value of TS as the detection criterion, the LCR em- 6 The null hypothesis is tested against the presence of one source at a known position (at the center of the ROI) ploys a Bayesian profile likelihood method7 to calculate the 95% confidence level upper limits for any interval that yields a TS ≤ 4, and also extracting a flux estima- tion of the target source for any interval with a TS ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The resulting best-fit values (or upper limits) for photon flux (cm−2 s−1), energy flux8 (GeV cm−2 s−1), and asso- ciated spectral shape are saved to the LCR database, for both the fixed and free photon index analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' This pro- cedure ultimately allows for a user-selectable detection threshold and spectral fit method, as the flux estimates and upper limits between 1 ≤ TS ≤ 4 are both recorded in the LCR database, as well as the results from both the fixed and free spectral analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 4 we compare, for the case of the bright quasar Ton 599, the best-fit flux values found using a fixed spec- tral index vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' leaving the spectral index free to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The colors in the plot refer to the three cadences analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The results are generally stable, with greater uncertain- ties for lower fluxes for higher cadences due to lower statistics in the ROI in each time bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Computational Strategy For analyses of a relatively small number (a few thou- sands) of photons, an unbinned likelihood analysis can be performed rapidly (a few minutes), but as the num- ber of events increases, the time to perform the analysis can become prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' This limitation becomes in- creasingly burdensome when the need arises to perform 7 A description of the Bayesian profile likelihood method employed by the LCR can be found at https://fermi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gov/ssc/ data/p7rep/analysis/scitools/python tutorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 8 The energy flux is obtained from the photon differential flux dN dE as � E dN dE dE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='00e-6 三 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='50e-6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='00e-6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='00e-7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='00e+0 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 Date (UTC)The Fermi-LAT Light Curve Repository 7 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Comparison of the fluxes [ph/cm2/s] in time bins obtained with the fitting pipeline keeping the spectral index fixed versus letting the spectral index be free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Empty circles indicate results from analyses that did not converge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=', MINUIT Return Code q ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' MINUIT (James 1994) is the optimizer adopted in the likelihood analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The plot shows the fluxes measured for the quasar Ton 599.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In this case, the fit for the highest-flux point (with TS∼ 4) converged when the spectral index was set free, but did not converge when the spectral index was fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' a source characterization over a large number of time bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' A binned likelihood analysis could alleviate this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' However, information is lost when binning data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The LCR tackles the computational overhead by paral- lelizing the process of performing a full unbinned likeli- hood analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In order to produce a high cadence light curve over the entire lifetime of the mission in a reason- able amount of time, the LCR distributes the analyses of the light curve bins to separate nodes in a computer clus- ter hosted at the SLAC National Accelerator Laboratory and utilizing the IBM Spectrum LSF workload manage- ment platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The parallelization allows for thousands of time bins to be analyzed simultaneously, with the net effect of drastically reducing the time to generate light curves over the entire duration of the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Exposure analysis The Fermi spacecraft has been executing a sky- scanning strategy for the LAT for the great majority of the mission, generally reaching an almost uniform full- sky coverage daily, in fact, every two orbits (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=', ap- proximately every three hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' During the mission the scanning strategy has been interrupted occasionally for targeted observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The longest-duration such obser- vation was a modified observing strategy executed from 2013 December to 2015 July that favored the Galactic center region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In March 2018, the seizing of the drive motor for one of the solar panels forced the temporary shut down of the LAT and a redefinition of the survey mode9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' This caused the exposure to be limited to a part of the sky for some period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The effects of these periods of uneven exposure can be seen in the 3-day light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Therefore we quantify and discuss these effects in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The number of photons in a given ROI is modulated by the exposure of the observation, which is given by the product of the LAT livetime (during the brief readout time of a photon or cosmic-ray interaction the LAT, the instrument is ‘dead’ to triggering on other interactions) and the energy and angle-dependent effective area of the LAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In general, analyses integrating over long ob- servation times can have small fractional exposure vari- ations across an ROI, but shorter-timescale analysis can be significantly impacted by low exposure for a particu- lar region of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In fact, since the beginning of the post-anomaly modification of the sky-scanning strategy in February 2019, a number of the 3-day and 7-day ca- dence time bins have no exposure in particular portions of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' This is due to the constraints on the direc- tion of the zenith of spacecraft with respect to the Sun10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 5 shows the range of values of the LAT exposure in the sky for each time bin of 3-day cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The maps in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 6 illustrate the positions of all the sources ana- lyzed in the LCR, overlaid on the counts map (top) and an exposure map (bottom) for an typical 3-day cadence time bin with a zero minimum exposure: some sources fall in the region of the sky with zero exposure, which naturally translates into zero events to analyze in the ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In these cases, or in those with minimum exposure orders of magnitude lower than the maximum (shaded pink regions in the maps in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 6), the pipeline typically returns an upper limit on the source flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' However, in some cases it could still find a solution that maximizes the likelihood, but warns of a poor fit quality or an un- reasonably low error value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' For this reason, we recom- mend LCR users use caution when using the 3-day and 7-day cadence light curve for these low-exposure inter- vals, or exclude the affected time bins entirely from their analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' To illustrate the effect of low exposure on the photon counts, we generated binned all-sky counts and related exposure maps in the same time bins used for the LCR data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' We consider events in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='1–100 GeV 9 https://fermi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gov/ssc/observations/types/ post anomaly/ 10 Note that the ‘gaps’ in the exposure for different time bins are not always in the same portion of the sky 4FGL J1159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='5+2914 LCR, daily LCR, daily, q±0 LCR, weekly LCR, weekly, q± 0 LCR, monthly8 The Fermi-LAT Collaboration Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The gray band marks the range of LAT on-sky exposures for each time bin of 3-day cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The pink bands, in order, mark time ranges of the original rocking angle mission profile(†), the Galactic center monitoring, and the time gap between the solar panel anomaly and the beginning of the new survey mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Note that some time bins have a minimum exposure value of zero: this means that some part of the sky was not observed during that time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The exposure maps are computed at the central energy in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='1–100 GeV band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' (†) Details on the Observatory sky-survey profiles can be found at https://fermi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gov/ssc/observations/types/allsky/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' energy range, and the maps are produced in HEALPix11 format with the tool gtexpcube212, using the same anal- ysis setup and IRFs as for the automated likelihood anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The HEALPix format allows us to easily extract the average exposure in every ROI, and the pixel res- olution (HEALPix order 6) matches the resolution of the pre-computed livetime cubes provided for LAT data analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' For each source considered in the LCR and each time bin of 3-day and 7-day cadences, we extract the average exposure and the total photon counts in the ROI cen- tered on the source between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='1 and 100 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 7 we illustrate the statistics of sources with a given num- ber of time bins in the light curve that had no photons in the ROI or had fewer than 20 photons (considered as an arbitrary threshold for low-statistics analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Hun- dreds of sources have up to 30 time bins with zero events in the light curve and/or less than 20 events, while tens of sources have fewer than 20 photons in more than 50 time bins (note that the total number of time bins in 11 http://healpix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='jpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gov/ (G´orski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2005) 12 The Fermitools, as part of the maximum likelihood calculation, automatically account for the exposure, as describet on the of- ficial mission web page at https://fermi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gov/ssc/data/ analysis/documentation/Cicerone/Cicerone Data Exploration/ livetime and exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The module gtexpmap is used to compute the exposure when performing unbinned analyses, as for the in the LCR analysis pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In this section, we compute the exposure through the gtexpcube2 module, which is used when performing binned likelihood analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' We stress that this is for purely illustrative purposes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' the exposure maps computed in this section were not used in the LCR likelihood analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' a 3-day cadence light curve is more than 1680 as of now).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The monthly cadence, due to the longer inte- gration time, should not be affected by this issue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' how- ever, we still recommend that users check the exposure of each bin used in their analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The exposures within the 12◦ radius ROIs for each source for the 3-day and 7-day cadences are provided in the LCR downloadable data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' USAGE CAVEATS In this section we provide a number of caveats to be mindful of when using the LCR data for scientific re- search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' This list is also available and will be updated periodically on the LCR data portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The LCR provides fit results from likelihood anal- yses that both did and did not converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' How- ever, it is important that the end user is aware that results from analyses that did not converge should be considered suspect and not be used for higher-level analyses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=', multi-frequency cross- correlation, power spectral density, or studies of variability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The convergence status for a partic- ular time bin is recorded in the MINUIT Return Code parameter and non-convergent analyses are hidden by default, but are optionally accessible to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' A review of the current version of the LCR fit results for the first fourteen years of mis- sion data shows that the analyses for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='7% of time bins, for all sources, did not converge when the spectral indices were held fixed, and that ∼35% did not converge when the spectral indices were free to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 108 [cm 107 OLD ROCKING 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' GC POINTING SOLAR PANEL ANOMALY 3-day exposure range in the sky 3 4 5 Time [MET] 1e8The Fermi-LAT Light Curve Repository 9 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Top: Counts map for a 3-day time bin with 0 min- imum exposure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Bottom: Exposure map for the same time bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The white pixels mark the positions of LCR sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The light blue and magenta circles in the maps highlight the cases of two ROIs lying in a region of the sky with zero and non-flat exposures, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' From its definition, equation 4, the TS is a man- ifestly positive quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' However, negative TS values can sometimes be obtained when the pa- rameters reach the limits of their allowed intervals without having maximized the likelihood profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Fit results obtained from intervals that resulted in negative TS values should be considered suspect and not used in higher-level analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The cur- rent version of the LCR data products for the first fourteen years of the mission for all LCR sources have only a few bins with negative TS results, per cadence, for both the fixed and free fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' While time intervals containing gamma-ray bursts (GRBs) and solar flares have been removed from the LAT data prior to the likelihood analyses, pos- sible contamination by the proximity of the qui- escent Sun has not been accounted for, nor have those time ranges been excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The angular sep- aration of the Sun from the target source is pro- vided for each time bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' A total of 175 GRBs and 266 solar flare time intervals were excluded from the data prior to performing the maximum like- Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In light blue, we show the distribution of LCR sources by number of time bins with zero photons within the ROI used for their analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In magenta, we show the distribution of LCR sources according the number of time bins with fewer than 20 photons within the ROI used for their analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Darker shades of the lines refer to the 3-day cadence, while lighter shades refer to the weekly cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The low-exposure time bins represent the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='1% of the total time bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' lihood analysis for the LCR first fourteen years of light curve data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' This affects less than about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='01% of the time bins for all the LCR sources per cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Because the LCR analyses are made available in real-time (new analysis results are generally made available within 24 hours of being processed), the results are not validated by the LAT Collaboration prior to release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Users are encouraged to perform sanity checks by examining the ratio of flux to flux uncertainty vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' the square root of the TS, and the distributions of fit results, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=', flux, flux un- certainties, spectral indices mean values (photon index Γ for PL, α for LP, and γ1 for the PLEC model) and their uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Some examples of these are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 8 for the specific case of the FSRQ 4C +28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='07 (4FGL J0237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='8+2848).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The flux over the flux uncertainty ratio is expected to be approximately proportional to the square root of the TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Any outliers should be either further investigated or removed before using the data for higher-level analyses, as should any extreme out- liers from the data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The free-spectral-index light curves provided by the LCR were produced using a model of the per- tinent region of the sky for which only the spectral index of the target source is set free, and those of Exp0sure map △T=491788801-492048001 MET 0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='21022e+083-day Counts < 20 3-day Counts = 0 Weekly Counts < 20 Weekly Counts = 0 Number of sources 102 101 0 5 10 15 20 25 30 35 Number of binsCounts map △T=491788801-492048001 MET 0 111510 The Fermi-LAT Collaboration Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Example of validation plots for the FSRQ 4C +28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The top panels show the case for which the spectral index (α for this source) is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In all the plots we are considering the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='1–100 GeV energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The bottom panels show the case for which the spectral index is free to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The middle panels show the distributions of flux statistical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Flux distributions are shown in the right panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In this example, distributions are good except for the outliers highlighted within green squares, which should be either further investigated or removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In the histograms, dashed lines represent results from all time bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Solid lines represent bins that did not converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Notice that outliers are not necessarily the results from analyses that did not converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' all the other sources were fixed to the 4FGL-DR2 catalog values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Therefore, contamination induced by possible changes in the spectral indices of the sources surrounding the target are not taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' For instance, sources undergoing bright flares have been seen to also experience dramatic changes in their spectral indices, including changes in the curvature, at the same time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=', harder- when-brighter behavior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Therefore, bright, vari- able sources in the ROI can induce this type of con- tamination and must be considered when a target source is in close proximity to any bright variable sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Some erroneously high flux values during periods of zero or low exposure, often associated to a very small (or zero) error, have been found for several sources (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' We recommend check- ing the exposure value indicated for each time bin and source ROI in the provided tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' If the light curve has fluxes with very small fractional uncer- tainties, rather than flux upper limit, in the time bins with zero or low exposure, the reported fluxes and uncertainties should be considered unreliable and those time bins should be excluded from con- sideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Time bins with zero error on the flux estimations are automatically hidden but are op- tionally available to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' A bug in the make4FGLxml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='py tool that the LCR uses to generate the ROI models was re- cently identified, in which the reported exten- sion of the extended sources with RadialGauss spatial profile is the 68% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' value instead of the sigma value used in the XML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Thir- teen 4FGL-DR2 sources have this spatial profile, namely, Crab IC, IC 443, Monoceros, HESS J1303- 631, FHES J1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='0−6310, FHES J1626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='9−2431, FHES J1723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='5−0501, HESS J1825−137, W 41, Cygnus Cocoon, FHES J2129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='9+5833, FHES J2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='4+6443 and FHES J2304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='0+5406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' None of these sources is deemed variable in the 4FGL-DR2 catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' However, they are present in the ROIs of 101 LCR sources, which might result in a system- atic bias in the target flux values across the whole light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Correcting this issue will require re- processing the data for these sources, which is al- ready underway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Meanwhile users should be mind- ful of this issue when considering sources in the vicinities of these extended sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Particular care should be taken when using the light curve for the synchrotron component of the Crab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The Crab has different components treated as separate entries in the 4FGL-DR2 catalog: the extended emission, the inverse Compton emission from the PWN and the synchrotron emission from the pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The LCR analysis generates only the light curve for the synchrotron (variable) emission, 4FGL J0237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='8+2848 LCR, daily LCR, monthly LCR, weekly103 bin 102 b 102 occurrence nce occurrer 101 101 100 100 12 10 8 7 5 4 log10(oF) log10(F)uig 102 102 occurrence occurrence 101 101 100 100 10 9 7 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='4 2 8 log10(oF) log10(F)The Fermi-LAT Light Curve Repository 11 while keeping the parameters of the other two com- ponents frozen in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' This could result in some contamination of the synchrotron component light curve deriving from any unaccounted-for variabil- ity of the other Crab components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' On the other hand, the LCR approach overrides the apparent (not real) variability reported for the pulsar in 4FGL-DR2 (and all other versions of the 4FGL catalog).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' PROSPECTS AND CONCLUSIONS The development of the Fermi-LAT LCR was moti- vated by the need for a coherent collection of light curves of variable gamma-ray sources observed by the Fermi- LAT in support of the time-domain and multi-messenger communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' By continuously reporting the flux evolu- tion and transition to high-flux states for many variable sources, the LCR is a valuable resource for triggering ob- servations of other observatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Furthermore, the LCR can be used to validate the study of variable activity in neighboring faint sources, helping to identify potential contamination from flaring activity of a bright source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In this manuscript we described the automated analysis pipeline which will continuously update the repository with new data as soon as new observations by the LAT become available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' We invite the community to use the LCR data products, and report any issues or suggestions to the LCR contacts at the Fermi Science Support Cen- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' A brief guide for navigating the LCR to navigate the web site is provided in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In the future, the LCR source list and the resulting light curve data will be updated with every new Fermi- LAT source catalog, with the next LCR version planned to be released alongside the 5FGL catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Due to the computational expense of re-analyzing the full-mission light curves for the entire LCR sample, the LCR sample will not be updated for each catalog sub-release (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=', the newly available 4FGL-DR3 (Abdollahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2022b) and the upcoming 4FGL-DR4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Currently, the LCR does not provide permanent identifiers that allow distinguishing between from different versions due to data reprocess- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' This will be implemented in a future version of the repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' In conclusion, in this era of large surveys, the Fermi- LAT is the only high-energy gamma-ray observatory to continuously monitor variable sources, providing the all-sky coverage to identify gamma-ray counterparts to transient events at other wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' We expect that the LCR will greatly enhance the usefulness of LAT data to the time-domain, multi-messenger and multi- wavelength communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' ACKNOWLEDGMENTS MN and JV acknowledge that the material is based upon work supported by NASA under award number 80GSFC21M0002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' DK and MN acknowledge support to this work from NASA Fermi GI Program under grant number 80NSSC23K0242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' AB is supported by the NASA Postdoctoral Program at NASA Goddard Space Flight Center, administered by Oak Ridge As- sociated Universities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' INFN and ASI personnel per- formed in part under ASI-INFN Agreements No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2021- 43-HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The Fermi LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the develop- ment and the operation of the LAT as well as scientific data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' These include the National Aeronautics and Space Administration and the Department of En- ergy in the United States,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' the Commissariat `a l’Energie Atomique and the Centre National de la Recherche Sci- entifique / Institut National de Physique Nucl´eaire et de Physique des Particules in France,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' the Ministry of Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Culture,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Sports,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Sci- ence and Technology (MEXT),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' High Energy Accelerator Research Organization (KEK) and Japan Aerospace Ex- ploration Agency (JAXA) in Japan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' and the K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Wal- lenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Additional support for science analysis during the op- erations phase is gratefully acknowledged 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=', Filippenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=', Kocevski, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=', & Zheng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2019, ApJ, 880, 32, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='3847/1538-4357/ab26b7 Mattox, J.' metadata={'source': 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Neyman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=', & Pearson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 1928, Biometrika, 20A, 175 Wilks, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 1938, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=', 9, 60, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='1214/aoms/1177732360 The Fermi-LAT Light Curve Repository 13 APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' QUICK USER GUIDE In this appendix we provide a comprehensive list of the main features of the LCR website13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The main page of the website features an interactive Catalog Map plotting the positions of all 4FGL-DR2 sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Optionally, additional data may be overlaid on the map , e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=', real time Sun or Moon position, positions of IceCube neutrino alerts, and GRB error circles as reported in the Second LAT GRB catalog (Ajello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The 1525 LCR sources are highlighted in dark gray, while by default the non-variable 4FGL-DR2 sources are marked in light gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Hovering over any source displays a tooltip box showing its name and key characteristics as well as linking to its 4FGL light curve and spectrum, related FAVA entry, and LCR light curve if applicable14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Below the map, a table is shown listing the 4FGL sources and important parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Clicking the name of a source included in the LCR opens a separate page dedicated to that source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' A Map Options menu provides numerous options related to the display of the Catalog Map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Options are provided to change the coordinate system and celestial projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The marker label and color can be changed, as can the meaning of its size to indicate the variability index, average significance, or time-resolved significance in 3-day bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' A Catalog Search toolbox allows the user to search for a specific source by name or Right Ascension and Declination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The results of the search are highlighted in the Catalog Map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Clicking on the linked name of the target source opens a dedicated page for the source in a new tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' A Data Overlays toolbox allows the user to visualize a number of additional catalog overlays in the sky map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' These catalogs include: the Fermi-LAT Gamma-ray Burst Catalog (2FLGC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Ajello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2019), the IceCube Neutrino Alerts15, and the FAVA Flare Catalog (2FAV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Abdollahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Activating any of the catalogs will also add the related table under the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The dedicated page for each source shows an interactive light curve displaying the detections and upper limits for that source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Options are provided to show the 3 day, 7 day (1 week), or 30 day (1 month) cadence light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' By default only the significant flux points (with TS≥4) are shown, while upper limits are shown for less-significant time bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' However the user can choose to change the minimum detection significance through the drop-down menu, by selecting among the available options (TSmin=4,3,2,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The Spectral Fitting option allows the user to choose to visualize either the best-fit values obtained with the fixed spectral index fit or the ones resulting from the fit with spectral index free to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' A table listing the main characteristics of the selected source is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Additional information about the fit convergence, fit tolerance and detection ratio are reported for diagnostics purposes below the light curve plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' A Data Download toolbox on each dedicated source page provides all data for that source for download.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' The data are provided in CSV and JSON formats16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' All data points are provided, potentially including unconstrained and possibly TS< 0 data points, or from analyses that did not converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Guidelines for cleaning the data before use in analysis are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' Finally, the LCR contains a Usage Notes page which reviews the data analysis and modeling details, fitting strategy, and caveats for usage (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 13 https://fermi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gov/ssc/data/access/lat/ LightCurveRepository/ 14 This option can be disabled by unchecking the Source info option from the Map Options menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content=' 15 https://gcn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gov/amon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='html 16 A description of the file formats can be found at https:// fermi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfo_3Q/content/2301.01607v1.pdf'} +page_content='gov/ssc/data/access/lat/LightCurveRepository/ table description.' 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We consider two types of potentials: +general Cantor and general Smith-Volterra-Cantor potential. The Cantor potential is an +example of fractal potential while the Smith-Volterra-Cantor potential doesn’t belong to +the category of a fractal system. The present study brings for the first time, the study of +quantum tunneling through fractal potential in fractional quantum mechanics. We report +several new features of scattering in the domain of space fractional quantum mechanics +including the emergence of energy-band like features from these systems and extremely +sharp transmission features. Further the scaling relation of the scattering amplitude with +wave vector k is presented analytically for both types of potentials. +1e-mail address: vibhav.ecc123@gmail.com +2e-mail address: pha212475@iitd.ac.in +3e-mail address: mhasan@isro.gov.in, mohammadhasan786@gmail.com +4e-mail address: bhabani.mandal@gmail.com, bhabani@bhu.ac.in +1 +arXiv:2301.00674v1 [quant-ph] 29 Dec 2022 + +1 +Introduction +Over the last two decades, fractional dynamics have been a diverse area of research. The +concept of fractional quantum mechanics was introduced by Laskin in the year 2000 [1,2]. +The motivation behind this work was to extend the path integral (PI) formulation of +quantum mechanics (QM) [3] to the more broader class of paths. In the PI formulation +of QM, the path integrals are taken over Brownian paths which lead to the Schrodinger +equation of motion. However, the Brownian paths are the subset of a broader general +class of paths known as Levy paths characterized by a Levy index α. For α = 2, all Levy +paths are Brownian paths. When the PI formulation of QM is extended to Levy paths, +one get the fractional Schrodinger equation [1,2] and the associated quantum mechanics +is known as space fractional quantum mechanics (SFQM). A time fractional Schrodinger +equation was proposed by Naber [4]. Later Wang and Xu [5] combined the two kinds of +fractional Schrodinger equation together to construct a space-time fractional Schrodinger +equation. These generalization of QM may help to describe more extensive phenomena +of the microscopic world. +The Levy paths has fractal dimension α. +In the case of SFQM, the range of α is +1 < α ≤ 2 [1, 2]. The domain of SFQM have grown fast over the last two decades and +various applications are discussed by different authors. Some of the notable work are +the energy band structure for the periodic potential [6], position-dependent mass frac- +tional Schrodinger equation [7], fractional quantum oscillator [8], nuclear dynamics of the +H+ +2 molecular ion [9], propagation dynamics of a light beam [10], spatial soliton prop- +agation [11], solitons in the fractional Schrodinger equation with parity-time-symmetric +lattice potential [12], gap solitons [13], Rabi oscillations in a fractional Schrodinger equa- +tion [14], self-focusing and wave collapse [15], elliptic solitons [16], light propagation in +a honeycomb lattice [17], scattering features in non-Hermitian SFQM [18], tunneling +time [19, 20] etc. +Different methods are used in such studies such as domain decom- +position method [21], energy conservative difference scheme [22], conservative finite el- +ement method [23], fractional Fan sub-equation method [24], split-step Fourier spectral +method [25], transfer-matrix method [26] etc. +The term fractal was first coined by Mandelbrot [27]. Fractals are geometric objects +which have self-similarity and homogeneity at all known scales. The geometric structures +of fractals at a given scale or stage are obtained through a basic mathematical operation +acting on the geometric object known as ‘initiator’. The process of mathematical opera- +tion is called ‘generator’ which can be repeated on multiple levels. Through ‘generator’, +a geometrical object with sub-units are created that resembles the structure of the entire +object (the initiator) [28]. Due to the fact that the real numbers can be divided arbitrarily, +the self-similarity of fractals hold at all scales. Since nature has many fractal structures, +regular and irregular fragmented structures can be understood/approximated in the con- +text of fractals [27,29,30]. However, in nature, the self-similarity doesn’t hold at all scales +2 + +and in general, there exists an upper and lower limit within which the self-similarity +applies. +One-dimensional scattering by a Cantor fractal potential is one of the simplest scat- +tering problems of quantum tunneling through fractal system. This problem have been +extensively studied in quantum mechanics by using the transfer matrix method to derive +various scattering properties [31–41]. The composition properties of the transfer matrix +have been used to derive the scattering coefficients and associated properties. In Cantor +fractal potential, scattering coefficients have been found to show scaling law and sharp +features of resonances k [31, 32, 36–38]. The tunneling amplitude from Cantor potential +can also be derived by using the concept of super periodic potential (SPP) [41]. +Despite the several advancement in the study of SFQM as well as quantum tunneling +from fractal potentials, at present tunneling properties from fractal potentials in SFQM +is not yet studied. It is expected that such studies will bring new features of scattering +properties in the domain of SFQM (α < 2) which are absent in the case of standard +QM (α = 2). In the present study we mainly focus on the simplest fractal system in +one dimension, Cantor fractal along with an another member of Cantor family potential +known as Smith-Volterra-Cantor (SVC) potential. The SVC potential is not a fractal +potential while the Cantor potential is a fractal. In order to keep the study more general in +nature, we consider the general Cantor (GC) and general SVC (GSVC) potential system. +These are constructed in such a way that for a given initial length L and height V of the +rectangular barrier potential, a fraction of 1 +3 from the middle is removed at every stage +‘G’ from the remaining segments for standard Cantor-3 potential. For GC potential (or +Cantor-ρ potential), instead of 1 +3, a fraction 1 +ρ is removed where ρ > 1 is a real positive +number. Similarly in GSVC potential (or SVC-ρ) potential, a fraction of +1 +ρG is removed +from the middle at each stage G instead of +1 +4G as in case of standard SVC-4 system. +Again ρ ∈ R+ and ρ > 1. A simple observation shows that SVC system doesn’t satisfy +the criteria for the same ‘self-similarity’ at each stage G and therefore is not a fractal +system. +In an earlier work, we have shown that Cantor-3 and SVC-4 potential system are +the special case of SPP [41]. This is also true for Cantor-ρ and SVC-ρ system. SPP +concept is the generalization of periodic potential having arbitrary number of internal +periodicity [41]. As we have not yet extended the concept of SPP in the domain of SFQM, +we use the fundamental principle to derive the expressions for transmission amplitude +using transfer matrix approach for both types of potential. We report new features of +scattering from these systems in the domain of SFQM. Notable features are emergence +of energy band structures from these potentials which are absent in standard QM and +extremely sharp transmission resonances. The scaling behavior with wave vector k is also +presented analytically. +3 + +This paper is organized as follows. In section 2, an overview of SFQM is presented. +The transfer matrix in SFQM for a localized and repeated potential is discussed in detail +in the section 3. In section 4 and 5, we provide a brief review of the symmetric fractal +potential of the cantor family and its repeated system. In next section 6, explicit ex- +pression of ‘ζj’ (argument of Chebyshev polynomial of second kind) is expressed in order +to get transmission amplitude in SFQM for general SVC and general Cantor potential. +Afterward, in section 7, we provide graphically a detailed analysis of the transmission +features for both the fractal potential in the domain of SFQM. Finally, at last, in section +8 results and discussion are mentioned. +2 +Space fractional Schrodinger equation +When the path integral formulation of quantum mechanics is generalized over Levy flight +paths, it results in space fractional quantum mechanics (SFQM). The governing equation +for SFQM is the space fractional Schrodinger equation. +The form of space fractional +Schrodinger equation is given by [1], +i¯h∂ψ(x, t) +∂t += Hα(x, t)ψ(x, t), +(1) +Where, Hα(x, t) is the fractional Hamiltonian operator. The Hamiltonian is expressed +through the use of Riesz fractional derivative (−¯h2∆)α/2 as, +Hα(x, t) = Dα(−¯h2∆)α/2 + V (x, t). +(2) +Here ‘α’ is the Levy index and ∆ = +∂2 +∂x2. In SFQM, the range of α is 1 < α ≤ 2 [2]. +Dα is a constant, also called as generalized diffusion coefficient and depends upon system +characteristics. The Riesz fractional derivative of the wave function ψ(x, t) is defined +through the use of Fourier transform of ψ(x, t) as, +(−¯h2∆)α/2ψ(x, t) = +1 +2π¯h +� ∞ +−∞ +˜ψ(p, t)|p|αeipx/¯hdp. +(3) +The Fourier transform of ψ(x, t) is given by, +˜ψ(p, t) = +� ∞ +−∞ +ψ(x, t)e−ipx/¯hdx. +(4) +and its inverse Fourier transform is, +ψ(x, t) = +1 +2π¯h +� ∞ +−∞ +˜ψ(p, t)eipx/¯hdp. +(5) +For the case when potential V (x, t) is time independent i.e., V (x, t) = V (x), we have the +time independent fractional Hamiltonian operator Hα(x) as, +Hα(x) = Dα(−¯h2∆)α/2 + V (x). +(6) +4 + +The time-independent space-fractional Schrodinger equation is, +Dα(−¯h2∆) +α +2 ψ(x) + V (x)ψ(x) = Eψ(x). +(7) +By using the concept of separation of variables, it can be shown that the time independent +wave function ψ(x) is related to ψ(x, t) as ψ(x, t) = ψ(x)e−iEt/¯h where E is the energy of +the particle. For a detail discussion on SFQM readers are referred to [42]. +In the next section, we briefly discuss the transfer matrix formulation of the tunneling +problem in SFQM. +3 +Transfer matrix in SFQM +Figure 1: Depiction of the scattering of the quantum wave from an arbitrary potential V (x) in +one dimension. +Consider a localized potential V (x) bounded in the region (−a, a) as shown in Fig. +1. The solution of time independent space fractional Schrodinger equation (Eq. 7) in all +the three regions x < −a, −a < x < a, and x > a are, +ϕ(x) = Aeikαx + Be−ikαx, +x < −a, +(8) +ϕ(x) = ϕab(x), +−a < x < a, +(9) +ϕ(x) = Ceikαx + De−ikαx, +x > a. +(10) +Where, +kα = +� +E +Dα¯hα +�1/α +(11) +and the coefficients A, B, C, and D are the amplitudes of the waves on either side of the +potential V (x). The solution of the space fractional Schrodinger equation provides two +5 + +linear equations in terms of the coefficients A, B, C, and D. The two linear equations +can be represented in matrix form as, +�A(kα) +B(kα) +� += M(kα) +�C(kα) +D(kα) +� +. +(12) +M(kα) is a 2 × 2 matrix, +M(kα) = +�M11(kα) +M12(kα) +M21(kα) +M22(kα) +� +, +(13) +which is known as the transfer matrix of the potential V (x). For the case when V (x) is +Hermitian, the time invariance property of Eq. 7 leads to +M11(kα) = M22(kα)∗, +M21(kα) = M12(kα)∗, +(14) +i.e., the diagonal and off-diagonal elements are complex conjugate to each other. The +determinant of the transfer matrix is always unity which together with the above property +implies |M11(kα)|2−|M12(kα)|2= 1. If the transfer matrix of a potential V (x) is known +then one can obtain the scattering coefficients for the potential V (x) through the following +expression, +tl(kα) = tr(kα) = +1 +M22(kα), +rl(kα) = −M21(kα) +M22(kα), rr(kα) = M12(kα) +M22(kα). +(15) +From the knowledge of the transfer matrix of a single localized potential V (x), one can +obtain the transfer matrix of the periodic potential when V (x) is periodically repeated +N1 times [43]. +Formulation of the transfer matrix of locally periodic media from the +knowledge of the transfer matrix of single ‘unit cell’ potential V (x) is also applicable in +space fractional quantum mechanics [26]. The transfer matrix MN1(kα) for the periodic +potential is given by, +MN1(kα) = +�[M11e−ikαsUN1−1(ζ1) − UN1−2(ζ1)]eikαN1s +M12UN1−1(ζ1)e−ikα(N1−1)s +M ∗ +12UN1−1(ζ1)eikα(N1−1)s +[M ∗ +11eikαsUN1−1(ζ1) − UN1−2(ζ1)]e−ikαN1s +� +. +(16) +In the above expression, ‘s’ is the separation between the starting points of two consecutive +‘unit cell’ potentials and UN(ζ1) is the Chebyshev polynomial of the second kind. The +argument of Chebyshev polynomial ‘ζ1’, which is the Bloch phase of the corresponding +fully developed periodic system, is computed from the knowledge of the ‘unit cell’ transfer +matrix and the separation ‘s’ as [43], +ζ1(kα) = 1 +2 +� +M11e−ikαs + M22eikαs� +. +(17) +Using the property M11(kα) = M22(kα)∗, the above equation can also be written as, +ζ1(kα) = Re[M22] cos(kαs) − Im[M22] sin(kαs) = |M22|cos(φ + kαs), +(18) +6 + +where φ is the argument of M22, i.e., M22 = |M22|eiφ. The transmission coefficient for the +periodic potential is the inverse of the lower diagonal element of the matrix given by 16. +Using the unitary properties of the transfer matrix, the transmission amplitude T = |tl,r|2 +can be obtained as [43], +T(N1) = +1 +1 + [|M12|UN1−1(ζ1)]2. +(19) +A few comments and the associated generalizations are in order. We can write the term +[|M12|UN1−1(ζ1)]2 appearing in the above equation as [|M12|UN1−1(ζ1)]2 = |M12UN1−1(ζ1)|2 += |(M12)N1|2 where (M12)N1 is the (1, 2) element of the transfer matrix (TM) given by +16. This can also be read as, +|M12 element of periodic system TM|= |M12 element of unit cell TM× +UN1−1(Bloch phase of the fully developed periodic system)|. +(20) +If we periodically repeat this periodic system N2 times with a different periodic distance +s2, then from Eq. 20, the modulus of (1, 2) element, |(M12)N1,N2| of the transfer matrix +of the new periodic system will be given by, +|(M12)N1,N2|= |(M12)N1UN2−1(ζ2)|= |M12UN1−1(ζ1)UN2−1(ζ2)|. +(21) +Where ζ2 is the Bloch phase for the new periodic system. If we periodically repeat the +systems with parameters Ni and si where i = 1, 2, 3, ..., G which yield ζ1, ζ2, .....ζG as the +respective Bloch phases, then Eq. 21 easily generalizes to +|(M12)N1,N2,N2,...,NG|= |M12 +G +� +i=1 +UNi−1(ζi)|. +(22) +The corresponding transmission amplitude can be obtained from +T(N1, N2, N3, ..., NG) = +1 +1 + |(M12)N1,N2,N2,...,NG|2 = +1 +1 + |M12|2�G +i=1 U 2 +Ni−1(ζi) +. +(23) +A rigorous proof of Eq. +23 based on the transfer matrix elements for super periodic +potential is presented in [41] for the case of standard QM. In particular, when Ni = 2, we +have UNi−1(ζi) = U1(ζi) = 2ζi. Substitution of this in Eq. 23 leads to +T(2, 2, 2, ..., G times) = TG = +1 +1 + 4G|M12|2�G +i=1 ζ2 +i +, +(24) +and the transfer matrix becomes, +MN1=2(kα) = +�2M ∗ +22ζ1eikαs − e2ikαs +2M12ζ1e−ikαs +2M ∗ +12ζ1eikαs +2M22ζ1e−ikαs − e−2ikαs +� +. +(25) +It is to be noted that the form of Eq. 24 is the general expression for tunneling amplitude +when a single potential cell is repeated only two times and that system as a whole is +7 + +further repeated two times and so on. We will extensively use Eq. 24 and Eq. 25 to +calculate tunneling amplitudes for the symmetric potential of Cantor family. It turns out +that tunneling amplitude for any symmetric potential which is generated by the division of +a real line in three parts and subsequent removal of the middle segment can be expressed +using Eq. 24. We will discuss this in detail in the subsequent sections. +4 +Symmetric potential of Cantor family +In one dimension, a fractal is generated by the division of a real line in a fashion which +preserves self-similarity. Similarly, a rectangular fractal potential can be generated by +dividing the length of the barrier in a self-similar fashion while keeping the height of the +barrier unchanged. Symmetric fractal potential obeys parity symmetry about the origin +i.e., the fractal potential is symmetric with respect to changing x → −x and −x → x. A +potential of Cantor family is generated when the line segments are divided into three parts +and the middle parts are removed at any stage G. A particular case of the symmetric +Cantor potential is when the removal of the middle part from the line segment leaves +the resultant two segments of equal sizes. This configuration of the system is always +symmetric. Starting from a length L and stage G = 0, symmetric Cantor potential can +be generated by the removal of a fraction +1 +ρa1+a2G from the middle segment(s) at each +stage G. Here ρ ∈ R+ and a1, a2 ∈ {0, R+}. When a1 = 1 and a2 = 0, we have general +Cantor potential (also, for a2 = 0, a1 can be absorbed by defining ρa1 = ρ1 for some real +ρ1 and we still have general Cantor potential). Similarly, for a1 = 0, we have general +Smith-Volterra-Cantor (SVC) potential. For the special case when a1 = 0, a2 = 1 and +ρ = 4 we have standard SVC potential system. Again when a1 = 0, we can absorb a2 by +defining an associated new ρ and the fractal potential is named an SVC-ρ system. The +geometrical construction of general Cantor and general SVC potential is illustrated in Fig. +2. At any stage G, both general Cantor and general SVC potential have 2G segments of +equal length lG. The value of lG are different for both types of potential. In the case of +general Cantor, +lG = +�ρ − 1 +2ρ +�G +L. +(26) +For the case of general SVC, lG can be obtained through the use of the q-Pochhammer +symbol as shown below. From Fig. 2, it is noted that, +l1 = L +2 +� +1 − 1 +ρ +� +. +(27) +Similarly, the segment length l2 for stage G = 2 is, +l2 = l1 +2 +� +1 − 1 +ρ2 +� += L +22 +� +1 − 1 +ρ +� � +1 − 1 +ρ2 +� +. +(28) +Similarly, +l3 = l2 +2 +� +1 − 1 +ρ3 +� += L +23 +� +1 − 1 +ρ +� � +1 − 1 +ρ2 +� � +1 − 1 +ρ3 +� +. +(29) +8 + +[H] +Figure 2: +Construction of Cantor-ρ and SVC-ρ potential. The white region shows the gap +between the potentials and the height of the opaque region is the potential height V . Here +G represents the stage of the system. In Cantor-ρ potential, a fraction 1/ρ is removed at +every stage while in SVC-ρ, a fraction +1 +ρG is removed at each stage G. +By continuing the same steps, the segment length ‘lG’ for arbitrary Gth order SVC-ρ 0 +potential is obtained as, +lG = L +2G +G +� +i=1 +� +1 − 1 +ρi +� +. +(30) +The product series can be recognized as, +G +� +i=1 +� +1 − 1 +ρi +� += q +�1 +ρ; 1 +ρ +� +G +. +(31) +Where, +q(a; λ)n = +n−1 +� +i=0 +(1 − a.λi) = (1 − a)(1 − a.λ)(1 − a.λ2).....(1 − a.λn−1) +(32) +9 + +GC +GSVC +GC +GSVC +GC +GSVCis q-Pochhammer symbol [44]. Therefore, through the use of the q-Pochhammer symbol, +we can express lG as, +lG = L +2Gq +�1 +ρ; 1 +ρ +� +G +. +(33) +5 +Symmetric Cantor family potentials as repeating +systems +In this section, we illustrate that a symmetric potential of the Cantor family can be +generated through a ‘unit cell’ by repeating it two times and then repeating the resultant +‘cell’ further two times and so on. Consider a rectangular barrier of height V and width +lG as shown in Fig. 3. We can repeat this barrier at a distance s1 > lG as shown in Fig. 3. +The resultant system of these two barriers are further repeated at a distance of s2 thereby +generating a system of four rectangular barriers which as a whole is further repeated at a +distance of s3 as shown in Fig. 3. This process of repeating the resultant barrier systems +two times at a specific distance can continue up-to an arbitrary stage G. As the Cantor +family systems are well defined mathematical structures, the value of ‘lG’ and various ‘si’ +can be easily identified for any arbitrary stage G for a particular system. +[H] +Figure 3: Construction of the symmetric Cantor family potential for the stage G = 4 as +periodic repetition of the periodic system of order 4. +10 + +First we present general expression of sj for Cantor-ρ fractal system. For this system +we have, +s1 = lG + lG−1 +ρ , +s2 = lG−1 + lG−2 +ρ , +s3 = lG−2 + lG−3 +ρ , +The above sequences show that, +sj = lG+1−j + lG−j +ρ . +(34) +Using Eq. 26, this can be simplified to, +sj = xG−jyL, +(35) +where, +x = ρ − 1 +2ρ , y = ρ + 1 +2ρ . +(36) +Similarly, it can be shown that for SVC-ρ potential, sj is given by, +sj = lG+1−p + +lG−p +ρG+1−p. +(37) +Using Eq. 30 in the above expression, we have after simplification +sj = +L +2G+1−j +� +1 + +1 +ρG+1−j +� +q +�1 +ρ; 1 +ρ +� +G−j +. +(38) +For a given G, by choosing a single barrier of length lG as given by Eq. 26 and placing +the barrier at various sj as given by Eq. 35, we get Cantor-ρ potential. Similarly, by +choosing lG from Eq. 33 and sj from Eq. 38 we get SVC-ρ potential. In the next section, +we calculate the transmission amplitudes from these two types of fractal potentials in +SFQM. +6 +Transmission amplitudes in SFQM +It is clear from the discussion in the previous section that (symmetric) Cantor-ρ (GC) +and SVC-ρ (GSVC) potentials are the special cases of systems that are repeated two +times and that configuration as a whole is further repeated two times and so on. The +number of such operations of repetitions is equal to the stage G of the GC and GSVC +potential. The general expression of the tunneling amplitude for such a potential system +in SFQM is given by Eq. 24. What remains is to calculate the general expressions for +ζi, i = 1, 2, 3, .., G for GC and GSVC potentials. We will derive the general expression +for ζi and then would specialize to calculate specific expressions for ζi for GC and GSVC +potentials. The calculations are illustrated below. +11 + +Let M22(kα) denotes the lower diagonal elements of the transfer matrix of rectangular +barrier of width b = lG and height V . This potential configuration is represented by P0 in +the Fig. 3. Similarly, let (M22)1, (M22)2, (M22)3 etc. denote the lower diagonal elements +of the transfer matrix of the combined system represented by P1, P2, P3 etc. as shown in +Fig. 3. The corresponding Bloch phases are ζ1, ζ2, ζ3 etc. respectively. From Eq. 25 we +can read that +(M22)j = 2(M22)j−1ζje−ikαsj − e−2ikαsj, +(39) +where j = 1, 2, 3, .., G and (M22)0 = M22. Now from the general Eq. 18 we can write, +ζ2(kα) = Re[(M22)1] cos kαs2 − Im[(M22)1] sin kαs2. +(40) +We can use Eq. 39 in the above equation so that, +ζ2(kα) = Re[(2 × M22.ζ1)e−ikαs1 − e−2.ikαs1] cos kαs2 +− Im[(2 × M22ζ1)e−ikαs1 − e−2.ikαs1] sin kαs2. +(41) +The simplification of the real and imaginary parts finally gives, +ζ2 = 2|M22|ζ1 cos [φ − kα{s1 − s2}] − cos [kα{2s1 − s2}]. +(42) +Similarly, repeating the above procedure to calculate ζ3 we have, +ζ3 = Re[(M22)2] cos kαs3 − Im[(M22)2] sin kαs3. +(43) +Again using Eq. 39 to simplify the above, we obtain for ζ3, +ζ3(kα) = 22|M22|ζ1ζ2 cos [φ − kα{s1 + s2 − s3}]− +2.ζ2 cos [kα{2s1 + s2 − s3}] − cos [kα{2s2 − s3}]. +(44) +Similarly, we have for ζ4 +ζ4(kα) = Re[(M22)3] cos kαs4 − Im[(M22)3] sin kαs4. +(45) +The repeated application of Eq. 39 and simplifications of the real and imaginary parts in +the above equation gives, +ζ4(kα) = 23|M22|ζ1ζ2ζ3 cos [φ − kα{s1 + s2 + s3 − s4}] +− 22ζ2ζ3 cos [kα{2s1 + s2 + s3 − s4}] − 2.ζ3 cos [kα{2s2 + s3 − s4}] +− cos [kα{2s3 − s4}]. +(46) +Similarly, the expression for ζ5 is given by, +ζ5(kα) = 24|M22|ζ1ζ2ζ3ζ4 cos [φ − kα{s1 + s2 + s3 + s4 − s5}] +− 23ζ2ζ3ζ4 cos [kα{2s1 + s2 + s3 + s4 − s5}] +− 22ζ3ζ4 cos [kα{2s2 + s3 + s4 − s5}] − 2.ζ4 cos [kα{2s3 + s4 − s5}] +− cos [kα{2s4 − s5}]. +(47) +12 + +We observe from the sequence of Eqs. 42, 44, 46, and 47 that the general expression for +ζj can be written in the following series form, +ζj(kα) = 2j−1|M22|cos [φ − kαη1(j)] +j−1 +� +p=1 +ζp − +j−1 +� +r=1 +� +2j−r−1 cos [kαη2(j, r)] +j−1 +� +p=r+1 +ζp +� +. +(48) +In the above equation, we have used the following notation, +η1(j) ≡ +� j−1 +� +p=1 +sp +� +− sj, +(49) +η2(j, r) ≡ +� +j +� +p=r +sp +� +− (2sj − sr). +(50) +It is easy to show that, +η2(j, r) ≡ η1(j) − η1(r). +(51) +Eq. 48 is the general expression for ζj, j = 1, 2, 3, .., G. However, it is important to note +here that in Eq. 48, we have to drop the terms when the running variable ‘r’ is more than +the upper limit for the summation operation and we take terms as unity when the running +variable is more than the upper limit for the product operation. From the knowledge of +ζ1, ζ2, ζ3, ..., ζG, we can calculate the tunneling amplitude from Eq. 24. Now we calculate +the values of η1,2 and their properties. +Figure 4: Symmetric Cantor family potential shows the length and gap between the seg- +ments. +From Fig. 3 and 4, we observe s1 = lG + gG, s2 = s1 + lG + gG−1, s3 = s1 + s2 + lG + gG−2, +13 + +L +G=0 +11 +11 +G= 1 +91 +12 +12 +12 +12 +G=2 +92 +91 +92 +13 +13 +I3 +I3 +13 +13 +I3 +13 +G=3 +93 +92 +91 +[g3 +93 +92 +g3s4 = s1 + s2 + s3 + lG + gG−3 and so on. Thus we arrive at, +sj = +� j−1 +� +p=1 +sp +� ++ lG + gG−j+1. +(52) +Therefore, η1(j) is given by, +η1(j) = −(lG + gG−j+1), +(53) +which shows that η1(j) is always negative. Combining Eq. 51 and 53 we get, +η2(j, r) = gG−r+1 − gG−j+1. +(54) +We see from Fig. 4 that for i > j, gi < gj. Also for r < j, G − r + 1 > G − j + 1 +which implies gG−r+1 < gG−j+1. Therefore η2(j, r) < 0 for r < j. Eq. 53 and 54 gives the +general expression for η1 and η2 respectively. Now we provide these expressions for GC +and GSVC cases. +6.1 +Case 1: General SVC potential +To calculate η1,2 for GSVC, we re-write Eq. 33 as +lj−1 = +L +2j−1q +�1 +ρ; 1 +ρ +� +j−1 +. +(55) +As we know, for GSVC a fraction +1 +ρj is removed from segment length lj−1 to generate the +system for stage G = j, therefore, gj = lj−1 +ρj +and hence, +gj = +L +ρj2j−1q +�1 +ρ; 1 +ρ +� +j−1 +. +(56) +Now we simplify for η1(j) by using Eq. 53, Eq. 33 and Eq. 56 to obtain, +η1(j) = − +� +L +2Gq +�1 +ρ; 1 +ρ +� +G ++ +L +2G−j q +�1 +ρ; 1 +ρ +� +G−j +1 +ρG−j+1 +� +. +(57) +Now we calculate η2(j, r) by using Eq. 51 and 57 to obtain, +η2(j, r) = +2L +(2ρ)G+1 +� +(2ρ)rq +�1 +ρ; 1 +ρ +� +G−r +− (2ρ)jq +�1 +ρ; 1 +ρ +� +G−j +� +(58) +Now we can substitute Eq. 57 and 58 in Eq. 48 to obtain the general expression for +‘ζj’ for GSVC potential. +14 + +6.2 +Case 2: General Cantor potential +We re-write Eq. 26 as, +lj−1 = +�ρ − 1 +2ρ +�j−1 +L. +(59) +As we know, in case of GC potential, a fraction 1 +ρ is taken from stage G = j − 1 to create +the fractal system for G = j stage, therefore gj = lj−1 +ρ +and thus, +gj = 1 +ρ +�ρ − 1 +2ρ +�j−1 +L. +(60) +Now, using Eq. 53 and 26 in Eq. 60, we simplify for η1(j) to get, +η1(j) = − +� +L. +�ρ − 1 +2ρ +�G ++ L +ρ . +�ρ − 1 +2ρ +�G−j� +. +(61) +Now using Eq. 51, η2(j, r) can be simplified as, +η2(j, r) = L +ρ +�ρ − 1 +2ρ +�G−r−j ��ρ − 1 +2ρ +�j +− +�ρ − 1 +2ρ +�r� +. +(62) +Substitution of Eq. +61 and 62 in Eq. +48 gives the general expression of ‘ζj’ for GC +potential. +7 +Transmission features +In the previous section, the analytical expressions of the tunneling amplitudes from two +types of Cantor potentials in SFQM have been derived. In this section, we study the +various features of transmission through these systems in SFQM. As the Cantor potentials +have been studied in detail in standard QM (α = 2) [31–41], therefore we largely focus +here to study the tunneling behavior in the domain of SFQM (i.e., the case of α < 2) as +well as the comparison with the case of standard QM (i.e., the case of α = 2). Fig. 5 shows +the comparison of the profiles of the transmission amplitudes for GC and GSVC potential +in standard QM and in SFQM for different stages G. In all plots of Fig. 5, it is noted +that the transmission resonances are much sharper in GSVC potential as compared to GC +potential. As these two types of potentials are different, they show different transmission +profiles which are not relatable (at the present level of investigations) though both are +special cases of repeating systems. Therefore, we present these two cases separately in +subsequent sections. +Subsection 7.1 discusses the case of GSVC while subsection 7.2 +details the case for GC potential. +15 + +G=3 +G=5 +α=2.00 +α=1.90 +α=1.80 +Figure 5: Plots showing the comparison of transmission amplitudes for GC (Red-curve) +and GSVC (Blue-curve) potential for two different stages of G (= 3 and 5) in SFQM (α += 2.00, 1.90 and 1.80). Here potential parameters are L = 1, V = 400 and ρ = 3.5. +From figures, it is observed that GSVC potential has sharper peaks as compared to general +Cantor potential. +16 + +0.8 +0.6 +0.4 +0.2 +40 +10 +20 +30 +k0.8 +0.6 +T +0.4 +0.2 +40 +10 +20 +30 +k0.8 +0.6 +T +0.4 +0.2 +10 +20 +30 +40 +k0.8 +0.6 +T +0.4 +0.2 +40 +10 +20 +30 +k0.8 +0.6 +T +0.4 +0.2 +40 +10 +20 +30 +k0.8 +0.6 +T +0.4 +0.2 +10 +20 +30 +40 +k7.1 +Transmission features of general SVC potential in SFQM +(a) +(b) +(c) +Figure 6: The transmission amplitude for GSVC potential in SFQM with α and k. The +potential parameters are V = 100, ρ = 3 and G = 3. The transmission peaks occurs at +lower k values with decreasing α. It is also evident from Fig. (c) that the sharpness of +the transmission peaks are increasing as α is lowered. +This section exclusively discusses the nature of the transmission profile from GSVC system +in SFQM. As the expression for the transmission amplitude is transcendental in nature +(Eq. +24), presently we rely on the numerical investigation towards investigating the +general features of tunneling amplitude. The transmission amplitude is plotted for stage +G = 3 in Fig. 6. To understand the behavior of transmission resonances with α, we +3D plot T(α, k) with α and k as shown in Fig. 6-a. Here the potential parameters are +V = 100, ρ = 3, and G = 3. A closer look at this figure is shown in Fig. 6-b for a +smaller range of k. From both these figures, it is seen that the locus of transmission +resonances has a positive slope with increasing α. This indicates that the transmission +peaks are red-shifted with decreasing values of α. This appears to be a general trend for +17 + +0.8 +α=2.00 +0.6 +α=1.98 +T +α=1.96 +0.4 +α=1.94 +0.2 +α=1.92 +α=1.90 +10 +12 +14 +16 +18 +20 +k2. +1.98 +α +1.96 +1.0 +1.94 +0.8 +1.92 +1.9 +0.6 +1. +0.8 +0.4 +0.6 +0.4 +0.2 +0.2 +0 +10 +0 +20 +30 +k +40 +502. +1.98 +α +1.96 +1.0 +1.94 +0.8 +1.92 +1.9 +0.6 +1. +0.8 +0.4 +0.6 +0.4 +0.2 +0.2 +0 +10 +0 +12 +14 +16 +k +18 +20the case when α is not far away from 2. However, much more complex behavior of the +locus of transmission resonances is seen when α is closer to 1 and is presented later in the +paper. Fig. 6-c shows the 2D plot depicting the variation of T for different α with the +same range of k as shown in Fig. 6-b. This figure shows that the transmission resonances +become sharper at lower values of α. This appears to be a general feature and will be +more evident in the later part of the discussion and associated graphical representations. +(a) +(b) +(c) +(d) +(e) +(f) +Figure 7: Plots showing several sharp transmission resonances near unity (i.e α = 1) for +GSVC potential of stage G = 5 in SFQM. The potential parameters are V = 450, L = 1 +and ρ = 3. +18 + +α=1.02 +0.8 +0.6 +T +0.4 +0.2 +27 +29 +31 +33 +35 +25 +kα=1.025 +0.8 +0.6 +T +0.4 +0.2 +0 +27 +29 +25 +31 +33 +35 +kα=1.03 +0.8 +0.6 +T +0.4 +0.2 +27 +29 +31 +33 +35 +25 +kα=1.005 +0.8 +0.6 +T +0.4 +0.2 +27 +29 +31 +33 +35 +25 +kα=1.01 +0.8 +0.6 +T +0.4 +0.2 +25 +27 +29 +31 +33 +35 +kα=1.015 +0.8 +0.6 +T +0.4 +0.2 +25 +27 +29 +31 +33 +35 +kAn interesting parameter region for the study of tunneling amplitude in SFQM is the +case when α is close to 1. In this regime, extreme behavior in the transmission amplitudes +is observed which is demonstrated graphically in Fig. 7. The figure shows the transmission +amplitude for stages G = 5, V = 450, L = 1, and different values of α near unity. In all +these figures, the emergence of several extremely sharp transmission resonances is observed +for both evanescent and non-evanescent waves. The transmission resonances are separated +by deep valleys in the T(k) profile such that T(k) vanishes over a range of k (it may be +noted that for any Hermitian potential, as in the present case transmission amplitudes are +never ideally zero [45]). Many transmission resonances in Fig. 7 are extremely sharp and +appear as the sudden jump from T = 0 to T = 1. Towards understanding these features +over a continuous range of α near 1, the transmission amplitudes are represented through +density plots in α − k plane for different stages of the potential in Fig. 8 and Fig. 9. For +both these figures L = 1, ρ = 3 while V = 300 and 450 for Fig. 8 and Fig. 9 respectively. +(a) +(b) +Figure 8: Density plot showing the variation of transmission amplitude T in α − k plane +for α close to 1 for GSVC potential of different stages G = 5 and 7. +The potential +parameters are V = 300, L = 1 and ρ = 3. Extreme sharp transmission resonances are +seen. For both the stages of the potential, extreme behavior of variations in T is observed +for wave energy E < V . However, for E > V , the transmission amplitudes are observed +to saturate with increasing G. +Further, deep minima in T occur in the transmission +profile for several finite ranges of k which indicates the presence of allowed and forbidden +band-like structures from this potential system in SFQM. +The different stages G are shown in the figures. Both these figures indicate the pres- +ence of extremely sharp transmission resonances as thin streaks of yellow lines. In some +cases, the lines are so thin that these are not captured graphically over the red regions. +The behavior of these T = 1 loci is challenging to understand analytically due to the +transcendental nature of the expression of the tunneling amplitudes. Extreme variations +for both α and k are observed for wave energy E < V while for E > V the transmission +19 + +G=5 +1.03 +1.0 +0.8 +1.025 +0.6 +α +1.02 +0.4 +1.015 +0.2 +1.01 +0 +5 +10 +15 +20 +25 +30 +35 +kG=7 +1.03 +1.0 +0.8 +1.025 +0.6 +α +1.02 +0.4 +1.015 +0.2 +1.01 +0 +5 +10 +15 +20 +25 +30 +35 +kprofile appears to saturate with increasing G. An apparent conclusion that may be drawn +from Fig. 8 and 9 is the presence of deep valleys in the transmission amplitudes for which +T is nearly vanishing in α−k plane for α in the vicinity of 1. The presence of these valleys +in the transmission amplitudes is noted as the precursor of the emergence of allowed and +forbidden energy bands [46] from locally periodic potential. This shows that the band +features emerge in the case of GSVC potential in SFQM. In the case of standard QM, the +band doesn’t appear from Cantor family potential to the best of our knowledge. However, +the emergence of band-like features from GSVC potential (and will be shown later for GC +potential) appears only when α is in the vicinity of 1. +(a) +(b) +(c) +(d) +Figure 9: Density plot showing the variation of transmission amplitude T in α−k plane for +α close to 1 for GSVC potential of different stage G. Here V = 450 and other parameters +are the same as Fig. 8. Again, the presence of extremely sharp transmission resonances +is noticed with extreme variations in T for wave energy E < V . It is also seen in the +figure that the transmission amplitude saturates with increasing G for E > V . Again, the +density plot shows the occurrence of band-like features. +20 + +G=9 +1.0 +1.030 +0.8 +1.025 +1.020 +0.6 +α 1.015 +0.4 +1.010 +1.005 +0.2 +1.000 +5 +10 +15 +20 +25 +30 +35 +40 +0 +k1.0 +G=11 +1.030 +0.8 +1.025 +1.020 +0.6 +α 1.015 +0.4 +1.010 +1.005 +0.2 +1.000 +5 +10 +15 +20 +25 +30 +35 +40 +0 +k1.0 +G=13 +1.030 +0.8 +1.025 +1.020 +0.6 +α 1.015 +0.4 +1.010 +1.005 +0.2 +1.000 +5 +10 +15 +20 +25 +30 +35 +40 +0 +k1.0 +G=15 +1.030 +0.8 +1.025 +1.020 +0.6 +α 1.015 +0.4 +1.010 +1.005 +0.2 +1.000 +5 +10 +15 +20 +25 +30 +35 +40 +0 +kA discussion is in order. The deep valleys in T(k) profile for a periodic potential for a +range ∆k means that the waves are reflected from the potential for k ∈ ∆k. From the +emergence of deep valleys in T for tunneling through locally periodic delta potential, it is +argued that the band-like structures emerge even when number of periodic delta barriers +are just five, N = 5 [43,46]. Based on the similar studies in SFQM, it is noted that the +band emerges even when N = 4 and are more prominently present for lower α values [26]. +In the present case, the repetitions are based on Ni = 2. As N = 2 system doesn’t show +band structure (and deep valleys in T for ranges of k) in standard QM, the type of Cantor +family system studied here don’t show allowed energy bands in standard QM. +(a) +(b) +(c) +(d) +Figure 10: Plot of transmission amplitude for G = 1 (double barrier system), above Fig. +(a) shows density plot for potential height V = 500 and Fig. (b) represent 2D plot for +potential height V = 648 with potential width L = 1 for both cases. Both these plots show +the occurrence of valleys for which T is very close to zero. Fig. (c) shows the density +plots for potential parameters V = 700, L = 1 in α − k plane for range of α from 1.001 +to 1.01 and Fig. (d) corresponding 2D plot when α = 1.004 for same potential parametrs +as Fig. (c) . The density plot clearly shows range of k for which T nearly vanishes. This +again indicates the presence of allowed bands for the double barrier system. Interesting +oscillations are seen in T over α − k plane for the evanescent waves. +21 + +G=1.V=500 +1.01 +1.0 +0.8 +1.008 +0.6 +1.006 +α +0.4 +1.004 +0.2 +1.002 +0 +25 +27 +29 +31 +33 +35 +kG=1.V=700 +1.01 +1.0 +0.8 +1.008 +0.6 +1.006 +α +0.4 +1.004 +0.2 +1.002 +0 +31 33 35 37 39 41 43 45 +kα=1.004 +0.8 +0.6 +T +0.4 +0.2 +38 +37 +39 +kα=1.004 +0.8 +0.6 +T +0.4 +0.2 +33 +32 +34 +kThus, a question may arise that if the present GSVC system, which is an arrangement +of Ni = 2 barriers, show band likes features for α → 1, would this also mean that a +double barrier system will show energy bands like features for α → 1. Surprisingly, we +find that this is indeed the case and are graphically shown in Fig. 10 for three different +double barrier potential systems in SFQM. It is an extraordinary fact to recognize that +there are allowed and forbidden bands for just double barrier systems in the domain of +SFQM. If the N = 2 barriers system could show band structures in SFQM, therefore the +present GSVC systems which are Ni = 2 repeating systems could also show energy band +structures. We will show in the later section that this is also true for GC potential in +SFQM for α near to 1. +General SVC +General Cantor +Figure 11: Plot of log10 (− log10 T) for the case of general SVC and general Cantor poten- +tial for G = 7 (red curve), G = 9 (dashed green curve), and G = 11 (dashed blue curve). +The potential parameters are V = 100, L = 1 and ρ = 3. As it is clearly visible from first +column (for general SVC) and second column (for general Cantor), that the tunnelling +saturates with increasing G in standard (i.e., α = 2) as well as in SFQM for general SVC +potential. However, this saturation behavior is not observed for general Cantor potential +in SFQM. +. +22 + +α=1.80 +5 +10 +-15 +10 +20 +30 +40 +50 +kα=1.90 +10 +-15 +10 +20 +30 +40 +50 +kα=2.00 +Log10(-Log10T) +10 +-15 +10 +20 +30 +40 +50 +kα=2.00 +Log10(-Log10T) +.5 +.10 +15 +10 +20 +30 +40 +50 +kα=1.80 +Log10(-Log10T) +5 +10 +-15 +10 +20 +30 +40 +50 +kα=1.90 +Log10(-Log10 T) +10 +-15 +10 +20 +30 +40 +50 +kAnother observation from Fig. +9 is the saturation of the transmission profile with +increasing G. From the definitions of GSVC system, a portion +1 +ρG is taken out from the +middle at each stage G. Thus progressively lesser fractions are taken from each stage +with increasing G. This would imply that for larger G, the transmission profile should +saturate with G as only very thin portions are removed from the segments of the previous +stages when G is large. This is illustrated graphically in Fig. 11 in which a function of +T is plotted for GSVC and GC potential for stages G = 7, 9 and 11 for different α. For +a better resolution in different T(k, G), we have plotted y = log10 (− log10 T) in y-axis. +As 0 < T ≤ 1, therefore log10 T ≤ 0 and thus − log10 T ≥ 0. This implies that the +function y = log10 (− log10 T) is well defined. Various plots with different α in Fig. 11 +show that the saturation in T(k) profile with G is observed in GSVC potential but not +in GC potential. However, for α → 1, this saturation is present only in the case for +wave energy E > V and not for E < V as shown in Fig. 12 for different potentials and +α = 1.01. +, V = 8.0 +, V = 18.0 +, V = 20.0 +, V = 32.0 +Figure 12: Plot of log10 (− log10 T) for the case of GSVC for G = 5 (red curve), G = 8 +(dashed green curve), and G = 11 (dashed blue curve). The potential parameters are +L = 1, ρ = 3, and the height V is indicated in each figure. It is seen from all these +plots that for values of α close to 1, T(k) profile saturates with G when wave energy +E(= k2/2m) > V . No saturation in the T(k) profile is observed when E < V . +23 + +α=1.01 +0 +Log10(-Log10T) +-5 +-10 +2 +6 +8 +0 +10 +4 +kα=1.01 +Log10(-Log10T) +5 +-10 +2 +8 +10 +0 +4 +6 +kα=1.01 +0 +Log10(-Log10T) +.5 +-10 +2 +6 +8 +10 +0 +4 +kα=1.01 +Log10(-Log10T) +-5 +-10 +2 +6 +8 +10 +0 +4 +k7.2 +Transmission features of general Cantor potential in SFQM +(a) +(b) +(c) +Figure 13: Plot showing the transmission profile for general Cantor of stage G = 3, ρ = 3, +L = 1 and V = 100 for different value of α. It is evident from the plots that as α reduces, +the transmission peaks shifts to lower values of wave numbers. (Right Image) 3D plot +showing the variation of T with α and k which again depict that the transmission peak +occurs at lower k values with reducing α. +In the previous section, we provided some general features of scattering such as emergence +of energy bands, increase in the sharpness of transmission resonances with reducing α, +extreme features of transmission for α near to 1 etc. for GSVC potential, In this section, +we show that such features also exists for GC potential in SFQM. In Fig. 13, we plot +T(α, k) with α and k with potential parameters as G = 3, V = 100, ρ = 3 and L = 1. A +closer look of this figure is shown in Fig. 13-b for a smaller range of k. Similar to the case +of GSVC for α near 2, it is seen that the locus of transmission resonances has a positive +slope with increasing α. This indicates that the transmission peaks are red-shifted with +decreasing values of α for GC potential in SFQM. An exception to this could occur when +24 + +2. +1.98 +α +1.96 +1.0 +1.94 +0.8 +1.92 +1.9 +0.6 +1. +0.8 +0.4 +0.6 +0.4 +0.2 +0.2 +10 +0 +20 +30 +k +40 +502. +1.98 +α +1.96 +1.0 +1.94 +0.8 +1.92 +1.9 +0.6 +0.8 +0.4 +0.6 +0.4 +0.2 +0.2 +10 +0 +k +200.8 +α=2.00 +α=1.98 +T 0.6 +α=1.96 +α=1.94 +0.4 +α=1.92 +α=1.90 +0.2 +10 +16 +12 +14 +18 +20 +kα is in the vicinity of 1 (see Fig. 14). Fig. 13-c shows the 2D plot depicting the variation +of T for different α with the same range of k as shown in Fig. 13-b. Again, similar to the +case of GSVC potential, it is observed that the transmission resonances become sharper +at lower values of α. Fig. 14 also shows extremely sharp loci of transmission resonances +as thin streaks of yellow lines. +(a) +(b) +Figure 14: Density plot showing the variation of transmission amplitude T in α − k plane +for α close to 1 for GC potential of different stages G. Here V = 450, L = 1 and ρ = 3. +Along with the presence of very sharp transmission peaks, the plots also shows the region +of deep valleys in α−k plane where transmission amplitude continuously vanishes. These +deep valleys in the transmission profile are the precursor of energy band structure. +These lines are also separated by deep valleys in T(k) profile which depict the presence +of energy band-like features. The deep valleys in T(k) profiles are also shown graphically in +Fig. 15 through the density of T(α, k) in α−k plane as well as 2D plots for discrete values +of α. It is to be noted from Fig. 15, that many sharp features of transmission resonances +that are not visualized due to graphic limitations of capturing very thin streaks of lines +are clearly seen in 2D plots. +25 + +G=2 +1.03 +1.0 +1.025 +0.8 +1.02 +0.6 +1.015 +α +0.4 +1.01 +1.005 +0.2 +1. +0 +10 +20 +30 +40 +50 +kG=3 +1.03 +1.0 +1.025 +0.8 +1.02 +0.6 +α 1.015 +0.4 +1.01 +1.005 +0.2 +1 +0 +10 +20 +30 +40 +50 +k(a) +(b) +Figure 15: (a) Density plot for GC potential of stage G = 4 showing sharp transmission +in α − k plane and energy band features when α is close to 1 and (b) more closer view of +density plot for α ranging from 1.001 to 1.005. 2D plots illustrate clearly sharp valley for +α = 1.002, 1.003, and 1.004. Here potential parameters are V = 450, L = 1 and ρ = 3. +7.3 +Scaling behavior +This section presents the scaling behavior of the reflection amplitudes R = |r|2 with k for +both types of Cantor potentials considered in the paper. This section also present on how +the reflection amplitude behaves when height of the potential V varies in specific manner +at each stage G. For larger k, reflection amplitude R is very small. In this limit, R can +be approximated as +R ∼ 4G|M12|2 +G +� +i=1 +ζ2 +i . +(63) +Again for larger k we have V +k2 << 1 and upon Taylor expanding, it can be shown in the +first order that +|M12|2∼ +�α − 1 +α +V lG +�2 +1 +k +4(α−1) +α +. +(64) +26 + +G=4, α=1.004 +0.8 +0.6 +T +0.4 +0.2 +25 +27 +29 +31 +33 +35 +kG=4 +1.03 +1.0 +1.025 +0.8 +1.02 +0.6 +α +1.015 +0.4 +1.01 +0.2 +1.005 +1.001 +0 +10 +20 +30 +40 +kG=4 +1.005 +1.0 +0.8 +1.004 +0.6 +1.003 +0.4 +1.002 +0.2 +1.001 +0 +25 +27 +29 +31 +33 +35 +kG=4, α=1.002 +0.8 +0.6 +T +0.4 +0.2 +25 +27 +29 +31 +33 +35 +kG=4,α=1.003 +0.8 +0.6 +T +0.4 +0.2 +25 +27 +29 +31 +33 +35 +kTherefore, the expression for R becomes, +R ∼ 4G +�α − 1 +α +V lG +�2 +1 +k +4(α−1) +α +G +� +i=1 +ζ2 +i . +(65) +If VG is the height of the potential at each stage G, then it can be shown that the +following value of VG keeps the total area of potential barrier (sum of the area of all +potential segment at stage G) as constant +VG = +L +2GlG +V0, +(66) +where V0 is the height of the potential barrier at G = 0. Substituting the value of lG for +GC and GSVC potentials, VG is given by +VG = +� +ρ +ρ − 1 +�G +V0, for GC and, VG = +V0 +q +� +1 +ρ; 1 +ρ +� +G +for GSVC. +(67) +If RG is the reflection amplitude at each stage G with potential height of each segment +as VG then, it can be shown that (valid for large k) +RG +L2V 2 +0 +∼ +�α − 1 +α +�2 +1 +k +4(α−1) +α +G +� +i=1 +ζ2 +i . +(68) +Figure 16: Plot showing the reflection amplitudes for GSVC potential for G = 5 and 10. +The potential height VG is determined from Eq. 67 for GSVC potential. Other potential +parameters are shown in the figure. The difference between the two plots is invisible. This +shows the convergence of the product term of Eq. 68. +In Fig. 16 we show the behavior of RGSV C +5 +and RGSV C +10 +. The difference between the +two plots is invisible which shows the fast convergence of product term of Eq. 68 with +27 + +p=3,Vo=10,L=1,α=1.8 +0.0015 +RGSVC +RGSVC +10 +R +0.0010 +0.0005 +0.0000 +30 +40 +50 +60 +70 +80 +90 +100 +kincreasing G for GSVC case in SFQM. Similar plots are shown for GC case for different +α values in Fig. 17. The result shows that the convergence of the product term occurs +for GC case in SFQM with increasing stage G. +(a) +(b) +Figure 17: Plots showing the reflection amplitudes for GC potential for G = 10 and 15 +for different α value. The potential height VG is determined from Eq. 67 for GC case. +Other potential parameters are shown in the figure. The difference between the two plots is +nearly invisible. This shows the convergence of product term of Eq. 68 for GC potential. +For Cantor potential in standard QM, this has already shown in earlier work [37]. +Again, due to the convergence nature of the product term (provided it is evaluated at VG) +with increasing G, it is evident from Eq. 68 that RG would scale as +1 +k +4(α−1) +α +for large G +and k values. For α = 2, RG will scale as +1 +k2 which is proven result for standard Cantor +potential in standard QM [37]. The scaling behavior of RG with k in SFQM is shown +graphically for GC potential in Fig. 18. An interesting region of interest is when α is near +to 1. In this case RG would scale horizontally with k which is indeed the case as shown +in Fig. 18-c. The scaling behavior of RG with k is shown in Fig. 19 for GSVC potential +for different α values. +28 + +0.0005 +p=3,Vo=5,L=1,α=1.8 +RGC +10 +0.0004 +RGC +15 +0.0003 +R +0.0002 +0.0001 +0.0000 +160 +180 +200 +220 +240 +k0.020 +RGC +10 +RGC +15 +0.015 +p=3,Vo=10, L=1, α=1.1 +R +0.010 +0.005 +0.000 +250 +252 +254 +256 +258 +260 +k(a) +(b) +(c) +Figure 18: +log − log Plots showing the scaling behavior of reflection amplitudes RG +V 2 +0 for +large k in SFQM in case of general Cantor potential. The dotted curve represent +1 +k +4(α−1) +α +.It +is observed that at large k, RG falls of according to this expression. The potential param- +eters are shown in the figures. +29 + +p=3.0, Vo=10, L=1.0, α=1.90 +(α-1) +10-5 +10-8 +100 +200 +500 +1000 +kp=3.0, Vo=10, L=1.0, α=1.10 +0.01 +10-6 +100 +200 +500 +1000 +kp=3.0, Vo=10, L=1.0, α=1.01 +0.1 +100 +200 +500 +1000 +k(a) +(b) +Figure 19: Plots shows reflection amplitude +RG +V 2 +0 +for large k in SFQM (α = 1.90 and +α = 1.50) for general SVC potential. The dotted curve represent +1 +k +4(α−1) +α +.It is observed +that at large k, RG falls of according to this expression. The potential parameters are +shown in the figures. +8 +Results and Discussions +Fractional quantum mechanics is a fast-developing domain with several applications. We +have studied the tunneling features from fractal (general Cantor) and non-fractal (gen- +eral SVC) potential in space fractional quantum mechanics (SFQM). To the best of our +knowledge, this is the first time that the quantum tunneling from fractal potentials in +the domain of SFQM are studied. We have considered the generalized form of two kinds +of potentials of Cantor family, namely general Cantor (GC) and general Smith-Volterra- +Cantor (GSVC) potential. For both the kind of potentials, we have provided close form +expressions of transmission amplitudes in SFQM. These close form expressions are ex- +pected to provide better understanding of various scattering features in the domain of +SFQM from fractal potentials. It is to be noted that the derived expressions are of gen- +30 + +0.1 +p=3.0, Vo=5, L=1.0, α=1.90 +10-4 +R +10-10 +10 +100 +200 +500 +1000 +k0.1 +p=3.0, Vo=5, L=1.0, α=1.50 +10-4 +GSV +10-10 +10-13 +10-16 +100 +200 +500 +1000 +keral type and valid for any potentials of GC and GSVC type in which the ‘unit cell’ is +not a rectangular barrier. As long as the transfer matrix of ‘unit cell’ potential is known, +the derived expressions can be used to obtain the tunneling amplitudes from such kind of +Cantor family potentials. +In the present study, we have found several new features of scattering, and are re- +ported graphically. The most striking feature is the appearance of energy band structures +from fractal potential in SFQM which are absent in the case of standard Cantor frac- +tal and standard SVC potentials in standard QM. Standard fractal potentials based on +1 +3 division of the real segments don’t show energy band-like features in standard QM. +More surprisingly we noted that a double barrier potential system display energy bands +in SFQM. We have reported the emergence of band structures for both the type, GC and +GSVC potentials. However, it is to be noted that these band like features appear in the +extreme range of Levy index α close to the vicinity of 1. In this range of α, extremely +sharp transmission resonances are found to occur for both type of potentials. +Fractal potentials are known to display sharp transmission resonances, This feature is +further amplified in the domain of SFQM. It is found that the sharpness of the transmission +resonances further increases with a decrease in Levy index α. +In comparison, GSVC +potential displays more sharp transmission resonances as compared to GC potential in +standard QM as well as in SFQM. Also for the case of GSVC potential, it is observed +that the profile of transmission amplitudes saturates with increasing stage G. The reason +for this is due to the fact that a consecutively smaller fraction of the remaining previous +segments is removed at each stage G for GSVC potentials Therefore for higher G, only +very thin portions are removed from previous segments as compared to the case when +G is small. This leads to the saturation of the tunneling profile with k for higher G. +However, this behavior is found to be different near α = 1. For α ∼ 1+, the tunneling +profile saturates only for E > V . +Another interesting feature is the scaling behavior of reflection amplitude. We have +shown analytically that for large k, the reflection coefficient scale as +1 +k +4(α−1) +α +for GC and +GSVC potential provided that total area of the potential regions remain constant at +different stages G. Also for such case, the reflection amplitude converges as G increases +for both GC and GSVC systems. +Acknowledgements: +The present investigation has been carried out under financial support from BHU RET +fellowships to VNS from Banaras Hindu University (BHU), Varanasi. BPM acknowledges +the support from the Research grant under IoE scheme (Number- 6031), UGC-Govt. of +India. MH acknowledges supports from SPO-ISRO HQ for the encouragement of research +activities. +31 + +References +[1] N. 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Taussig, Scattering from a locally periodic potential, American +journal of physics 60 (10) (1992) 883–888. +35 + diff --git a/itAyT4oBgHgl3EQfxvny/content/tmp_files/load_file.txt b/itAyT4oBgHgl3EQfxvny/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..830ce29e2231417499959738198aa0c48758bd3a --- /dev/null +++ b/itAyT4oBgHgl3EQfxvny/content/tmp_files/load_file.txt @@ -0,0 +1,1143 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf,len=1142 +page_content='Quantum tunneling from family of Cantor potentials in fractional quantum mechanics Vibhav Narayan Singh1, Mohammad Umar2, Mohammad Hasan3, Bhabani Prasad Mandal4 1,4 Department of Physics, Banaras Hindu University, Varanasi-221005, INDIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 2 Indian Institute of Technology, Delhi-110016, INDIA 3Indian Space Research Organisation, Bangalore-560094, INDIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Abstract We explore the features of non-relativistic quantum tunneling in space fractional quantum mechanics through a family of Cantor potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' We consider two types of potentials: general Cantor and general Smith-Volterra-Cantor potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The Cantor potential is an example of fractal potential while the Smith-Volterra-Cantor potential doesn’t belong to the category of a fractal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The present study brings for the first time, the study of quantum tunneling through fractal potential in fractional quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' We report several new features of scattering in the domain of space fractional quantum mechanics including the emergence of energy-band like features from these systems and extremely sharp transmission features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Further the scaling relation of the scattering amplitude with wave vector k is presented analytically for both types of potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 1e-mail address: vibhav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='ecc123@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='com 2e-mail address: pha212475@iitd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='in 3e-mail address: mhasan@isro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='in, mohammadhasan786@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='com 4e-mail address: bhabani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='mandal@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='com, bhabani@bhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='in 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='00674v1 [quant-ph] 29 Dec 2022 1 Introduction Over the last two decades, fractional dynamics have been a diverse area of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The concept of fractional quantum mechanics was introduced by Laskin in the year 2000 [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The motivation behind this work was to extend the path integral (PI) formulation of quantum mechanics (QM) [3] to the more broader class of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In the PI formulation of QM, the path integrals are taken over Brownian paths which lead to the Schrodinger equation of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' However, the Brownian paths are the subset of a broader general class of paths known as Levy paths characterized by a Levy index α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' For α = 2, all Levy paths are Brownian paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' When the PI formulation of QM is extended to Levy paths, one get the fractional Schrodinger equation [1,2] and the associated quantum mechanics is known as space fractional quantum mechanics (SFQM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' A time fractional Schrodinger equation was proposed by Naber [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Later Wang and Xu [5] combined the two kinds of fractional Schrodinger equation together to construct a space-time fractional Schrodinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' These generalization of QM may help to describe more extensive phenomena of the microscopic world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The Levy paths has fractal dimension α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In the case of SFQM, the range of α is 1 < α ≤ 2 [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The domain of SFQM have grown fast over the last two decades and various applications are discussed by different authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Some of the notable work are the energy band structure for the periodic potential [6],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' position-dependent mass frac- tional Schrodinger equation [7],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' fractional quantum oscillator [8],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' nuclear dynamics of the H+ 2 molecular ion [9],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' propagation dynamics of a light beam [10],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' spatial soliton prop- agation [11],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' solitons in the fractional Schrodinger equation with parity-time-symmetric lattice potential [12],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' gap solitons [13],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Rabi oscillations in a fractional Schrodinger equa- tion [14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' self-focusing and wave collapse [15],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' elliptic solitons [16],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' light propagation in a honeycomb lattice [17],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' scattering features in non-Hermitian SFQM [18],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' tunneling time [19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 20] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Different methods are used in such studies such as domain decom- position method [21], energy conservative difference scheme [22], conservative finite el- ement method [23], fractional Fan sub-equation method [24], split-step Fourier spectral method [25], transfer-matrix method [26] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The term fractal was first coined by Mandelbrot [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Fractals are geometric objects which have self-similarity and homogeneity at all known scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The geometric structures of fractals at a given scale or stage are obtained through a basic mathematical operation acting on the geometric object known as ‘initiator’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The process of mathematical opera- tion is called ‘generator’ which can be repeated on multiple levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Through ‘generator’, a geometrical object with sub-units are created that resembles the structure of the entire object (the initiator) [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Due to the fact that the real numbers can be divided arbitrarily, the self-similarity of fractals hold at all scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Since nature has many fractal structures, regular and irregular fragmented structures can be understood/approximated in the con- text of fractals [27,29,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' However, in nature, the self-similarity doesn’t hold at all scales 2 and in general, there exists an upper and lower limit within which the self-similarity applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' One-dimensional scattering by a Cantor fractal potential is one of the simplest scat- tering problems of quantum tunneling through fractal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This problem have been extensively studied in quantum mechanics by using the transfer matrix method to derive various scattering properties [31–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The composition properties of the transfer matrix have been used to derive the scattering coefficients and associated properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In Cantor fractal potential, scattering coefficients have been found to show scaling law and sharp features of resonances k [31, 32, 36–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The tunneling amplitude from Cantor potential can also be derived by using the concept of super periodic potential (SPP) [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Despite the several advancement in the study of SFQM as well as quantum tunneling from fractal potentials, at present tunneling properties from fractal potentials in SFQM is not yet studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' It is expected that such studies will bring new features of scattering properties in the domain of SFQM (α < 2) which are absent in the case of standard QM (α = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In the present study we mainly focus on the simplest fractal system in one dimension, Cantor fractal along with an another member of Cantor family potential known as Smith-Volterra-Cantor (SVC) potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The SVC potential is not a fractal potential while the Cantor potential is a fractal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In order to keep the study more general in nature, we consider the general Cantor (GC) and general SVC (GSVC) potential system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' These are constructed in such a way that for a given initial length L and height V of the rectangular barrier potential, a fraction of 1 3 from the middle is removed at every stage ‘G’ from the remaining segments for standard Cantor-3 potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' For GC potential (or Cantor-ρ potential), instead of 1 3, a fraction 1 ρ is removed where ρ > 1 is a real positive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Similarly in GSVC potential (or SVC-ρ) potential, a fraction of 1 ρG is removed from the middle at each stage G instead of 1 4G as in case of standard SVC-4 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Again ρ ∈ R+ and ρ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' A simple observation shows that SVC system doesn’t satisfy the criteria for the same ‘self-similarity’ at each stage G and therefore is not a fractal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In an earlier work, we have shown that Cantor-3 and SVC-4 potential system are the special case of SPP [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This is also true for Cantor-ρ and SVC-ρ system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' SPP concept is the generalization of periodic potential having arbitrary number of internal periodicity [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' As we have not yet extended the concept of SPP in the domain of SFQM, we use the fundamental principle to derive the expressions for transmission amplitude using transfer matrix approach for both types of potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' We report new features of scattering from these systems in the domain of SFQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Notable features are emergence of energy band structures from these potentials which are absent in standard QM and extremely sharp transmission resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The scaling behavior with wave vector k is also presented analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 3 This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In section 2, an overview of SFQM is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The transfer matrix in SFQM for a localized and repeated potential is discussed in detail in the section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In section 4 and 5, we provide a brief review of the symmetric fractal potential of the cantor family and its repeated system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In next section 6, explicit ex- pression of ‘ζj’ (argument of Chebyshev polynomial of second kind) is expressed in order to get transmission amplitude in SFQM for general SVC and general Cantor potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Afterward, in section 7, we provide graphically a detailed analysis of the transmission features for both the fractal potential in the domain of SFQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Finally, at last, in section 8 results and discussion are mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 2 Space fractional Schrodinger equation When the path integral formulation of quantum mechanics is generalized over Levy flight paths, it results in space fractional quantum mechanics (SFQM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The governing equation for SFQM is the space fractional Schrodinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The form of space fractional Schrodinger equation is given by [1], i¯h∂ψ(x, t) ∂t = Hα(x, t)ψ(x, t), (1) Where, Hα(x, t) is the fractional Hamiltonian operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The Hamiltonian is expressed through the use of Riesz fractional derivative (−¯h2∆)α/2 as, Hα(x, t) = Dα(−¯h2∆)α/2 + V (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (2) Here ‘α’ is the Levy index and ∆ = ∂2 ∂x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In SFQM, the range of α is 1 < α ≤ 2 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Dα is a constant, also called as generalized diffusion coefficient and depends upon system characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The Riesz fractional derivative of the wave function ψ(x, t) is defined through the use of Fourier transform of ψ(x, t) as, (−¯h2∆)α/2ψ(x, t) = 1 2π¯h � ∞ −∞ ˜ψ(p, t)|p|αeipx/¯hdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (3) The Fourier transform of ψ(x, t) is given by, ˜ψ(p, t) = � ∞ −∞ ψ(x, t)e−ipx/¯hdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (4) and its inverse Fourier transform is, ψ(x, t) = 1 2π¯h � ∞ −∞ ˜ψ(p, t)eipx/¯hdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (5) For the case when potential V (x, t) is time independent i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=', V (x, t) = V (x), we have the time independent fractional Hamiltonian operator Hα(x) as, Hα(x) = Dα(−¯h2∆)α/2 + V (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (6) 4 The time-independent space-fractional Schrodinger equation is, Dα(−¯h2∆) α 2 ψ(x) + V (x)ψ(x) = Eψ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (7) By using the concept of separation of variables, it can be shown that the time independent wave function ψ(x) is related to ψ(x, t) as ψ(x, t) = ψ(x)e−iEt/¯h where E is the energy of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' For a detail discussion on SFQM readers are referred to [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In the next section, we briefly discuss the transfer matrix formulation of the tunneling problem in SFQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 3 Transfer matrix in SFQM Figure 1: Depiction of the scattering of the quantum wave from an arbitrary potential V (x) in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Consider a localized potential V (x) bounded in the region (−a, a) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The solution of time independent space fractional Schrodinger equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 7) in all the three regions x < −a, −a < x < a, and x > a are, ϕ(x) = Aeikαx + Be−ikαx, x < −a, (8) ϕ(x) = ϕab(x), −a < x < a, (9) ϕ(x) = Ceikαx + De−ikαx, x > a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (10) Where, kα = � E Dα¯hα �1/α (11) and the coefficients A, B, C, and D are the amplitudes of the waves on either side of the potential V (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The solution of the space fractional Schrodinger equation provides two 5 linear equations in terms of the coefficients A, B, C, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The two linear equations can be represented in matrix form as, �A(kα) B(kα) � = M(kα) �C(kα) D(kα) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (12) M(kα) is a 2 × 2 matrix, M(kα) = �M11(kα) M12(kα) M21(kα) M22(kα) � , (13) which is known as the transfer matrix of the potential V (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' For the case when V (x) is Hermitian, the time invariance property of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 7 leads to M11(kα) = M22(kα)∗, M21(kα) = M12(kα)∗, (14) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=', the diagonal and off-diagonal elements are complex conjugate to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The determinant of the transfer matrix is always unity which together with the above property implies |M11(kα)|2−|M12(kα)|2= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' If the transfer matrix of a potential V (x) is known then one can obtain the scattering coefficients for the potential V (x) through the following expression, tl(kα) = tr(kα) = 1 M22(kα), rl(kα) = −M21(kα) M22(kα), rr(kα) = M12(kα) M22(kα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (15) From the knowledge of the transfer matrix of a single localized potential V (x), one can obtain the transfer matrix of the periodic potential when V (x) is periodically repeated N1 times [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Formulation of the transfer matrix of locally periodic media from the knowledge of the transfer matrix of single ‘unit cell’ potential V (x) is also applicable in space fractional quantum mechanics [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The transfer matrix MN1(kα) for the periodic potential is given by, MN1(kα) = �[M11e−ikαsUN1−1(ζ1) − UN1−2(ζ1)]eikαN1s M12UN1−1(ζ1)e−ikα(N1−1)s M ∗ 12UN1−1(ζ1)eikα(N1−1)s [M ∗ 11eikαsUN1−1(ζ1) − UN1−2(ζ1)]e−ikαN1s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (16) In the above expression, ‘s’ is the separation between the starting points of two consecutive ‘unit cell’ potentials and UN(ζ1) is the Chebyshev polynomial of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The argument of Chebyshev polynomial ‘ζ1’, which is the Bloch phase of the corresponding fully developed periodic system, is computed from the knowledge of the ‘unit cell’ transfer matrix and the separation ‘s’ as [43], ζ1(kα) = 1 2 � M11e−ikαs + M22eikαs� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (17) Using the property M11(kα) = M22(kα)∗, the above equation can also be written as, ζ1(kα) = Re[M22] cos(kαs) − Im[M22] sin(kαs) = |M22|cos(φ + kαs), (18) 6 where φ is the argument of M22, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=', M22 = |M22|eiφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The transmission coefficient for the periodic potential is the inverse of the lower diagonal element of the matrix given by 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Using the unitary properties of the transfer matrix, the transmission amplitude T = |tl,r|2 can be obtained as [43], T(N1) = 1 1 + [|M12|UN1−1(ζ1)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (19) A few comments and the associated generalizations are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' We can write the term [|M12|UN1−1(ζ1)]2 appearing in the above equation as [|M12|UN1−1(ζ1)]2 = |M12UN1−1(ζ1)|2 = |(M12)N1|2 where (M12)N1 is the (1, 2) element of the transfer matrix (TM) given by 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This can also be read as, |M12 element of periodic system TM|= |M12 element of unit cell TM× UN1−1(Bloch phase of the fully developed periodic system)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (20) If we periodically repeat this periodic system N2 times with a different periodic distance s2, then from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 20, the modulus of (1, 2) element, |(M12)N1,N2| of the transfer matrix of the new periodic system will be given by, |(M12)N1,N2|= |(M12)N1UN2−1(ζ2)|= |M12UN1−1(ζ1)UN2−1(ζ2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (21) Where ζ2 is the Bloch phase for the new periodic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' If we periodically repeat the systems with parameters Ni and si where i = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=', G which yield ζ1, ζ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='ζG as the respective Bloch phases, then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 21 easily generalizes to |(M12)N1,N2,N2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=',NG|= |M12 G � i=1 UNi−1(ζi)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (22) The corresponding transmission amplitude can be obtained from T(N1, N2, N3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=', NG) = 1 1 + |(M12)N1,N2,N2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=',NG|2 = 1 1 + |M12|2�G i=1 U 2 Ni−1(ζi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (23) A rigorous proof of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 23 based on the transfer matrix elements for super periodic potential is presented in [41] for the case of standard QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In particular, when Ni = 2, we have UNi−1(ζi) = U1(ζi) = 2ζi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Substitution of this in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 23 leads to T(2, 2, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=', G times) = TG = 1 1 + 4G|M12|2�G i=1 ζ2 i , (24) and the transfer matrix becomes, MN1=2(kα) = �2M ∗ 22ζ1eikαs − e2ikαs 2M12ζ1e−ikαs 2M ∗ 12ζ1eikαs 2M22ζ1e−ikαs − e−2ikαs � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (25) It is to be noted that the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 24 is the general expression for tunneling amplitude when a single potential cell is repeated only two times and that system as a whole is 7 further repeated two times and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' We will extensively use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 24 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 25 to calculate tunneling amplitudes for the symmetric potential of Cantor family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' It turns out that tunneling amplitude for any symmetric potential which is generated by the division of a real line in three parts and subsequent removal of the middle segment can be expressed using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' We will discuss this in detail in the subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 4 Symmetric potential of Cantor family In one dimension, a fractal is generated by the division of a real line in a fashion which preserves self-similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Similarly, a rectangular fractal potential can be generated by dividing the length of the barrier in a self-similar fashion while keeping the height of the barrier unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Symmetric fractal potential obeys parity symmetry about the origin i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=', the fractal potential is symmetric with respect to changing x → −x and −x → x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' A potential of Cantor family is generated when the line segments are divided into three parts and the middle parts are removed at any stage G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' A particular case of the symmetric Cantor potential is when the removal of the middle part from the line segment leaves the resultant two segments of equal sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This configuration of the system is always symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Starting from a length L and stage G = 0, symmetric Cantor potential can be generated by the removal of a fraction 1 ρa1+a2G from the middle segment(s) at each stage G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Here ρ ∈ R+ and a1, a2 ∈ {0, R+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' When a1 = 1 and a2 = 0, we have general Cantor potential (also, for a2 = 0, a1 can be absorbed by defining ρa1 = ρ1 for some real ρ1 and we still have general Cantor potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Similarly, for a1 = 0, we have general Smith-Volterra-Cantor (SVC) potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' For the special case when a1 = 0, a2 = 1 and ρ = 4 we have standard SVC potential system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Again when a1 = 0, we can absorb a2 by defining an associated new ρ and the fractal potential is named an SVC-ρ system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The geometrical construction of general Cantor and general SVC potential is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' At any stage G, both general Cantor and general SVC potential have 2G segments of equal length lG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The value of lG are different for both types of potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In the case of general Cantor, lG = �ρ − 1 2ρ �G L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (26) For the case of general SVC, lG can be obtained through the use of the q-Pochhammer symbol as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 2, it is noted that, l1 = L 2 � 1 − 1 ρ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (27) Similarly, the segment length l2 for stage G = 2 is, l2 = l1 2 � 1 − 1 ρ2 � = L 22 � 1 − 1 ρ � � 1 − 1 ρ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (28) Similarly, l3 = l2 2 � 1 − 1 ρ3 � = L 23 � 1 − 1 ρ � � 1 − 1 ρ2 � � 1 − 1 ρ3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (29) 8 [H] Figure 2: Construction of Cantor-ρ and SVC-ρ potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The white region shows the gap between the potentials and the height of the opaque region is the potential height V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Here G represents the stage of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In Cantor-ρ potential, a fraction 1/ρ is removed at every stage while in SVC-ρ, a fraction 1 ρG is removed at each stage G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' By continuing the same steps, the segment length ‘lG’ for arbitrary Gth order SVC-ρ 0 potential is obtained as, lG = L 2G G � i=1 � 1 − 1 ρi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (30) The product series can be recognized as, G � i=1 � 1 − 1 ρi � = q �1 ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 1 ρ � G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (31) Where, q(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' λ)n = n−1 � i=0 (1 − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='λi) = (1 − a)(1 − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='λ)(1 − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='(1 − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='λn−1) (32) 9 GC GSVC GC GSVC GC GSVCis q-Pochhammer symbol [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Therefore, through the use of the q-Pochhammer symbol, we can express lG as, lG = L 2Gq �1 ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 1 ρ � G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (33) 5 Symmetric Cantor family potentials as repeating systems In this section, we illustrate that a symmetric potential of the Cantor family can be generated through a ‘unit cell’ by repeating it two times and then repeating the resultant ‘cell’ further two times and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Consider a rectangular barrier of height V and width lG as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' We can repeat this barrier at a distance s1 > lG as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The resultant system of these two barriers are further repeated at a distance of s2 thereby generating a system of four rectangular barriers which as a whole is further repeated at a distance of s3 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This process of repeating the resultant barrier systems two times at a specific distance can continue up-to an arbitrary stage G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' As the Cantor family systems are well defined mathematical structures, the value of ‘lG’ and various ‘si’ can be easily identified for any arbitrary stage G for a particular system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' [H] Figure 3: Construction of the symmetric Cantor family potential for the stage G = 4 as periodic repetition of the periodic system of order 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 10 First we present general expression of sj for Cantor-ρ fractal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' For this system we have, s1 = lG + lG−1 ρ , s2 = lG−1 + lG−2 ρ , s3 = lG−2 + lG−3 ρ , The above sequences show that, sj = lG+1−j + lG−j ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (34) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 26, this can be simplified to, sj = xG−jyL, (35) where, x = ρ − 1 2ρ , y = ρ + 1 2ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (36) Similarly, it can be shown that for SVC-ρ potential, sj is given by, sj = lG+1−p + lG−p ρG+1−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (37) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 30 in the above expression, we have after simplification sj = L 2G+1−j � 1 + 1 ρG+1−j � q �1 ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 1 ρ � G−j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (38) For a given G, by choosing a single barrier of length lG as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 26 and placing the barrier at various sj as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 35, we get Cantor-ρ potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Similarly, by choosing lG from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 33 and sj from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 38 we get SVC-ρ potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In the next section, we calculate the transmission amplitudes from these two types of fractal potentials in SFQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 6 Transmission amplitudes in SFQM It is clear from the discussion in the previous section that (symmetric) Cantor-ρ (GC) and SVC-ρ (GSVC) potentials are the special cases of systems that are repeated two times and that configuration as a whole is further repeated two times and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The number of such operations of repetitions is equal to the stage G of the GC and GSVC potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The general expression of the tunneling amplitude for such a potential system in SFQM is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' What remains is to calculate the general expressions for ζi, i = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='., G for GC and GSVC potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' We will derive the general expression for ζi and then would specialize to calculate specific expressions for ζi for GC and GSVC potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The calculations are illustrated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 11 Let M22(kα) denotes the lower diagonal elements of the transfer matrix of rectangular barrier of width b = lG and height V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This potential configuration is represented by P0 in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Similarly, let (M22)1, (M22)2, (M22)3 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' denote the lower diagonal elements of the transfer matrix of the combined system represented by P1, P2, P3 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The corresponding Bloch phases are ζ1, ζ2, ζ3 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 25 we can read that (M22)j = 2(M22)j−1ζje−ikαsj − e−2ikαsj, (39) where j = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='., G and (M22)0 = M22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Now from the general Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 18 we can write, ζ2(kα) = Re[(M22)1] cos kαs2 − Im[(M22)1] sin kαs2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (40) We can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 39 in the above equation so that, ζ2(kα) = Re[(2 × M22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='ζ1)e−ikαs1 − e−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='ikαs1] cos kαs2 − Im[(2 × M22ζ1)e−ikαs1 − e−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='ikαs1] sin kαs2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (41) The simplification of the real and imaginary parts finally gives, ζ2 = 2|M22|ζ1 cos [φ − kα{s1 − s2}] − cos [kα{2s1 − s2}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (42) Similarly, repeating the above procedure to calculate ζ3 we have, ζ3 = Re[(M22)2] cos kαs3 − Im[(M22)2] sin kαs3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (43) Again using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 39 to simplify the above, we obtain for ζ3, ζ3(kα) = 22|M22|ζ1ζ2 cos [φ − kα{s1 + s2 − s3}]− 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='ζ2 cos [kα{2s1 + s2 − s3}] − cos [kα{2s2 − s3}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (44) Similarly, we have for ζ4 ζ4(kα) = Re[(M22)3] cos kαs4 − Im[(M22)3] sin kαs4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (45) The repeated application of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 39 and simplifications of the real and imaginary parts in the above equation gives, ζ4(kα) = 23|M22|ζ1ζ2ζ3 cos [φ − kα{s1 + s2 + s3 − s4}] − 22ζ2ζ3 cos [kα{2s1 + s2 + s3 − s4}] − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='ζ3 cos [kα{2s2 + s3 − s4}] − cos [kα{2s3 − s4}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (46) Similarly, the expression for ζ5 is given by, ζ5(kα) = 24|M22|ζ1ζ2ζ3ζ4 cos [φ − kα{s1 + s2 + s3 + s4 − s5}] − 23ζ2ζ3ζ4 cos [kα{2s1 + s2 + s3 + s4 − s5}] − 22ζ3ζ4 cos [kα{2s2 + s3 + s4 − s5}] − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='ζ4 cos [kα{2s3 + s4 − s5}] − cos [kα{2s4 − s5}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (47) 12 We observe from the sequence of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 42, 44, 46, and 47 that the general expression for ζj can be written in the following series form, ζj(kα) = 2j−1|M22|cos [φ − kαη1(j)] j−1 � p=1 ζp − j−1 � r=1 � 2j−r−1 cos [kαη2(j, r)] j−1 � p=r+1 ζp � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (48) In the above equation, we have used the following notation, η1(j) ≡ � j−1 � p=1 sp � − sj, (49) η2(j, r) ≡ � j � p=r sp � − (2sj − sr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (50) It is easy to show that, η2(j, r) ≡ η1(j) − η1(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (51) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 48 is the general expression for ζj, j = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='., G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' However, it is important to note here that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 48, we have to drop the terms when the running variable ‘r’ is more than the upper limit for the summation operation and we take terms as unity when the running variable is more than the upper limit for the product operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' From the knowledge of ζ1, ζ2, ζ3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=', ζG, we can calculate the tunneling amplitude from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Now we calculate the values of η1,2 and their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Figure 4: Symmetric Cantor family potential shows the length and gap between the seg- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 3 and 4, we observe s1 = lG + gG, s2 = s1 + lG + gG−1, s3 = s1 + s2 + lG + gG−2, 13 L G=0 11 11 G= 1 91 12 12 12 12 G=2 92 91 92 13 13 I3 I3 13 13 I3 13 G=3 93 92 91 [g3 93 92 g3s4 = s1 + s2 + s3 + lG + gG−3 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Thus we arrive at, sj = � j−1 � p=1 sp � + lG + gG−j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (52) Therefore, η1(j) is given by, η1(j) = −(lG + gG−j+1), (53) which shows that η1(j) is always negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 51 and 53 we get, η2(j, r) = gG−r+1 − gG−j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (54) We see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 4 that for i > j, gi < gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Also for r < j, G − r + 1 > G − j + 1 which implies gG−r+1 < gG−j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Therefore η2(j, r) < 0 for r < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 53 and 54 gives the general expression for η1 and η2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Now we provide these expressions for GC and GSVC cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='1 Case 1: General SVC potential To calculate η1,2 for GSVC, we re-write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 33 as lj−1 = L 2j−1q �1 ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 1 ρ � j−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (55) As we know, for GSVC a fraction 1 ρj is removed from segment length lj−1 to generate the system for stage G = j, therefore, gj = lj−1 ρj and hence, gj = L ρj2j−1q �1 ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 1 ρ � j−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (56) Now we simplify for η1(j) by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 53, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 33 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 56 to obtain, η1(j) = − � L 2Gq �1 ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 1 ρ � G + L 2G−j q �1 ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 1 ρ � G−j 1 ρG−j+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (57) Now we calculate η2(j, r) by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 51 and 57 to obtain, η2(j, r) = 2L (2ρ)G+1 � (2ρ)rq �1 ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 1 ρ � G−r − (2ρ)jq �1 ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 1 ρ � G−j � (58) Now we can substitute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 57 and 58 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 48 to obtain the general expression for ‘ζj’ for GSVC potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 14 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 Case 2: General Cantor potential We re-write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 26 as, lj−1 = �ρ − 1 2ρ �j−1 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (59) As we know, in case of GC potential, a fraction 1 ρ is taken from stage G = j − 1 to create the fractal system for G = j stage, therefore gj = lj−1 ρ and thus, gj = 1 ρ �ρ − 1 2ρ �j−1 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (60) Now, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 53 and 26 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 60, we simplify for η1(j) to get, η1(j) = − � L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' �ρ − 1 2ρ �G + L ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' �ρ − 1 2ρ �G−j� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (61) Now using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 51, η2(j, r) can be simplified as, η2(j, r) = L ρ �ρ − 1 2ρ �G−r−j ��ρ − 1 2ρ �j − �ρ − 1 2ρ �r� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (62) Substitution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 61 and 62 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 48 gives the general expression of ‘ζj’ for GC potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 7 Transmission features In the previous section, the analytical expressions of the tunneling amplitudes from two types of Cantor potentials in SFQM have been derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In this section, we study the various features of transmission through these systems in SFQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' As the Cantor potentials have been studied in detail in standard QM (α = 2) [31–41], therefore we largely focus here to study the tunneling behavior in the domain of SFQM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=', the case of α < 2) as well as the comparison with the case of standard QM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=', the case of α = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 5 shows the comparison of the profiles of the transmission amplitudes for GC and GSVC potential in standard QM and in SFQM for different stages G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In all plots of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 5, it is noted that the transmission resonances are much sharper in GSVC potential as compared to GC potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' As these two types of potentials are different, they show different transmission profiles which are not relatable (at the present level of investigations) though both are special cases of repeating systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Therefore, we present these two cases separately in subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='1 discusses the case of GSVC while subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 details the case for GC potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 15 G=3 G=5 α=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='00 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='90 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='80 Figure 5: Plots showing the comparison of transmission amplitudes for GC (Red-curve) and GSVC (Blue-curve) potential for two different stages of G (= 3 and 5) in SFQM (α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='00, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='90 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Here potential parameters are L = 1, V = 400 and ρ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' From figures, it is observed that GSVC potential has sharper peaks as compared to general Cantor potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 40 10 20 30 k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 40 10 20 30 k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 10 20 30 40 k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 40 10 20 30 k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 40 10 20 30 k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 10 20 30 40 k7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='1 Transmission features of general SVC potential in SFQM (a) (b) (c) Figure 6: The transmission amplitude for GSVC potential in SFQM with α and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The potential parameters are V = 100, ρ = 3 and G = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The transmission peaks occurs at lower k values with decreasing α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' It is also evident from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (c) that the sharpness of the transmission peaks are increasing as α is lowered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This section exclusively discusses the nature of the transmission profile from GSVC system in SFQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' As the expression for the transmission amplitude is transcendental in nature (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 24), presently we rely on the numerical investigation towards investigating the general features of tunneling amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The transmission amplitude is plotted for stage G = 3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' To understand the behavior of transmission resonances with α, we 3D plot T(α, k) with α and k as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 6-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Here the potential parameters are V = 100, ρ = 3, and G = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' A closer look at this figure is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 6-b for a smaller range of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' From both these figures, it is seen that the locus of transmission resonances has a positive slope with increasing α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This indicates that the transmission peaks are red-shifted with decreasing values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This appears to be a general trend for 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 α=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='98 T α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='92 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='90 10 12 14 16 18 20 k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='98 α 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='92 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 0 10 0 20 30 k 40 502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='98 α 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='92 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 0 10 0 12 14 16 k 18 20the case when α is not far away from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' However, much more complex behavior of the locus of transmission resonances is seen when α is closer to 1 and is presented later in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 6-c shows the 2D plot depicting the variation of T for different α with the same range of k as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 6-b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This figure shows that the transmission resonances become sharper at lower values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This appears to be a general feature and will be more evident in the later part of the discussion and associated graphical representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (a) (b) (c) (d) (e) (f) Figure 7: Plots showing several sharp transmission resonances near unity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='e α = 1) for GSVC potential of stage G = 5 in SFQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The potential parameters are V = 450, L = 1 and ρ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 18 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 27 29 31 33 35 25 kα=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 0 27 29 25 31 33 35 kα=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 27 29 31 33 35 25 kα=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 27 29 31 33 35 25 kα=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 25 27 29 31 33 35 kα=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 25 27 29 31 33 35 kAn interesting parameter region for the study of tunneling amplitude in SFQM is the case when α is close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In this regime, extreme behavior in the transmission amplitudes is observed which is demonstrated graphically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The figure shows the transmission amplitude for stages G = 5, V = 450, L = 1, and different values of α near unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In all these figures, the emergence of several extremely sharp transmission resonances is observed for both evanescent and non-evanescent waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The transmission resonances are separated by deep valleys in the T(k) profile such that T(k) vanishes over a range of k (it may be noted that for any Hermitian potential, as in the present case transmission amplitudes are never ideally zero [45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Many transmission resonances in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 7 are extremely sharp and appear as the sudden jump from T = 0 to T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Towards understanding these features over a continuous range of α near 1, the transmission amplitudes are represented through density plots in α − k plane for different stages of the potential in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 8 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' For both these figures L = 1, ρ = 3 while V = 300 and 450 for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 8 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 9 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (a) (b) Figure 8: Density plot showing the variation of transmission amplitude T in α − k plane for α close to 1 for GSVC potential of different stages G = 5 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The potential parameters are V = 300, L = 1 and ρ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Extreme sharp transmission resonances are seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' For both the stages of the potential, extreme behavior of variations in T is observed for wave energy E < V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' However, for E > V , the transmission amplitudes are observed to saturate with increasing G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Further, deep minima in T occur in the transmission profile for several finite ranges of k which indicates the presence of allowed and forbidden band-like structures from this potential system in SFQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The different stages G are shown in the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Both these figures indicate the pres- ence of extremely sharp transmission resonances as thin streaks of yellow lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In some cases, the lines are so thin that these are not captured graphically over the red regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The behavior of these T = 1 loci is challenging to understand analytically due to the transcendental nature of the expression of the tunneling amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Extreme variations for both α and k are observed for wave energy E < V while for E > V the transmission 19 G=5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 α 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='01 0 5 10 15 20 25 30 35 kG=7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 α 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='01 0 5 10 15 20 25 30 35 kprofile appears to saturate with increasing G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' An apparent conclusion that may be drawn from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 8 and 9 is the presence of deep valleys in the transmission amplitudes for which T is nearly vanishing in α−k plane for α in the vicinity of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The presence of these valleys in the transmission amplitudes is noted as the precursor of the emergence of allowed and forbidden energy bands [46] from locally periodic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This shows that the band features emerge in the case of GSVC potential in SFQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In the case of standard QM, the band doesn’t appear from Cantor family potential to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' However, the emergence of band-like features from GSVC potential (and will be shown later for GC potential) appears only when α is in the vicinity of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (a) (b) (c) (d) Figure 9: Density plot showing the variation of transmission amplitude T in α−k plane for α close to 1 for GSVC potential of different stage G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Here V = 450 and other parameters are the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Again, the presence of extremely sharp transmission resonances is noticed with extreme variations in T for wave energy E < V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' It is also seen in the figure that the transmission amplitude saturates with increasing G for E > V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Again, the density plot shows the occurrence of band-like features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 20 G=9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 α 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='000 5 10 15 20 25 30 35 40 0 k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0 G=11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 α 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='000 5 10 15 20 25 30 35 40 0 k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0 G=13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 α 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='000 5 10 15 20 25 30 35 40 0 k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0 G=15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 α 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='000 5 10 15 20 25 30 35 40 0 kA discussion is in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The deep valleys in T(k) profile for a periodic potential for a range ∆k means that the waves are reflected from the potential for k ∈ ∆k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' From the emergence of deep valleys in T for tunneling through locally periodic delta potential, it is argued that the band-like structures emerge even when number of periodic delta barriers are just five, N = 5 [43,46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Based on the similar studies in SFQM, it is noted that the band emerges even when N = 4 and are more prominently present for lower α values [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In the present case, the repetitions are based on Ni = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' As N = 2 system doesn’t show band structure (and deep valleys in T for ranges of k) in standard QM, the type of Cantor family system studied here don’t show allowed energy bands in standard QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (a) (b) (c) (d) Figure 10: Plot of transmission amplitude for G = 1 (double barrier system), above Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (a) shows density plot for potential height V = 500 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (b) represent 2D plot for potential height V = 648 with potential width L = 1 for both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Both these plots show the occurrence of valleys for which T is very close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (c) shows the density plots for potential parameters V = 700, L = 1 in α − k plane for range of α from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='001 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='01 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (d) corresponding 2D plot when α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='004 for same potential parametrs as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (c) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The density plot clearly shows range of k for which T nearly vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This again indicates the presence of allowed bands for the double barrier system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Interesting oscillations are seen in T over α − k plane for the evanescent waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 21 G=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='V=500 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='006 α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='002 0 25 27 29 31 33 35 kG=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='V=700 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='006 α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='002 0 31 33 35 37 39 41 43 45 kα=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 38 37 39 kα=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 33 32 34 kThus, a question may arise that if the present GSVC system, which is an arrangement of Ni = 2 barriers, show band likes features for α → 1, would this also mean that a double barrier system will show energy bands like features for α → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Surprisingly, we find that this is indeed the case and are graphically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 10 for three different double barrier potential systems in SFQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' It is an extraordinary fact to recognize that there are allowed and forbidden bands for just double barrier systems in the domain of SFQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' If the N = 2 barriers system could show band structures in SFQM, therefore the present GSVC systems which are Ni = 2 repeating systems could also show energy band structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' We will show in the later section that this is also true for GC potential in SFQM for α near to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' General SVC General Cantor Figure 11: Plot of log10 (− log10 T) for the case of general SVC and general Cantor poten- tial for G = 7 (red curve), G = 9 (dashed green curve), and G = 11 (dashed blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The potential parameters are V = 100, L = 1 and ρ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' As it is clearly visible from first column (for general SVC) and second column (for general Cantor), that the tunnelling saturates with increasing G in standard (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=', α = 2) as well as in SFQM for general SVC potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' However, this saturation behavior is not observed for general Cantor potential in SFQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 22 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='80 5 10 15 10 20 30 40 50 kα=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='90 10 15 10 20 30 40 50 kα=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='00 Log10(-Log10T) 10 15 10 20 30 40 50 kα=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='00 Log10(-Log10T) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='10 15 10 20 30 40 50 kα=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='80 Log10(-Log10T) 5 10 15 10 20 30 40 50 kα=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='90 Log10(-Log10 T) 10 15 10 20 30 40 50 kAnother observation from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 9 is the saturation of the transmission profile with increasing G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' From the definitions of GSVC system, a portion 1 ρG is taken out from the middle at each stage G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Thus progressively lesser fractions are taken from each stage with increasing G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This would imply that for larger G, the transmission profile should saturate with G as only very thin portions are removed from the segments of the previous stages when G is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This is illustrated graphically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 11 in which a function of T is plotted for GSVC and GC potential for stages G = 7, 9 and 11 for different α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' For a better resolution in different T(k, G), we have plotted y = log10 (− log10 T) in y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' As 0 < T ≤ 1, therefore log10 T ≤ 0 and thus − log10 T ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This implies that the function y = log10 (− log10 T) is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Various plots with different α in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 11 show that the saturation in T(k) profile with G is observed in GSVC potential but not in GC potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' However, for α → 1, this saturation is present only in the case for wave energy E > V and not for E < V as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 12 for different potentials and α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' , V = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0 , V = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0 , V = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0 , V = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0 Figure 12: Plot of log10 (− log10 T) for the case of GSVC for G = 5 (red curve), G = 8 (dashed green curve), and G = 11 (dashed blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The potential parameters are L = 1, ρ = 3, and the height V is indicated in each figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' It is seen from all these plots that for values of α close to 1, T(k) profile saturates with G when wave energy E(= k2/2m) > V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' No saturation in the T(k) profile is observed when E < V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 23 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='01 0 Log10(-Log10T) 5 10 2 6 8 0 10 4 kα=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='01 Log10(-Log10T) 5 10 2 8 10 0 4 6 kα=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='01 0 Log10(-Log10T) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='5 10 2 6 8 10 0 4 kα=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='01 Log10(-Log10T) 5 10 2 6 8 10 0 4 k7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 Transmission features of general Cantor potential in SFQM (a) (b) (c) Figure 13: Plot showing the transmission profile for general Cantor of stage G = 3, ρ = 3, L = 1 and V = 100 for different value of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' It is evident from the plots that as α reduces, the transmission peaks shifts to lower values of wave numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (Right Image) 3D plot showing the variation of T with α and k which again depict that the transmission peak occurs at lower k values with reducing α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In the previous section, we provided some general features of scattering such as emergence of energy bands, increase in the sharpness of transmission resonances with reducing α, extreme features of transmission for α near to 1 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' for GSVC potential, In this section, we show that such features also exists for GC potential in SFQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 13, we plot T(α, k) with α and k with potential parameters as G = 3, V = 100, ρ = 3 and L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' A closer look of this figure is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 13-b for a smaller range of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Similar to the case of GSVC for α near 2, it is seen that the locus of transmission resonances has a positive slope with increasing α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This indicates that the transmission peaks are red-shifted with decreasing values of α for GC potential in SFQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' An exception to this could occur when 24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='98 α 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='92 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 10 0 20 30 k 40 502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='98 α 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='92 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 10 0 k 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 α=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='00 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='98 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='96 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='92 α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 10 16 12 14 18 20 kα is in the vicinity of 1 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 13-c shows the 2D plot depicting the variation of T for different α with the same range of k as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 13-b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Again, similar to the case of GSVC potential, it is observed that the transmission resonances become sharper at lower values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 14 also shows extremely sharp loci of transmission resonances as thin streaks of yellow lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (a) (b) Figure 14: Density plot showing the variation of transmission amplitude T in α − k plane for α close to 1 for GC potential of different stages G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Here V = 450, L = 1 and ρ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Along with the presence of very sharp transmission peaks, the plots also shows the region of deep valleys in α−k plane where transmission amplitude continuously vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' These deep valleys in the transmission profile are the precursor of energy band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' These lines are also separated by deep valleys in T(k) profile which depict the presence of energy band-like features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The deep valleys in T(k) profiles are also shown graphically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 15 through the density of T(α, k) in α−k plane as well as 2D plots for discrete values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' It is to be noted from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 15, that many sharp features of transmission resonances that are not visualized due to graphic limitations of capturing very thin streaks of lines are clearly seen in 2D plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 25 G=2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='015 α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 0 10 20 30 40 50 kG=3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 α 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 1 0 10 20 30 40 50 k(a) (b) Figure 15: (a) Density plot for GC potential of stage G = 4 showing sharp transmission in α − k plane and energy band features when α is close to 1 and (b) more closer view of density plot for α ranging from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='001 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 2D plots illustrate clearly sharp valley for α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='002, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='003, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Here potential parameters are V = 450, L = 1 and ρ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='3 Scaling behavior This section presents the scaling behavior of the reflection amplitudes R = |r|2 with k for both types of Cantor potentials considered in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This section also present on how the reflection amplitude behaves when height of the potential V varies in specific manner at each stage G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' For larger k, reflection amplitude R is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In this limit, R can be approximated as R ∼ 4G|M12|2 G � i=1 ζ2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (63) Again for larger k we have V k2 << 1 and upon Taylor expanding, it can be shown in the first order that |M12|2∼ �α − 1 α V lG �2 1 k 4(α−1) α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (64) 26 G=4, α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 25 27 29 31 33 35 kG=4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 α 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='001 0 10 20 30 40 kG=4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='001 0 25 27 29 31 33 35 kG=4, α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 25 27 29 31 33 35 kG=4,α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='6 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='2 25 27 29 31 33 35 kTherefore, the expression for R becomes, R ∼ 4G �α − 1 α V lG �2 1 k 4(α−1) α G � i=1 ζ2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (65) If VG is the height of the potential at each stage G, then it can be shown that the following value of VG keeps the total area of potential barrier (sum of the area of all potential segment at stage G) as constant VG = L 2GlG V0, (66) where V0 is the height of the potential barrier at G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Substituting the value of lG for GC and GSVC potentials, VG is given by VG = � ρ ρ − 1 �G V0, for GC and, VG = V0 q � 1 ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 1 ρ � G for GSVC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (67) If RG is the reflection amplitude at each stage G with potential height of each segment as VG then, it can be shown that (valid for large k) RG L2V 2 0 ∼ �α − 1 α �2 1 k 4(α−1) α G � i=1 ζ2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (68) Figure 16: Plot showing the reflection amplitudes for GSVC potential for G = 5 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The potential height VG is determined from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 67 for GSVC potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Other potential parameters are shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The difference between the two plots is invisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This shows the convergence of the product term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 16 we show the behavior of RGSV C 5 and RGSV C 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The difference between the two plots is invisible which shows the fast convergence of product term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 68 with 27 p=3,Vo=10,L=1,α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0015 RGSVC RGSVC 10 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0000 30 40 50 60 70 80 90 100 kincreasing G for GSVC case in SFQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Similar plots are shown for GC case for different α values in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The result shows that the convergence of the product term occurs for GC case in SFQM with increasing stage G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' (a) (b) Figure 17: Plots showing the reflection amplitudes for GC potential for G = 10 and 15 for different α value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The potential height VG is determined from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 67 for GC case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Other potential parameters are shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The difference between the two plots is nearly invisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This shows the convergence of product term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 68 for GC potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' For Cantor potential in standard QM, this has already shown in earlier work [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Again, due to the convergence nature of the product term (provided it is evaluated at VG) with increasing G, it is evident from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 68 that RG would scale as 1 k 4(α−1) α for large G and k values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' For α = 2, RG will scale as 1 k2 which is proven result for standard Cantor potential in standard QM [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The scaling behavior of RG with k in SFQM is shown graphically for GC potential in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' An interesting region of interest is when α is near to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In this case RG would scale horizontally with k which is indeed the case as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 18-c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The scaling behavior of RG with k is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 19 for GSVC potential for different α values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0005 p=3,Vo=5,L=1,α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='8 RGC 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0004 RGC 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0003 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0000 160 180 200 220 240 k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='020 RGC 10 RGC 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='015 p=3,Vo=10, L=1, α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='1 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='000 250 252 254 256 258 260 k(a) (b) (c) Figure 18: log − log Plots showing the scaling behavior of reflection amplitudes RG V 2 0 for large k in SFQM in case of general Cantor potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The dotted curve represent 1 k 4(α−1) α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='It is observed that at large k, RG falls of according to this expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The potential param- eters are shown in the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 29 p=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0, Vo=10, L=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0, α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='90 (α-1) 10-5 10-8 100 200 500 1000 kp=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0, Vo=10, L=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0, α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='01 10-6 100 200 500 1000 kp=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0, Vo=10, L=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0, α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='1 100 200 500 1000 k(a) (b) Figure 19: Plots shows reflection amplitude RG V 2 0 for large k in SFQM (α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='90 and α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='50) for general SVC potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The dotted curve represent 1 k 4(α−1) α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='It is observed that at large k, RG falls of according to this expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The potential parameters are shown in the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 8 Results and Discussions Fractional quantum mechanics is a fast-developing domain with several applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' We have studied the tunneling features from fractal (general Cantor) and non-fractal (gen- eral SVC) potential in space fractional quantum mechanics (SFQM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' To the best of our knowledge, this is the first time that the quantum tunneling from fractal potentials in the domain of SFQM are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' We have considered the generalized form of two kinds of potentials of Cantor family, namely general Cantor (GC) and general Smith-Volterra- Cantor (GSVC) potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' For both the kind of potentials, we have provided close form expressions of transmission amplitudes in SFQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' These close form expressions are ex- pected to provide better understanding of various scattering features in the domain of SFQM from fractal potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' It is to be noted that the derived expressions are of gen- 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='1 p=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0, Vo=5, L=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0, α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='90 10-4 R 10-10 10 100 200 500 1000 k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='1 p=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0, Vo=5, L=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='0, α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content='50 10-4 GSV 10-10 10-13 10-16 100 200 500 1000 keral type and valid for any potentials of GC and GSVC type in which the ‘unit cell’ is not a rectangular barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' As long as the transfer matrix of ‘unit cell’ potential is known, the derived expressions can be used to obtain the tunneling amplitudes from such kind of Cantor family potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In the present study, we have found several new features of scattering, and are re- ported graphically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The most striking feature is the appearance of energy band structures from fractal potential in SFQM which are absent in the case of standard Cantor frac- tal and standard SVC potentials in standard QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Standard fractal potentials based on 1 3 division of the real segments don’t show energy band-like features in standard QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' More surprisingly we noted that a double barrier potential system display energy bands in SFQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' We have reported the emergence of band structures for both the type, GC and GSVC potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' However, it is to be noted that these band like features appear in the extreme range of Levy index α close to the vicinity of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In this range of α, extremely sharp transmission resonances are found to occur for both type of potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Fractal potentials are known to display sharp transmission resonances, This feature is further amplified in the domain of SFQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' It is found that the sharpness of the transmission resonances further increases with a decrease in Levy index α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' In comparison, GSVC potential displays more sharp transmission resonances as compared to GC potential in standard QM as well as in SFQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Also for the case of GSVC potential, it is observed that the profile of transmission amplitudes saturates with increasing stage G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' The reason for this is due to the fact that a consecutively smaller fraction of the remaining previous segments is removed at each stage G for GSVC potentials Therefore for higher G, only very thin portions are removed from previous segments as compared to the case when G is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' This leads to the saturation of the tunneling profile with k for higher G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' However, this behavior is found to be different near α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' For α ∼ 1+, the tunneling profile saturates only for E > V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Another interesting feature is the scaling behavior of reflection amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' We have shown analytically that for large k, the reflection coefficient scale as 1 k 4(α−1) α for GC and GSVC potential provided that total area of the potential regions remain constant at different stages G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Also for such case, the reflection amplitude converges as G increases for both GC and GSVC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Acknowledgements: The present investigation has been carried out under financial support from BHU RET fellowships to VNS from Banaras Hindu University (BHU), Varanasi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' BPM acknowledges the support from the Research grant under IoE scheme (Number- 6031), UGC-Govt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' MH acknowledges supports from SPO-ISRO HQ for the encouragement of research activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' 31 References [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' Laskin, Fractional quantum mechanics and l´evy path integrals, Physics Letters A 268 (4-6) (2000) 298–305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} +page_content=' [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAyT4oBgHgl3EQfxvny/content/2301.00674v1.pdf'} 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+1,1544 @@ +StereoDistill: Pick the Cream from LiDAR for Distilling +Stereo-based 3D Object Detection +Zhe Liu1, Xiaoqing Ye2, Xiao Tan2, Errui Ding2, Xiang Bai1* +1Huazhong University of Science and Technology, 2Baidu Inc., China +zheliu1994@hust.edu.cn, yxq@whu.edu.cn, tanxchong@gmail.com, dingerrui@baidu.com, xbai@hust.edu.cn +Abstract +In this paper, we propose a cross-modal distillation method +named StereoDistill to narrow the gap between the stereo +and LiDAR-based approaches via distilling the stereo detec- +tors from the superior LiDAR model at the response level, +which is usually overlooked in 3D object detection distilla- +tion. The key designs of StereoDistill are: the X-component +Guided Distillation (XGD) for regression and the Cross- +anchor Logit Distillation (CLD) for classification. In XGD, +instead of empirically adopting a threshold to select the high- +quality teacher predictions as soft targets, we decompose the +predicted 3D box into sub-components and retain the corre- +sponding part for distillation if the teacher component pilot +is consistent with ground truth to largely boost the number +of positive predictions and alleviate the mimicking difficulty +of the student model. For CLD, we aggregate the probability +distribution of all anchors at the same position to encourage +the highest probability anchor rather than individually distill +the distribution at the anchor level. Finally, our StereoDistill +achieves state-of-the-art results for stereo-based 3D detection +on the KITTI test benchmark and extensive experiments on +KITTI and Argoverse Dataset validate the effectiveness. +Introduction +3D detectors equipped with LiDAR points (Shi, Wang, and +Li 2019; Yang et al. 2020; Deng et al. 2020; Chen et al. +2017b; Huang et al. 2020; Liu et al. 2022) for autonomous +driving have presented outperforming performance. How- +ever, LiDAR sensors usually have a high cost and sensitiv- +ity to weather, which limit their application. Alternatively, +stereo cameras are capturing increasing interest thanks to +their good trade-off in low cost and accuracy. There is still +a huge performance gap between stereo-based and cutting- +edge LiDAR-based 3D detection methods due to the inaccu- +rate depth estimation by stereo matching. A question natu- +rally arises: can the LiDAR model help to improve the per- +formance of the stereo model? +Knowledge distillation (KD) (Hinton et al. 2015) might +be a promising solution for this question, which guides the +student model to mimic the knowledge of the teacher model +for performance improvement or model compression. The +*Corresponding Author. +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +35 +40 +45 +50 +55 +60 +65 +70 +75 +80 +Car +Pedestrian +Cyclist +Stereo Model +Repalce Classfication +Replace Regression +Gains of StereoDistill +Figure 1: 3D detection performance (3D mAP) on KITTI valida- +tion set of LIGA (Guo et al. 2021) by replacing the regression and +classification results of the stereo model (student) with the teacher +LiDAR model SECOND (Yan, Mao, and Li 2018). +current KD methods of object detection can be mainly clas- +sified into the feature-based and response-based streams, +in which the former carry out distillation at the feature +level (Zagoruyko and Komodakis 2017; Romero et al. 2014; +Huang and Wang 2017; Heo et al. 2019; Ye et al. 2020; Du +et al. 2020) for enforcing the consistency of feature repre- +sentations between the teacher-student pair whereas the lat- +ter adopts the confident prediction from the teacher model +as soft targets in addition to the hard ground truth supervi- +sion (Yuan et al. 2020; Zheng et al. 2022; Dai et al. 2021). +However, directly migrating the existing KD methods to +LiDAR-to-stereo cross-modal distillation is less effective +due to the huge gap between the two extremely different +modalities. The pioneering work LIGA (Guo et al. 2021) +boosted the performance of stereo-based models by apply- +ing fine-grained feature-level distillation under the guidance +of LiDAR-based models. However, it found little benefit +from the response-based distillation due to the erroneous and +noisy predictions of the LiDAR teacher. +On the contrary, we argue that the response-level distilla- +tion is promising to shrink the gap in the cross-modal do- +main (e.g., LiDAR point cloud and binocular images). For +illustration, we first obtain the upper bound of the stereo +model by replacing its prediction of 3D box regression and +classification with the corresponding outputs of the LiDAR +model (teacher). As shown in Figure 1, the stereo model +with the replaced regression or classification predictions +produces impressive results, demonstrating the potential of +response-based distillation in the cross-modal domain. How- +arXiv:2301.01615v1 [cs.CV] 4 Jan 2023 + +ever, directly applying the vanilla response-level distilla- +tion is less effective, either by selecting the high-confident +((Yang et al. 2022)) or high-IoU 3D boxes (box-level) pre- +dicted from the LiDAR model as soft targets (Sun et al. +2020c). The reasons are two-fold: 1) unlike dense 2D im- +ages, much fewer high-IoU or high-confident boxes can be +adopted as soft labels in a 3D scene due to the high sparsity +of LiDAR point cloud; 2) the low-quality boxes discarded +by one-size-fits-all thresholds contain underlying beneficial +components (e.g., center, size, or orientation angle) that have +been overlooked. +To tackle the problem, we propose a novel X-component +Guided Distillation (XGD) from the response level. The key +idea of XGD is to first decompose a 3D box into sub-X- +components (X can be center, size, or orientation angle) and +retain the beneficial subcomponent as the soft targets if the +vector between the teacher’s X-component and the student’s +component is consistent with the vector between the ground +truth and the student’s, i.e., the two vectors are acute-angled. +Moreover, we find that only one out of all anchors at the +same position can be selected as being responsible for a fore- +ground object in most cases due to the fact that there is usu- +ally no overlap among objects in real autonomous driving +scenarios, which is different in the 2D domain. Motivated by +this observation, we propose a simple and effective Cross- +anchor Logit Distillation (CLD) for classification distillation +in our StereoDistill to distill by aggregating the confidence +distribution of all anchors to a unified distribution so as to +highlight the highest probability anchor. +To summarize, our key contributions are as follows. +• We validate that the cross-modal knowledge distillation +at the response level can boost the performance of stereo- +based 3D object detection. The proposed X-component +Guided Distillation (XGD) for regression avoids the neg- +ative effect of erroneous 3D boxes from the LiDAR +model by keeping the beneficial X-component as soft tar- +gets under the guidance of acute-angled vectors. +• Given the fact that there is no overlap among objects in +autonomous driving scenarios, we introduce the simple +yet effective Cross-anchor Logit Distillation (CLD) for +classification to aggregate the probability distribution of +all anchors at the same position rather than distilling the +distribution at anchor level. +Related Works +Stereo-based 3D Object Detection. The earlier meth- +ods (Li, Chen, and Shen 2019; Sun et al. 2020a; Xu et al. +2020) achieve stereo 3D detection based on a strong 2D de- +tector (Ren et al. 2015; He et al. 2017), which does not fully +explore the 3D information, leading to suboptimal perfor- +mance. To introduce more 3D information, +(Wang et al. +2019; You et al. 2020; Qian et al. 2020) try to convert the +estimated depth maps combined with the corresponding im- +age to pseudo point clouds and then can apply the existing +LiDAR-based 3D detectors (Yan, Mao, and Li 2018; Lang +et al. 2019) to detect 3D boxes. However, directly applying +pseudo point clouds for 3D detection might bring erroneous +localization due to the limitation of depth estimation, lead- +ing to sub-optimal performance. To tackle this problem, the +recent methods DSGN (Chen et al. 2020), CDN (Garg et al. +2020), DSGN++ (Chen et al. 2022) and PLUME (Wang +et al. 2021) build cost volume (Flynn et al. 2016) to encode +the implicit 3D geometry features instead of the raw pseudo +point representations for 3D object detection. In this paper, +we select the prominent DSGN as our stereo model and keep +the same configuration with LIGA (Guo et al. 2021). +LiDAR-based 3D Object Detection. Due to the plentiful +geometric structure information and accurate depth informa- +tion from LiDAR sensors, LiDAR-based 3D detectors (Shi, +Wang, and Li 2019; Yan, Mao, and Li 2018) usually achieve +superior performance than the camera-based (Brazil and Liu +2019; Chen et al. 2016; Li et al. 2020; Simonelli et al. 2019; +Chen et al. 2020). At present, the mainstream 3D detection +methods are divided into two types according to the input +data format, including point-based and voxel-based detec- +tors. The point-based methods (Shi, Wang, and Li 2019; +Yang et al. 2020) usually apply PointNets (Qi et al. 2017a,b) +to deal with this problem of permutation invariance. The +voxel-based methods (Yan, Mao, and Li 2018; Zhou and +Tuzel 2018; Lang et al. 2019; Liu et al. 2020; Deng et al. +2020) convert the irregular 3D points to the regular voxel +grids and employ 2D/3D convolution operation to estimate +the final 3D boxes. In this paper, to better align the pre- +dictions with the stereo model DSGN (Chen et al. 2020), +we choose the popular voxel-based detector SECOND (Yan, +Mao, and Li 2018) as the LiDAR model. +Knowledge Distillation. Knowledge distillation (KD) is ini- +tially proposed by (Hinton et al. 2015), which can trans- +fer knowledge from a larger network to a small network +to promote the performance or achieve model compression +for lightweight devices. Recently, (Dai et al. 2021; Yang +et al. 2021; Chen et al. 2021; Zhang and Ma 2020) achieve +feature-based distillation by focusing on the foreground area +or considering a weight matrix for the features. LD (Zheng +et al. 2022) implements the difficult problem of localiza- +tion distillation from the response level by converting the +regression of bounding boxes to the probability distribution +representation. Besides, Cross-modal feature distillation ap- +proaches (Chong et al. 2022; Guo et al. 2021) are gaining +popularity as a way to take advantage of the complementar- +ity between different modalities. LIGA (Guo et al. 2021) is +the first attempt to explore the fine-grained feature distilla- +tion from LiDAR to stereo 3D detector. However, LIGA fails +to benefit the stereo model through the response-based dis- +tillation due to the erroneous targets from the LiDAR model. +In this paper, we propose an X-component Guided Distilla- +tion (XGD) to deal with this problem by retaining the bene- +ficial component which is consistent with ground truth. +Method +In this part, we introduce the proposed cross-modal dis- +tillation StereoDistill, which consists of the X-component +Guided Distillation (XGD) and Cross-anchor Logit Distilla- +tion (CLD) at the response level. As shown in Figure 2, we +present the pipeline of our StereoDistill, which employs a +stereo model, DSGN (Chen et al. 2020) for instance, as the +student network and a LiDAR model, SECOND (Yan, Mao, + +Feature-level +Distillation +Image +Backbone +LiDAR +Backbone +3D +Head +(a) Student Network (or Stereo Model) +(b) Teacher Network (or LiDAR Model) +Stereo Images +LiDAR Points +3D +Head +Only Training +Class +3D Box +Class +3D Box +W +H +L +! +XYZ +Decompose +" +# +$ +! +Flatten +CLD: +Cross-anchor +Logit-D +Decompose +Flatten +XGD: X-component Guided-D +Figure 2: The pipeline of our proposed StereoDistill method. The student and the teacher model take the stereo images and +LiDAR points as inputs, respectively. At the response level, X-component Guided Distillation (XGD) and Cross-anchor Logit +Distillation (CLD) are applied to the 3D box regression and classification head, respectively. In XGD, we decompose the 3D +box into sub-components, i.e., size (HWL), center (XYZ) and rotation angle (θ) and keep the components as soft targets if the +vectorial angle between teacher-student and GT-student pair is acute. In CLD, we flatten the confidence scores of all anchors +falling in the same position and convert them to a unified distribution to highlight the most valuable anchor. +and Li 2018) for instance, as the teacher network only for +training. Although StereoDistill contains the feature-level +and response-level distillations, our main contribution fo- +cuses on the response-level distillation since the effective- +ness on the feature-level has been illustrated in LIGA (Guo +et al. 2021). For the feature level, we mainly revise the fea- +ture distillation in LIGA (Guo et al. 2021) by introducing +the attention weight of features (Zagoruyko and Komodakis +2017) and the relationship among instance features (Hou +et al. 2020) to further improve the performance, which is +regarded as our baseline (named Improved LIGA). For more +details, please refer to our supplementary materials. +For the response-based distillation, however, the predicted +boxes (box-level) from a teacher network inevitably contain +false predictions. Therefore, using all predicted boxes di- +rectly without any purifying process is likely harmful to the +student network and results in a sub-optimal solution (Guo +et al. 2021). To resolve this problem, we propose a novel +XGD to preserve the beneficial X-component (e.g., center, +size and angle) decomposed from a box through the pro- +posed positive component updating algorithm. In addition, +we notice that only one out of all anchors at the same po- +sition can usually be selected as being responsible for a +foreground object in autonomous driving scenarios. Thus, +CLD is proposed to highlight the highest probability anchor +across all anchors at the same position. Next, we introduce +the proposed XGD and CLD in detail. +X-component Guided Distillation. As we all know, the Li- +DAR model has an inherent advantage in localization since +the LiDAR sensor can provide more accurate geometrical +information and depth information. However, the final pre- +dictions from the teacher model benefit little from training +the stereo network (Guo et al. 2021). The main reason is that +the erroneous regression of the teacher model may guide the +Right!! +j +c + +j +c + +j +c + +Harmful center +Beneficial center +(a) +(b) +(b) +Student +Ground-Truth +Teacher +Figure 3: Our X-component Guided Distillation (take the +center component as an example to illustrate whether the +teacher’s prediction is beneficial to the student. Case (a) de- +picts an obtuse angle between the student-to-GT vector and +the student-to-teacher vector, showing that the teacher is in- +consistent with the GT. Conversely, in Case (b) we observe +an acute angle between the two vectors, validating that it is +beneficial to be adopted as soft targets to guide the student +to regress towards the direction of GT. +student model to learn in a detrimental direction. Although +an available solution is to only keep these high-quality boxes +for distillation, it brings two flaws. One is that high-quality +boxes are too few, resulting in inefficient distillation. The +other is that some low-quality discarded boxes can also pro- +vide the estimated beneficial component through further de- +composing a 3D box into three components (the center po- +sition, the size, and the orientation angle). To be more in- +tuitive, we take the center position as an example and show +the harmful and beneficial predicted center position from the +teacher model in Figure 3 (a) and (b), respectively. + +E +KeH10KA.GB974Algorithm 1: positive component updating +Input: Boxes of teacher Bt = (Tc, Ts, To), Boxes of student +Bs = (Sc, Ss, So), Boxes of GT Bg = (Gc, Gs, Go). The +number of assigned positive boxes Npos. +Output: Updated boxes of teacher Bt∗. +1: Let Tc∗, Ts∗, To∗ =[], [], [] +2: for j ∈ {1, 2, ..., Npos} do +3: +Compute cos βj +c, cos βj +s and cos βj +o by the formula (1) +4: +for x ∈ {c, s, o} do +5: +if cos βj +x > 0 then +6: +T j +x∗ ← T j +x +7: +else +8: +T j +x∗ ← Sj +x; # Disable the harmful X-component +9: +end if +10: +end for +11: end for +Bt∗ = (Tc∗, Ts∗, To∗) +12: return Bt∗ +Motivated by the above observation, we propose a novel +component Guided Distillation (XGD) to tackle this prob- +lem. XGD first obtains all the boxes predicted from the as- +signed positive anchors to keep more valuable 3D boxes. +Then XGD selects the ‘soft boxes’ from these predicted +boxes at the X-component level rather than the box level. +Specifically, for the predicted jth 3D box of the teacher net- +work, we decompose Bj +t = (T j +c , T j +s , T j +o ) into three com- +ponents, where T j +c = (xj +t, yj +t , zj +t ), T j +s = (wj +t, hj +t, lj +t), and +T j +o = θj +t, where T j +c is the center position of the box along +X, Y and Z axes, T j +s represents the size including the width, +height and length of the 3D box and T j +o means the orien- +tation angle of the 3D box. Similarly, we can define the +predicted box Bj +s = (Sj +c, Sj +s, Sj +o) from the student network +and the corresponding ground-truth (GT) assigned boxes of +Bj +g = (Gj +c, Gj +s, Gj +o). Then, we can judge whether the esti- +mated center T j +c from the LiDAR model is beneficial to the +stereo model by measuring the cosine value of T j +c − Sj +c and +Gj +c − Sj +c, which can be formulated as: +cos βj +c = +(T j +c − Sj +c)(Gj +c − Sj +c)T +∥T j +c − Sj +c∥2∥Gj +c − Sj +c)T ∥2 +(1) +Where βj +c is the angle between the vector of T j +c − Sj +c and +Gj +c − Sj +c. Here, when βj +c is an acute angle (or cos βj +c > 0), +we think the provided center regression T j +c from the teacher +model can guide the student model regress a more accu- +rate center position. Similarly, we can obtain the βj +s and +βj +o for the size and angle components following the formu- +lation (1). Then, the final ‘soft boxes’ Bt∗ is produced by +our positive component updating in Algorithm 1. Finally, +we employ 3D IoU loss (Zhou et al. 2019) with rotation as +the soft regression term since 3D IoU can comprehensively +evaluate the quality of a bounding box. The XGD loss can +be computed as: +Lxgd = +Npos +� +j=1 +(1 − IoU3D(Bj +s, Bj +t∗)), +(2) +… +… +0.25 +0.12 +0.02 +0.73 +0.05 +0.42 +0.17 +0.08 +0.83 +… +… +… +… +… +1 +2 + ������ +… +(1) Flatten +(2) Softmax +0.25 +0.12 +0.02 +0.73 +0.05 +0.42 +0.17 +0.08 +0.83 +… +… +… +0.25 +0.25 0.12 0.02 +0.17 0.08 0.83 + ������������ + ������ +0.12 0.04 0.01 +0.06 0.02 0.55 +0.25 +0.12 +0.02 +0.73 +0.05 +0.42 +0.17 +0.08 +0.83 +… +… +… +… +… +1 +2 + ������ +… + (1) +Flatten + (2) + Softmax +0.25 0.12 0.02 +0.17 0.08 +0.12 0.04 0.01 +0.06 0.02 + ������������ +������ +0.25 +0.12 +0.02 +0.73 +0.05 +0.42 +0.17 +0.08 +0.83 +… +… +… +1 +2 + ������ +… +������ +KL loss + ������ +������ + ������ +������ +Flatten + ������ +������ + ������ +KL loss +Softmax +KL loss +(a) Classical Logit Distillation (LD) +(b) Cross-anchor Logit Distillation (CLD) + ������������ +(Student) +(Teacher) +Figure 4: The process of the Classical Logit Distillation +and our Cross-anchor Logit Distillation. The confidence +scores from the student network and the teacher network are +marked in blue and green, respectively. And the darker the +color, the higher the confidence. +where Npos is the number of the positive anchors in the +stereo model and IoU3D(Bj +s, Bj +t∗) denotes the 3D IoU be- +tween Bj +s and Bj +t∗. +Cross-anchor Logit Distillation. Some distillation meth- +ods (Chen et al. 2017a; Dai et al. 2021; Sun et al. 2020c) +via the classification probability usually bring benefits to fi- +nal results for the 2D detection task, where these distillations +are only carried out for positive boxes. However, our distil- +lation is carried out in all foreground regions since the Li- +DAR model generates fewer positive 3D samples compared +with 2D detection counterparts. Moreover, another distinct +characteristic of the 3D detection task against 2D detection +lies in the fact that it is rare to find a conflict or overlapping +among 3D boxes in autonomous driving scenarios. That is to +say, distinct anchors lying in the same position are designed +for different objects with different scales and aspect ratios, +and hence only one out of these anchors can be selected +as being responsible for a foreground object in most cases. +However, these classical logit distillation approaches (Chen +et al. 2017a; Dai et al. 2021; Sun et al. 2020c) designed for +2D detection tasks treat anchors separately and do not work +well in the 3D detection task, shown in Figure. 4 (a). Given +that, we propose a Cross-anchor Logit Distillation (CLD) +approach to highlight the most representative anchor from +all anchors in the same position by converting the confi- +dence distribution of each anchor to a unified distribution, +whose process is described in Figure. 4 (b). Specifically, +we first reshape the output confidence map of the teacher +network Pt ∈ RNfore×Kc as P ′ +t ∈ RMfore×(KcKa), where +Nfore = MforeKa. Here, Mfore, Ka and Kc represent the +number of all foreground positions, the pre-defined anchors +for each position and the categories on the 3D object detec- +tion task, respectively. Then, the softmax function is applied + +Modality +Method +Car AP0.7 +Pedestrian AP0.5 +Cyclist AP0.5 +Easy +Mod. +Hard +Easy +Mod. +Hard +Easy +Mod. +Hard +LiDAR +MV3D (Chen et al. 2017b) +74.97 +63.63 +54.00 +– +– +- +– +– +– +SECOND (Yan, Mao, and Li 2018) +83.34 +72.55 +65.82 +– +– +– +– +– +– +AVOD-FPN (Ku et al. 2018) +83.07 +71.76 +65.73 +50.46 +42.27 +39.04 +63.76 +50.55 +44.93 +Stereo +Stereo R-CNN (Li, Chen, and Shen 2019) +47.58 +30.23 +23.72 +– +– +– +– +– +– +Pseudo-Lidar (Wang et al. 2019) +54.53 +34.05 +28.25 +– +– +– +– +– +– +ZoomNet (Xu et al. 2020) +55.98 +38.64 +30.97 +– +– +– +– +– +– +Pseudo-LiDAR++ (You et al. 2020) +61.11 +42.43 +36.99 +– +– +– +– +– +– +CDN (Garg et al. 2020) +74.52 +54.22 +46.36 +– +– +– +– +– +– +SNVC (Li et al. 2022) +78.54 +61.34 +54.23 +– +– +– +– +– +– +OC-Stereo (Pon et al. 2020) +55.15 +37.60 +30.25 +24.48 +17.58 +15.60 +29.40 +16.63 +14.72 +YOLOStereo3D (Liu, Wang, and Liu 2021) +65.68 +41.25 +30.42 +28.49 +19.75 +16.48 +– +– +– +Disp-RCNN (Sun et al. 2020a) +68.21 +45.78 +37.73 +37.12 +25.80 +22.04 +40.05 +24.40 +21.12 +DSGN (Chen et al. 2020) +73.50 +52.18 +45.14 +20.53 +15.55 +14.15 +27.76 +18.17 +16.21 +CG-Stereo (Li, Ku, and Waslander 2020) +74.39 +53.58 +46.50 +33.22 +24.31 +20.95 +47.40 +30.89 +27.23 +LIGA (Guo et al. 2021) +81.39 +64.66 +57.22 +40.46 +30.00 +27.07 +54.44 +36.86 +32.06 +StereoDistill (Ours) +81.66 +66.39 +57.39 +44.12 +32.23 +28.95 +63.96 +44.02 +39.19 +Table 1: 3D Detection results on the KITTI test benchmark. APthr means the threshold value of 3D IoU between the prediction +and the ground truth as thr. ‘Mod.’ is short for Moderate. +Method +Car AP0.5 +Pedestrian AP0.25 +3D +BEV +3D +BEV +SECOND (2018) (T) +80.67 +87.06 +48.96 +49.14 +DSGN (2020) (S) +33.68 +42.76 +8.58 +8.95 +LIGA (2021) +34.37 +45.47 +10.76 +11.01 +Ours +37.55 +46.99 +13.70 +14.04 +Table 2: Car and Pedestrian detection results on Argoverse +validation set with the evaluation metric of 11 recall posi- +tions. T and S denote the teacher and the student. +to further normalize the flattened confidence scores P ′ +t along +the dimension of KcKa and obtain the unified confidence +distribution P ∗ +t across all anchors at the same position: +P ∗ +t = softmax(P ′ +t) +(3) +Similarly, we can get the confidence distribution P ∗ +s for the +student network. Finally, the CLD loss can be computed by +KL divergence: +Lcld = KL(P ∗ +t , P ∗ +s ), +(4) +Total Loss Function. We train the stereo model in an end- +to-end manner, and the total loss function is as follows: +Ltotal = Lori + Lxgd + Lcld, +(5) +where Lori denotes the loss function except the feature dis- +tillation loss in the LIGA (Guo et al. 2021). For training the +LiDAR model, we adopt the same loss function with SEC- +OND (Yan, Mao, and Li 2018). +Experiments +Experimental Datasets and Evaluation Metrics +KITTI. The KITTI dataset (Geiger, Lenz, and Urtasun +2012) includes 7,481 training and 7,518 testing stereo image +pairs with the corresponding LiDAR point clouds. We fur- +ther split the training data into training set with 3712 sam- +ples and a validation set with 3769 samples following (Chen +et al. 2020; Qi et al. 2018; Shi, Wang, and Li 2019). The +evaluation metric (Simonelli et al. 2019) adopts the mean +Average Precision (mAP) with 40 recall positions. If not +specified, the metric of all results in the following tables uses +40 recall positions. In this paper, we evaluate our method on +the validation set and the test benchmark for three categories +of Cars, Pedestrians and Cyclists under three difficulty lev- +els (e.g., Easy, Moderate, and Hard). +Argoverse. The Argoverse dataset (Chang et al. 2019) con- +tains 3D detection and tracking annotations from 113 scenes. +Different from the Waymo (Sun et al. 2020b) and Nusc- +nes (Caesar et al. 2020) datasets, Argoverse provides stereo +image pairs, which can be adopted to verify the generality +of our method. For convenience, we convert the Argoverse +dataset to the format of KITTI following (Wang et al. 2020) +and obtain a training set with 13122 samples and a valida- +tion set with 5015 samples. We adopt the same evaluation +metric with KITTI. +Implementation Details +For the stereo model DSDN (Chen et al. 2020) and the Li- +DAR model SECOND (Yan, Mao, and Li 2018), we use the +same network structure with LIGA (Guo et al. 2021) for +fair comparisons. The stereo model is trained on 4 NVIDIA +V100 GPUs with a batch size of 4 and is optimized by Adap- +tive Momentum Estimation (Adam) (Kingma and Ba 2014) +with the initial learning rate, weight decay, and momentum +factor set to 0.003, 0.01, and 0.9, respectively. Random hor- +izontal flipping is adopted for data augmentation. For both +the KITTI dataset and the Argoverse dataset, we employ the +range of the detection area to [-30, 30], [-1, 3], [2, 59.6] +meters along the X (right), Y (down), Z (forward) axis in +the camera coordinate. The voxel size of the LiDAR model +is (0.2, 0.2, 0.2) meters and the volume size of the stereo +model is (0.05, 0.1, 0.05) meters. All experiments are con- +ducted on a single model for multiple categories. For more +details, please refer to our supplementary materials. + +# +XGD +CLD +Car AP3D +Pedestrian AP3D +Cyclist AP3D +Easy +Mod. +Hard +Easy +Mod. +Hard +Easy +Mod. +Hard +SECOND⋆ (2018) (teacher) +90.72 +81.66 +78.78 +69.45 +62.39 +56.49 +84.97 +64.06 +60.21 +DSGN† (2020) (student) +83.27 +64.21 +58.61 +40.45 +34.33 +29.07 +54.76 +32.91 +30.04 +LIGA⋆ (2021) (paper) +86.84 +67.71 +62.02 +45.54 +37.80 +32.09 +60.00 +37.31 +34.25 +LIGA† (2021) (reproduced) +84.32 +67.14 +61.93 +47.16 +38.97 +34.09 +63.98 +38.49 +36.01 +Improved LIGA† (2021) (baseline) +86.62 +67.03 +61.94 +47.77 +40.11 +35.19 +65.02 +40.90 +37.81 +I +✓ +– +86.78 +67.65 +62.43 +52.89 +45.37 +39.40 +66.25 +41.38 +38.27 +II +– +✓ +86.67 +67.57 +62.19 +47.81 +40.69 +35.84 +67.77 +41.65 +38.62 +III +✓ +✓ +87.57 +69.75 +62.92 +55.19 +46.76 +40.42 +69.43 +42.31 +39.10 +Improvement over baseline ++0.95 ++2.72 ++0.98 ++7.42 ++6.65 ++5.23 ++4.41 ++1.41 ++1.29 +Table 3: Ablation studies for our proposed XGD and CLD on the KITTI validation set. ⋆ and † in this table denote the results +reported in the paper and our reproduced results. ‘Mod.’ is short for Moderate. +Methods +Car AP3D +Pedestrian AP3D +Cyclist AP3D +Easy +Mod. +Hard +Easy +Mod. +Hard +Easy +Mod. +Hard +PointPillars (2019) (teacher) +88.89 +78.47 +75.38 +62.84 +56.01 +51.87 +82.58 +62.06 +58.37 +DSGN (2020) (student) +83.27 +64.21 +58.61 +40.45 +34.33 +29.07 +54.76 +32.91 +30.04 +LIGA (2021) +83.46 +63.40 +58.29 +41.68 +35.76 +30.38 +62.75 +37.28 +34.25 +XGD + CLD +84.74 +65.49 +60.13 +45.98 +40.18 +34.75 +66.67 +41.00 +37.95 +Improvement ++1.28 ++2.09 ++1.84 ++4.30 ++4.42 ++4.37 ++3.92 ++3.72 ++3.70 +Improved LIGA (2021) +83.94 +64.27 +59.00 +42.37 +36.84 +31.54 +62.84 +37.71 +35.12 +XGD + CLD +85.24 +67.62 +60.72 +47.81 +40.69 +34.78 +67.10 +41.19 +38.13 +Improvement ++1.30 ++3.35 ++1.72 ++5.44 ++3.85 ++3.24 ++4.26 ++3.48 ++3.01 +Table 4: Generality of our StereoDistill (XGD + CLD) on the KITTI validation set. We select the popular LiDAR model +PointPillars (Lang et al. 2019) as the teacher model. ‘Mod.’ is short for Moderate. +Comparisons with State-of-the-art Methods +Evaluation on KITTI. In Table 1, we present quantitative +comparison with the leading stereo-based 3D detectors and +several popular LiDAR-based 3D detectors on the KITTI +test benchmark. Our method outperforms the SOTA model +LIGA (Guo et al. 2021) with 1.73%, 2.23% and 7.16% 3D +mAP on Cars, Pedestrian and Cyclists at the moderate dif- +ficulty level, without introducing any extra cost during in- +ference. Our StereoDistill even surpasses the LiDAR-based +method MV3D (Chen et al. 2017b) with 3D mAP of 2.76% +on Cars. These superior results demonstrate the effective- +ness of our StereoDistill. For visualization, please refer to +our supplementary materials. +Evaluation on Argoverse. To further verify the generality +of our proposed method, we conduct experiments on the Ar- +goverse dataset. For fair comparisons, we adopt the same +network of student and teacher models with LIGA (Guo +et al. 2021) and also re-implement LIGA under the same +setting on the Argoverse dataset. In Table 2, we present the +results with the 3D IoU thresholds of 0.5 and 0.25 for both +the BEV and 3D detection on moderate Cars and Pedestri- +ans. Our method exceeds LIGA with 3D mAPs of 3.18% +and 2.94% and BEV mAPs of 1.52% and 3.03% on Cars and +Pedestrians, which validates the generality of our method. +Ablation Studies +Ablation Studies on StereoDistill. In this part, we verify +the effectiveness of the proposed compositions, including +the XGD for regression, CLD for classification, and their +combinations in StereoDistill. The baseline model is our im- +proved LIGA (Guo et al. 2021) by further enhancing the +feature distillation (refer to our supplementary materials). +In Table 3, by comparing (I), (III) with the baseline model, +the proposed XGD and CLD bring consistent improvements +over the baseline on all difficulty levels for three categories, +which demonstrates their effectiveness. Note that the pro- +posed XGD greatly boosts the detection performance on +small objects (e.g., Pedestrians), which requires more accu- +rate regression. It illustrates that the X-component guided +distillation can indeed transfer superior location awareness +from the LiDAR model to the stereo model so as to obtain +better performance. Integrating these two ingredients, Stere- +oDistill in (III) outperforms the baseline with remarkable +margins of 2.72%, 6.65% and 1.41% 3D mAP on the mod- +erate Cars, Pedestrians and Cyclists, respectively. +Generality of StereoDistill. To verify the generality of +StereoDistill, we replace the common teacher network SEC- +OND (Yan, Mao, and Li 2018) with the other popular 3D de- +tector Pointpillars (Lang et al. 2019). In Table 4, we provide +two baseline settings: the original LIGA (Guo et al. 2021) +in Line 4 and the improved LIGA in Line 7. Not surpris- +ingly, our StereoDistill yields obvious performance gains on +all difficulty levels for three categories, which further the su- +periority and generality of our proposed XGD and CLD. +Analysis of XGD. In Table 5, we conduct extensive abla- +tion studies to analyze the effectiveness of XGD. The base- + +# +Methods +Cars +Pedestrians +Cyclists +I +Baseline +67.57 +40.69 +41.65 +II +XGD-Center +68.22 +46.33 +42.19 +III +XGD-Size +68.02 +42.46 +40.71 +IV +XGD-Angle +67.84 +44.11 +41.18 +V +XGD +69.75 +46.76 +42.31 +VI +High-quality boxes +67.72 +43.65 +42.11 +VII +Positive anchors (Ours) +69.75 +46.76 +42.31 +Table 5: Ablation studies for XGD. The results are evalu- +ated with 3D mAP on the moderate difficulty level for Cars, +Pedestrians and Cyclists, respectively. XGD-* means the +manner of only adopting * for computing XGD loss. +# +Methods +Cars +Pedestrians +Cyclists +I +Baseline +67.65 +45.37 +41.38 +II +Positive Anchors +67.66 +45.71 +42.32 +III +All Foregrounds (Ours) +69.75 +46.76 +42.31 +IV +Classical (Chen et al. 2017a) +67.74 +46.18 +40.69 +V +CLD (Ours) +69.75 +46.76 +42.31 +Table 6: Ablation studies for CLD. The results are evalu- +ated with 3D mAP on the moderate Cars, Pedestrians and +Cyclists, respectively. +line (I) is our StereoDistill without XGD loss. Then, we +decompose the regression of 3D boxes into three compo- +nents including the center position, the size, and the orienta- +tion angle to analyze the effect of each component on XGD. +We observe that XGD with only the center position (II) ex- +hibits the most competitive performance of the three (II, III, +IV), which illustrates the positive guidance of the center po- +sition is the crucial component to help the student model +to acquire more beneficial localization information. Finally, +combined with these three components, XGD (V) exceeds +the baseline (I) with 3D mAP of 2.08%, 6.07% and 0.66% +on moderate Cars, Pedestrians, and Cyclists, demonstrating +the superiority of XGD. Moreover, compared with retain- +ing high-quality boxes in (VI) whose confidence scores are +greater than 0.3 (a proper threshold), our manner of adopting +all positive anchors in (VII) has obvious gains on Cars and +Pedestrians. In the real scene, these two categories usually +occupy a much larger number than Cyclists, which means +that there may be more false positives on Cars and Pedestri- +ans. This demonstrates that our XGD provides a reasonable +workaround to deal with some low-quality boxes by retain- +ing the beneficial X-component but discarding the harmful +X-component decomposed from a 3D box. +Effectiveness of CLD. In Table 6, we first present the re- +sults of distilling the confidence distribution from two alter- +native regions. The way based on the foreground (III) ex- +ceeds the approach by only considering the positions from +positive anchors (II) on average, which illustrates the im- +portance of introducing the useful foreground positions be- +yond the positions of the positive anchors. Furthermore, we +provide a classical logit distillation (IV) (Sun et al. 2020c) +as a comparison, which individually treats the confidence +distribution of each anchor from each position. It can be +observed that our CLD boosts the performance with mAP +Methods +Cars +Pedestrians +Cyclists +Mean +DSGN (LIGA wo FD) +63.32 +34.23 +30.26 +42.60 ++ FD +67.14 +38.97 +38.49 +48.20 +Improvement ++3.82 ++4.74 ++8.23 ++5.60 ++ our RD +67.42 +45.28 +37.66 +50.12 +Improvement ++4.10 ++11.05 ++7.4 ++7.52 +StereoDistill +69.75 +46.76 +42.31 +52.94 +Improvement ++6.43 ++12.53 ++12.05 ++10.34 +Table 7: Comparisons for the feature-based and response- +based distillation. The results are evaluated with 3D mAP +on moderate Cars, Pedestrians and Cyclists. FD and RD are +short for feature-based distillation in LIGA (Guo et al. 2021) +and the response-based distillation in StereoDistill. +of 2.01%, 0.58% and 1.62% on Cars, Pedestrian and Cy- +clists, clearly demonstrating the effectiveness of underlining +the confidence distribution for the best competitive anchor +from all anchors in a position. +Comparison of Different Distillation. In Table 7, we indi- +vidually present the comparisons for adopting the feature- +based in LIGA (Guo et al. 2021) or response-based distilla- +tion in StereoDistill based on the stereo model DSGN (Chen +et al. 2020). First, these two distillations can consistently +boost performance over the baseline DSGN (Chen et al. +2020). Then, the proposed response-based distillation of +our XGD and CLD in StereoDistill even outperforms the +feature-based distillation in LIGA (Guo et al. 2021) with 3D +mAP of 1.92% on average under the same setting, which +further demonstrates the effectiveness of the proposed XGD +and CLD. Moreover, combined with the response-based dis- +tillation and the feature-based distillation, our StereoDis- +till produces superior performance over the baseline model +DSGN with a 3D mAP of 10.34% on average. This reveals +that superior feature representations often lead to better re- +sponses, which in turn can further facilitate feature learning. +Conclusions +This paper presents an effective cross-modal distillation ap- +proach termed StereoDistill from the response levels for the +stereo 3D detection task. The core designs of StereoDistill +are the proposed X-component Guided Distillation (XGD) +for regression and the Cross-anchor Logit Distillation (CLD) +for classification. The extension ablation studies demon- +strate the superiority of our proposed XGD for regression +and CLD for classification. Finally, StereoDistill achieves +state-of-the-art performance among stereo-based detectors +without introducing extra cost in the inference process com- +pared to our stereo model on the KITTI test benchmark and +the large-scale Argoverse dataset. 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In CVPR. + diff --git a/kNAzT4oBgHgl3EQfpf3m/content/tmp_files/load_file.txt b/kNAzT4oBgHgl3EQfpf3m/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a57acd14ed84e16c1ca1783ae38da8ee2bdf54fc --- /dev/null +++ b/kNAzT4oBgHgl3EQfpf3m/content/tmp_files/load_file.txt @@ -0,0 +1,1605 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf,len=1604 +page_content='StereoDistill: Pick the Cream from LiDAR for Distilling Stereo-based 3D Object Detection Zhe Liu1, Xiaoqing Ye2, Xiao Tan2, Errui Ding2, Xiang Bai1* 1Huazhong University of Science and Technology, 2Baidu Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=', China zheliu1994@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='cn, yxq@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='cn, tanxchong@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='com, dingerrui@baidu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='com, xbai@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='cn Abstract In this paper, we propose a cross-modal distillation method named StereoDistill to narrow the gap between the stereo and LiDAR-based approaches via distilling the stereo detec- tors from the superior LiDAR model at the response level, which is usually overlooked in 3D object detection distilla- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The key designs of StereoDistill are: the X-component Guided Distillation (XGD) for regression and the Cross- anchor Logit Distillation (CLD) for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' In XGD, instead of empirically adopting a threshold to select the high- quality teacher predictions as soft targets, we decompose the predicted 3D box into sub-components and retain the corre- sponding part for distillation if the teacher component pilot is consistent with ground truth to largely boost the number of positive predictions and alleviate the mimicking difficulty of the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' For CLD, we aggregate the probability distribution of all anchors at the same position to encourage the highest probability anchor rather than individually distill the distribution at the anchor level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Finally, our StereoDistill achieves state-of-the-art results for stereo-based 3D detection on the KITTI test benchmark and extensive experiments on KITTI and Argoverse Dataset validate the effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Introduction 3D detectors equipped with LiDAR points (Shi, Wang, and Li 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2017b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2022) for autonomous driving have presented outperforming performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' How- ever, LiDAR sensors usually have a high cost and sensitiv- ity to weather, which limit their application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Alternatively, stereo cameras are capturing increasing interest thanks to their good trade-off in low cost and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' There is still a huge performance gap between stereo-based and cutting- edge LiDAR-based 3D detection methods due to the inaccu- rate depth estimation by stereo matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' A question natu- rally arises: can the LiDAR model help to improve the per- formance of the stereo model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Knowledge distillation (KD) (Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2015) might be a promising solution for this question, which guides the student model to mimic the knowledge of the teacher model for performance improvement or model compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The Corresponding Author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 35 40 45 50 55 60 65 70 75 80 Car Pedestrian Cyclist Stereo Model Repalce Classfication Replace Regression Gains of StereoDistill Figure 1: 3D detection performance (3D mAP) on KITTI valida- tion set of LIGA (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021) by replacing the regression and classification results of the stereo model (student) with the teacher LiDAR model SECOND (Yan, Mao, and Li 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' current KD methods of object detection can be mainly clas- sified into the feature-based and response-based streams, in which the former carry out distillation at the feature level (Zagoruyko and Komodakis 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Romero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Huang and Wang 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Heo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020) for enforcing the consistency of feature repre- sentations between the teacher-student pair whereas the lat- ter adopts the confident prediction from the teacher model as soft targets in addition to the hard ground truth supervi- sion (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' However, directly migrating the existing KD methods to LiDAR-to-stereo cross-modal distillation is less effective due to the huge gap between the two extremely different modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The pioneering work LIGA (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021) boosted the performance of stereo-based models by apply- ing fine-grained feature-level distillation under the guidance of LiDAR-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' However, it found little benefit from the response-based distillation due to the erroneous and noisy predictions of the LiDAR teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' On the contrary, we argue that the response-level distilla- tion is promising to shrink the gap in the cross-modal do- main (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=', LiDAR point cloud and binocular images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' For illustration, we first obtain the upper bound of the stereo model by replacing its prediction of 3D box regression and classification with the corresponding outputs of the LiDAR model (teacher).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' As shown in Figure 1, the stereo model with the replaced regression or classification predictions produces impressive results, demonstrating the potential of response-based distillation in the cross-modal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' How- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='01615v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='CV] 4 Jan 2023 ever, directly applying the vanilla response-level distilla- tion is less effective, either by selecting the high-confident ((Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2022)) or high-IoU 3D boxes (box-level) pre- dicted from the LiDAR model as soft targets (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The reasons are two-fold: 1) unlike dense 2D im- ages, much fewer high-IoU or high-confident boxes can be adopted as soft labels in a 3D scene due to the high sparsity of LiDAR point cloud;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2) the low-quality boxes discarded by one-size-fits-all thresholds contain underlying beneficial components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=', center, size, or orientation angle) that have been overlooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' To tackle the problem, we propose a novel X-component Guided Distillation (XGD) from the response level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The key idea of XGD is to first decompose a 3D box into sub-X- components (X can be center, size, or orientation angle) and retain the beneficial subcomponent as the soft targets if the vector between the teacher’s X-component and the student’s component is consistent with the vector between the ground truth and the student’s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=', the two vectors are acute-angled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Moreover, we find that only one out of all anchors at the same position can be selected as being responsible for a fore- ground object in most cases due to the fact that there is usu- ally no overlap among objects in real autonomous driving scenarios, which is different in the 2D domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Motivated by this observation, we propose a simple and effective Cross- anchor Logit Distillation (CLD) for classification distillation in our StereoDistill to distill by aggregating the confidence distribution of all anchors to a unified distribution so as to highlight the highest probability anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' To summarize, our key contributions are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' We validate that the cross-modal knowledge distillation at the response level can boost the performance of stereo- based 3D object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The proposed X-component Guided Distillation (XGD) for regression avoids the neg- ative effect of erroneous 3D boxes from the LiDAR model by keeping the beneficial X-component as soft tar- gets under the guidance of acute-angled vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Given the fact that there is no overlap among objects in autonomous driving scenarios, we introduce the simple yet effective Cross-anchor Logit Distillation (CLD) for classification to aggregate the probability distribution of all anchors at the same position rather than distilling the distribution at anchor level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Related Works Stereo-based 3D Object Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The earlier meth- ods (Li, Chen, and Shen 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020) achieve stereo 3D detection based on a strong 2D de- tector (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2017), which does not fully explore the 3D information, leading to suboptimal perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' To introduce more 3D information, (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' You et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Qian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020) try to convert the estimated depth maps combined with the corresponding im- age to pseudo point clouds and then can apply the existing LiDAR-based 3D detectors (Yan, Mao, and Li 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Lang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2019) to detect 3D boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' However, directly applying pseudo point clouds for 3D detection might bring erroneous localization due to the limitation of depth estimation, lead- ing to sub-optimal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' To tackle this problem, the recent methods DSGN (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020), CDN (Garg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020), DSGN++ (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2022) and PLUME (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021) build cost volume (Flynn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2016) to encode the implicit 3D geometry features instead of the raw pseudo point representations for 3D object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' In this paper, we select the prominent DSGN as our stereo model and keep the same configuration with LIGA (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' LiDAR-based 3D Object Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Due to the plentiful geometric structure information and accurate depth informa- tion from LiDAR sensors, LiDAR-based 3D detectors (Shi, Wang, and Li 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Yan, Mao, and Li 2018) usually achieve superior performance than the camera-based (Brazil and Liu 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Simonelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' At present, the mainstream 3D detection methods are divided into two types according to the input data format, including point-based and voxel-based detec- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The point-based methods (Shi, Wang, and Li 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020) usually apply PointNets (Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2017a,b) to deal with this problem of permutation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The voxel-based methods (Yan, Mao, and Li 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Zhou and Tuzel 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Lang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020) convert the irregular 3D points to the regular voxel grids and employ 2D/3D convolution operation to estimate the final 3D boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' In this paper, to better align the pre- dictions with the stereo model DSGN (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020), we choose the popular voxel-based detector SECOND (Yan, Mao, and Li 2018) as the LiDAR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Knowledge Distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Knowledge distillation (KD) is ini- tially proposed by (Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2015), which can trans- fer knowledge from a larger network to a small network to promote the performance or achieve model compression for lightweight devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Recently, (Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Zhang and Ma 2020) achieve feature-based distillation by focusing on the foreground area or considering a weight matrix for the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' LD (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2022) implements the difficult problem of localiza- tion distillation from the response level by converting the regression of bounding boxes to the probability distribution representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Besides, Cross-modal feature distillation ap- proaches (Chong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021) are gaining popularity as a way to take advantage of the complementar- ity between different modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' LIGA (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021) is the first attempt to explore the fine-grained feature distilla- tion from LiDAR to stereo 3D detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' However, LIGA fails to benefit the stereo model through the response-based dis- tillation due to the erroneous targets from the LiDAR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' In this paper, we propose an X-component Guided Distilla- tion (XGD) to deal with this problem by retaining the bene- ficial component which is consistent with ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Method In this part, we introduce the proposed cross-modal dis- tillation StereoDistill, which consists of the X-component Guided Distillation (XGD) and Cross-anchor Logit Distilla- tion (CLD) at the response level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' As shown in Figure 2, we present the pipeline of our StereoDistill, which employs a stereo model, DSGN (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020) for instance, as the student network and a LiDAR model, SECOND (Yan, Mao, Feature-level Distillation Image Backbone LiDAR Backbone 3D Head (a) Student Network (or Stereo Model) (b) Teacher Network (or LiDAR Model) Stereo Images LiDAR Points 3D Head Only Training Class 3D Box Class 3D Box W H L !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' XYZ Decompose " # $ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Flatten CLD: Cross-anchor Logit-D Decompose Flatten XGD: X-component Guided-D Figure 2: The pipeline of our proposed StereoDistill method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The student and the teacher model take the stereo images and LiDAR points as inputs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' At the response level, X-component Guided Distillation (XGD) and Cross-anchor Logit Distillation (CLD) are applied to the 3D box regression and classification head, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' In XGD, we decompose the 3D box into sub-components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=', size (HWL), center (XYZ) and rotation angle (θ) and keep the components as soft targets if the vectorial angle between teacher-student and GT-student pair is acute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' In CLD, we flatten the confidence scores of all anchors falling in the same position and convert them to a unified distribution to highlight the most valuable anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' and Li 2018) for instance, as the teacher network only for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Although StereoDistill contains the feature-level and response-level distillations, our main contribution fo- cuses on the response-level distillation since the effective- ness on the feature-level has been illustrated in LIGA (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' For the feature level, we mainly revise the fea- ture distillation in LIGA (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021) by introducing the attention weight of features (Zagoruyko and Komodakis 2017) and the relationship among instance features (Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020) to further improve the performance, which is regarded as our baseline (named Improved LIGA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' For more details, please refer to our supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' For the response-based distillation, however, the predicted boxes (box-level) from a teacher network inevitably contain false predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Therefore, using all predicted boxes di- rectly without any purifying process is likely harmful to the student network and results in a sub-optimal solution (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' To resolve this problem, we propose a novel XGD to preserve the beneficial X-component (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=', center, size and angle) decomposed from a box through the pro- posed positive component updating algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' In addition, we notice that only one out of all anchors at the same po- sition can usually be selected as being responsible for a foreground object in autonomous driving scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Thus, CLD is proposed to highlight the highest probability anchor across all anchors at the same position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Next, we introduce the proposed XGD and CLD in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' X-component Guided Distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' As we all know, the Li- DAR model has an inherent advantage in localization since the LiDAR sensor can provide more accurate geometrical information and depth information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' However, the final pre- dictions from the teacher model benefit little from training the stereo network (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The main reason is that the erroneous regression of the teacher model may guide the Right!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' j c \uf062 j c \uf062 j c \uf062 Harmful center Beneficial center (a) (b) (b) Student Ground-Truth Teacher Figure 3: Our X-component Guided Distillation (take the center component as an example to illustrate whether the teacher’s prediction is beneficial to the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Case (a) de- picts an obtuse angle between the student-to-GT vector and the student-to-teacher vector, showing that the teacher is in- consistent with the GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Conversely, in Case (b) we observe an acute angle between the two vectors, validating that it is beneficial to be adopted as soft targets to guide the student to regress towards the direction of GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' student model to learn in a detrimental direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Although an available solution is to only keep these high-quality boxes for distillation, it brings two flaws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' One is that high-quality boxes are too few, resulting in inefficient distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The other is that some low-quality discarded boxes can also pro- vide the estimated beneficial component through further de- composing a 3D box into three components (the center po- sition, the size, and the orientation angle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' To be more in- tuitive, we take the center position as an example and show the harmful and beneficial predicted center position from the teacher model in Figure 3 (a) and (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' E KeH10KA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='GB974Algorithm 1: positive component updating Input: Boxes of teacher Bt = (Tc, Ts, To), Boxes of student Bs = (Sc, Ss, So), Boxes of GT Bg = (Gc, Gs, Go).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The number of assigned positive boxes Npos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Output: Updated boxes of teacher Bt∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 1: Let Tc∗, Ts∗, To∗ =[], [], [] 2: for j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=', Npos} do 3: Compute cos βj c, cos βj s and cos βj o by the formula (1) 4: for x ∈ {c, s, o} do 5: if cos βj x > 0 then 6: T j x∗ ← T j x 7: else 8: T j x∗ ← Sj x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' # Disable the harmful X-component 9: end if 10: end for 11: end for Bt∗ = (Tc∗, Ts∗, To∗) 12: return Bt∗ Motivated by the above observation, we propose a novel component Guided Distillation (XGD) to tackle this prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' XGD first obtains all the boxes predicted from the as- signed positive anchors to keep more valuable 3D boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Then XGD selects the ‘soft boxes’ from these predicted boxes at the X-component level rather than the box level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Specifically, for the predicted jth 3D box of the teacher net- work, we decompose Bj t = (T j c , T j s , T j o ) into three com- ponents, where T j c = (xj t, yj t , zj t ), T j s = (wj t, hj t, lj t), and T j o = θj t, where T j c is the center position of the box along X, Y and Z axes, T j s represents the size including the width, height and length of the 3D box and T j o means the orien- tation angle of the 3D box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Similarly, we can define the predicted box Bj s = (Sj c, Sj s, Sj o) from the student network and the corresponding ground-truth (GT) assigned boxes of Bj g = (Gj c, Gj s, Gj o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Then, we can judge whether the esti- mated center T j c from the LiDAR model is beneficial to the stereo model by measuring the cosine value of T j c − Sj c and Gj c − Sj c, which can be formulated as: cos βj c = (T j c − Sj c)(Gj c − Sj c)T ∥T j c − Sj c∥2∥Gj c − Sj c)T ∥2 (1) Where βj c is the angle between the vector of T j c − Sj c and Gj c − Sj c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Here, when βj c is an acute angle (or cos βj c > 0), we think the provided center regression T j c from the teacher model can guide the student model regress a more accu- rate center position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Similarly, we can obtain the βj s and βj o for the size and angle components following the formu- lation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Then, the final ‘soft boxes’ Bt∗ is produced by our positive component updating in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Finally, we employ 3D IoU loss (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2019) with rotation as the soft regression term since 3D IoU can comprehensively evaluate the quality of a bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The XGD loss can be computed as: Lxgd = Npos � j=1 (1 − IoU3D(Bj s, Bj t∗)), (2) … … 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='05 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='02 ������������ ������ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='83 … … … 1 2 ������ … ������ KL loss ������ ������ ������ ������ Flatten ������ ������ ������ KL loss Softmax KL loss (a) Classical Logit Distillation (LD) (b) Cross-anchor Logit Distillation (CLD) ������������ (Student) (Teacher) Figure 4: The process of the Classical Logit Distillation and our Cross-anchor Logit Distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The confidence scores from the student network and the teacher network are marked in blue and green, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' And the darker the color, the higher the confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' where Npos is the number of the positive anchors in the stereo model and IoU3D(Bj s, Bj t∗) denotes the 3D IoU be- tween Bj s and Bj t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Cross-anchor Logit Distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Some distillation meth- ods (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020c) via the classification probability usually bring benefits to fi- nal results for the 2D detection task, where these distillations are only carried out for positive boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' However, our distil- lation is carried out in all foreground regions since the Li- DAR model generates fewer positive 3D samples compared with 2D detection counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Moreover, another distinct characteristic of the 3D detection task against 2D detection lies in the fact that it is rare to find a conflict or overlapping among 3D boxes in autonomous driving scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' That is to say, distinct anchors lying in the same position are designed for different objects with different scales and aspect ratios, and hence only one out of these anchors can be selected as being responsible for a foreground object in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' However, these classical logit distillation approaches (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020c) designed for 2D detection tasks treat anchors separately and do not work well in the 3D detection task, shown in Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 4 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Given that, we propose a Cross-anchor Logit Distillation (CLD) approach to highlight the most representative anchor from all anchors in the same position by converting the confi- dence distribution of each anchor to a unified distribution, whose process is described in Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 4 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Specifically, we first reshape the output confidence map of the teacher network Pt ∈ RNfore×Kc as P ′ t ∈ RMfore×(KcKa), where Nfore = MforeKa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Here, Mfore, Ka and Kc represent the number of all foreground positions, the pre-defined anchors for each position and the categories on the 3D object detec- tion task, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Then, the softmax function is applied Modality Method Car AP0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='7 Pedestrian AP0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='5 Cyclist AP0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='5 Easy Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Hard Easy Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Hard Easy Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Hard LiDAR MV3D (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2017b) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='97 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='63 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='00 – – – – – SECOND (Yan, Mao, and Li 2018) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='34 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='55 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='82 – – – – – – AVOD-FPN (Ku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2018) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='07 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='76 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='73 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='46 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='27 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='04 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='76 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='55 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='93 Stereo Stereo R-CNN (Li, Chen, and Shen 2019) 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='58 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='23 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='72 – – – – – – Pseudo-Lidar (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2019) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='53 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='05 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='25 – – – – – – ZoomNet (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020) 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='98 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='64 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='97 – – – – – – Pseudo-LiDAR++ (You et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='11 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='43 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='99 – – – – – – CDN (Garg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='52 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='22 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='36 – – – – – – SNVC (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2022) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='54 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='34 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='23 – – – – – – OC-Stereo (Pon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020) 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='15 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='60 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='25 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='48 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='58 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='60 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='40 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='63 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='72 YOLOStereo3D (Liu, Wang, and Liu 2021) 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='68 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='25 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='42 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='49 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='75 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='48 – – – Disp-RCNN (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020a) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='21 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='78 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='73 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='12 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='80 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='04 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='05 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='40 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='12 DSGN (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='50 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='18 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='14 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='53 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='55 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='15 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='76 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='17 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='21 CG-Stereo (Li, Ku, and Waslander 2020) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='39 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='58 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='50 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='22 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='31 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='95 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='40 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='89 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='23 LIGA (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='39 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='66 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='22 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='46 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='00 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='07 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='44 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='86 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='06 StereoDistill (Ours) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='66 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='39 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='39 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='12 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='23 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='95 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='96 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='02 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='19 Table 1: 3D Detection results on the KITTI test benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' APthr means the threshold value of 3D IoU between the prediction and the ground truth as thr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' ‘Mod.’ is short for Moderate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Method Car AP0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='5 Pedestrian AP0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='25 3D BEV 3D BEV SECOND (2018) (T) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='67 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='06 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='96 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='14 DSGN (2020) (S) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='68 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='76 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='58 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='95 LIGA (2021) 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='37 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='47 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='76 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='01 Ours 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='55 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='99 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='70 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='04 Table 2: Car and Pedestrian detection results on Argoverse validation set with the evaluation metric of 11 recall posi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' T and S denote the teacher and the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' to further normalize the flattened confidence scores P ′ t along the dimension of KcKa and obtain the unified confidence distribution P ∗ t across all anchors at the same position: P ∗ t = softmax(P ′ t) (3) Similarly, we can get the confidence distribution P ∗ s for the student network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Finally, the CLD loss can be computed by KL divergence: Lcld = KL(P ∗ t , P ∗ s ), (4) Total Loss Function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' We train the stereo model in an end- to-end manner, and the total loss function is as follows: Ltotal = Lori + Lxgd + Lcld, (5) where Lori denotes the loss function except the feature dis- tillation loss in the LIGA (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' For training the LiDAR model, we adopt the same loss function with SEC- OND (Yan, Mao, and Li 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Experiments Experimental Datasets and Evaluation Metrics KITTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The KITTI dataset (Geiger, Lenz, and Urtasun 2012) includes 7,481 training and 7,518 testing stereo image pairs with the corresponding LiDAR point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' We fur- ther split the training data into training set with 3712 sam- ples and a validation set with 3769 samples following (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Shi, Wang, and Li 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The evaluation metric (Simonelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2019) adopts the mean Average Precision (mAP) with 40 recall positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' If not specified, the metric of all results in the following tables uses 40 recall positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' In this paper, we evaluate our method on the validation set and the test benchmark for three categories of Cars, Pedestrians and Cyclists under three difficulty lev- els (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=', Easy, Moderate, and Hard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Argoverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The Argoverse dataset (Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2019) con- tains 3D detection and tracking annotations from 113 scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Different from the Waymo (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020b) and Nusc- nes (Caesar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020) datasets, Argoverse provides stereo image pairs, which can be adopted to verify the generality of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' For convenience, we convert the Argoverse dataset to the format of KITTI following (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020) and obtain a training set with 13122 samples and a valida- tion set with 5015 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' We adopt the same evaluation metric with KITTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Implementation Details For the stereo model DSDN (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020) and the Li- DAR model SECOND (Yan, Mao, and Li 2018), we use the same network structure with LIGA (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021) for fair comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The stereo model is trained on 4 NVIDIA V100 GPUs with a batch size of 4 and is optimized by Adap- tive Momentum Estimation (Adam) (Kingma and Ba 2014) with the initial learning rate, weight decay, and momentum factor set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='003, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='01, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='9, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Random hor- izontal flipping is adopted for data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' For both the KITTI dataset and the Argoverse dataset, we employ the range of the detection area to [-30, 30], [-1, 3], [2, 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='6] meters along the X (right), Y (down), Z (forward) axis in the camera coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The voxel size of the LiDAR model is (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='2) meters and the volume size of the stereo model is (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='05) meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' All experiments are con- ducted on a single model for multiple categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' For more details, please refer to our supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' # XGD CLD Car AP3D Pedestrian AP3D Cyclist AP3D Easy Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Hard Easy Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Hard Easy Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Hard SECOND⋆ (2018) (teacher) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='72 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='66 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='78 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='45 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='39 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='49 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='97 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='06 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='21 DSGN† (2020) (student) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='27 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='21 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='61 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='45 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='33 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='07 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='76 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='91 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='04 LIGA⋆ (2021) (paper) 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='84 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='71 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='02 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='54 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='80 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='09 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='00 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='31 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='25 LIGA† (2021) (reproduced) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='32 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='14 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='93 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='16 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='97 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='09 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='98 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='49 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='01 Improved LIGA† (2021) (baseline) 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='62 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='03 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='94 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='77 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='11 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='19 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='02 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='90 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='81 I ✓ – 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='78 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='65 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='43 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='89 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='37 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='40 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='25 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='38 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='27 II – ✓ 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='67 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='57 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='19 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='81 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='69 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='84 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='77 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='65 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='62 III ✓ ✓ 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='57 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='75 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='92 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='19 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='76 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='42 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='43 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='31 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='10 Improvement over baseline +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='95 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='72 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='98 +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='42 +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='65 +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='23 +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='41 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='41 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='29 Table 3: Ablation studies for our proposed XGD and CLD on the KITTI validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' ⋆ and † in this table denote the results reported in the paper and our reproduced results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' ‘Mod.’ is short for Moderate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Methods Car AP3D Pedestrian AP3D Cyclist AP3D Easy Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Hard Easy Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Hard Easy Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Hard PointPillars (2019) (teacher) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='89 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='47 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='38 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='84 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='01 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='87 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='58 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='06 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='37 DSGN (2020) (student) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='27 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='21 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='61 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='45 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='33 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='07 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='76 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='91 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='04 LIGA (2021) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='46 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='40 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='29 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='68 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='76 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='38 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='75 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='28 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='25 XGD + CLD 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='74 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='49 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='13 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='98 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='18 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='75 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='67 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='00 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='95 Improvement +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='28 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='09 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='84 +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='30 +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='42 +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='37 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='92 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='72 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='70 Improved LIGA (2021) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='94 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='27 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='00 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='37 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='84 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='54 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='84 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='71 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='12 XGD + CLD 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='24 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='62 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='72 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='81 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='69 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='78 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='10 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='19 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='13 Improvement +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='30 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='35 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='72 +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='44 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='85 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='24 +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='26 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='48 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='01 Table 4: Generality of our StereoDistill (XGD + CLD) on the KITTI validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' We select the popular LiDAR model PointPillars (Lang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2019) as the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' ‘Mod.’ is short for Moderate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Comparisons with State-of-the-art Methods Evaluation on KITTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' In Table 1, we present quantitative comparison with the leading stereo-based 3D detectors and several popular LiDAR-based 3D detectors on the KITTI test benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Our method outperforms the SOTA model LIGA (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021) with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='73%, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='23% and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='16% 3D mAP on Cars, Pedestrian and Cyclists at the moderate dif- ficulty level, without introducing any extra cost during in- ference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Our StereoDistill even surpasses the LiDAR-based method MV3D (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2017b) with 3D mAP of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='76% on Cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' These superior results demonstrate the effective- ness of our StereoDistill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' For visualization, please refer to our supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Evaluation on Argoverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' To further verify the generality of our proposed method, we conduct experiments on the Ar- goverse dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' For fair comparisons, we adopt the same network of student and teacher models with LIGA (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021) and also re-implement LIGA under the same setting on the Argoverse dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' In Table 2, we present the results with the 3D IoU thresholds of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='25 for both the BEV and 3D detection on moderate Cars and Pedestri- ans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Our method exceeds LIGA with 3D mAPs of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='18% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='94% and BEV mAPs of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='52% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='03% on Cars and Pedestrians, which validates the generality of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Ablation Studies Ablation Studies on StereoDistill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' In this part, we verify the effectiveness of the proposed compositions, including the XGD for regression, CLD for classification, and their combinations in StereoDistill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The baseline model is our im- proved LIGA (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021) by further enhancing the feature distillation (refer to our supplementary materials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' In Table 3, by comparing (I), (III) with the baseline model, the proposed XGD and CLD bring consistent improvements over the baseline on all difficulty levels for three categories, which demonstrates their effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Note that the pro- posed XGD greatly boosts the detection performance on small objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=', Pedestrians), which requires more accu- rate regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' It illustrates that the X-component guided distillation can indeed transfer superior location awareness from the LiDAR model to the stereo model so as to obtain better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Integrating these two ingredients, Stere- oDistill in (III) outperforms the baseline with remarkable margins of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='72%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='65% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='41% 3D mAP on the mod- erate Cars, Pedestrians and Cyclists, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Generality of StereoDistill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' To verify the generality of StereoDistill, we replace the common teacher network SEC- OND (Yan, Mao, and Li 2018) with the other popular 3D de- tector Pointpillars (Lang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' In Table 4, we provide two baseline settings: the original LIGA (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021) in Line 4 and the improved LIGA in Line 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Not surpris- ingly, our StereoDistill yields obvious performance gains on all difficulty levels for three categories, which further the su- periority and generality of our proposed XGD and CLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Analysis of XGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' In Table 5, we conduct extensive abla- tion studies to analyze the effectiveness of XGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The base- # Methods Cars Pedestrians Cyclists I Baseline 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='57 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='69 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='65 II XGD-Center 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='22 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='33 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='19 III XGD-Size 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='02 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='46 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='71 IV XGD-Angle 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='84 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='11 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='18 V XGD 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='75 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='76 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='31 VI High-quality boxes 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='72 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='65 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='11 VII Positive anchors (Ours) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='75 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='76 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='31 Table 5: Ablation studies for XGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The results are evalu- ated with 3D mAP on the moderate difficulty level for Cars, Pedestrians and Cyclists, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' XGD-* means the manner of only adopting * for computing XGD loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' # Methods Cars Pedestrians Cyclists I Baseline 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='65 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='37 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='38 II Positive Anchors 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='66 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='71 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='32 III All Foregrounds (Ours) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='75 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='76 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='31 IV Classical (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2017a) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='74 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='18 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='69 V CLD (Ours) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='75 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='76 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='31 Table 6: Ablation studies for CLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The results are evalu- ated with 3D mAP on the moderate Cars, Pedestrians and Cyclists, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' line (I) is our StereoDistill without XGD loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Then, we decompose the regression of 3D boxes into three compo- nents including the center position, the size, and the orienta- tion angle to analyze the effect of each component on XGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' We observe that XGD with only the center position (II) ex- hibits the most competitive performance of the three (II, III, IV), which illustrates the positive guidance of the center po- sition is the crucial component to help the student model to acquire more beneficial localization information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Finally, combined with these three components, XGD (V) exceeds the baseline (I) with 3D mAP of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='08%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='07% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='66% on moderate Cars, Pedestrians, and Cyclists, demonstrating the superiority of XGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Moreover, compared with retain- ing high-quality boxes in (VI) whose confidence scores are greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='3 (a proper threshold), our manner of adopting all positive anchors in (VII) has obvious gains on Cars and Pedestrians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' In the real scene, these two categories usually occupy a much larger number than Cyclists, which means that there may be more false positives on Cars and Pedestri- ans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' This demonstrates that our XGD provides a reasonable workaround to deal with some low-quality boxes by retain- ing the beneficial X-component but discarding the harmful X-component decomposed from a 3D box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Effectiveness of CLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' In Table 6, we first present the re- sults of distilling the confidence distribution from two alter- native regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The way based on the foreground (III) ex- ceeds the approach by only considering the positions from positive anchors (II) on average, which illustrates the im- portance of introducing the useful foreground positions be- yond the positions of the positive anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Furthermore, we provide a classical logit distillation (IV) (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020c) as a comparison, which individually treats the confidence distribution of each anchor from each position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' It can be observed that our CLD boosts the performance with mAP Methods Cars Pedestrians Cyclists Mean DSGN (LIGA wo FD) 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='32 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='23 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='26 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='60 + FD 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='14 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='97 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='49 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='20 Improvement +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='82 +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='74 +8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='23 +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='60 + our RD 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='42 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='28 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='66 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='12 Improvement +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='10 +11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='05 +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='4 +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='52 StereoDistill 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='75 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='76 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='31 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='94 Improvement +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='43 +12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='53 +12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='05 +10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='34 Table 7: Comparisons for the feature-based and response- based distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The results are evaluated with 3D mAP on moderate Cars, Pedestrians and Cyclists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' FD and RD are short for feature-based distillation in LIGA (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021) and the response-based distillation in StereoDistill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='01%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='58% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='62% on Cars, Pedestrian and Cy- clists, clearly demonstrating the effectiveness of underlining the confidence distribution for the best competitive anchor from all anchors in a position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Comparison of Different Distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' In Table 7, we indi- vidually present the comparisons for adopting the feature- based in LIGA (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021) or response-based distilla- tion in StereoDistill based on the stereo model DSGN (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' First, these two distillations can consistently boost performance over the baseline DSGN (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Then, the proposed response-based distillation of our XGD and CLD in StereoDistill even outperforms the feature-based distillation in LIGA (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2021) with 3D mAP of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='92% on average under the same setting, which further demonstrates the effectiveness of the proposed XGD and CLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Moreover, combined with the response-based dis- tillation and the feature-based distillation, our StereoDis- till produces superior performance over the baseline model DSGN with a 3D mAP of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content='34% on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' This reveals that superior feature representations often lead to better re- sponses, which in turn can further facilitate feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Conclusions This paper presents an effective cross-modal distillation ap- proach termed StereoDistill from the response levels for the stereo 3D detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The core designs of StereoDistill are the proposed X-component Guided Distillation (XGD) for regression and the Cross-anchor Logit Distillation (CLD) for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' The extension ablation studies demon- strate the superiority of our proposed XGD for regression and CLD for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Finally, StereoDistill achieves state-of-the-art performance among stereo-based detectors without introducing extra cost in the inference process com- pared to our stereo model on the KITTI test benchmark and the large-scale Argoverse dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' In the future, we wish StereoDistill can be applied to more 3D detectors to improve their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Acknowledgement This work was supported by the National Science Fund of China for Distinguished Young Scholars (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 62225603).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' References Brazil, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=';' metadata={'source': 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A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Carr, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Lucey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=';' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' and Chandraker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' 2017a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Learning efficient object detection models with knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' NeurIPS, 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=' Chen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} +page_content=';' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNAzT4oBgHgl3EQfpf3m/content/2301.01615v1.pdf'} diff --git a/ktE1T4oBgHgl3EQfggRt/content/tmp_files/2301.03230v1.pdf.txt b/ktE1T4oBgHgl3EQfggRt/content/tmp_files/2301.03230v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..492755a432aa91a358995d1b7970cdad35063dbe --- /dev/null +++ b/ktE1T4oBgHgl3EQfggRt/content/tmp_files/2301.03230v1.pdf.txt @@ -0,0 +1,1817 @@ +arXiv:2301.03230v1 [math.CO] 9 Jan 2023 +January 10, 2023 +1:37 +Combination +COMBINATORIAL PROPERTIES FOR A CLASS OF +SIMPLICIAL COMPLEXES EXTENDED FROM +PSEUDO-FRACTAL SCALE-FREE WEB +ZIXUAN XIE1,2, YUCHENG WANG1,3, WANYUE XU1,3, LIWANG ZHU1,3, WEI LI4 and ZHONGZHI +ZHANG1,3 +1Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, +China +2School of Software, Fudan University, Shanghai 200433, China +3School of Computer Science, Fudan University, Shanghai 200433, China +4Academy for Engineering and Technology, Fudan University, Shanghai, 200433, China +{20302010061,15307130038,xuwy,19210240147,fd liwei,zhangzz}@fudan.edu.cn +Simplicial complexes are a popular tool used to model higher-order interactions between elements +of complex social and biological systems. In this paper, we study some combinatorial aspects +of a class of simplicial complexes created by a graph product, which is an extension of the +pseudo-fractal scale-free web. We determine explicitly the independence number, the domination +number, and the chromatic number. Moreover, we derive closed-form expressions for the number +of acyclic orientations, the number of root-connected acyclic orientations, the number of spanning +trees, as well as the number of perfect matchings for some particular cases. +Keywords: simplicial complex; pseudo-fractal; graph product; combinatorial problem; domina- +tion number; independence number; chromatic number; acyclic orientations; perfect matching; +spanning trees +aCorresponding author: Wei Li and Zhongzhi Zhang. +1 + +January 10, 2023 +1:37 +Combination +Z. Xie, Y. Wang, W. Xu, W. Zhu, W. Li & Z. Zhang +1. +INTRODUCTION +Complex networks have become a popular and pow- +erful formalism for describing diverse types of real- +world complex interactive systems in nature and +society, whose nodes and edges represent, respec- +tively, the elements and their interactions in real +systems 1. In a large majority of previous stud- +ies 2, the authors consider only pairwise interactions +between elements in complex systems, overlooking +other interactions such as higher-order ones among +multiple elements. Some recent works 3; 4; 5; 6; 7 +demonstrate that many real-life systems involve not +only dyadic interactions but also interactions among +more than two elements at a time. Such multi- +way interactions among elements are usually called +higher-order interactions or simplicial interactions. +For example, in a scientific collaboration network 8, +for a paper with more than two authors, the in- +teractions among the authors are not pairwise but +higher-order. Similar higher-order interactions are +also ubiquitous in neuronal spiking activities 9; 10, +proteins 11, and other real-life systems. +Since the function and various dynamics of a +complex system rely to a large extent on the way +of interactions between its elements, it is expected +that higher-order interactions have a substantial +impact on collective dynamics of complex systems +with simplicial structure. The last several years +have seen some important progress about profound +influences of higher-order interactions on different +dynamical processes 12, including percolation 13, +public goods game 14, synchronization 15; 16, and +epidemic spreading 17. For example, in compari- +son with pairwise interactions, three-way interac- +tions can lead to many novel phenomena, such +as Berezinkii-Kosterlitz-Thouless percolation tran- +sition 13, abrupt desynchronization 15, as well as +abrupt phase transition of epidemic spreading 17. +In order to describe the widespread higher- +order interactions observed in various real-world +complex systems, a lot of models have been pro- +posed 12, based on some popular mathematical +tools, such as simplicial complexes 18; 19; 7; 20. A +simplex of dimension d, called d-simplex, represents +a single high-order interaction among d+1 nodes 21, +which can be described by a complete graph of d+1 +nodes. For example, a 0-simplex is a node, a 1- +simplex is a link, a 3-simplex is a triangle, while +a 4-simplex is a tetrahedron. For a d-simplex α, +a δ-dimensional face α′ of α is a δ-simplex with +0 ⩽ δ < d formed by a subset of the nodes in α, +i.e., α′ ⊆ α. For instance, the faces of a 4-simplex +include four nodes, six links, and four triangles. A +simplicial complex is a collection of simplices, which +is formed by simplices glued along their faces. A +simplicial complex is called d-dimensional if its con- +stituent simplices are those of dimension at most d. +Thus, simplicial complexes describe higher-order in- +teractions in a natural way. +Most of existing models are stochastically, +which makes it a challengeable task to exactly an- +alyze their topological and dynamical properties. +Very recently, leveraging the edge corona product of +graphs 22; 23, a family of iteratively growing deter- +ministic network model was developed 24; 25 to de- +scribe higher-order interactions. They are called de- +terministic simplicial networks, since they are con- +sist of simplexes. This network family subsumes +the pseudo-fractal scale-free web 26 as a partic- +ular case, which has received considerable atten- +tion from the scientific community, including frac- +tals 27; 28; 29, physics 30; 31; 32; 33; 34; 35, and +cybernetics 36; 37. The deterministic construction +allow to study exactly at least analytically rele- +vant properties: They display the remarkable scale- +free 38 and small-world 39 properties that are ob- +served in most real-world networks 1, and all the +eigenvalues and their multiplicities of their nor- +malized Laplacian matrices can be exactly deter- +mined 24. +Although some structural and algebraic prop- +erties for the deterministic simplicial networks have +been studied, their combinatorial properties are less +explored or not well understood. In this paper, we +present an in-depth study on several combinato- +rial problems for deterministic simplicial networks. +Our main contributions are as follows. We first pro- +vide an alternative construction of the networks, +which shows that the networks are self-similar. We +then determine explicitly the domination number, +the independence number, as well as the chromatic +matching number. Finally, we provide exact formu- +las for the number of acyclic orientations, the num- +ber of root-connected acyclic orientations, the num- +ber of spanning trees, and the number of perfect +matchings for some special cases. Our exact formu- +lae for the independence number, the domination +number, and the number of spanning trees general- +ize the results 40; 41; 33 previously obtained for the +pseudo-fractal scale-free web. +The main reasons for studying the above com- +2 + +January 10, 2023 +1:37 +Combination +Combinatorial properties for a class of simplicial complexes extended from pseudo-fractal scale-free web +binatorial problems lie in at least two aspects. +The first one is their inherent theoretical inter- +est 42; 43; 44; 45, because it is a theoretical chal- +lenge to solve these problems. For example, count- +ing all perfect matchings in a graph is #P- +complete 46; 47. In view of the hardness, Lov´asz 48 +pointed out that it is of great interest to construct +or find special graphs for which these combinato- +rial problems can be exactly solved. The determin- +istic simplicial networks are in such graph category. +The other justification lies in the relevance of the +studied combinatorial problems to practical appli- +cations. For example, minimum dominating sets 49 +and maximum matchings 50 can be applied to study +structural controllability of networks 51, while max- +imum independent set problem is closely related +to graph data mining 52; 53. Thus, our work pro- +vides useful insight into understanding higher-order +structures in the application scenarios of these com- +binatorial problems. +2. +NETWORK CONSTRUCTIONS +AND PROPERTIES +The family of simplicial networks under considera- +tion was proposed in 24, which is constructed based +on the edge corona product of graphs 22; 23. Let +G1 and G2 be two graphs with disjoint node sets, +where G1 have n1 nodes and m1 edges. The edge +corona product G1 ⊚ G2 of G1 and G2 is a graph ob- +tained by taking one replica of G1 and m1 replicas +of G2, and connecting both end nodes of the i-th +edge of G1 to every node in the i-th replica of G2 +for i = 1, 2, . . . , m1. Let Kq (q ⩾ 1) denote the q- +node complete graph, with K1 being a graph with an +isolate node. Let Gq(g) = (V(Gq(g)), E(Gq(g))) de- +note the studied networks after g iterations, where +V(Gq(g)) and E(Gq(g)) are the sets of nodes and +edges, respectively. Then, Gq(g) is constructed as +follows, controlled by two parameters q and g with +q ⩾ 1 and g ⩾ 0. +Definition 1. +For g = 0, Gq(0) is the complete +graph Kq+2. For g ⩾ 0, Gq(g + 1) is obtained from +Gq(g) by performing the following operation: for +every existing edge of Gq(g), one creates a copy +of the complete graph Kq and connects all its q +nodes to both end nodes of the edge. That is, +Gq(g + 1) = Gq(g) ⊚ Kq. +Figure 1 illustrates the operation obtaining +Gq(g+1) from Gq(g), while Fig. 2 illustrates the net- +work construction processes for two cases of q = 1 +and q = 2. +⇒ +q = 1 +q = 2 +q = 3 +Fig. 1. +Network construction approach. For each existing +edge in network Gq(g), performing the operation on the right- +hand side of the arrow generates network Gq(g+1). The filled +circles stand for the nodes constructing the complete graph +Kq, and all the filled circles which appeared in step g +1 link +to both end open nodes of the edge that already exist in the +previous step g. +G1(2) +G1(1) +G1(0) +G2(0) +G2(1) +Fig. 2. +The first several iterations of Gq(g) for q = 1 and 2. +The nodes generated at different iterations are marked with +different colors. +Let Ng,q = |V(Gq(g))| and Mg,q = |E(Gq(g))| +denote, respectively, the number of nodes and the +number of edges in Gq(g). By construction, one ob- +tains the following recursion relations for Mg,q and +Ng,q: +Mg+1,q = (q + 1)(q + 2) +2 +Mg,q +(1) +and +Ng+1,q = q Mg,q + Ng,q, +(2) +which, together with N0,q = q + 2 and M0,q = +(q + 1)(q + 2)/2, lead to +Mg,q = +�(q + 1)(q + 2) +2 +�g+1 +(3) +and +Ng,q = +2 +q + 3 +�(q + 1)(q + 2) +2 +�g+1 ++ 2(q + 2) +q + 3 . (4) +3 + +January 10, 2023 +1:37 +Combination +Z. Xie, Y. Wang, W. Xu, W. Zhu, W. Li & Z. Zhang +Thus, the average degree of nodes in graph Gq(g) is +2Mg,q/Ng,q, which tends to q + 3 when g is large, +implying that Gq(g) is sparse. +For graph Gq(g), let Wq(g) = V(Gq(g))\V(Gq(g− +1)) represent the set of new nodes generated at it- +eration g. Then, +|Wq(g)| = q +�(q + 1)(q + 2) +2 +�g +. +(5) +Let d(g,q) +v +denote the degree of node v in graph +Gq(g), which was created at iteration gv. Then, +d(g,q) +v += (q + 1)g−gv+1. It is easy to verify that +in graph Gq(g), there are q + 2 nodes with degree +(q + 1)g+1 and q +� +(q+1)(q+2) +2 +�tv nodes with degree +(q + 1)g−gv+1 for 0 < tv < g. +In graph Gq(g), the q + 2 nodes with the high- +est degree (q + 1)g+1 are called hub nodes, which +are generated at the initial interaction g = 0. Let +hk(g), k = 1, 2, . . . , q + 2, denote the q + 2 hub +nodes of graph Gq(g), and let Vh(Gq(g)) denote +the set of these hub nodes, that is, Vh(Gq(g)) = +{h1(g), h2(g), . . . , hq+2(g)}. +Then, the simplicial +networks can be generated in an alternative way, +highlighting the self-similarity. +Proposition 2. +Given graph Gq(g), graph Gq(g+1) +can be obtained by joining (q+1)(q+2) +2 +copies of Gq(g), +denoted as G(i,j) +q +(g) (1 ≤ i < j ≤ q + 2), the k- +th (k = 1, 2, . . . , q + 2) hub node of which is de- +noted by h(i,j) +k +(g). Concretely, in the merging pro- +cess, for each k = 1, 2, . . . , q+2, the q+1 hub nodes +h(1,k) +k +(g), h(2,k) +k +(g), . . ., h(k−1,k) +k +(g), h(k,k+1) +k +(g), . . ., +h(k,q+2) +k +(g) in the corresponding replicas of Gq(g) are +identified as the hub node hk(g + 1) of Gq(g + 1). +Proof. We prove this proposition by induction on +g. For g = 0, the proof is trivial. For g > 0, +assume that the conclusion holds for Gq(g), i.e: +Gq(g) can be obtained as joining (q+1)(q+2) +2 +copies +of Gq(g −1), which are denoted by G(i,j) +q +(g −1) with +1 ≤ i < j ≤ q + 2. During the amalgamation pro- +cess, for each k ∈ {1, 2, . . . , q + 2}, the q + 1 hub +nodes h(1,k) +k +(g −1), h(2,k) +k +(g −1), . . ., h(k−1,k) +k +(g −1), +h(k,k+1) +k +(g−1), . . ., h(k,q+2) +k +(g−1) in the correspond- +ing q + 1 replicas of Gq(g − 1) are identified as the +hub node hk(g) of Gq(g). For convenient description, +we use +Gq(g) =J +� +G(1,2) +q +(g − 1), G(1,3) +q +(g − 1), . . . , +G(1,q+2) +q +(g − 1), . . . , G(q+1,q−1) +q +(g − 1), +G(q+1,q+2) +q +(g − 1); Vh(Gq(g)) +� +, +to denote the above process merging G(i,j) +q +(g − 1) +(1 ≤ i < j ≤ q + 2) to Gq(g), where Vh(Gq(g))) +is the set of the q + 2 hub nodes in Gq(g), which +are identified from the hub nodes in the (q+1)(q+2) +2 +copies G(i,j) +q +(g − 1) of Gq(g − 1). +Next we will prove that the conclusion holds for +Gq(g+1). Note that for any graph G with the degree +of its hub nodes larger than 1, Vh(G ⊚ Kq) = Vh(G). +By Definition 1, Gq(g+1) = Gq(g)⊚Kq. Thus, using +the definition of edge corona product and inductive +hypothesis, we have +Gq(g + 1) = Gq(g) ⊚ Kq += J +� +G(1,2) +q +(g − 1), G(1,3) +q +(g − 1), . . . , +G(q+1,q+2) +q +(g − 1); Vh(Gq(g)) +� +⊚ Kq += J +� +G(1,2) +q +(g − 1) ⊚ Kq, G(1,3) +q +(g − 1) ⊚ Kq, . . . , +G(q+1,q+2) +q +(g − 1) ⊚ Kq; Vh(Gq(g)) +� += J +� +G(1,2) +q +(g), G(1,3) +q +(g), . . . , G(q+1,q+2) +q +(g); +Vh(Gq(g)) +� += J +� +G(1,2) +q +(g), G(1,3) +q +(g), . . . , G(q+1,q+2) +q +(g); +Vh(Gq(g + 1)) +� +. +This finishes the proof. +Figure 3 illustrates the second construction way +of graph Gq(g + 1) for q = 1 and q = 2. +The simplicial networks display some remark- +able properties 24 as observed in most real net- +works 1. They are scale-free, since their node de- +grees obey a power-law distribution P(d) ∼ d−γq +with γq = 2 + ln(q+2) +ln(q+1) − +ln 2 +ln(q+1). They are small- +world, since their diameters grow logarithmically +with the number of nodes and their mean cluster- +ing coefficients approach a large constant q2+3q+3 +q2+3q+5. +Moreover, they have a finite spectral dimension +2[ln(q2+3q+3)−ln 2] +ln(q+1) +. +After introducing the two construction methods +of the simplicial networks Gq(g) and their relevant +properties, in the sequel, we will study analytically +some combinatorial properties of the networks. +4 + +January 10, 2023 +1:37 +Combination +Combinatorial properties for a class of simplicial complexes extended from pseudo-fractal scale-free web +G(1,2) +1 +(g) +G(1,3) +1 +(g) +G(2,3) +1 +(g) +h1(g+1) +h2(g+1) +h3(g+1) +G1(g+1) +G(1,2) +1 +(g) +G(1,3) +1 +(g) +G(2,3) +1 +(g) +h(1,2) +2 +(g) +h(1,2) +1 +(g) +h(1,3) +1 +(g) +h(1,3) +3 +(g) +h(2,3) +3 +(g) +h(2,3) +2 +(g) +⇒ +h1(g+1) +h2(g+1) +h3(g+1) +h4(g+1) +G(1,2) +2 +(g) +G(1,4) +2 +(g) +G(3,4) +2 +(g) +G(2,3) +2 +(g) +G(1,3) +2 +(g) +G(2,4) +2 +(g) +G2(g+1) +G(1,2) +2 +(g) +h(1,2) +2 +(g) +h(1,2) +1 +(g) +G(1,4) +2 +(g) +h(1,4) +1 +(g) +h(1,4) +4 +(g) +G(3,4) +2 +(g) +h(3,4) +4 +(g) +h(3,4) +3 +(g) +G(2,3) +2 +(g) +h(2,3) +2 +(g) +h(2,3) +3 +(g) +G(1,3) +2 +(g) +h(1,3) +1 +(g) +h(1,3) +3 +(g) +G(2,4) +2 +(g) +h(2,4) +4 +(g) +h(2,4) +2 +(g) +⇒ +Fig. 3. +Second construction means for the simiplical networks for two special cases of q = 1 and q = 2. For each +k ∈ {1, 2, . . . , q + 2}, the q + 1 hub nodes h(1,k) +k +(g − 1), h(2,k) +k +(g − 1), . . ., h(k−1,k) +k +(g − 1), h(k,k+1) +k +(g − 1), . . ., h(k,q+2) +k +(g − 1) +in the corresponding q + 1 replicas of Gq(g − 1) are identified as the hub node hk(g) of Gq(g). +3. +INDEPENDENCE NUMBER +For a simple connected graph G = (V(G), E(G)), +abbreviated as G = (V, E), an independent set of G +is a proper subset I of V satisfying that each pair +of nodes in I is not adjacent. An independent set +is called a maximal independent set if it is not a +subset of any other independent set. A maximal in- +dependent set is called a maximum independent set +if it has the largest possible cardinality or size. The +cardinality of any maximum independent set for a +graph G is called the independence number of G and +is denoted by α(G). We now study the independence +number of graph Gq(g), denoted by αq(g). +Theorem 3. +For all g ≥ 0, the independence num- +ber of Gq(g) is: +αq(g) = +�(q + 1)(q + 2) +2 +�g +. +(6) +Proof. For g = 0, Gq(0) is a complete graph of q + 2 +nodes. It is obvious that αq(0) = 1, which is consis- +tent with Eq. (6). +For g ≥ 1, by Definition 1, Gq(g) is obtained +from Gq(g − 1) through replacing each of Mg−1,q +edges in Gq(g − 1) by a (q + 2)-node complete +graph, which includes the edge and its two end +nodes. Let K(1) +q+2, K(2) +q+2, . . ., K(Mg−1,q) +q+2 +denote the +Mg−1,q complete graphs, respectively, correspond- +ing to the Mg−1,q edges in graph Gq(g − 1). Then, +for any independent set I of Gq(g), there is at most +one node in K(i) +q+2, i = 1, 2, . . . , Mg−1,q. In other +words, +���I ∩ K(i) +q+2 +��� ≤ 1 for i = 1, 2, . . . , Mg−1,q. +Therefore, |I| ≤ Mg−1,q = +� +(q+1)(q+2) +2 +�g +, implying +αq(g) ≤ +� +(q+1)(q+2) +2 +�g +. +On the other hand, for every node u in Gq(g), +which is created at generation g, it belongs to a +certain clique K(i) +q+2 (namely, u ∈ K(i) +q+2), 1 ≤ i ≤ +Mg−1,q, and is only connected to the other q + 1 +nodes in this (q+2)-clique. Therefore, by arbitrarily +selecting one newly created node from each K(i) +q+2, +i = 1, 2, . . . , Mg−1,q, one obtains an independent set +5 + +January 10, 2023 +1:37 +Combination +Z. Xie, Y. Wang, W. Xu, W. Zhu, W. Li & Z. Zhang +for Gq(g) with size equal to Mg−1,q = +� +(q+1)(q+2) +2 +�g +, +which leads to αq(g) ≥ +� +(q+1)(q+2) +2 +�g +. +Combining the above arguments leads to the +statement. +Equation (6) generalizes the result in 41 for q = 1 +to positive integer q. +4. +DOMINATION NUMBER +For a graph G = (V, E), a dominating set for G is +a subset D of V such that every node not in D is +adjacent to at least one node in D. A dominating +set is called a minimal dominating set if it is not a +proper subset of any other dominating set. A dom- +inating set is called a minimum dominating set if it +has the smallest cardinality among all dominating +sets. The cardinality of any minimum dominating +set for graph G is called the domination number +of G, denoted by γ(G). For graph G, the relation +γ(G) ≤ α(G) always holds 54; 55. +Let γq(g) denote the domination number of +graph Gq(g). When g is small, γq(g) is easily de- +termined. Since Gq(0) is a complete graph of q + 2 +nodes, γq(0) = 1, and every node can be considered +as a minimum dominating set for Gq(0). For Gq(1), +the domination number is obtained in the following +lemma. +Lemma 4. +The domination number of graph Gq(1) +is γq(1) = q + 1. And each subset of the hub node +set Vh(Gq(1)) containing q + 1 nodes is a minimum +dominating set for Gq(1). +Proof. By Proposition 2, Gq(1) can be generated by +joining (q+1)(q+2) +2 +copies of Kq+2 at the q + 2 hub +nodes, which are denoted by K(i,j) +q+2, 1 ≤ i < j ≤ +q + 2. Now we show that for any dominating set +D of Gq(1), we can construct a dominating set D′ +including only hub nodes in Gq(1). +Suppose that v ∈ D is not a hub node. By con- +struction, the neighbors of v are all from a single +complete graph K(i,j) +q+2. We can replace v by hub node +hi(1) or hj(1) to obtain a dominating set Dv. In a +similar way, we can replace other non-hub nodes in +Dv to obtain a dominating set D′. Thus, to find the +domination number for Gq(1), one can only choose +hub nodes to form a minimum dominating set. +We continue to show that for any minimum +dominating set D of Gq(1) containing only hub +nodes, |D| = q + 1. By contradiction, assume that +|D| < q + 1, which means that there exist at least +two hub nodes hi(1) and hj(1) not in D, which be- +long to the complete graph K(i,j) +q+2. Then, the non- +hub nodes in K(i,j) +q+2 are not dominated, implying the +D is not a dominating set. Therefore, |D| ≥ q + 1 +and γq(1) ≥ q + 1. On the other hand, it is easy to +check that any q + 1 hub nodes can dominate all +nodes in Gq(1). Hence, γq(1) = q + 1. +Figure 4 illustrates the minimum dominating +sets of G1(1) and G2(1). +h1(1) +h2(1) +h3(1) +G1(1) +h1(1) +h2(1) +h3(1) +h4(1) +G2(1) +Fig. 4. +Examples of the minimum dominating sets for G1(1) +and G2(1). +For a subset C of V(Gq(g)), if a node v in +V(Gq(g))\C is adjacent to at least one node in C, +we say that v is dominated by C. Thus, if all nodes +in V(Gq(g))\C are dominated by C, then C is a dom- +inating set of Gq(g). +Lemma 5. +For a subset C of V(Gq(g)) correspond- +ing to graph Gq(g) with g ≥ 1, if all the non-hub +nodes of Gq(g) are in C or dominated by C, then C +is a dominating set of Gq(g). +Proof. We only need to prove that all the q + 2 hub +nodes of Gq(g) are either dominated by C or included +in C. For any hub node hi(g) (i = 1, 2, . . . , q + 2) of +graph Gq(g), it and another hub node hj(g) (i ̸= j) +create a clique Kq with q nodes at generation g, +which, together with hi(g) and hj(g), form a (q+2)- +clique in Gq(g). If hi(g) ∈ C, the lemma holds. For +the case that hi(g) /∈ C, we show below that hi(g) is +dominated by a node in C. Since the q newly intro- +duced nodes in Kq are non-hub nodes in Gq(g), for +any node v ∈ Kq, it is either included in C or dom- +inated by C. If v ∈ C, then hi(g) is dominated by +C. If v is not in C but dominated by hj(g) or other +non-hub nodes in its generating (q + 2)-clique, then +hi(g) is dominated by C. +We are now in position to determine the domi- +nation number γq(g) of graph Gq(g) for the case of +g ≥ 1. +6 + +January 10, 2023 +1:37 +Combination +Combinatorial properties for a class of simplicial complexes extended from pseudo-fractal scale-free web +Theorem 6. +For all g ≥ 1, the domination number +of Gq(g) is: +γq(g) = q2 + 2q − 1 +q + 3 +�(q + 1)(q + 2) +2 +�g−1 ++ 2(q + 2) +q + 3 . +Proof. From Proposition 2, Gq(g + 1) can be gen- +erated by joining (q+1)(q+2) +2 +copies of Gq(g) at the +q + 2 hub nodes, denoted by G(1) +q (g), G(2) +q (g), . . ., +G +( (q+1)(q+2) +2 +) +q +(g), respectively. Let D(Gq(g + 1)) rep- +resent a dominating set of Gq(g + 1). For any i = +1, 2, . . . , (q+1)(q+2) +2 +, the non-hub nodes of G(i) +q (g) are +dominated or belong to D(Gq(g + 1)) ∩ G(i) +q (g). By +Lemma 5, D(Gq(g +1))∩G(i) +q (g) is a dominating set +of G(i) +q (g). Then, |D(Gq(g + 1))| can be computed in +terms of |D(Gq(g + 1)) ∩ G(i) +q (g)| as +|D(Gq(g + 1))| = −q|D(Gq(g + 1)) ∩ Vh(Gq(g + 1))|+ +(q+1)(q+2) +2 +� +i=1 +|D(Gq(g + 1)) ∩ G(i) +q (g)|, +where the first term on the right-hand side compen- +sates for the overcounting of the hub nodes chosen +in D(Gq(g + 1)). Since |Vh(Gq(g + 1))| = q + 2, +|D(Gq(g + 1))| ≥ +(q+1)(q+2) +2 +� +i=1 +|D(Gq(g + 1)) ∩ G(i) +q (g)| +− q(q + 2). +Particularly, when D(Gq(g +1)) is a minimum dom- +inating set of Gq(g+1), denoted by Dmin(Gq(g+1)), +|Dmin(Gq(g + 1))| +≥ −q(q + 2) + +(q+1)(q+2) +2 +� +i=1 +|Dmin(Gq(g + 1)) ∩ G(i) +q (g)| +≥ −q(q + 2) + (q + 1)(q + 2) +2 +|Dmin(Gq(g))|, +(7) +where the second inequality is due to the fact that +Dmin(Gq(g + 1)) ∩ G(i) +q (g) is a dominating set of +G(i) +q (g), with cardinality larger than that of a mini- +mum dominating set for G(i) +q (g). +Note that the terms on both sides of Eq. (7) +are equal to each other, when all hub nodes of +Gq(g + 1) are in Dmin(Gq(g + 1)) and the inter- +section of Dmin(Gq(g + 1)) with each G(i) +q (g) forms +a minimum dominating set of G(i) +q (g). In other +words, Dmin(Gq(g + 1)) is the union of the mini- +mum dominating sets D(i) +min(Gq(g)) of G(i) +q (g), such +that Dmin(Gq(g + 1)) includes all hub nodes of +Gq(g +1). Thus, for each D(i) +min(Gq(g)), it contains as +many hub nodes in G(i) +q (g) as possible. For g = 1, +all hub nodes of Gq(2) and arbitrary other q − 1 +hub nodes for each G(i) +q (1) constitute a minimum +dominating set Dmin(Gq(2)) of Gq(2). By Lemma 5, +Dmin(Gq(2)) ∩ G(i) +q (1) forms a minimum dominating +set of G(i) +q (1). It is easy to verify that for g = 1, +the terms on both sides of Eq. (7) are equal to +each other. Using Dmin(Gq(2)), we can construct a +minimum dominating set Dmin(Gq(3)) of Gq(3) by +merging the minimum dominating sets D(i) +min(Gq(2)) +for each G(i) +q (2) and removing those duplicate hub +nodes of Gq(3). In a similar way, we can iteratively +construct a minimum dominating set Dmin(Gq(g)) +for Gq(g) when g ≥ 4, for which the equal mark +holds for Eq. (7). Then, we have +γq(g + 1) = −q(q + 2) + (q + 1)(q + 2) +2 +γq(g) +(8) +for all g ≥ 1. With initial condition γq(1) = q + 1, +the above equation is solved to obtain: +γq(g) = q2 + 2q − 1 +q + 3 +�(q + 1)(q + 2) +2 +�g−1 ++ 2(q + 2) +q + 3 . +This finishes the proof. +Theorem 6 is reduced to the result in 40 obtained +for q = 1. +5. +CHROMATIC NUMBER +Node coloring of a graph G = (V, E) is a way of +coloring the nodes of G such that no two adjacent +nodes in V are of the same color. The chromatic +number of a graph G, denoted by χ(G), is the small- +est number of colors needed to color the nodes of G. +For graph G, node coloring is closely related to its +chromatic polynomial P(G, λ), which is a polyno- +mial counting the number of distinct ways to color +G with λ or fewer colors. The chromatic polynomial +was first introduced by George David Birkhoff 56. It +contains at least as much information about the col- +orability of graph G as does the chromatic number. +Indeed, χ(G) is the smallest positive integer that is +not a root of the chromatic polynomial, that is, +χ(G) = min{λ : P(G, λ) > 0}. +(9) +7 + +January 10, 2023 +1:37 +Combination +Z. Xie, Y. Wang, W. Xu, W. Zhu, W. Li & Z. Zhang +The chromatic polynomial for the q-node complete +graph Kq is +P(Kq, λ) = λ(λ − 1)(λ − 2) · · · (λ − q + 1). +(10) +For two graphs G += +(V(G), E(G)) and G′ += +(V(G′), E(G′)), let G ∪ G′ represent their union with +node set V(G) ∪ V(G′) and edge set E(G) ∪ E(G′), +and let G ∩ G′ denote their intersection with node +set V(G) ∩ V(G′) and edge set E(G) ∩ E(G′). Then, +the chromatic polynomial of graph G ∪ G′ is 57 +P(G ∪ G′, λ) = P(G, λ) · P(G′, λ) +P(G ∩ G′, λ) +. +(11) +Lemma 7. +For all g ≥ 0, the chromatic polynomial +of Gq(g) is +P(Gq(g), λ) += λ(λ − 1) +q+1 +� +i=2 +(λ − i) +2 +q(q+3) +� +(q+1)(q+2) +2 +�g+1 +− +2 +q(q+3) . +(12) +Proof. Proposition 2 shows that Gq(g + 1) is in fact +an amalgamation of (q+1)(q+2) +2 +copies of Gq(g) at the +q + 2 hub nodes, denoted by G(1) +q (g), G(2) +q (g), . . ., +G +( (q+1)(q+2) +2 +) +q +(g), respectively. Since the hub nodes of +Gq(g + 1) are linked to each other, Gq(g + 1) can be +also obtained from the (q+1)(q+2) +2 +copies of Gq(g) by +merging them at the (q+1)(q+2) +2 +edges of the com- +plete graph Kq+2 formed by their q + 2 hub nodes. +That is, +Gq(g+1) = Kq+2∪G(1) +q (g)∪G(2) +q (g)∪. . .∪G +(q+1)(q+2) +2 +q +(g), +where +Kq+2 ∩ G(1) +q (g) = K2 +and +� +Kq+2∪G(1) +q (g)∪G(2) +q (g) . . .∪G(j) +q (g) +� +∩G(j+1) +q +(g) = K2, +for all 1 ≤ j < (q+1)(q+2) +2 +. Thus, by using Eq. (11) +(q+1)(q+2) +2 +times, we establish a recursion relation +between P(Gq(g + 1), λ) and P(Gq(g), λ) as +P(Gq(g+1), λ) = (P(Gq(g), λ)) +(q+1)(q+2) +2 +· P(Kq+2, λ) +(P(K2, λ)) +(q+1)(q+2) +2 +. +(13) +With the initial condition P(Gq(0), λ) = P(Kq+2, λ) +and Eq. (10), Eq. (13) is solved to obtain Eq. (12). +Combining Eq. (9) and Lemma 7, one obtains +the following theorem. +Theorem 8. +For all g ≥ 0, the chromatic number +of Gq(g) is χ(Gq(g)) = q + 2. +6. +ENUMERATION OF ACYCLIC +ORIENTATIONS +For an undirected graph G = (V, E), an acyclic ori- +entation of G is to assign a direction to each edge +in G to make it into a directed acyclic graph 58. +An acyclic orientation of G is called an acyclic root- +connected orientation when there exists a distinct +root node reachable from every node in G in the +resulting directed graph 59. +This section is devoted to the determination +of the number of acyclic orientations, as well as +the number of acyclic root-connected orientations +in graph Gq(g). To achieves this goal, we resort to +the tool of Tutte polynomial 60. For graph G = +(V(G), E(G)), its Tutte polynomial T(G; x, y) is de- +fined as +T(G; x, y) = +� +H⊆G +(x − 1)r(G)−r(H)(y − 1)n(H), (14) +where the sum runs over all the spanning subgraphs +H of G, r(G) = |V(G)| − k(G) is the rank of G, +n(G) = |E(G)| − |V(G)| + k(G) is the nullity of G, +and k(G) is the number of components of G. +The evaluation of the Tutte polynomial of +graph G at a particular point on (x, y)-plane is re- +lated to many combinatorial aspects of G 61. It +has been shown that T(G; 2, 0) equals the number +of acyclic orientations of G 58, while T(G; 1, 0) is +equivalent to the number of root-connected acyclic +orientations of G 59. Moreover, the Tutte polyno- +mial T(G; x, y) is also relevant to the chromatic +polynomial P(G, λ) of graph G. Specifically, P(G, λ) +can be represented in terms of T(G; x, y) at y = 0 +as +P(G, λ) = (−λ)k(G)(−1)n(G)T(G; 1 − λ, 0). +(15) +This connection between the Tutte polynomial and +the chromatic polynomial allows to determine the +number of acyclic orientations and root-connected +acyclic orientations for Gq(g). +Theorem 9. +For graph Gq(g) with g ≥ 0, the num- +8 + +January 10, 2023 +1:37 +Combination +Combinatorial properties for a class of simplicial complexes extended from pseudo-fractal scale-free web +ber of acyclic orientations is +Nao(Gq(g)) = 2 +�(q + 2)! +2 +� +2 +q(q+3) +� +(q+1)(q+2) +2 +�g+1 +− +2 +q(q+3) +, +(16) +and the number of root-connected acyclic orienta- +tions is +Nrao(Gq(g)) = [(q + 1)!] +2 +q(q+3) +� +(q+1)(q+2) +2 +�g+1 +− +2 +q(q+3) . +(17) +Proof. Lemma 7 gives the chromatic polynomial for +graph Gq(g), which together with Eq. (15) governing +the Tutte polynomial and the chromatic polynomial +yields +T(Gq(g); x, 0) = x +q� +i=1 +(x + i) +2 +q(q+3) +� +(q+1)(q+2) +2 +�g+1 +− +2 +q(q+3) . +(18) +By replacing x = 2 and x = 1 in Eq. (18), we obtain +the number of acyclic orientations and the number +of acyclic root-connected orientations of Gq(g), as +given by Eqs. (16) and (17), respectively. +7. +NUMBER OF PERFECT +MATCHINGS +For a graph G = (V, E), a matching M of G is a +subset of E such that no two edges in M share a +common node. A node is called matched or covered +by a matching, if it is an endpoint of one of the edges +in the matching. For a graph G with even number of +nodes, a perfect matching of G is a matching which +matches all nodes in the graph. It was shown that +for a complete graph with even q nodes, the number +of its perfect matchings is (q − 1)!! 62. +For graph Gq(g + 1) and Gq(g), by the sec- +ond construction approach given in Proposition 2, +the number of their nodes satisfies Ng+1,q += +(q+1)(q+2) +2 +Ng,q −q(q +2). Hence, Ng,q is always even +when q is an even number; but Ng,q may be an odd +number when q is odd. Then, for odd q, a perfect +matching may not exist for Gq(g). Below, we will +show that for an even q, perfect matchings always +exist in Gq(g). +Theorem 10. +When q is even and not less than 2, +perfect matchings always exist in Gq(g) for all g ≥ 0. +Proof. By induction on g. For g = 0, Gq(0) is a com- +plete graph with q + 2 nodes. There exist (q + 1)!! +perfect matchings in Gq(0). Thus, the result is true +for g = 0. Suppose that there is a perfect matchings +in Gq(g). Let Mg be a perfect matching of Gq(g) for +g ≥ 0. By construction in Definition 1, Gq(g + 1) is +obtained from Gq(g) by replacing each of Mg,q edges +in Gq(g) by a complete graph having (q + 2) nodes, +which includes the edge and its two end nodes. Let +K(1) +q+2, K(2) +q+2, . . ., K(Mg,q) +q+2 +denote the Mg,q complete +graphs. For each of these complete graphs, corre- +sponding to an edge in Gq(g), since the two end +nodes are covered by Mg, one can chose q/2 inde- +pendent edges in the complete graph to cover the +remaining nodes. In this way, one obtains a perfect +matchings for Gq(g + 1). +We proceed to determine the number of perfect +matchings in Gq(g) for even q, denoted by Nper(g, q). +For this purpose, we first define some intermediary +variables for graph Gq(g). Let Υ0(Gq(g)) denote the +set of matchings of Gq(g) such that the two hub +nodes h1(g) and h2(g) are vacant, while all the other +nodes in Gq(g) are matched. Let Aq(g) be the cardi- +nality of Υ0(Gq(g)). Let Υ1(Gq(g)) be the set of per- +fect matchings of Gq(g), and let Bq(g) = Nper(g, q) +be the cardinality of Υ1(Gq(g)). Note that in Gq(g), +there does not exist such a match that either h1(g) +or h2(g) is vacant, while all other nodes are covered. +Theorem 11. +For even q ≥ 2 and g ≥ 0, the num- +ber of perfect matchings in Gq(g) is +Nper(g, q) = [(q + 1)!!] +2 +q(q+3) +� +(q+1)(q+2) +2 +�g+1 +− +2 +q(q+3) × +(q + 1) +− 2(q+2) +q(q+3)2 +� +(q+1)(q+2) +2 +�g+1 ++ q+2 +q+3 g+ (q+1)(q+2)2 +q(q+3)2 +. +(19) +Proof. Note that Bq(g) = Nper(g, q). We now de- +rive recursive relations for Aq(g + 1) and Bq(g + 1), +on the basis of which we further determine an ex- +act expression for Bq(g). From Proposition 2 and +the definitions of Aq(g) and Bq(g), one obtains the +following recursion relations for Aq(g) and Bq(g): +Aq(g + 1) = (q − 1)!!B +q +2q (g)A +q2+2q+2 +2 +q +(g) , +(20) +Bq(g + 1) = (q + 1)!!B +q +2+1 +q +(g)A +q(q+2) +2 +q +(g) . +(21) +Equation (20) is explained as follows. By +Proposition 2, any matching for graph Gq(g + 1) +can be constructed by joining the matchings of the +(q+1)(q+2) +2 +copies of Gq(g) at the q + 2 hub vertices. +According to the aforementioned analysis, for any +matching of Gq(g + 1), the matched hub vertices of +9 + +January 10, 2023 +1:37 +Combination +Z. Xie, Y. Wang, W. Xu, W. Zhu, W. Li & Z. Zhang +Gq(g+1) must come in pairs. Then, any matching in +Υ0(Gq(g+1)) can be obtained recursively from those +in Υ0(Gq(g)) and Υ1(Gq(g)) for the (q+1)(q+2) +2 +copies +of Gq(g), denoted by G(i) +q (g), i = 1, 2, . . . , (q+1)(q+2) +2 +. +Moreover, related quantities in Eq. (20) are ac- +counted for as follows: (q−1)!! indicates the ways of +pairing the q matched hub vertices, which is equal +to the number of perfect matchings of Kq; q +2 denotes +the pairs of matched hub vertices h1(g) or h2(g) in +G(i) +q (g); while q2+2q+2 +2 += (q+1)(q+2) +2 +− q +2 represents +the cases that h1(g) or h2(g) are vacant in G(i) +q (g). +Analogously, we can interpret Eq. (21). +Figure 5 gives a graphic illustration of Eqs. (20) +and (21) for a particular case of q = 2. +Dividing Eq. (21) by Eq. (20) yields a recursive +relation for Bq(g) +Aq(g) as +Bq(g + 1) +Aq(g + 1) = (q + 1)Bq(g) +Aq(g) . +(22) +With the initial condition Bq(0) +Aq(0) = (q+1)!! +(q−1)!! = q + 1, +Eq. (22) is solved to obtain +Bq(g) +Aq(g) = (q + 1)g+1, +(23) +which implies +Aq(g) = +Bq(g) +(q + 1)g+1 . +(24) +Plugging this expression into Eq. (21) leads to +Bq(g + 1) = +(q + 1)!! +(q + 1) +q(q+2)(g+1) +2 +B +(q+1)(q+2) +2 +q +(g) . (25) +Under the initial condition Bq(0) = (q + 1)!!, +Eq. (25) is solved to obtain Eq. (19). +B2(g)A5 +2(g) +A2(g + 1) +: +(1) +B2 +2(g)A4 +2(g) +B2 +2(g)A4 +2(g) +B2 +2(g)A4 +2(g) +B2(g + 1) +: +(2) +Fig. 5. +Illustration of recursive configurations for matchings +in Υ0(Gq(g + 1)) and Υ1(Gq(g + 1)) for graph Gq(g + 1) with +q = 2. Each filled node represents a covered node, while each +empty node represents a vacant node. +8. +NUMBER OF SPANNING +TREES +For a graph G = (V, E) with |V| nodes, a spanning +tree of G is a connected subgraph of G that has a +node set V and |V| − 1 edges. The number of span- +ning trees in a graph G is called the complexity of +G, which is an important graph invariant 63. +Lemma 12. +Let u and v denote two distinct nodes +in a complete graph Kq with q ≥ 3 nodes. Let +Nu,v(Kq) be the number of spanning forests for Kq, +each of which consists two trees such that u and v +belong to the two different trees. Then, Nu,v(Kq) = +2qq−3. +Proof. Obviously, Nu,v(Kq) equals the number of +those spanning trees in Kq, in each of which u and +v are adjacent to each. By Cayley’s formula 64, +the number of spanning trees of a q-node com- +plete graph is qq−2. For a randomly chosen span- +ning tree T of Kq, the expected degree of node +u is +2(q−1) +q +. Then, in T the probability that u is +a neighbor of v is +2(q−1) +q +· +1 +q−1 = +2 +q. Therefore, +Nu,v(Kq) = qq−2 · 2 +q = 2qq−3. +Theorem 13. +Let Nst(g, q) be the number of span- +ning trees in graph Gq(g) with g ≥ 0 and q ≥ 1. +Then, +Nst(g, q) = 2 +2(q+1) +q(q+3)2 +� (q+1)(q+2) +2 +�g+1 +− +� +q+1 +q+3 +� +g− (q+1)2(q+2) +q(q+3)2 +× +(q + 2) +2(q2+2q−1) +q(q+3)2 +� +(q+1)(q+2) +2 +�g+1 ++ +� +q+1 +q+3 +� +g+ q3+2q2−q+2 +q(q+3)2 +. +Proof. By Definition 1, Gq(g + 1) is obtained from +Gq(g) through replacing every edge in Gq(g) by a +Kq+2 clique, which contains the edge and its two +end nodes. Accordingly, for any spanning tree Tg for +Gq(g), one can construct a spanning tree for Gq(g+1) +in the following way: replace each edge in Tg by a +spanning tree of the Kq+2 clique and replace each +edge (u, v) in Gq(g) but absent in Tg by a spanning +forest for Kq+2, which consists two trees such that u +and v are in different trees. Thus, using Lemma 12, +one obtains +Nst(g + 1, q) += [(q + 2)q]Ng,q−1 [2(q + 2)q−1]Mg,q−Ng,q+1 Nst(g, q) += 2Mg,q−Ng,q+1 (q + 2)(q−1)Mg,q+Ng,q−1 Nst(g, q), +(26) +where Ng,q − 1 is the number of edges in the span- +ning tree Tg in Gq(g), and Mg,q − Ng,q + 1 is the +10 + +January 10, 2023 +1:37 +Combination +Combinatorial properties for a class of simplicial complexes extended from pseudo-fractal scale-free web +number of edges which exist in Gq(g) but is absent +in Tg, in other words, the number of edges in Gq(g) +minus the number of edges in Tg. Plugging the ex- +pressions for Ng,q and Mg,q in Eqs. (4) and (3) into +Eq. (26) gives +Nst(g + 1, q) += Nst(g, q)2 +q+1 +q+3 +� +(q+1)(q+2) +2 +�g+1 +− q+1 +q+3 +!‘!‘ · (q + 2) +q2+2q−1 +q+3 +� +(q+1)(q+2) +2 +�g+1 ++ q+1 +q+3. +(27) +With the initial condition Nst(0, q) = (q + 2)q, one +obtains +Nst(g, q) +=2 +q+1 +q+3 +�g +i=1 +� +(q+1)(q+2) +2 +�i +− q+1 +q+3g × (q + 2)q× +(q + 2) +q2+2q−1 +q+3 +�g +i=1 +� (q+1)(q+2) +2 +�i ++ q+1 +q+3g +=2 +2(q+1) +q(q+3)2 +� +(q+1)(q+2) +2 +�g+1 +− +� +q+1 +q+3 +� +g− (q+1)2(q+2) +q(q+3)2 +× +(q + 2) +2(q2+2q−1) +q(q+3)2 +� +(q+1)(q+2) +2 +�g+1 ++ +� +q+1 +q+3 +� +g+ q3+2q2−q+2 +q(q+3)2 +, +which completes the proof. +Note that Theorem 13 is consistent with result +in 24, obtained using different technique, and re- +duces to the result 33 for the special case of q = 1. +9. +CONCLUSIONS +In this paper, we presented a systematic analytical +study of combinatorial properties for a class of itera- +tively generating simplicial networks, based on their +particular construction and self-similar structure. +We derived the domination number, the indepen- +dence number, and the chromatic number. More- +over, we obtained exact expressions for the number +of spanning trees, the number of perfect matchings +for even q, the number of acyclic orientations, and +the number of root-connected acyclic orientations. +The considered combinatorial problems are a +fundamental research subject of theoretical com- +puter science, many of which are NP-hard and even +#P-complete for a general graph. It is thus of great +interest to study the special family of graphs for +which these challenging combinatorial problems can +be exactly solved. In addition, since the considered +combinatorial problems are relevant to various prac- +tical application in the aspects of network science +and graph data miming, this work provides insight +into understanding the applications of these combi- +natorial problems for simplicial complexes. +Finally, it is worth mentioning that the stud- +ied simplicial networks are in fact constructed it- +eratively by edge corona product, which leads to +the self-similarity of the resulting graphs. It is ex- +pected that our computation approach and pro- +cess for relevant problems are also applicable to +other graph families 65; 66; 67; 68; 69; 70; 71 with +self-similar properties, built by other graph opera- +tions. We note that our techniques also have some +limitations. For example, they do not apply to re- +cursive graphs with stochasticity 72; 73. +ACKNOWLEDGEMENT +The work was supported by the Shanghai Munic- +ipal Science and Technology Major Project (Nos. +2018SHZDZX01 and 2021SHZDZX0103), the Na- +tional Natural Science Foundation of China (Nos. +61872093 and U20B2051), Ji Hua Laboratory, Fos- +han, China (No.X190011TB190), +ZJ Lab, and +Shanghai Center for Brain Science and Brain- +Inspired Technology. Zixuan Xie was also supported +by Fudan’s Undergraduate Research Opportunities +Program (FDUROP) under Grant No. 22099. +REFERENCES +1. M. E. Newman, The structure and function of com- +plex networks, SIAM Rev. 45(2) (2003) 167–256. +2. A.-L. Barab´asi, Network science (Cambridge Uni- +versity Press, 2016). +3. A. R. Benson, D. F. Gleich and J. Leskovec, Higher- +order organization of complex networks, Science +353(6295) (2016) 163–166. +4. J. Grilli, G. Barab´as, M. J. Michalska-Smith and +S. 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Rev. +E 79(3) (2009) p. 031110. +13 + diff --git a/ktE1T4oBgHgl3EQfggRt/content/tmp_files/load_file.txt b/ktE1T4oBgHgl3EQfggRt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..29b3a0a22e9abd2975c1c43c372d6efb4e35127e --- /dev/null +++ b/ktE1T4oBgHgl3EQfggRt/content/tmp_files/load_file.txt @@ -0,0 +1,1245 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf,len=1244 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='03230v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='CO] 9 Jan 2023 January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 2023 1:37 Combination COMBINATORIAL PROPERTIES FOR A CLASS OF SIMPLICIAL COMPLEXES EXTENDED FROM PSEUDO-FRACTAL SCALE-FREE WEB ZIXUAN XIE1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' YUCHENG WANG1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' WANYUE XU1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' LIWANG ZHU1,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Fudan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Shanghai 200433,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' China 3School of Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Fudan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Shanghai 200433,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' China 4Academy for Engineering and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Fudan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Shanghai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 200433,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' China {20302010061,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='15307130038,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='xuwy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='19210240147,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='fd liwei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='zhangzz}@fudan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='cn Simplicial complexes are a popular tool used to model higher-order interactions between elements of complex social and biological systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' In this paper, we study some combinatorial aspects of a class of simplicial complexes created by a graph product, which is an extension of the pseudo-fractal scale-free web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' We determine explicitly the independence number, the domination number, and the chromatic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Moreover, we derive closed-form expressions for the number of acyclic orientations, the number of root-connected acyclic orientations, the number of spanning trees, as well as the number of perfect matchings for some particular cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Keywords: simplicial complex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' pseudo-fractal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' graph product;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' combinatorial problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' domina- tion number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' independence number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' chromatic number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' acyclic orientations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' perfect matching;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' spanning trees aCorresponding author: Wei Li and Zhongzhi Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 1 January 10, 2023 1:37 Combination Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Xie, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Xu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Zhu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Li & Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Zhang 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' INTRODUCTION Complex networks have become a popular and pow- erful formalism for describing diverse types of real- world complex interactive systems in nature and society, whose nodes and edges represent, respec- tively, the elements and their interactions in real systems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' In a large majority of previous stud- ies 2, the authors consider only pairwise interactions between elements in complex systems, overlooking other interactions such as higher-order ones among multiple elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Some recent works 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 7 demonstrate that many real-life systems involve not only dyadic interactions but also interactions among more than two elements at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Such multi- way interactions among elements are usually called higher-order interactions or simplicial interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For example, in a scientific collaboration network 8, for a paper with more than two authors, the in- teractions among the authors are not pairwise but higher-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Similar higher-order interactions are also ubiquitous in neuronal spiking activities 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 10, proteins 11, and other real-life systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Since the function and various dynamics of a complex system rely to a large extent on the way of interactions between its elements, it is expected that higher-order interactions have a substantial impact on collective dynamics of complex systems with simplicial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' The last several years have seen some important progress about profound influences of higher-order interactions on different dynamical processes 12, including percolation 13, public goods game 14, synchronization 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 16, and epidemic spreading 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For example, in compari- son with pairwise interactions, three-way interac- tions can lead to many novel phenomena, such as Berezinkii-Kosterlitz-Thouless percolation tran- sition 13, abrupt desynchronization 15, as well as abrupt phase transition of epidemic spreading 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' In order to describe the widespread higher- order interactions observed in various real-world complex systems, a lot of models have been pro- posed 12, based on some popular mathematical tools, such as simplicial complexes 18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' A simplex of dimension d, called d-simplex, represents a single high-order interaction among d+1 nodes 21, which can be described by a complete graph of d+1 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For example, a 0-simplex is a node, a 1- simplex is a link, a 3-simplex is a triangle, while a 4-simplex is a tetrahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For a d-simplex α, a δ-dimensional face α′ of α is a δ-simplex with 0 ⩽ δ < d formed by a subset of the nodes in α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=', α′ ⊆ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For instance, the faces of a 4-simplex include four nodes, six links, and four triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' A simplicial complex is a collection of simplices, which is formed by simplices glued along their faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' A simplicial complex is called d-dimensional if its con- stituent simplices are those of dimension at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Thus, simplicial complexes describe higher-order in- teractions in a natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Most of existing models are stochastically, which makes it a challengeable task to exactly an- alyze their topological and dynamical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Very recently, leveraging the edge corona product of graphs 22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 23, a family of iteratively growing deter- ministic network model was developed 24;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 25 to de- scribe higher-order interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' They are called de- terministic simplicial networks, since they are con- sist of simplexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' This network family subsumes the pseudo-fractal scale-free web 26 as a partic- ular case, which has received considerable atten- tion from the scientific community, including frac- tals 27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 28;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 29, physics 30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 31;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 33;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 35, and cybernetics 36;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' The deterministic construction allow to study exactly at least analytically rele- vant properties: They display the remarkable scale- free 38 and small-world 39 properties that are ob- served in most real-world networks 1, and all the eigenvalues and their multiplicities of their nor- malized Laplacian matrices can be exactly deter- mined 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Although some structural and algebraic prop- erties for the deterministic simplicial networks have been studied, their combinatorial properties are less explored or not well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' In this paper, we present an in-depth study on several combinato- rial problems for deterministic simplicial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Our main contributions are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' We first pro- vide an alternative construction of the networks, which shows that the networks are self-similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' We then determine explicitly the domination number, the independence number, as well as the chromatic matching number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Finally, we provide exact formu- las for the number of acyclic orientations, the num- ber of root-connected acyclic orientations, the num- ber of spanning trees, and the number of perfect matchings for some special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Our exact formu- lae for the independence number, the domination number, and the number of spanning trees general- ize the results 40;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 41;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 33 previously obtained for the pseudo-fractal scale-free web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' The main reasons for studying the above com- 2 January 10, 2023 1:37 Combination Combinatorial properties for a class of simplicial complexes extended from pseudo-fractal scale-free web binatorial problems lie in at least two aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' The first one is their inherent theoretical inter- est 42;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 43;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 44;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 45, because it is a theoretical chal- lenge to solve these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For example, count- ing all perfect matchings in a graph is #P- complete 46;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' In view of the hardness, Lov´asz 48 pointed out that it is of great interest to construct or find special graphs for which these combinato- rial problems can be exactly solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' The determin- istic simplicial networks are in such graph category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' The other justification lies in the relevance of the studied combinatorial problems to practical appli- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For example, minimum dominating sets 49 and maximum matchings 50 can be applied to study structural controllability of networks 51, while max- imum independent set problem is closely related to graph data mining 52;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Thus, our work pro- vides useful insight into understanding higher-order structures in the application scenarios of these com- binatorial problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' NETWORK CONSTRUCTIONS AND PROPERTIES The family of simplicial networks under considera- tion was proposed in 24, which is constructed based on the edge corona product of graphs 22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Let G1 and G2 be two graphs with disjoint node sets, where G1 have n1 nodes and m1 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' The edge corona product G1 ⊚ G2 of G1 and G2 is a graph ob- tained by taking one replica of G1 and m1 replicas of G2, and connecting both end nodes of the i-th edge of G1 to every node in the i-th replica of G2 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' , m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Let Kq (q ⩾ 1) denote the q- node complete graph, with K1 being a graph with an isolate node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Let Gq(g) = (V(Gq(g)), E(Gq(g))) de- note the studied networks after g iterations, where V(Gq(g)) and E(Gq(g)) are the sets of nodes and edges, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Then, Gq(g) is constructed as follows, controlled by two parameters q and g with q ⩾ 1 and g ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For g = 0, Gq(0) is the complete graph Kq+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For g ⩾ 0, Gq(g + 1) is obtained from Gq(g) by performing the following operation: for every existing edge of Gq(g), one creates a copy of the complete graph Kq and connects all its q nodes to both end nodes of the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' That is, Gq(g + 1) = Gq(g) ⊚ Kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Figure 1 illustrates the operation obtaining Gq(g+1) from Gq(g), while Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 2 illustrates the net- work construction processes for two cases of q = 1 and q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' ⇒ q = 1 q = 2 q = 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Network construction approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For each existing edge in network Gq(g), performing the operation on the right- hand side of the arrow generates network Gq(g+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' The filled circles stand for the nodes constructing the complete graph Kq, and all the filled circles which appeared in step g +1 link to both end open nodes of the edge that already exist in the previous step g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' G1(2) G1(1) G1(0) G2(0) G2(1) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' The first several iterations of Gq(g) for q = 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' The nodes generated at different iterations are marked with different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Let Ng,q = |V(Gq(g))| and Mg,q = |E(Gq(g))| denote, respectively, the number of nodes and the number of edges in Gq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' By construction, one ob- tains the following recursion relations for Mg,q and Ng,q: Mg+1,q = (q + 1)(q + 2) 2 Mg,q (1) and Ng+1,q = q Mg,q + Ng,q, (2) which, together with N0,q = q + 2 and M0,q = (q + 1)(q + 2)/2, lead to Mg,q = �(q + 1)(q + 2) 2 �g+1 (3) and Ng,q = 2 q + 3 �(q + 1)(q + 2) 2 �g+1 + 2(q + 2) q + 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (4) 3 January 10, 2023 1:37 Combination Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Xie, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Xu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Zhu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Li & Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Zhang Thus, the average degree of nodes in graph Gq(g) is 2Mg,q/Ng,q, which tends to q + 3 when g is large, implying that Gq(g) is sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For graph Gq(g), let Wq(g) = V(Gq(g))\\V(Gq(g− 1)) represent the set of new nodes generated at it- eration g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Then, |Wq(g)| = q �(q + 1)(q + 2) 2 �g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (5) Let d(g,q) v denote the degree of node v in graph Gq(g), which was created at iteration gv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Then, d(g,q) v = (q + 1)g−gv+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' It is easy to verify that in graph Gq(g), there are q + 2 nodes with degree (q + 1)g+1 and q � (q+1)(q+2) 2 �tv nodes with degree (q + 1)g−gv+1 for 0 < tv < g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' In graph Gq(g), the q + 2 nodes with the high- est degree (q + 1)g+1 are called hub nodes, which are generated at the initial interaction g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Let hk(g), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' , q + 2, denote the q + 2 hub nodes of graph Gq(g), and let Vh(Gq(g)) denote the set of these hub nodes, that is, Vh(Gq(g)) = {h1(g), h2(g), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' , hq+2(g)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Then, the simplicial networks can be generated in an alternative way, highlighting the self-similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Given graph Gq(g), graph Gq(g+1) can be obtained by joining (q+1)(q+2) 2 copies of Gq(g), denoted as G(i,j) q (g) (1 ≤ i < j ≤ q + 2), the k- th (k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' , q + 2) hub node of which is de- noted by h(i,j) k (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Concretely, in the merging pro- cess, for each k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' , q+2, the q+1 hub nodes h(1,k) k (g), h(2,k) k (g), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=', h(k−1,k) k (g), h(k,k+1) k (g), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=', h(k,q+2) k (g) in the corresponding replicas of Gq(g) are identified as the hub node hk(g + 1) of Gq(g + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' We prove this proposition by induction on g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For g = 0, the proof is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For g > 0, assume that the conclusion holds for Gq(g), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='e: Gq(g) can be obtained as joining (q+1)(q+2) 2 copies of Gq(g −1), which are denoted by G(i,j) q (g −1) with 1 ≤ i < j ≤ q + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' During the amalgamation pro- cess, for each k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' , q + 2}, the q + 1 hub nodes h(1,k) k (g −1), h(2,k) k (g −1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=', h(k−1,k) k (g −1), h(k,k+1) k (g−1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=', h(k,q+2) k (g−1) in the correspond- ing q + 1 replicas of Gq(g − 1) are identified as the hub node hk(g) of Gq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For convenient description, we use Gq(g) =J � G(1,2) q (g − 1), G(1,3) q (g − 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' , G(1,q+2) q (g − 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' , G(q+1,q−1) q (g − 1), G(q+1,q+2) q (g − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Vh(Gq(g)) � , to denote the above process merging G(i,j) q (g − 1) (1 ≤ i < j ≤ q + 2) to Gq(g), where Vh(Gq(g))) is the set of the q + 2 hub nodes in Gq(g), which are identified from the hub nodes in the (q+1)(q+2) 2 copies G(i,j) q (g − 1) of Gq(g − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Next we will prove that the conclusion holds for Gq(g+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Note that for any graph G with the degree of its hub nodes larger than 1, Vh(G ⊚ Kq) = Vh(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' By Definition 1, Gq(g+1) = Gq(g)⊚Kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Thus, using the definition of edge corona product and inductive hypothesis, we have Gq(g + 1) = Gq(g) ⊚ Kq = J � G(1,2) q (g − 1), G(1,3) q (g − 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' , G(q+1,q+2) q (g − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Vh(Gq(g)) � ⊚ Kq = J � G(1,2) q (g − 1) ⊚ Kq, G(1,3) q (g − 1) ⊚ Kq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' , G(q+1,q+2) q (g − 1) ⊚ Kq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Vh(Gq(g)) � = J � G(1,2) q (g), G(1,3) q (g), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' , G(q+1,q+2) q (g);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Vh(Gq(g)) � = J � G(1,2) q (g), G(1,3) q (g), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' , G(q+1,q+2) q (g);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Vh(Gq(g + 1)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Figure 3 illustrates the second construction way of graph Gq(g + 1) for q = 1 and q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' The simplicial networks display some remark- able properties 24 as observed in most real net- works 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' They are scale-free, since their node de- grees obey a power-law distribution P(d) ∼ d−γq with γq = 2 + ln(q+2) ln(q+1) − ln 2 ln(q+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' They are small- world, since their diameters grow logarithmically with the number of nodes and their mean cluster- ing coefficients approach a large constant q2+3q+3 q2+3q+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Moreover, they have a finite spectral dimension 2[ln(q2+3q+3)−ln 2] ln(q+1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' After introducing the two construction methods of the simplicial networks Gq(g) and their relevant properties, in the sequel, we will study analytically some combinatorial properties of the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 4 January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 2023 1:37 Combination Combinatorial properties for a class of simplicial complexes extended from pseudo-fractal scale-free web G(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='2) 1 (g) G(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='3) 1 (g) G(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='3) 1 (g) h1(g+1) h2(g+1) h3(g+1) G1(g+1) G(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='2) 1 (g) G(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='3) 1 (g) G(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='3) 1 (g) h(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='2) 2 (g) h(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='2) 1 (g) h(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='3) 1 (g) h(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='3) 3 (g) h(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='3) 3 (g) h(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='3) 2 (g) ⇒ h1(g+1) h2(g+1) h3(g+1) h4(g+1) G(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='2) 2 (g) G(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='4) 2 (g) G(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='4) 2 (g) G(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='3) 2 (g) G(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='3) 2 (g) G(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='4) 2 (g) G2(g+1) G(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='2) 2 (g) h(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='2) 2 (g) h(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='2) 1 (g) G(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='4) 2 (g) h(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='4) 1 (g) h(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='4) 4 (g) G(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='4) 2 (g) h(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='4) 4 (g) h(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='4) 3 (g) G(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='3) 2 (g) h(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='3) 2 (g) h(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='3) 3 (g) G(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='3) 2 (g) h(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='3) 1 (g) h(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='3) 3 (g) G(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='4) 2 (g) h(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='4) 4 (g) h(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='4) 2 (g) ⇒ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Second construction means for the simiplical networks for two special cases of q = 1 and q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For each k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' , q + 2}, the q + 1 hub nodes h(1,k) k (g − 1), h(2,k) k (g − 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=', h(k−1,k) k (g − 1), h(k,k+1) k (g − 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=', h(k,q+2) k (g − 1) in the corresponding q + 1 replicas of Gq(g − 1) are identified as the hub node hk(g) of Gq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' INDEPENDENCE NUMBER For a simple connected graph G = (V(G), E(G)), abbreviated as G = (V, E), an independent set of G is a proper subset I of V satisfying that each pair of nodes in I is not adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' An independent set is called a maximal independent set if it is not a subset of any other independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' A maximal in- dependent set is called a maximum independent set if it has the largest possible cardinality or size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' The cardinality of any maximum independent set for a graph G is called the independence number of G and is denoted by α(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' We now study the independence number of graph Gq(g), denoted by αq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For all g ≥ 0, the independence num- ber of Gq(g) is: αq(g) = �(q + 1)(q + 2) 2 �g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For g = 0, Gq(0) is a complete graph of q + 2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' It is obvious that αq(0) = 1, which is consis- tent with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For g ≥ 1, by Definition 1, Gq(g) is obtained from Gq(g − 1) through replacing each of Mg−1,q edges in Gq(g − 1) by a (q + 2)-node complete graph, which includes the edge and its two end nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Let K(1) q+2, K(2) q+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=', K(Mg−1,q) q+2 denote the Mg−1,q complete graphs, respectively, correspond- ing to the Mg−1,q edges in graph Gq(g − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Then, for any independent set I of Gq(g), there is at most one node in K(i) q+2, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' , Mg−1,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' In other words, ���I ∩ K(i) q+2 ��� ≤ 1 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' , Mg−1,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Therefore, |I| ≤ Mg−1,q = � (q+1)(q+2) 2 �g , implying αq(g) ≤ � (q+1)(q+2) 2 �g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' On the other hand, for every node u in Gq(g), which is created at generation g, it belongs to a certain clique K(i) q+2 (namely, u ∈ K(i) q+2), 1 ≤ i ≤ Mg−1,q, and is only connected to the other q + 1 nodes in this (q+2)-clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Therefore, by arbitrarily selecting one newly created node from each K(i) q+2, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' , Mg−1,q, one obtains an independent set 5 January 10, 2023 1:37 Combination Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Xie, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Xu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Zhu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Li & Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Zhang for Gq(g) with size equal to Mg−1,q = � (q+1)(q+2) 2 �g , which leads to αq(g) ≥ � (q+1)(q+2) 2 �g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Combining the above arguments leads to the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Equation (6) generalizes the result in 41 for q = 1 to positive integer q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' DOMINATION NUMBER For a graph G = (V, E), a dominating set for G is a subset D of V such that every node not in D is adjacent to at least one node in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' A dominating set is called a minimal dominating set if it is not a proper subset of any other dominating set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' A dom- inating set is called a minimum dominating set if it has the smallest cardinality among all dominating sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' The cardinality of any minimum dominating set for graph G is called the domination number of G, denoted by γ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For graph G, the relation γ(G) ≤ α(G) always holds 54;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Let γq(g) denote the domination number of graph Gq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' When g is small, γq(g) is easily de- termined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Since Gq(0) is a complete graph of q + 2 nodes, γq(0) = 1, and every node can be considered as a minimum dominating set for Gq(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For Gq(1), the domination number is obtained in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' The domination number of graph Gq(1) is γq(1) = q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' And each subset of the hub node set Vh(Gq(1)) containing q + 1 nodes is a minimum dominating set for Gq(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' By Proposition 2, Gq(1) can be generated by joining (q+1)(q+2) 2 copies of Kq+2 at the q + 2 hub nodes, which are denoted by K(i,j) q+2, 1 ≤ i < j ≤ q + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Now we show that for any dominating set D of Gq(1), we can construct a dominating set D′ including only hub nodes in Gq(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Suppose that v ∈ D is not a hub node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' By con- struction, the neighbors of v are all from a single complete graph K(i,j) q+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' We can replace v by hub node hi(1) or hj(1) to obtain a dominating set Dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' In a similar way, we can replace other non-hub nodes in Dv to obtain a dominating set D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Thus, to find the domination number for Gq(1), one can only choose hub nodes to form a minimum dominating set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' We continue to show that for any minimum dominating set D of Gq(1) containing only hub nodes, |D| = q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' By contradiction, assume that |D| < q + 1, which means that there exist at least two hub nodes hi(1) and hj(1) not in D, which be- long to the complete graph K(i,j) q+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Then, the non- hub nodes in K(i,j) q+2 are not dominated, implying the D is not a dominating set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Therefore, |D| ≥ q + 1 and γq(1) ≥ q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' On the other hand, it is easy to check that any q + 1 hub nodes can dominate all nodes in Gq(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Hence, γq(1) = q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Figure 4 illustrates the minimum dominating sets of G1(1) and G2(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' h1(1) h2(1) h3(1) G1(1) h1(1) h2(1) h3(1) h4(1) G2(1) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Examples of the minimum dominating sets for G1(1) and G2(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For a subset C of V(Gq(g)), if a node v in V(Gq(g))\\C is adjacent to at least one node in C, we say that v is dominated by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Thus, if all nodes in V(Gq(g))\\C are dominated by C, then C is a dom- inating set of Gq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For a subset C of V(Gq(g)) correspond- ing to graph Gq(g) with g ≥ 1, if all the non-hub nodes of Gq(g) are in C or dominated by C, then C is a dominating set of Gq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' We only need to prove that all the q + 2 hub nodes of Gq(g) are either dominated by C or included in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For any hub node hi(g) (i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' , q + 2) of graph Gq(g), it and another hub node hj(g) (i ̸= j) create a clique Kq with q nodes at generation g, which, together with hi(g) and hj(g), form a (q+2)- clique in Gq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' If hi(g) ∈ C, the lemma holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For the case that hi(g) /∈ C, we show below that hi(g) is dominated by a node in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Since the q newly intro- duced nodes in Kq are non-hub nodes in Gq(g), for any node v ∈ Kq, it is either included in C or dom- inated by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' If v ∈ C, then hi(g) is dominated by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' If v is not in C but dominated by hj(g) or other non-hub nodes in its generating (q + 2)-clique, then hi(g) is dominated by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' We are now in position to determine the domi- nation number γq(g) of graph Gq(g) for the case of g ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 6 January 10, 2023 1:37 Combination Combinatorial properties for a class of simplicial complexes extended from pseudo-fractal scale-free web Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For all g ≥ 1, the domination number of Gq(g) is: γq(g) = q2 + 2q − 1 q + 3 �(q + 1)(q + 2) 2 �g−1 + 2(q + 2) q + 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' From Proposition 2, Gq(g + 1) can be gen- erated by joining (q+1)(q+2) 2 copies of Gq(g) at the q + 2 hub nodes, denoted by G(1) q (g), G(2) q (g), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=', G ( (q+1)(q+2) 2 ) q (g), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Let D(Gq(g + 1)) rep- resent a dominating set of Gq(g + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For any i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' , (q+1)(q+2) 2 , the non-hub nodes of G(i) q (g) are dominated or belong to D(Gq(g + 1)) ∩ G(i) q (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' By Lemma 5, D(Gq(g +1))∩G(i) q (g) is a dominating set of G(i) q (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Then, |D(Gq(g + 1))| can be computed in terms of |D(Gq(g + 1)) ∩ G(i) q (g)| as |D(Gq(g + 1))| = −q|D(Gq(g + 1)) ∩ Vh(Gq(g + 1))|+ (q+1)(q+2) 2 � i=1 |D(Gq(g + 1)) ∩ G(i) q (g)|, where the first term on the right-hand side compen- sates for the overcounting of the hub nodes chosen in D(Gq(g + 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Since |Vh(Gq(g + 1))| = q + 2, |D(Gq(g + 1))| ≥ (q+1)(q+2) 2 � i=1 |D(Gq(g + 1)) ∩ G(i) q (g)| − q(q + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Particularly, when D(Gq(g +1)) is a minimum dom- inating set of Gq(g+1), denoted by Dmin(Gq(g+1)), |Dmin(Gq(g + 1))| ≥ −q(q + 2) + (q+1)(q+2) 2 � i=1 |Dmin(Gq(g + 1)) ∩ G(i) q (g)| ≥ −q(q + 2) + (q + 1)(q + 2) 2 |Dmin(Gq(g))|, (7) where the second inequality is due to the fact that Dmin(Gq(g + 1)) ∩ G(i) q (g) is a dominating set of G(i) q (g), with cardinality larger than that of a mini- mum dominating set for G(i) q (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Note that the terms on both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (7) are equal to each other, when all hub nodes of Gq(g + 1) are in Dmin(Gq(g + 1)) and the inter- section of Dmin(Gq(g + 1)) with each G(i) q (g) forms a minimum dominating set of G(i) q (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' In other words, Dmin(Gq(g + 1)) is the union of the mini- mum dominating sets D(i) min(Gq(g)) of G(i) q (g), such that Dmin(Gq(g + 1)) includes all hub nodes of Gq(g +1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Thus, for each D(i) min(Gq(g)), it contains as many hub nodes in G(i) q (g) as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For g = 1, all hub nodes of Gq(2) and arbitrary other q − 1 hub nodes for each G(i) q (1) constitute a minimum dominating set Dmin(Gq(2)) of Gq(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' By Lemma 5, Dmin(Gq(2)) ∩ G(i) q (1) forms a minimum dominating set of G(i) q (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' It is easy to verify that for g = 1, the terms on both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (7) are equal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Using Dmin(Gq(2)), we can construct a minimum dominating set Dmin(Gq(3)) of Gq(3) by merging the minimum dominating sets D(i) min(Gq(2)) for each G(i) q (2) and removing those duplicate hub nodes of Gq(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' In a similar way, we can iteratively construct a minimum dominating set Dmin(Gq(g)) for Gq(g) when g ≥ 4, for which the equal mark holds for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Then, we have γq(g + 1) = −q(q + 2) + (q + 1)(q + 2) 2 γq(g) (8) for all g ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' With initial condition γq(1) = q + 1, the above equation is solved to obtain: γq(g) = q2 + 2q − 1 q + 3 �(q + 1)(q + 2) 2 �g−1 + 2(q + 2) q + 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Theorem 6 is reduced to the result in 40 obtained for q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' CHROMATIC NUMBER Node coloring of a graph G = (V, E) is a way of coloring the nodes of G such that no two adjacent nodes in V are of the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' The chromatic number of a graph G, denoted by χ(G), is the small- est number of colors needed to color the nodes of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For graph G, node coloring is closely related to its chromatic polynomial P(G, λ), which is a polyno- mial counting the number of distinct ways to color G with λ or fewer colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' The chromatic polynomial was first introduced by George David Birkhoff 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' It contains at least as much information about the col- orability of graph G as does the chromatic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Indeed, χ(G) is the smallest positive integer that is not a root of the chromatic polynomial, that is, χ(G) = min{λ : P(G, λ) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (9) 7 January 10, 2023 1:37 Combination Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Xie, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Xu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Zhu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Li & Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Zhang The chromatic polynomial for the q-node complete graph Kq is P(Kq, λ) = λ(λ − 1)(λ − 2) · · · (λ − q + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (10) For two graphs G = (V(G), E(G)) and G′ = (V(G′), E(G′)), let G ∪ G′ represent their union with node set V(G) ∪ V(G′) and edge set E(G) ∪ E(G′), and let G ∩ G′ denote their intersection with node set V(G) ∩ V(G′) and edge set E(G) ∩ E(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Then, the chromatic polynomial of graph G ∪ G′ is 57 P(G ∪ G′, λ) = P(G, λ) · P(G′, λ) P(G ∩ G′, λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (11) Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For all g ≥ 0, the chromatic polynomial of Gq(g) is P(Gq(g), λ) = λ(λ − 1) q+1 � i=2 (λ − i) 2 q(q+3) � (q+1)(q+2) 2 �g+1 − 2 q(q+3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (12) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Proposition 2 shows that Gq(g + 1) is in fact an amalgamation of (q+1)(q+2) 2 copies of Gq(g) at the q + 2 hub nodes, denoted by G(1) q (g), G(2) q (g), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=', G ( (q+1)(q+2) 2 ) q (g), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Since the hub nodes of Gq(g + 1) are linked to each other, Gq(g + 1) can be also obtained from the (q+1)(q+2) 2 copies of Gq(g) by merging them at the (q+1)(q+2) 2 edges of the com- plete graph Kq+2 formed by their q + 2 hub nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' That is, Gq(g+1) = Kq+2∪G(1) q (g)∪G(2) q (g)∪.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='∪G (q+1)(q+2) 2 q (g), where Kq+2 ∩ G(1) q (g) = K2 and � Kq+2∪G(1) q (g)∪G(2) q (g) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='∪G(j) q (g) � ∩G(j+1) q (g) = K2, for all 1 ≤ j < (q+1)(q+2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Thus, by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (11) (q+1)(q+2) 2 times, we establish a recursion relation between P(Gq(g + 1), λ) and P(Gq(g), λ) as P(Gq(g+1), λ) = (P(Gq(g), λ)) (q+1)(q+2) 2 P(Kq+2, λ) (P(K2, λ)) (q+1)(q+2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (13) With the initial condition P(Gq(0), λ) = P(Kq+2, λ) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (10), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (13) is solved to obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (9) and Lemma 7, one obtains the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For all g ≥ 0, the chromatic number of Gq(g) is χ(Gq(g)) = q + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' ENUMERATION OF ACYCLIC ORIENTATIONS For an undirected graph G = (V, E), an acyclic ori- entation of G is to assign a direction to each edge in G to make it into a directed acyclic graph 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' An acyclic orientation of G is called an acyclic root- connected orientation when there exists a distinct root node reachable from every node in G in the resulting directed graph 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' This section is devoted to the determination of the number of acyclic orientations, as well as the number of acyclic root-connected orientations in graph Gq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' To achieves this goal, we resort to the tool of Tutte polynomial 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For graph G = (V(G), E(G)), its Tutte polynomial T(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' x, y) is de- fined as T(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' x, y) = � H⊆G (x − 1)r(G)−r(H)(y − 1)n(H), (14) where the sum runs over all the spanning subgraphs H of G, r(G) = |V(G)| − k(G) is the rank of G, n(G) = |E(G)| − |V(G)| + k(G) is the nullity of G, and k(G) is the number of components of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' The evaluation of the Tutte polynomial of graph G at a particular point on (x, y)-plane is re- lated to many combinatorial aspects of G 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' It has been shown that T(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 2, 0) equals the number of acyclic orientations of G 58, while T(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 1, 0) is equivalent to the number of root-connected acyclic orientations of G 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Moreover, the Tutte polyno- mial T(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' x, y) is also relevant to the chromatic polynomial P(G, λ) of graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Specifically, P(G, λ) can be represented in terms of T(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' x, y) at y = 0 as P(G, λ) = (−λ)k(G)(−1)n(G)T(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 1 − λ, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (15) This connection between the Tutte polynomial and the chromatic polynomial allows to determine the number of acyclic orientations and root-connected acyclic orientations for Gq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For graph Gq(g) with g ≥ 0, the num- 8 January 10, 2023 1:37 Combination Combinatorial properties for a class of simplicial complexes extended from pseudo-fractal scale-free web ber of acyclic orientations is Nao(Gq(g)) = 2 �(q + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 2 � 2 q(q+3) � (q+1)(q+2) 2 �g+1 − 2 q(q+3) , (16) and the number of root-connected acyclic orienta- tions is Nrao(Gq(g)) = [(q + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='] 2 q(q+3) � (q+1)(q+2) 2 �g+1 − 2 q(q+3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (17) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Lemma 7 gives the chromatic polynomial for graph Gq(g), which together with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (15) governing the Tutte polynomial and the chromatic polynomial yields T(Gq(g);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' x, 0) = x q� i=1 (x + i) 2 q(q+3) � (q+1)(q+2) 2 �g+1 − 2 q(q+3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (18) By replacing x = 2 and x = 1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (18), we obtain the number of acyclic orientations and the number of acyclic root-connected orientations of Gq(g), as given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (16) and (17), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' NUMBER OF PERFECT MATCHINGS For a graph G = (V, E), a matching M of G is a subset of E such that no two edges in M share a common node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' A node is called matched or covered by a matching, if it is an endpoint of one of the edges in the matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For a graph G with even number of nodes, a perfect matching of G is a matching which matches all nodes in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' It was shown that for a complete graph with even q nodes, the number of its perfect matchings is (q − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For graph Gq(g + 1) and Gq(g), by the sec- ond construction approach given in Proposition 2, the number of their nodes satisfies Ng+1,q = (q+1)(q+2) 2 Ng,q −q(q +2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Hence, Ng,q is always even when q is an even number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' but Ng,q may be an odd number when q is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Then, for odd q, a perfect matching may not exist for Gq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Below, we will show that for an even q, perfect matchings always exist in Gq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' When q is even and not less than 2, perfect matchings always exist in Gq(g) for all g ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' By induction on g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For g = 0, Gq(0) is a com- plete graph with q + 2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' There exist (q + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' perfect matchings in Gq(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Thus, the result is true for g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Suppose that there is a perfect matchings in Gq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Let Mg be a perfect matching of Gq(g) for g ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' By construction in Definition 1, Gq(g + 1) is obtained from Gq(g) by replacing each of Mg,q edges in Gq(g) by a complete graph having (q + 2) nodes, which includes the edge and its two end nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Let K(1) q+2, K(2) q+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=', K(Mg,q) q+2 denote the Mg,q complete graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For each of these complete graphs, corre- sponding to an edge in Gq(g), since the two end nodes are covered by Mg, one can chose q/2 inde- pendent edges in the complete graph to cover the remaining nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' In this way, one obtains a perfect matchings for Gq(g + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' We proceed to determine the number of perfect matchings in Gq(g) for even q, denoted by Nper(g, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For this purpose, we first define some intermediary variables for graph Gq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Let Υ0(Gq(g)) denote the set of matchings of Gq(g) such that the two hub nodes h1(g) and h2(g) are vacant, while all the other nodes in Gq(g) are matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Let Aq(g) be the cardi- nality of Υ0(Gq(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Let Υ1(Gq(g)) be the set of per- fect matchings of Gq(g), and let Bq(g) = Nper(g, q) be the cardinality of Υ1(Gq(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Note that in Gq(g), there does not exist such a match that either h1(g) or h2(g) is vacant, while all other nodes are covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For even q ≥ 2 and g ≥ 0, the num- ber of perfect matchings in Gq(g) is Nper(g, q) = [(q + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='] 2 q(q+3) � (q+1)(q+2) 2 �g+1 − 2 q(q+3) × (q + 1) − 2(q+2) q(q+3)2 � (q+1)(q+2) 2 �g+1 + q+2 q+3 g+ (q+1)(q+2)2 q(q+3)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (19) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Note that Bq(g) = Nper(g, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' We now de- rive recursive relations for Aq(g + 1) and Bq(g + 1), on the basis of which we further determine an ex- act expression for Bq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' From Proposition 2 and the definitions of Aq(g) and Bq(g), one obtains the following recursion relations for Aq(g) and Bq(g): Aq(g + 1) = (q − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='B q 2q (g)A q2+2q+2 2 q (g) , (20) Bq(g + 1) = (q + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='B q 2+1 q (g)A q(q+2) 2 q (g) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (21) Equation (20) is explained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' By Proposition 2, any matching for graph Gq(g + 1) can be constructed by joining the matchings of the (q+1)(q+2) 2 copies of Gq(g) at the q + 2 hub vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' According to the aforementioned analysis, for any matching of Gq(g + 1), the matched hub vertices of 9 January 10, 2023 1:37 Combination Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Xie, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Xu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Zhu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Li & Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Zhang Gq(g+1) must come in pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Then, any matching in Υ0(Gq(g+1)) can be obtained recursively from those in Υ0(Gq(g)) and Υ1(Gq(g)) for the (q+1)(q+2) 2 copies of Gq(g), denoted by G(i) q (g), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' , (q+1)(q+2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Moreover, related quantities in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (20) are ac- counted for as follows: (q−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' indicates the ways of pairing the q matched hub vertices, which is equal to the number of perfect matchings of Kq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' q 2 denotes the pairs of matched hub vertices h1(g) or h2(g) in G(i) q (g);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' while q2+2q+2 2 = (q+1)(q+2) 2 − q 2 represents the cases that h1(g) or h2(g) are vacant in G(i) q (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Analogously, we can interpret Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Figure 5 gives a graphic illustration of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (20) and (21) for a particular case of q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Dividing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (21) by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (20) yields a recursive relation for Bq(g) Aq(g) as Bq(g + 1) Aq(g + 1) = (q + 1)Bq(g) Aq(g) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (22) With the initial condition Bq(0) Aq(0) = (q+1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (q−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' = q + 1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (22) is solved to obtain Bq(g) Aq(g) = (q + 1)g+1, (23) which implies Aq(g) = Bq(g) (q + 1)g+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (24) Plugging this expression into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (21) leads to Bq(g + 1) = (q + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (q + 1) q(q+2)(g+1) 2 B (q+1)(q+2) 2 q (g) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (25) Under the initial condition Bq(0) = (q + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (25) is solved to obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' B2(g)A5 2(g) A2(g + 1) : (1) B2 2(g)A4 2(g) B2 2(g)A4 2(g) B2 2(g)A4 2(g) B2(g + 1) : (2) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Illustration of recursive configurations for matchings in Υ0(Gq(g + 1)) and Υ1(Gq(g + 1)) for graph Gq(g + 1) with q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Each filled node represents a covered node, while each empty node represents a vacant node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' NUMBER OF SPANNING TREES For a graph G = (V, E) with |V| nodes, a spanning tree of G is a connected subgraph of G that has a node set V and |V| − 1 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' The number of span- ning trees in a graph G is called the complexity of G, which is an important graph invariant 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Let u and v denote two distinct nodes in a complete graph Kq with q ≥ 3 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Let Nu,v(Kq) be the number of spanning forests for Kq, each of which consists two trees such that u and v belong to the two different trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Then, Nu,v(Kq) = 2qq−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Obviously, Nu,v(Kq) equals the number of those spanning trees in Kq, in each of which u and v are adjacent to each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' By Cayley’s formula 64, the number of spanning trees of a q-node com- plete graph is qq−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For a randomly chosen span- ning tree T of Kq, the expected degree of node u is 2(q−1) q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Then, in T the probability that u is a neighbor of v is 2(q−1) q 1 q−1 = 2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Therefore, Nu,v(Kq) = qq−2 · 2 q = 2qq−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Let Nst(g, q) be the number of span- ning trees in graph Gq(g) with g ≥ 0 and q ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Then, Nst(g, q) = 2 2(q+1) q(q+3)2 � (q+1)(q+2) 2 �g+1 − � q+1 q+3 � g− (q+1)2(q+2) q(q+3)2 × (q + 2) 2(q2+2q−1) q(q+3)2 � (q+1)(q+2) 2 �g+1 + � q+1 q+3 � g+ q3+2q2−q+2 q(q+3)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' By Definition 1, Gq(g + 1) is obtained from Gq(g) through replacing every edge in Gq(g) by a Kq+2 clique, which contains the edge and its two end nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Accordingly, for any spanning tree Tg for Gq(g), one can construct a spanning tree for Gq(g+1) in the following way: replace each edge in Tg by a spanning tree of the Kq+2 clique and replace each edge (u, v) in Gq(g) but absent in Tg by a spanning forest for Kq+2, which consists two trees such that u and v are in different trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' using Lemma 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' one obtains Nst(g + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' q) = [(q + 2)q]Ng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='q−1 [2(q + 2)q−1]Mg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='q−Ng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='q+1 Nst(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' q) = 2Mg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='q−Ng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='q+1 (q + 2)(q−1)Mg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='q+Ng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='q−1 Nst(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (26) where Ng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='q − 1 is the number of edges in the span- ning tree Tg in Gq(g),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' and Mg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='q − Ng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='q + 1 is the 10 January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 2023 1:37 Combination Combinatorial properties for a class of simplicial complexes extended from pseudo-fractal scale-free web number of edges which exist in Gq(g) but is absent in Tg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' in other words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' the number of edges in Gq(g) minus the number of edges in Tg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Plugging the ex- pressions for Ng,q and Mg,q in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (4) and (3) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (26) gives Nst(g + 1, q) = Nst(g, q)2 q+1 q+3 � (q+1)(q+2) 2 �g+1 − q+1 q+3 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='‘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='‘ · (q + 2) q2+2q−1 q+3 � (q+1)(q+2) 2 �g+1 + q+1 q+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' (27) With the initial condition Nst(0, q) = (q + 2)q, one obtains Nst(g, q) =2 q+1 q+3 �g i=1 � (q+1)(q+2) 2 �i − q+1 q+3g × (q + 2)q× (q + 2) q2+2q−1 q+3 �g i=1 � (q+1)(q+2) 2 �i + q+1 q+3g =2 2(q+1) q(q+3)2 � (q+1)(q+2) 2 �g+1 − � q+1 q+3 � g− (q+1)2(q+2) q(q+3)2 × (q + 2) 2(q2+2q−1) q(q+3)2 � (q+1)(q+2) 2 �g+1 + � q+1 q+3 � g+ q3+2q2−q+2 q(q+3)2 , which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Note that Theorem 13 is consistent with result in 24, obtained using different technique, and re- duces to the result 33 for the special case of q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' CONCLUSIONS In this paper, we presented a systematic analytical study of combinatorial properties for a class of itera- tively generating simplicial networks, based on their particular construction and self-similar structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' We derived the domination number, the indepen- dence number, and the chromatic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' More- over, we obtained exact expressions for the number of spanning trees, the number of perfect matchings for even q, the number of acyclic orientations, and the number of root-connected acyclic orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' The considered combinatorial problems are a fundamental research subject of theoretical com- puter science, many of which are NP-hard and even #P-complete for a general graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' It is thus of great interest to study the special family of graphs for which these challenging combinatorial problems can be exactly solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' In addition, since the considered combinatorial problems are relevant to various prac- tical application in the aspects of network science and graph data miming, this work provides insight into understanding the applications of these combi- natorial problems for simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Finally, it is worth mentioning that the stud- ied simplicial networks are in fact constructed it- eratively by edge corona product, which leads to the self-similarity of the resulting graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' It is ex- pected that our computation approach and pro- cess for relevant problems are also applicable to other graph families 65;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 66;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 67;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 68;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 69;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 70;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 71 with self-similar properties, built by other graph opera- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' We note that our techniques also have some limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' For example, they do not apply to re- cursive graphs with stochasticity 72;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' ACKNOWLEDGEMENT The work was supported by the Shanghai Munic- ipal Science and Technology Major Project (Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 2018SHZDZX01 and 2021SHZDZX0103), the Na- tional Natural Science Foundation of China (Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 61872093 and U20B2051), Ji Hua Laboratory, Fos- han, China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content='X190011TB190), ZJ Lab, and Shanghai Center for Brain Science and Brain- Inspired Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' Zixuan Xie was also supported by Fudan’s Undergraduate Research Opportunities Program (FDUROP) under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' 22099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE1T4oBgHgl3EQfggRt/content/2301.03230v1.pdf'} +page_content=' REFERENCES 1.' metadata={'source': 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Yajnik∗, +Physics Department, Indian Institute of Technology Bombay, Mumbai 400076 +Abstract +The laws of quantum mechanics are couched in subtle mathematical language. +The laws are not usually stated in a compact pedagogical form. Here I present a +possible way to correct this. Essential facts can be distilled into seven statements that +are easy to remember and easily referred back. Also, the current teaching of quantum +mechanics is laden with words of negative connotations, originating as they did during +the early decades of the subject when the subject was intellectually still puzzling. +A wide variety of experiments in the intervening decades, not least those that were +awarded the Nobel Prize of 2022 amply affirm the validity and substantial “reality” +of Quantum Mechanics as a theory. +I take a few of the inadequacies of classical +framework to illustrate that some of the complaints against Quantum Mechanics +are patently misplaced. Finally I discuss the bogeys such as “wave article duality”, +“uncertainty”, “indistinguishability” “statistics” and “entanglement” and advocate +adopting better terminology to save new learners from the old biases. +Contents +1 +Why number the laws? +2 +1.1 +Welcoming the new principle . . . . . . . . . . . . . . . . . . . . . . . . . . +2 +2 +Characteristic experiments +4 +3 +The postulates +6 +3.1 +Remarks on the postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +4 +Is the Classical all reasonable? +13 +5 +Exorcising the bogeys +15 +6 +Conclusion +17 +∗Email : yajnik@iitb.ac.in +1 + +1 +Why number the laws? +We teach three laws of Newton, three laws of Thermodynamics and then include a zeroth +law as well, Four Maxwell’s equations of Electromagnetism etc. This simple mnemonic +pedagogical device is missing from the teaching of Quantum Mechanics. Stating them as +a numbered set of laws would make it easier to remember them. This can also be used to +set logical primacy and separation between them. Then the critical comments would be +easier to make, as in discussing the subtlety of “Newton’s third law”. +Clearly differentiated laws also expose possible pitfalls. For example Newton’s first law +is essentially due to Galileo, and after Newton’s second law, the former can be taken to +correspond to the special case of no applied force. Is it logically independent? Or does it +still have a logical primacy due to direct observability? Furthermore, substantial discussion +is needed to explicate whether the second law defines mass or defines force[1]. Thus, setting +apart the First Law asserts its validity independent of the dynamical framework of Newton. +We hope that in the following discussion on Quantum Mechanics we are able to place +logically independent facts and rules under independent laws. After doing that we will be +able to identify the source of various concerns people have about Quantum Mechanics. It is +also hoped that such laws would be useful to new learners. Considering the strides quantum +science is making, we may soon be teaching such laws in the high school. Clear laws stated +without the negativity imposed by the older generation would reduce the hesitation and +suspicion in accepting and applying the laws. Until some completely new phenomenon +demanding a revision comes to the fore we can be at peace with the laws of Quantum +Mechanics, now known for a century. +This article is mostly motivated by Dirac’s elegant exposition in his famous textbook[2]. +This book is most widely praised but least widely read. Teachers and even experts are +heard saying “.. but that is a difficult book to read”. It happens that Dirac’s exposition +is mathematical, but its style is physics. It makes physicists feel it is too abstract while +the mathematicians need more sophisticated underpinnings. However the book is really +precious for its clarity and elegance. Among many other points of great clarity in this book +is the introduction to many-body quantum mechanics, laying bare the mistaken origins of +the term “second quantisation”. +1.1 +Welcoming the new principle +To be specific, our motivation in what follows is to propose that the language of the “wave +function” has been the slow poison of quantum mechanics. At its inception in 1926 this +language was the easiest to follow for physicists. Further, it serves a very good tool for +visualising electronic orbitals and shall remain a very useful language for many settings. +However its overuse, and debating many points of principle in that language, has lead +to misconceptions and a feeling of incompleteness of the subject and inadequacy of the +framework. +The facts look rather different if we grasp Heisenberg’s seminal contribution. Heisen- +berg’s 1925 paper [3] called for complete refurbishing of mechanics : +2 + +“... one realises that ... even for the simplest quantum-theoretical problems +the validity of classical mechanics simply cannot be maintained. In this situ- +ation it seems sensible to discard all hope of observing hitherto unobservable +quantities such as the position and the period of the electron, ... Instead it +seems more reasonable to try to establish theoretical quantum mechanics ... in +which only relations between observable quantities occur.” +According to this point of view the notion of classical trajectory X(t) needs to be aban- +doned. What can be really ascertained physically is only the transition amplitude Xab for +a particle to have been once seen at a and then at b. We must thus forgo the knowledge +of what may have happened in between as inaccessible in principle. What does not exist +does not need any organising principle. No ontology no epistemology. +Heisenberg replaces the trajectory by an array of numbers Xab and additionally ˙Xab +for the velocity [4]. He then proceeds to check the mutual commutability of these arrays +of numbers. This is what we call matrix mechanics now. It was Dirac’s insight[5] to note +that instead of velocity one should use the canonically conjugate Pab and when this is done +we get the very elegant analogy of the commutation relations (CR) needed in Quantum +Mechanics to the Poisson brackets (PB) of Classical Mechanics. However, unlike the PB +which are based on real analysis, the quantum mechanical brackets CR imply that the +dynamical variables cannot be represented by mere numbers, but need matrices for their +representation. +We may immediately feel very anxious about this radical framework to be used for +something as obvious as a trajectory. But we shall return in a later section to argue that +the real culprit is the Newtonian notion of instantaneous velocity, which is at least as +“unreasonable” as these quantum hypotheses. However for the moment let us consider the +two positive thoughts put across by Dirac. The existence of ¯h is the natural demarcation +between the “macroscopic” and the “microscopic”. +It sets the scale below which the +microscopic world begins. Otherwise the world should remain self-similar under infinite +subdivision. +But Dalton, Avogadro, Boltzmann, Thomson, Rutherford and others had +already investigated that the microscopic world is atomistic and rather different. +Dirac’s second point is that quantum principles need not be viewed negatively, as a +loss of familiar concepts. Rather, there is a new positive principle as a compensation so +to speak. This is the principle of linear superposition. Agreed, there is an added layer of +abstraction. “State of a system” in quantum mechanics is not given by a list of values that +are guaranteed to be the outcome of measurement. However, this abstract state vector +does obey the Principle of Superposition. +The uncanny nature of the situation is best brought home to students by the example +of the position and velocity of a ball in a playing field. +Common sense says that the +possible states of the ball are given by positions ⃗RA and corresponding velocities ⃗vA where +A denotes the point in space and ⃗RA its position vector. But in quantum mechanics we +get new valid “states” of the ball which are linear “sum”s of states at A1, A2, ...An. The +possible number of states increases manifolds, as the relative weightages vary from 0 to 1 +in magnitude, can have relative complex phases, and can include any number n of classical +3 + +states. Thus the Quantum World offers multitudes of possibilities far beyond the classical +imagination. Yet this is not a disaster. Dirac exhorts us to admire the simplicity of the +linear superposition principle rather than be baffled by it. +Thus the basic attitudes to quantum mechanics need to be changed. Presenting the set +of arguments that substantiate this appeal is the goal of this article. In the following in Sec. +2 we summarise the classic experiments that signal the novel behaviour, Sec. 3 presents +the statement of the laws, not in truly pedagogical form but as notes for knowledgeable +peers, Sec. 4 contains a discussion of a few key classical conceptions, where we argue that +the latter are, to wit, as technical as the ones involved in quantum mechanics and indeed +not in accord with observed reality. In Sec. 5 I comment on the prevalent terminology, +identifying the bogeys that need to be banished. Sec. 6 contains the concluding remarks. +2 +Characteristic experiments +It will be important to distinguish between what is truly novel in the quantum phenomena +themselves, versus what makes us uneasy about the mathematical framework. We shall +try to show that whatever is novel is indeed rooted in the phenomena themselves. More +interestingly, as argued later, whatever makes us uneasy about the formalism is perhaps +no worse than the state of affairs in the classical framework. +The enigmas of quantum systems can be summarised as +a) The observer is free to choose what to measure, though the choices are limited in any +one attempt at measurement. +b) The system will produce probabilistic outcomes. +c) Subsequently the system either ceases to exist completely, or the measured attribute +becomes predictable with certainty. +Note the most uncanny feature of the last statement. In macroscopic world we only see +systems change or transform or redistribute. But the quantum world allows entities to +vanish forever, like a photon absorbed by an atom or the neutrino by inverse beta decay in +nuclei. Then the case of measured attributes becoming certain in values may be understood +as one form of extinction of the other values of the attribute. +These points can be stated succinctly as +a’) Subjectivity in the choice of measurement, +b’) probabilistic outcomes, +c’) objectivity of the post measurement state +We now summarise the well known phenomena which we may keep in mind as intrinsic +quantum behaviour +4 + +1. Complementarity of description - The Davisson-Germer experiment idealised as +double slit experiment. +The electrons although particles can get redistributed to give the interference pattern +displayed by wave phenomena. +Extensive work has shown that one can recover +particle like properties or wave like properties but not both in the same measurement. +2. Quantised values in case of some observables - The Stern-Gerlach experiment +A beam of polarised Silver atoms passes through a region of magnetic field pointing +along z-axis. +The emerging beam is split into precisely two streams. +The only +explanation is that the component of intrinsic spin of the Silver atoms along the +applied field can have only two possible values. Further only one of these precise +values can emerge in any measurement, and not an averaged value. +3. Probabilistic outcomes +Outcomes of measurements on identically prepared systems are not identical. We can +at best associate a probability to any outcome. If the Stern-Gerlach type experiment +is performed with a rarefied beam of atoms so that only one atom is passing the +magnetic field region, we cannot predict which of the two orientations the atom will +finally emerge in, only relative probabilities for the two outcomes. +4. Indeterminate evolution during measurement process +Related to the previous point is the independent fact that there is no theory for how +the quantum state evolves “during” the measurement. +The imprint of the quan- +tum system is recorded and the quantum nature of that attribute then terminates. +However there is no theoretical framework for describing the evolution of a quan- +tum system into a residual system with specific value as it leaves its imprint in an +apparatus. +More generally the system itself may disappear such as charged particles in a Geiger +counter. The result is a macroscopic current, and we have a quantum theory of how +the single particle cascades into a current, but not a deterministic theory. +5. Non-locality of states +The quantum state can be spread over a macroscopic region. A state of two photons +can be stretched over many meters. +But as soon as the attributes of one of the +photons are measured, the state evolves through that indeterminate evolution and +the attributes of the other photon are instantaneously determined. +One version of +it is the famous EPR paradox. However unreasonable classically, the validity of this +outcome has finally been accepted and recognised by the Nobel Prize of 2022[6]. +The non-locality is also manifested over time. We might suspect that once we have +decided on which attribute to measure, the two photons mutually encode which one is +to manifest which value before they are far apart. But since there is complementarity +of which attribute to measure, we need not set up which attribute we wish to measure +5 + +until the photons are really far apart. This is called “delayed choice”. However even +under delayed choice, the effect of measuring an attribute on one photon immediately +determines the outcome of that attribute on the other photon. +6. Bose condensation and Pauli Exclusion Principle +Perhaps the most radical departure from classical conceptions arises in the very notion +of identity of the basic entities, particles, or more correctly, quanta. There are two +fundamental aspects to this. One is that there are identical particles : the primary +units are endowed with very few attributes, and all quanta of a particular species have +just a few possibilities for these attribute. For example all electrons have exactly the +same values of charge and mass, and can have one of two values for the projection of +their spin along any measured direction. This is not encountered classically even in +what we think are identical objects, say balls of same size, color and shine of polish. +Improving the precision of measurement always reveals the differences, often taking +continuum values. +Further the quanta obey peculiar rules for collective states. Atomic physics verifies +most directly that two electrons cannot occupy the exact same state of energy and +spin. Likewise, the spectrum of “light gas” under ideal conditions of Black Body +obeys Planck spectrum, which can be understood only if photons occupy their avail- +able energy states according to the rules discovered by Planck, Einstein and Bose. +In summary there are identical particles and they obey strange rules for their col- +lective states. +These facts are foundational to the quantum world and the laws +pertaining to them need to be formalised into the core of Quantum Mechanics. The +law should not be postponed as auxiliary rules to be learnt in more advanced courses. +3 +The postulates +In this section we enunciate the laws in simple naive language. These are stated more as +pointers to what all practitioners of Quantum Mechanics are quite familiar with. The idea +is to lay out these rules in an order from the more basic or elementary, moving towards +those that build the structure further. We skirt several subtleties about the Hilbert space +and precise meaning and varieties of measurement at the level of this presentations. Stating +the more detailed version will however not require a change to this basic list. +Let us make a brief qualitative statement about what is at stake, ´a la Dirac. We have +a quantum system and we have some apparatus that will record various clicks and ticks. +The requirement is to set up a mathematical framework that will make predictions about +what clicks and ticks can result as outcomes. The dynamical quantities that can be thus +measured will be called observables. Examples are charge, mass, spin projections, binding +energies, lifetimes, etc. This list needs to be established empirically for every new quantum +system one encounters. +The program of quantum mechanics is to identify a core set of variables which should +satisfy relatively simple kinematic conditions, the CR’s, and all the other observables should +6 + +be expressible in terms of them. Fortuitously, but with no guarantee, this set happens to +be the same as the set of classical canonical variables, satisfying the CR’s that are, upto +the fundamental unit ¯h the same as the classical PB’s. Indeed all the observables, all the +relevant symmetry operators, and the dynamical evolution operator, can be constructed +out of this canonical set. This is not at all obvious, and indeed it fails in a few major +exceptional cases such as spin which has no canonical representation. A failure also shows +up in the occurrence of anomalies in advanced implementation of quantum principles, such +as in Quantum Field Theory and String Theory. But majority of quantum observables do +have simple classical analogy and that is what has greatly facilitated the prediction and +control of the quantum world. +The rules stated below essentially address the logical structure of these constructs and +the mathematics required to implement them. They are stated in the simplest context of +a single variable and a single observable etc, to keep the statements compact. +We shall use the convention +|Ψ⟩, |ϕ⟩ +A generic state +|Ψt⟩, |ϕt⟩ +A generic state displaying its time dependence +|x⟩, |p⟩ +A basis state labeled by a canonical variable +|α⟩, |l, m⟩, |n⟩ +Eigenstates of general observables labeled by their +eigenvalues +We are dispensing with the convention of putting a hat or a caret above an operator, so +long as there is no ambiguity. +I. State functions constitute a Hilbert space. +The states of a quantum system obey linear superposition principle and have the +structure of a complex vector space. Further, for physical interpretation we need to +endow this space with a hermitian inner product. +⟨ψ|ϕ⟩ = ⟨ϕ|ψ⟩∗ +The framework of Hilbert spaces is applicable with some caveats. +II. Observables are realised as Hermitian operators. +For an operator A, the hermitian conjugate or adjoint operator is defined as : +⟨Aψ|ϕ⟩ ≡ ⟨ψ|A†ϕ⟩ +Hermitian operators are self-adjoint. All observables are represented by hermitian +operators. Their eigenvalues will be the possible list of answers we get upon observa- +tion. The eigenstates corresponding to these eigenvalues can be used to construct a +basis for the Hilbert space. There can be several independent choices of bases. +III. Change of basis is implemented by Unitary operators +7 + +If we change from a basis constructed using an observable n to a basis constructed +using an observable α, then the change is implemented by a unitary transformation +|α⟩ = +� +n +Uαn|n⟩ +� +β +(U†)mβUβn = δm,n +� +m +Uαm(U†)mβ = δα,β +where the Kronecker deltas need to be replaced by Dirac delta functions for continuum +eigenvalues. +IV. Observations and probabilities are described by projection operators. +In an observation process we can only predict the probability for the system to emerge +in a particular eigenvalue of the relevant observable. If the observable is α and the +initial normalised state vector is represented in the |αi⟩ basis as +|Ψ⟩ = +� +i +Ci|αi⟩, +then the probability of getting the outcome αr is |Cr|2. In Hilbert space this amounts +to the state being subjected to a projection operator. +This leads to a simple rule about the average measured value of that observable under +repeated measurement of identically prepared systems. +⟨A⟩ ≡ ⟨Ψ|A|Ψ⟩ ≡ ⟨Ψ|AΨ⟩ = +� +i +|Ci|2αi +where the second equivalence connects a Physics convention with Hilbert space oper- +ation. +V. Quantum kinematics. +The observables and other dynamical quantities introduced are operators and need +not commute. With a sufficiently exhaustive set of dynamical quantities Oi we find +that they satisfy a symplectic algebra +[Oi, Oj] ≡ OiOj − OjOi = +� +k +CijkOk +which has closure and obeys the Jacobi identity. This algebra sets up the quantum +kinematics. +Further, a great simplification is afforded by a deep classical analogy. Corresponding +to the classical canonical variables {x1 . . . xN, p1 . . . pN} there exists a set of quantum +variables such that +[xi, pj] ≡ xipj − pjxi = i¯h{xi, pj}PB = i¯hδij +8 + +where the xi, pj in the third expression are classical variables and PB denotes the +Poisson bracket. +Most operators Oi, hermitian and unitary, can be algebraically +constructed out of the canonical set, with a few notable exceptions such as spin. +Another important feature of quantum theory is that in Quantum Field Theory one +also needs anti-commutator kinematic conditions. These have no classical analogue. +VI. Quantum dynamics. +a) In analogy with Classical Mechanics, there exists a distinguished hermitian opera- +tor, the Hamiltonian. In the Heisenberg picture dynamical evolution is expressed in +terms of time dependent operators which can be observables or other operators +i¯h d +dtO(t) = [O(t), H] +In particular we have the analogues of Hamilton’s equations of motion for the canon- +ical variables, +i¯h d +dtp(t) = [p(t), H]; +i¯h d +dtx(t) = [x(t), H] +from which the evolution equation for any operator on the phase space can be worked +out using the canonical commutation rules. +b) A convenient alternative is the Schr¨odinger picture in which the state vector is +time dependent, |Ψt⟩. In this case the operators are not to be evolved in time, and +the equation of motion for the state function is +i¯h ∂ +∂t|Ψt⟩ = H|Ψt⟩ +In practice we do not work with the abstract state |Ψt⟩, but the “wave function” +Ψ(x, t) ≡ ⟨x|Ψt⟩ and express H(x, p) ≡ H(x, −i¯hd/dx). +c) A third very elegant and fruitful formulation of the dynamics is due to Dirac and +Feynman, the Path Integral version. +Usually the basis set |x⟩ is treated as time +independent. Now define a ”moving basis” (see [2], Sec. 32) which we might call +Dirac picture basis, +|xt⟩D = eiHt/¯h|x⟩ +Then the Path Integral formula gives the amplitude for going to xf at time tf given +that the system was at xi at ti : +D⟨xftf|xiti⟩D = +� +Dx(t)Dp(t) exp +� i +¯h +� tf +ti +dt(p ˙x − H) +� +where the action integral in the exponent is on the phase space and the symbolic +integration Dx(t)Dp(t) is over all possible paths connecting xf and xi. Then we can +obtain Ψ(x, t) from Ψ(xi, ti) for t > ti as +Ψ(x, t) = +� +dxi D⟨xt|xiti⟩DΨ(xi, ti) +9 + +VII. Bosons and fermions. +Another deep and non-classical feature of the quantum world is the existence of +“identical quanta”. These bits of nature have just a few attributes such as mass, spin +and a few charges. In weakly coupled systems, the full multi-quanta Hilbert space +can be constructed out of repeated tensor product of the one-quantum Hilbert space. +This is called the Fock space. +The admissible multi-quanta states are only the symmetrised ones in an assembly of +integer spin quanta, while the admissible states are only the anti-symmetrised ones +for half-integer quanta. For the case of two quanta, with the states labeled by the +values of the observable α, these tensorial constructions are +|Ψ⟩B = +1 +√ +2{|α(1) +1 ⟩|α(2) +2 ⟩ + |α(2) +1 ⟩|α(1) +2 ⟩} +|Ψ⟩F = 1 +√ +2 +{|α(1) +1 ⟩|α(2) +2 ⟩ − |α(2) +1 ⟩|α(1) +2 ⟩} +where subscripts B and F refer to Bose and Fermi respectively. +This ends the list of the postulates incorporating the most essential rules. To repeat in a +nutshell, the postulates refer to +I. Superposition principle and Hilbert space, +II. Observables as hermitian operators, +III. Change of basis as unitary operators, +IV. Observation as projection operator, +V. Kinematics as commutation rules, +VI. Dynamics via a special unitary operator, and +VII. Multi-quanta states via Bose-Einstein and Fermi-Dirac rules +3.1 +Remarks on the postulates +1. The Hilbert space postulate captures two important things in one. +Firstly there is +superposition principle for the states, the deepest non-classical feature of the quantum +world. Secondly we introduced the inner product on the space, as required for physical +interpretation. +2. Many cases of change of basis correspond to classical symmetry operations. Wigner’s +theorem proves that such transformations are indeed represented by unitary or anti- +unitary operators. Thus Postulate III is not entirely an independent postulate, but it +is important enough to be listed here. +10 + +3. The issue of measurement has been a source of creative proposals and long standing +debate among the finest minds. There are two aspects – the outcome is probabilistic and +there is no satisfactory description of the evolution from the unmeasured to the measured +state. Here my first caveat is that a conscious observer may not be a key component +of the measurement paradox. Scattering processes and spontaneous decay are directly +observed phenomena which capture most of the unsettling aspect of “collapse of the +state”, with purely quantum evolution. +i. Intentional measurement has a lot of resemblance to scattering. In Rutherford type +scattering we send in a stream of projectiles, which are momentum eigenstates. +The expected out state obeys the symmetries of the scatterer, for example the az- +imuthal rotation symmetry along the direction of the incoming projectile. However, +a particular scattered particle can emerge only in one fixed direction. Thus while +the evolution operator is unitary, the outcome for a single scattered particle is in a +momentum eigenstate projected out from the evolved state. +Only when a large number of the same projectiles with the same impact parameter +is studied do we recover the azimuthal symmetry. This can be considered to be +a “collapse” into the eigenvector representing that momentum value. Note that +scattering goes on in locations remote from any conscious observer all over the +Universe. The emergence in any one direction of the scattered particle is a generic +event. +ii. Spontaneous decay is a similar phenomenon. The emitted final state may be ex- +pected to obey the symmetries of the decaying parent ( for example rotational +symmetries of an atom or a molecule). And indeed this is so on the average. How- +ever any particular decay results in the particle emerging only in a specific direction, +and the symmetry can be recovered only through repeated experiments. Again, this +process happens exactly thus, with no observer needed, though there could be one, +light years away. +It is interesting to note that this issue is implicit in Einstein’s 1905 paper thus[8] : +According to the assumption considered here, in the propagation of a +light ray emitted from a point source, the energy is not distributed continu- +ously over ever increasing volumes of space, but consists of a finite number +of energy quanta localised at points of space that move without dividing, +and can be absorbed or generated only as complete units”. +It is clear that in the emission of any individual quantum, the rotational symmetries +of the source can not be respected. In turn, if the symmetry is to be recovered over +a large number of observations, it should not be a surprise that the question of +which direction is determined only by a probabilistic law. In hindsight one may +wonder why the bearer of so lucid and profound an insight shied away from the +collateral logical consequences. +4. Returning to the broader issues of measurement, the problem seems to lie in the inability +to characterise where the validity of classical paradigm ends and quantum regime begins. +11 + +In some sense this is due to the fact that ¯h dimensionally involves space, time and mass +and it is difficult to demarcate the transition between the two frameworks purely in +terms of length or time or mass scales. Indeed Bohr’s Correspondence Principle relies +on largeness of quantum numbers to recover a classical description. +In the Schr¨odinger’s Cat paradox the presumption is that all systems must rightly +be considered as quantum. In other words taking the observer to be entangled with +the system being observed. Further, the cat box plus its observer can be a combined +system being observed by another observer, thus demanding a combined description +with indefinite recursion. This is the ”Wigner’s Brother” paradox. +There are two possibilities for a resolution. +Perhaps more delicate experiments will +demand a more sophisticated formalism for their description. But if not, we only need +improved semantics. A characteristic of macroscopic systems is that they are highly +complex, say a Geiger counter or a bubble chamber, making it clear that they can +be simply treated classically, thus at least avoid the recursion paradox. However the +postulate as presented here has been verified in a variety of experiments directly or +indirectly over the past century. While the paradox may persist, there is no contradiction +with the proposed postulate. +5. The Einstein-Podolski-Rosen paradox epitomises another aspect that is counter-intuitive +about measurement. Suppose we have a two-electron state with net spin = 0. Suppose +an observer Alice at some remote point x makes an observation at time t′ +1 observing only +one of the electrons, and that in spin-up state, destroying that electron in the process, +then causality demands that at all subsequent times t > t′ +1, the other observer Bob can +only find one electron and that in spin-down state. According to Special Relativity, this +latter fact cannot in principle be known to Bob during the time t′ +1 < t < t′ +1 + |x|/c. In +case Bob makes an observation during the time t′ +1 < t < t′ +1+|x|/c they may legitimately +ascribe the outcome to the weightage factor 1/ +√ +2 for the spin-down state, and nothing +will go wrong. The main lesson is that quantum states are intrinsically non-local and yet +consistent with Special Relativity. The advanced framework of Relativistic Quantum +Field Theory does not throw up any contradictions either. It does require the existence +of anti-particles in order to preserve causality. More on the possible fallacy in EPR in +the last section. +6. We make a big leap of faith in assuming that all the observables and other operators +can be expressed in terms of the canonical set. We propose that the Oi have the same +algebraic dependence on the canonical variables as the corresponding classical variables. +This is an immense simplification. But it requires a price to be paid +− It results in operator ambiguity, which however is resolved by simple prescriptions. +− One may encounter observables with no classical analogue, e.g., spin +− For fermion fields one needs anti-commutators instead of commutators. There is +no classical limit available for this operation in the normal sense. +12 + +Fermion bilinears do have classical limits and one arranges to set up correct commutation +relations between them with desirable classical limits1. +Despite all these exceptions, this principle must be viewed as of special significance. It +suggests that the macroscopic canonical structure of Hamiltonian dynamics is firmly +rooted in microscopic principles. +7. Quantum kinematics can be equivalently expressed by the overlap between the bases +labeled by canonical observables. This is given by the fundamental relation +⟨x|p⟩ = +1 +√ +2π¯h +eipx/¯h +This is the key building block of the Path Integral formula. +8. Postulate VII is most likely not a logically independent one. When Lorentz invariance +is imposed on a quantum system, the “spin-statistics theorem” can be proved. We state +this here as a law to highlight its importance. That is, that in Quantum Mechanics +states are fundamental, not quanta. And to remind that “quanta are not particles”. +9. Quantum Field Theory provides the comprehensive framework of calculation in Fock +Space based on weakly coupled quanta. However, the Quantum Field Theory framework +has a far greater reach, in a wide variety of strongly coupled systems. +4 +Is the Classical all reasonable? +The framework of Quantum Mechanics couched in the language of the Hilbert space is +at once forbidding to those unfamiliar with advanced Mathematics. However there exists +several quantum systems where this Hilbert space may be no more forbidding than a finite +dimensional complex vector space for a finite number of spins. This is very promising +because it may become possible to teach simplified basics to even highschool students who +may be able to understand the technologies that are now emerging. +Despite this, the framework has become notorious for resorting to purely abstract and +therefore perhaps auxiliary constructs, and for being difficult to grasp instinctively. As a +counter to this, I take up here to argue that a few key concepts of Classical framework are +equally counter-intuitive, and it is more the familiarity developed over several centuries +that makes them so easily acceptable. +1. Instantaneous velocity : The most commonly held drawback of Quantum Mechanics is +the loss of the detailed trajectory of the particle, specified by position and velocity at +any given instant of time. It is however worth pondering how justified this expectation +was in the first place. The notion that an object is at a place and is moving while at +1One might wonder whether this and the preceding point about spin have any mutual connection since +fermions are prime examples of particles with spin. But integer spin particles also exist, and therefore the +need to insert spin by hand will persist and is independent from the enigmatic quantisation of fermions. +13 + +that place actually defies common sense. In a philosophical vain one is led to wonder +how something can be at a place if it is moving. Newton solved this problem through +the Calculus notion of limits. One defines +v = lim +∆t→0 +∆x +∆t +The conceptual proposal however has never been verified empirically. +No one, in +Galileo’s vain, took a meter stick and a stopwatch watch of unlimited precision and +allowed the ∆x and ∆t to physically go to zero. If they did they would discover Quan- +tum phenomena. Thus there was no loss of paradise of unlimited precision, it was never +there in the first place. +To repeat Dirac’s point of view quoted in the Introduction, ¯h indeed sets a scale sep- +arating the Classical from the Quantum. In the limit the action function governing +our moving particle reaches this level, the Newtonian paradise withers away and the +Quantum emerges. +This does draw our attention further to the carefully constructed notion of the contin- +uum in Mathematics which buttresses Newton’s conception. Development of science +may have reached a point making it worth revisiting the tenets of the continuum. And +in turn we need to develop mathematics in which quantum constructs take an intuitive +centre stage. +2. The abstract nature of the wave function is often contrasted with other physical quan- +tities of immediate empirical meaning. The notions of electric and Magnetic fields are +however worth re-inspecting. They are only a step away from being something palpa- +ble, since you need to hold a point charge or a small dipole in that region of space +to immediately see their manifestation as field lines as visualised by Faraday. However +when we propose that these fields oscillate and they carry energy, and they migrate over +light years, in vacuum as Einstein taught us, then we may wonder “what” is actually +propagating in the empirical sense. These fields therefore have a status comparable to +that of the wave function. +What is intriguing then is that according to the Sudarshan-Glauber theory, the quan- +tum theory of light subsumes all its classical states without any change. Perhaps the +validity of the conception of electromagnetic radiation is ultimately derived from the +more fundamental quantum property. +3. Classical conception is willing to live with point particles as well as continuum fields. +However there is an internal contradiction in the Electrodynamics of a point charge. +An oscillating point charge radiates, but the field of the charge cannot act upon itself. +Thus the charge loses energy without any force slowing it down. Lorentz worked to +give a finite size and an internal structure to the electron without success. If the field +of a point charge is allowed to act upon itself we get an infinite answer. The issue +occupied many stalwarts, Dirac, Wheeler and Feynman, and others. Its futility became +14 + +clear after it was realised that the electron can be a point particle, yet obey quantum +rules and never be localised to an ideal point. The questions in the revised version got +transferred to those of a calculable framework of Quantum Electrodynamics, eventually +addressed by renormalisation. +5 +Exorcising the bogeys +A large number of professionals putting Quantum Mechanics successfully to use remain +unsure about its consistency and validity. +To the extent these may only be anxieties +transmitted by the previous generation it is worth examining the various enigmatic issues, +and to finally banish the bogeys which can be resolved in the light of the postulates +formulated here. +To this end I go through a list of terms commonly used which continue to reinforce a +negative perception. It is my appeal that it is appropriate to replace some of the description +with better words and if this is not possible, at least to stop using the terms in that form. +1. The uncertainty principle. This essential novelty of quantum phenomena was gleaned +by Heisenberg before arriving at full quantum mechanics. It holds a central role in our +understanding of quantum systems. It provides a handy tool for making important esti- +mates, also extended to energy-time uncertainty relation. However the term “uncertain” +brings in serious negative connotation, which is worth avoiding. +As all textbooks discuss [7], this principle is a direct consequence of the superposition +principle and the kinematics of canonically conjugate variables, postulates I and V. +2. Wave function. +The concept of the wave function is of great utility especially in +visualising bound states and the electron probability density distribution in an atomic +orbital. But the concept seems to get confused with waves on water or on stretched +membranes. +In Dirac notation, Ψ(x) = ⟨x|Ψ⟩, or it is nothing but the component of the state along +the basis vector |x⟩ +|Ψ⟩ = +� +x +|x⟩⟨x|Ψ⟩ ≡ +� +x +Cx|x⟩ ≡ +� +x +Ψ(x)|x⟩ +showing that it has the status of component Cx. However the wave function can be +obtained in any convenient basis such as the momentum basis Φ(p, t) ≡ ⟨p|Ψt⟩, and +convenient for bound states in spherically symmetric potentials, Φ(l, m, t) ≡ ⟨l, m|Ψt⟩. +These can be interpreted as the corresponding +“components” of the abstract state +vector in that basis. Indeed position space has a special status, more than any other +observable. +One then accords great primacy to this description, leading to mental +pictures of catastrophic “collapse” of the wave function all over space. +What about the fact that we have de Broglie wavelength associated with a particle? A +free particle is in a momentum eigenstate, with some value p. Due to the existence of +15 + +the fundamental constant ¯h, we can come up with a quantity of dimension of length, +λ = 2π¯h/p. In hindsight, once the pilot waves are abandoned, there is nothing more +to the “wave picture” than this convenient re-expression of the momentum eigenvalue. +The argument extends to a “wave packet” which is a linear superposition of momentum +eigenstates. +3. Wave particle duality. The term duality immediately suggests ambiguity of descrip- +tion. The classic experimental situation where wave particle duality can be demon- +strated is the double slit experiment. If we can trace the electron as streaming through +one of the two slits we call it the particle manifestation, if we forgo the knowledge of +which slit it traveled through we obtain a fringe pattern characteristic of wave phenom- +ena. +Let us be clear that even in the “wave” avatar, the electron is going to be actually +measured as a particle, by a click counter. The number of clicks will now be distributed +in the fringe pattern. Further, the wave is only a mental construct obtained by relabeling +momentum eigenstates in terms of “wavelengths” λ as discussed in Wave Function topic +above. +But don’t we still face the paradoxical situation? Not if we properly digest postulate +V, the canonical commutation relation. Effectively we are switching between position +observable ( which slit) and momentum observable (free streaming to interfere) and the +what we observe is easily deduced from Kinematics. We can summarise the situation by +saying that the double slit experiment will remain a key part of pedagogical discussion +as included in Sec. 2, being the most direct manifestation of quantum phenomena. But +it needs formalisation as Postulate V and accepted as a part of the basic tenets rather +than be mystified. +4. Collapse of the wavefunction. We have discussed above the convenience but also +the secret of the wavefunction. +Due to primacy of space variable, our postulate IV +gets construed as “collapse of the wave function”. As we remarked in 3, Sec 3.1, the +enigmatic situation with respect to probabilistic outcomes will remain or get resolved +with more experiments. In the meantime we need to banish this term and live with this +postulate. +5. Indistinguishability. We tell students about indistinguishable quanta. The fallacy is +that there was nothing to distinguish to begin with. The term is only a hang over of +the classical ideas about billiard balls. Two of them machined with great care may be +impossible to distinguish. It is with respect to this classical expectation that we dub +the quanta to fall into the same descriptive category. +More carefully stated, in weakly coupled systems number is an observable quantity and +these are eigenstates of that observable with value 2 or more. For photons the number +is not conserved, but it is a valid observable. Note that regardless of how many quanta +it has, the state is normalised to unity. +16 + +We can trace two sources of the misnomer. +One is that the many-quanta states in +weakly coupled systems are most conveniently obtained as tensor products of 1-particle +or 1-quantum states. This accords an unjustified primacy to the 1-quantum state, and +one thinks of the purely mathematical construct as the physical one of two or more +quanta being brought together, whereas it is a single state in the Hilbert space. +Another is a more empirical source. A state of two quanta reduces to a state of one +quantum if one of them is absorbed or undergoes observation. And “which” quantum +got observed and “which” remained is a meaningless question. But after the observation +of one of them we have two distinct ones. It is with respect to this putative final state +that we call that system to be contain indistinguishable quanta. +6. Statistics. It is common to refer to the Bose-Einstein and Fermi-Dirac prescriptions +to be “statistics”. This may leave the student with the idea that some randomness or +incompleteness is present and we are sampling partial information, as would be done in +macroscopic setting. +The prescriptions are only providing the correct enumeration of states as tensor prod- +ucts of 1-quantum states, with nothing statistical about the prescription. +The first +manifestation of these uncanny rules was deduced in the context of thermodynamics of +quantum gases. It is perhaps a left over terminology from that. +7. Entanglement. A state of several quanta is merely the correctly identified basis state +of the weakly coupled system. There are really no entities undergoing “tangling” with +each other. Only if we refer to a post facto state of quanta subjected to observation +that we can say there were several different entities which could have been entangled. +Per se there are just many-quanta states as per B-E or F-D rules with nothing tangled +up inside them. Correlations is a less drastic term, though still with respect to post +observation states. +8. The EPR paradox. This is now a composite of several of the issues already described. +The conceptual error is to think that there are “two” systems. It is actually one quantum +state normalised to unity, with two quanta, the latter number being the eigenstate of +the number operator. When evolved, it can have configuration space component (in the +sense of point 2, Wavefunction, above) which is highly non-local. Observation process +breaks the quantum system, with the number operator eigenvalue having reduced to +one in the remainder state. The remaining system needs to be freshly normalised to +unity. The uncanny situation should not be held up as mysterious or unsatisfactory. It +only needs to be referred to postulate IV on Observation. +6 +Conclusion +We have summarised an essential core of the well known principles of Quantum Mechanics +itemised as seven postulates. They are not logically independent in the axiomatic sense +17 + +but are of primary importance to be listed independently. It is necessary that students +develop direct intuition about raw quantum facts, and understand them as the sources of +the mathematical laws. As such we discussed relevant basic experiments. I have tried to +dispel the negative perception in which a variety of quantum phenomena are held. After +one century of success of the theory one must jettison the baggage of the past and attempt +to come up with better terminology, or at least avoid using its negative contents. I have +tried to argue that if one properly digests the truth of the postulates, most of the negative +connotations and the sense of spooky mystery should get dispelled. +Finally and most +importantly, I have pointed out the fallacy of instantaneous velocity which gives rise to the +illusion of perfect predictive power of classical mechanics. Indeed with hindsight one may +note that the continuum is a purely logical construct, a description not borne out by the +material media once assumed to be continua. Functional analysis nevertheless continues to +be the bulwark of Quantum Mechanics, albeit after due tweaking. Perhaps there are more +subtle formulations of the continuum that make contact with the reality of fundamental +quantum phenomena more easily. +References +[1] T. W. B. Kibble, “Classical Mechanics“, 3rd Edition, Longman 1985 +[2] P. A. M. Dirac, “Principles of Quantum Mechanics”, 4th Ed., Oxford University Press, +1958 +[3] Paper 12 in B. L. van der Waerden, ”Source of Quantum Mechanics“ North Holland +Pub. Co. 1967 +[4] I. J. R. Aitchison, D. A. MacManus and T. M. Snyder, Am. J. Phys. 72, no.11, +1370-1379 (2004) [arXiv:quant-ph/0404009 [quant-ph]]. +[5] K. Gottfried, Am. J. Phys. 79, 261 (2011) [arXiv:1006.4610 [physics.hist-ph]]. +[6] https://www.nobelprize.org/prizes/physics/2022/summary/ +[7] See for example L. I. Schiff, “Quantum Mechanics”, 3rd Ed., McGraw Hill Book Co., +1968; E. Merzbacher, “Quantum Mechanics”, 3rd Ed. John Wiley and Sons Inc., 1999 +[8] John Stachel, Ed., “Einstein’s Miraculous Year”, Princeton University Press, 1998 +18 + diff --git a/lNE5T4oBgHgl3EQfHA6S/content/tmp_files/load_file.txt b/lNE5T4oBgHgl3EQfHA6S/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0f754c718b519bb81dee13bd610c066517fb5010 --- /dev/null +++ b/lNE5T4oBgHgl3EQfHA6S/content/tmp_files/load_file.txt @@ -0,0 +1,555 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf,len=554 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content='05436v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content='hist-ph] 13 Jan 2023 The seven laws of Quantum Mechanics : banishing the bogeys Urjit A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Yajnik∗, Physics Department, Indian Institute of Technology Bombay, Mumbai 400076 Abstract The laws of quantum mechanics are couched in subtle mathematical language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The laws are not usually stated in a compact pedagogical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Here I present a possible way to correct this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Essential facts can be distilled into seven statements that are easy to remember and easily referred back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Also, the current teaching of quantum mechanics is laden with words of negative connotations, originating as they did during the early decades of the subject when the subject was intellectually still puzzling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' A wide variety of experiments in the intervening decades, not least those that were awarded the Nobel Prize of 2022 amply affirm the validity and substantial “reality” of Quantum Mechanics as a theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' I take a few of the inadequacies of classical framework to illustrate that some of the complaints against Quantum Mechanics are patently misplaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Finally I discuss the bogeys such as “wave article duality”, “uncertainty”, “indistinguishability” “statistics” and “entanglement” and advocate adopting better terminology to save new learners from the old biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Contents 1 Why number the laws?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content='1 Welcoming the new principle .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 2 2 Characteristic experiments 4 3 The postulates 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content='1 Remarks on the postulates .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 10 4 Is the Classical all reasonable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 13 5 Exorcising the bogeys 15 6 Conclusion 17 ∗Email : yajnik@iitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content='in 1 1 Why number the laws?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' We teach three laws of Newton, three laws of Thermodynamics and then include a zeroth law as well, Four Maxwell’s equations of Electromagnetism etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This simple mnemonic pedagogical device is missing from the teaching of Quantum Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Stating them as a numbered set of laws would make it easier to remember them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This can also be used to set logical primacy and separation between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Then the critical comments would be easier to make, as in discussing the subtlety of “Newton’s third law”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Clearly differentiated laws also expose possible pitfalls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' For example Newton’s first law is essentially due to Galileo, and after Newton’s second law, the former can be taken to correspond to the special case of no applied force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Is it logically independent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Or does it still have a logical primacy due to direct observability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Furthermore, substantial discussion is needed to explicate whether the second law defines mass or defines force[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Thus, setting apart the First Law asserts its validity independent of the dynamical framework of Newton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' We hope that in the following discussion on Quantum Mechanics we are able to place logically independent facts and rules under independent laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' After doing that we will be able to identify the source of various concerns people have about Quantum Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' It is also hoped that such laws would be useful to new learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Considering the strides quantum science is making, we may soon be teaching such laws in the high school.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Clear laws stated without the negativity imposed by the older generation would reduce the hesitation and suspicion in accepting and applying the laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Until some completely new phenomenon demanding a revision comes to the fore we can be at peace with the laws of Quantum Mechanics, now known for a century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This article is mostly motivated by Dirac’s elegant exposition in his famous textbook[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This book is most widely praised but least widely read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Teachers and even experts are heard saying “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content='. but that is a difficult book to read”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' It happens that Dirac’s exposition is mathematical, but its style is physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' It makes physicists feel it is too abstract while the mathematicians need more sophisticated underpinnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' However the book is really precious for its clarity and elegance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Among many other points of great clarity in this book is the introduction to many-body quantum mechanics, laying bare the mistaken origins of the term “second quantisation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content='1 Welcoming the new principle To be specific, our motivation in what follows is to propose that the language of the “wave function” has been the slow poison of quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' At its inception in 1926 this language was the easiest to follow for physicists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Further, it serves a very good tool for visualising electronic orbitals and shall remain a very useful language for many settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' However its overuse, and debating many points of principle in that language, has lead to misconceptions and a feeling of incompleteness of the subject and inadequacy of the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The facts look rather different if we grasp Heisenberg’s seminal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Heisen- berg’s 1925 paper [3] called for complete refurbishing of mechanics : 2 “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' one realises that .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' even for the simplest quantum-theoretical problems the validity of classical mechanics simply cannot be maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In this situ- ation it seems sensible to discard all hope of observing hitherto unobservable quantities such as the position and the period of the electron, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Instead it seems more reasonable to try to establish theoretical quantum mechanics .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' in which only relations between observable quantities occur.” According to this point of view the notion of classical trajectory X(t) needs to be aban- doned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' What can be really ascertained physically is only the transition amplitude Xab for a particle to have been once seen at a and then at b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' We must thus forgo the knowledge of what may have happened in between as inaccessible in principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' What does not exist does not need any organising principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' No ontology no epistemology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Heisenberg replaces the trajectory by an array of numbers Xab and additionally ˙Xab for the velocity [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' He then proceeds to check the mutual commutability of these arrays of numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This is what we call matrix mechanics now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' It was Dirac’s insight[5] to note that instead of velocity one should use the canonically conjugate Pab and when this is done we get the very elegant analogy of the commutation relations (CR) needed in Quantum Mechanics to the Poisson brackets (PB) of Classical Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' However, unlike the PB which are based on real analysis, the quantum mechanical brackets CR imply that the dynamical variables cannot be represented by mere numbers, but need matrices for their representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' We may immediately feel very anxious about this radical framework to be used for something as obvious as a trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' But we shall return in a later section to argue that the real culprit is the Newtonian notion of instantaneous velocity, which is at least as “unreasonable” as these quantum hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' However for the moment let us consider the two positive thoughts put across by Dirac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The existence of ¯h is the natural demarcation between the “macroscopic” and the “microscopic”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' It sets the scale below which the microscopic world begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Otherwise the world should remain self-similar under infinite subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' But Dalton, Avogadro, Boltzmann, Thomson, Rutherford and others had already investigated that the microscopic world is atomistic and rather different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Dirac’s second point is that quantum principles need not be viewed negatively, as a loss of familiar concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Rather, there is a new positive principle as a compensation so to speak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This is the principle of linear superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Agreed, there is an added layer of abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' “State of a system” in quantum mechanics is not given by a list of values that are guaranteed to be the outcome of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' However, this abstract state vector does obey the Principle of Superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The uncanny nature of the situation is best brought home to students by the example of the position and velocity of a ball in a playing field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Common sense says that the possible states of the ball are given by positions ⃗RA and corresponding velocities ⃗vA where A denotes the point in space and ⃗RA its position vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' But in quantum mechanics we get new valid “states” of the ball which are linear “sum”s of states at A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content='An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The possible number of states increases manifolds, as the relative weightages vary from 0 to 1 in magnitude, can have relative complex phases, and can include any number n of classical 3 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Thus the Quantum World offers multitudes of possibilities far beyond the classical imagination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Yet this is not a disaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Dirac exhorts us to admire the simplicity of the linear superposition principle rather than be baffled by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Thus the basic attitudes to quantum mechanics need to be changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Presenting the set of arguments that substantiate this appeal is the goal of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In the following in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 2 we summarise the classic experiments that signal the novel behaviour, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 3 presents the statement of the laws, not in truly pedagogical form but as notes for knowledgeable peers, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 4 contains a discussion of a few key classical conceptions, where we argue that the latter are, to wit, as technical as the ones involved in quantum mechanics and indeed not in accord with observed reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 5 I comment on the prevalent terminology, identifying the bogeys that need to be banished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 6 contains the concluding remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 2 Characteristic experiments It will be important to distinguish between what is truly novel in the quantum phenomena themselves, versus what makes us uneasy about the mathematical framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' We shall try to show that whatever is novel is indeed rooted in the phenomena themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' More interestingly, as argued later, whatever makes us uneasy about the formalism is perhaps no worse than the state of affairs in the classical framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The enigmas of quantum systems can be summarised as a) The observer is free to choose what to measure, though the choices are limited in any one attempt at measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' b) The system will produce probabilistic outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' c) Subsequently the system either ceases to exist completely, or the measured attribute becomes predictable with certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Note the most uncanny feature of the last statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In macroscopic world we only see systems change or transform or redistribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' But the quantum world allows entities to vanish forever, like a photon absorbed by an atom or the neutrino by inverse beta decay in nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Then the case of measured attributes becoming certain in values may be understood as one form of extinction of the other values of the attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' These points can be stated succinctly as a’) Subjectivity in the choice of measurement, b’) probabilistic outcomes, c’) objectivity of the post measurement state We now summarise the well known phenomena which we may keep in mind as intrinsic quantum behaviour 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Complementarity of description - The Davisson-Germer experiment idealised as double slit experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The electrons although particles can get redistributed to give the interference pattern displayed by wave phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Extensive work has shown that one can recover particle like properties or wave like properties but not both in the same measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Quantised values in case of some observables - The Stern-Gerlach experiment A beam of polarised Silver atoms passes through a region of magnetic field pointing along z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The emerging beam is split into precisely two streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The only explanation is that the component of intrinsic spin of the Silver atoms along the applied field can have only two possible values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Further only one of these precise values can emerge in any measurement, and not an averaged value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Probabilistic outcomes Outcomes of measurements on identically prepared systems are not identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' We can at best associate a probability to any outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' If the Stern-Gerlach type experiment is performed with a rarefied beam of atoms so that only one atom is passing the magnetic field region, we cannot predict which of the two orientations the atom will finally emerge in, only relative probabilities for the two outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Indeterminate evolution during measurement process Related to the previous point is the independent fact that there is no theory for how the quantum state evolves “during” the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The imprint of the quan- tum system is recorded and the quantum nature of that attribute then terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' However there is no theoretical framework for describing the evolution of a quan- tum system into a residual system with specific value as it leaves its imprint in an apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' More generally the system itself may disappear such as charged particles in a Geiger counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The result is a macroscopic current, and we have a quantum theory of how the single particle cascades into a current, but not a deterministic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Non-locality of states The quantum state can be spread over a macroscopic region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' A state of two photons can be stretched over many meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' But as soon as the attributes of one of the photons are measured, the state evolves through that indeterminate evolution and the attributes of the other photon are instantaneously determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' One version of it is the famous EPR paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' However unreasonable classically, the validity of this outcome has finally been accepted and recognised by the Nobel Prize of 2022[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The non-locality is also manifested over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' We might suspect that once we have decided on which attribute to measure, the two photons mutually encode which one is to manifest which value before they are far apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' But since there is complementarity of which attribute to measure, we need not set up which attribute we wish to measure 5 until the photons are really far apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This is called “delayed choice”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' However even under delayed choice, the effect of measuring an attribute on one photon immediately determines the outcome of that attribute on the other photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Bose condensation and Pauli Exclusion Principle Perhaps the most radical departure from classical conceptions arises in the very notion of identity of the basic entities, particles, or more correctly, quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' There are two fundamental aspects to this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' One is that there are identical particles : the primary units are endowed with very few attributes, and all quanta of a particular species have just a few possibilities for these attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' For example all electrons have exactly the same values of charge and mass, and can have one of two values for the projection of their spin along any measured direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This is not encountered classically even in what we think are identical objects, say balls of same size, color and shine of polish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Improving the precision of measurement always reveals the differences, often taking continuum values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Further the quanta obey peculiar rules for collective states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Atomic physics verifies most directly that two electrons cannot occupy the exact same state of energy and spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Likewise, the spectrum of “light gas” under ideal conditions of Black Body obeys Planck spectrum, which can be understood only if photons occupy their avail- able energy states according to the rules discovered by Planck, Einstein and Bose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In summary there are identical particles and they obey strange rules for their col- lective states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' These facts are foundational to the quantum world and the laws pertaining to them need to be formalised into the core of Quantum Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The law should not be postponed as auxiliary rules to be learnt in more advanced courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 3 The postulates In this section we enunciate the laws in simple naive language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' These are stated more as pointers to what all practitioners of Quantum Mechanics are quite familiar with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The idea is to lay out these rules in an order from the more basic or elementary, moving towards those that build the structure further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' We skirt several subtleties about the Hilbert space and precise meaning and varieties of measurement at the level of this presentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Stating the more detailed version will however not require a change to this basic list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Let us make a brief qualitative statement about what is at stake, ´a la Dirac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' We have a quantum system and we have some apparatus that will record various clicks and ticks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The requirement is to set up a mathematical framework that will make predictions about what clicks and ticks can result as outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The dynamical quantities that can be thus measured will be called observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Examples are charge, mass, spin projections, binding energies, lifetimes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This list needs to be established empirically for every new quantum system one encounters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The program of quantum mechanics is to identify a core set of variables which should satisfy relatively simple kinematic conditions, the CR’s, and all the other observables should 6 be expressible in terms of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Fortuitously, but with no guarantee, this set happens to be the same as the set of classical canonical variables, satisfying the CR’s that are, upto the fundamental unit ¯h the same as the classical PB’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Indeed all the observables, all the relevant symmetry operators, and the dynamical evolution operator, can be constructed out of this canonical set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This is not at all obvious, and indeed it fails in a few major exceptional cases such as spin which has no canonical representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' A failure also shows up in the occurrence of anomalies in advanced implementation of quantum principles, such as in Quantum Field Theory and String Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' But majority of quantum observables do have simple classical analogy and that is what has greatly facilitated the prediction and control of the quantum world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The rules stated below essentially address the logical structure of these constructs and the mathematics required to implement them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' They are stated in the simplest context of a single variable and a single observable etc, to keep the statements compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' We shall use the convention |Ψ⟩, |ϕ⟩ A generic state |Ψt⟩, |ϕt⟩ A generic state displaying its time dependence |x⟩, |p⟩ A basis state labeled by a canonical variable |α⟩, |l, m⟩, |n⟩ Eigenstates of general observables labeled by their eigenvalues We are dispensing with the convention of putting a hat or a caret above an operator, so long as there is no ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' State functions constitute a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The states of a quantum system obey linear superposition principle and have the structure of a complex vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Further, for physical interpretation we need to endow this space with a hermitian inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' ⟨ψ|ϕ⟩ = ⟨ϕ|ψ⟩∗ The framework of Hilbert spaces is applicable with some caveats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Observables are realised as Hermitian operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' For an operator A, the hermitian conjugate or adjoint operator is defined as : ⟨Aψ|ϕ⟩ ≡ ⟨ψ|A†ϕ⟩ Hermitian operators are self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' All observables are represented by hermitian operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Their eigenvalues will be the possible list of answers we get upon observa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The eigenstates corresponding to these eigenvalues can be used to construct a basis for the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' There can be several independent choices of bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Change of basis is implemented by Unitary operators 7 If we change from a basis constructed using an observable n to a basis constructed using an observable α, then the change is implemented by a unitary transformation |α⟩ = � n Uαn|n⟩ � β (U†)mβUβn = δm,n � m Uαm(U†)mβ = δα,β where the Kronecker deltas need to be replaced by Dirac delta functions for continuum eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Observations and probabilities are described by projection operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In an observation process we can only predict the probability for the system to emerge in a particular eigenvalue of the relevant observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' If the observable is α and the initial normalised state vector is represented in the |αi⟩ basis as |Ψ⟩ = � i Ci|αi⟩, then the probability of getting the outcome αr is |Cr|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In Hilbert space this amounts to the state being subjected to a projection operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This leads to a simple rule about the average measured value of that observable under repeated measurement of identically prepared systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' ⟨A⟩ ≡ ⟨Ψ|A|Ψ⟩ ≡ ⟨Ψ|AΨ⟩ = � i |Ci|2αi where the second equivalence connects a Physics convention with Hilbert space oper- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Quantum kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The observables and other dynamical quantities introduced are operators and need not commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' With a sufficiently exhaustive set of dynamical quantities Oi we find that they satisfy a symplectic algebra [Oi, Oj] ≡ OiOj − OjOi = � k CijkOk which has closure and obeys the Jacobi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This algebra sets up the quantum kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Further, a great simplification is afforded by a deep classical analogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Corresponding to the classical canonical variables {x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' xN, p1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' pN} there exists a set of quantum variables such that [xi, pj] ≡ xipj − pjxi = i¯h{xi, pj}PB = i¯hδij 8 where the xi, pj in the third expression are classical variables and PB denotes the Poisson bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Most operators Oi, hermitian and unitary, can be algebraically constructed out of the canonical set, with a few notable exceptions such as spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Another important feature of quantum theory is that in Quantum Field Theory one also needs anti-commutator kinematic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' These have no classical analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Quantum dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' a) In analogy with Classical Mechanics, there exists a distinguished hermitian opera- tor, the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In the Heisenberg picture dynamical evolution is expressed in terms of time dependent operators which can be observables or other operators i¯h d dtO(t) = [O(t), H] In particular we have the analogues of Hamilton’s equations of motion for the canon- ical variables, i¯h d dtp(t) = [p(t), H];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' i¯h d dtx(t) = [x(t), H] from which the evolution equation for any operator on the phase space can be worked out using the canonical commutation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' b) A convenient alternative is the Schr¨odinger picture in which the state vector is time dependent, |Ψt⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In this case the operators are not to be evolved in time, and the equation of motion for the state function is i¯h ∂ ∂t|Ψt⟩ = H|Ψt⟩ In practice we do not work with the abstract state |Ψt⟩, but the “wave function” Ψ(x, t) ≡ ⟨x|Ψt⟩ and express H(x, p) ≡ H(x, −i¯hd/dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' c) A third very elegant and fruitful formulation of the dynamics is due to Dirac and Feynman, the Path Integral version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Usually the basis set |x⟩ is treated as time independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Now define a ”moving basis” (see [2], Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 32) which we might call Dirac picture basis, |xt⟩D = eiHt/¯h|x⟩ Then the Path Integral formula gives the amplitude for going to xf at time tf given that the system was at xi at ti : D⟨xftf|xiti⟩D = � Dx(t)Dp(t) exp � i ¯h � tf ti dt(p ˙x − H) � where the action integral in the exponent is on the phase space and the symbolic integration Dx(t)Dp(t) is over all possible paths connecting xf and xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Then we can obtain Ψ(x, t) from Ψ(xi, ti) for t > ti as Ψ(x, t) = � dxi D⟨xt|xiti⟩DΨ(xi, ti) 9 VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Bosons and fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Another deep and non-classical feature of the quantum world is the existence of “identical quanta”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' These bits of nature have just a few attributes such as mass, spin and a few charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In weakly coupled systems, the full multi-quanta Hilbert space can be constructed out of repeated tensor product of the one-quantum Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This is called the Fock space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The admissible multi-quanta states are only the symmetrised ones in an assembly of integer spin quanta, while the admissible states are only the anti-symmetrised ones for half-integer quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' For the case of two quanta, with the states labeled by the values of the observable α, these tensorial constructions are |Ψ⟩B = 1 √ 2{|α(1) 1 ⟩|α(2) 2 ⟩ + |α(2) 1 ⟩|α(1) 2 ⟩} |Ψ⟩F = 1 √ 2 {|α(1) 1 ⟩|α(2) 2 ⟩ − |α(2) 1 ⟩|α(1) 2 ⟩} where subscripts B and F refer to Bose and Fermi respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This ends the list of the postulates incorporating the most essential rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' To repeat in a nutshell, the postulates refer to I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Superposition principle and Hilbert space, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Observables as hermitian operators, III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Change of basis as unitary operators, IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Observation as projection operator, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Kinematics as commutation rules, VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Dynamics via a special unitary operator, and VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Multi-quanta states via Bose-Einstein and Fermi-Dirac rules 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content='1 Remarks on the postulates 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The Hilbert space postulate captures two important things in one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Firstly there is superposition principle for the states, the deepest non-classical feature of the quantum world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Secondly we introduced the inner product on the space, as required for physical interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Many cases of change of basis correspond to classical symmetry operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Wigner’s theorem proves that such transformations are indeed represented by unitary or anti- unitary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Thus Postulate III is not entirely an independent postulate, but it is important enough to be listed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The issue of measurement has been a source of creative proposals and long standing debate among the finest minds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' There are two aspects – the outcome is probabilistic and there is no satisfactory description of the evolution from the unmeasured to the measured state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Here my first caveat is that a conscious observer may not be a key component of the measurement paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Scattering processes and spontaneous decay are directly observed phenomena which capture most of the unsettling aspect of “collapse of the state”, with purely quantum evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Intentional measurement has a lot of resemblance to scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In Rutherford type scattering we send in a stream of projectiles, which are momentum eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The expected out state obeys the symmetries of the scatterer, for example the az- imuthal rotation symmetry along the direction of the incoming projectile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' However, a particular scattered particle can emerge only in one fixed direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Thus while the evolution operator is unitary, the outcome for a single scattered particle is in a momentum eigenstate projected out from the evolved state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Only when a large number of the same projectiles with the same impact parameter is studied do we recover the azimuthal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This can be considered to be a “collapse” into the eigenvector representing that momentum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Note that scattering goes on in locations remote from any conscious observer all over the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The emergence in any one direction of the scattered particle is a generic event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Spontaneous decay is a similar phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The emitted final state may be ex- pected to obey the symmetries of the decaying parent ( for example rotational symmetries of an atom or a molecule).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' And indeed this is so on the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' How- ever any particular decay results in the particle emerging only in a specific direction, and the symmetry can be recovered only through repeated experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Again, this process happens exactly thus, with no observer needed, though there could be one, light years away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' It is interesting to note that this issue is implicit in Einstein’s 1905 paper thus[8] : According to the assumption considered here, in the propagation of a light ray emitted from a point source, the energy is not distributed continu- ously over ever increasing volumes of space, but consists of a finite number of energy quanta localised at points of space that move without dividing, and can be absorbed or generated only as complete units”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' It is clear that in the emission of any individual quantum, the rotational symmetries of the source can not be respected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In turn, if the symmetry is to be recovered over a large number of observations, it should not be a surprise that the question of which direction is determined only by a probabilistic law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In hindsight one may wonder why the bearer of so lucid and profound an insight shied away from the collateral logical consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Returning to the broader issues of measurement, the problem seems to lie in the inability to characterise where the validity of classical paradigm ends and quantum regime begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 11 In some sense this is due to the fact that ¯h dimensionally involves space, time and mass and it is difficult to demarcate the transition between the two frameworks purely in terms of length or time or mass scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Indeed Bohr’s Correspondence Principle relies on largeness of quantum numbers to recover a classical description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In the Schr¨odinger’s Cat paradox the presumption is that all systems must rightly be considered as quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In other words taking the observer to be entangled with the system being observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Further, the cat box plus its observer can be a combined system being observed by another observer, thus demanding a combined description with indefinite recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This is the ”Wigner’s Brother” paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' There are two possibilities for a resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Perhaps more delicate experiments will demand a more sophisticated formalism for their description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' But if not, we only need improved semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' A characteristic of macroscopic systems is that they are highly complex, say a Geiger counter or a bubble chamber, making it clear that they can be simply treated classically, thus at least avoid the recursion paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' However the postulate as presented here has been verified in a variety of experiments directly or indirectly over the past century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' While the paradox may persist, there is no contradiction with the proposed postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The Einstein-Podolski-Rosen paradox epitomises another aspect that is counter-intuitive about measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Suppose we have a two-electron state with net spin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Suppose an observer Alice at some remote point x makes an observation at time t′ 1 observing only one of the electrons, and that in spin-up state, destroying that electron in the process, then causality demands that at all subsequent times t > t′ 1, the other observer Bob can only find one electron and that in spin-down state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' According to Special Relativity, this latter fact cannot in principle be known to Bob during the time t′ 1 < t < t′ 1 + |x|/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In case Bob makes an observation during the time t′ 1 < t < t′ 1+|x|/c they may legitimately ascribe the outcome to the weightage factor 1/ √ 2 for the spin-down state, and nothing will go wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The main lesson is that quantum states are intrinsically non-local and yet consistent with Special Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The advanced framework of Relativistic Quantum Field Theory does not throw up any contradictions either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' It does require the existence of anti-particles in order to preserve causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' More on the possible fallacy in EPR in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' We make a big leap of faith in assuming that all the observables and other operators can be expressed in terms of the canonical set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' We propose that the Oi have the same algebraic dependence on the canonical variables as the corresponding classical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This is an immense simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' But it requires a price to be paid − It results in operator ambiguity, which however is resolved by simple prescriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' − One may encounter observables with no classical analogue, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=', spin − For fermion fields one needs anti-commutators instead of commutators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' There is no classical limit available for this operation in the normal sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 12 Fermion bilinears do have classical limits and one arranges to set up correct commutation relations between them with desirable classical limits1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Despite all these exceptions, this principle must be viewed as of special significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' It suggests that the macroscopic canonical structure of Hamiltonian dynamics is firmly rooted in microscopic principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Quantum kinematics can be equivalently expressed by the overlap between the bases labeled by canonical observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This is given by the fundamental relation ⟨x|p⟩ = 1 √ 2π¯h eipx/¯h This is the key building block of the Path Integral formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Postulate VII is most likely not a logically independent one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' When Lorentz invariance is imposed on a quantum system, the “spin-statistics theorem” can be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' We state this here as a law to highlight its importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' That is, that in Quantum Mechanics states are fundamental, not quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' And to remind that “quanta are not particles”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Quantum Field Theory provides the comprehensive framework of calculation in Fock Space based on weakly coupled quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' However, the Quantum Field Theory framework has a far greater reach, in a wide variety of strongly coupled systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 4 Is the Classical all reasonable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The framework of Quantum Mechanics couched in the language of the Hilbert space is at once forbidding to those unfamiliar with advanced Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' However there exists several quantum systems where this Hilbert space may be no more forbidding than a finite dimensional complex vector space for a finite number of spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This is very promising because it may become possible to teach simplified basics to even highschool students who may be able to understand the technologies that are now emerging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Despite this, the framework has become notorious for resorting to purely abstract and therefore perhaps auxiliary constructs, and for being difficult to grasp instinctively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' As a counter to this, I take up here to argue that a few key concepts of Classical framework are equally counter-intuitive, and it is more the familiarity developed over several centuries that makes them so easily acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Instantaneous velocity : The most commonly held drawback of Quantum Mechanics is the loss of the detailed trajectory of the particle, specified by position and velocity at any given instant of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' It is however worth pondering how justified this expectation was in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The notion that an object is at a place and is moving while at 1One might wonder whether this and the preceding point about spin have any mutual connection since fermions are prime examples of particles with spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' But integer spin particles also exist, and therefore the need to insert spin by hand will persist and is independent from the enigmatic quantisation of fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 13 that place actually defies common sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In a philosophical vain one is led to wonder how something can be at a place if it is moving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Newton solved this problem through the Calculus notion of limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' One defines v = lim ∆t→0 ∆x ∆t The conceptual proposal however has never been verified empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' No one, in Galileo’s vain, took a meter stick and a stopwatch watch of unlimited precision and allowed the ∆x and ∆t to physically go to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' If they did they would discover Quan- tum phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Thus there was no loss of paradise of unlimited precision, it was never there in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' To repeat Dirac’s point of view quoted in the Introduction, ¯h indeed sets a scale sep- arating the Classical from the Quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In the limit the action function governing our moving particle reaches this level, the Newtonian paradise withers away and the Quantum emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This does draw our attention further to the carefully constructed notion of the contin- uum in Mathematics which buttresses Newton’s conception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Development of science may have reached a point making it worth revisiting the tenets of the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' And in turn we need to develop mathematics in which quantum constructs take an intuitive centre stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The abstract nature of the wave function is often contrasted with other physical quan- tities of immediate empirical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The notions of electric and Magnetic fields are however worth re-inspecting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' They are only a step away from being something palpa- ble, since you need to hold a point charge or a small dipole in that region of space to immediately see their manifestation as field lines as visualised by Faraday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' However when we propose that these fields oscillate and they carry energy, and they migrate over light years, in vacuum as Einstein taught us, then we may wonder “what” is actually propagating in the empirical sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' These fields therefore have a status comparable to that of the wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' What is intriguing then is that according to the Sudarshan-Glauber theory, the quan- tum theory of light subsumes all its classical states without any change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Perhaps the validity of the conception of electromagnetic radiation is ultimately derived from the more fundamental quantum property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Classical conception is willing to live with point particles as well as continuum fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' However there is an internal contradiction in the Electrodynamics of a point charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' An oscillating point charge radiates, but the field of the charge cannot act upon itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Thus the charge loses energy without any force slowing it down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Lorentz worked to give a finite size and an internal structure to the electron without success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' If the field of a point charge is allowed to act upon itself we get an infinite answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The issue occupied many stalwarts, Dirac, Wheeler and Feynman, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Its futility became 14 clear after it was realised that the electron can be a point particle, yet obey quantum rules and never be localised to an ideal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The questions in the revised version got transferred to those of a calculable framework of Quantum Electrodynamics, eventually addressed by renormalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 5 Exorcising the bogeys A large number of professionals putting Quantum Mechanics successfully to use remain unsure about its consistency and validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' To the extent these may only be anxieties transmitted by the previous generation it is worth examining the various enigmatic issues, and to finally banish the bogeys which can be resolved in the light of the postulates formulated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' To this end I go through a list of terms commonly used which continue to reinforce a negative perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' It is my appeal that it is appropriate to replace some of the description with better words and if this is not possible, at least to stop using the terms in that form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The uncertainty principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This essential novelty of quantum phenomena was gleaned by Heisenberg before arriving at full quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' It holds a central role in our understanding of quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' It provides a handy tool for making important esti- mates, also extended to energy-time uncertainty relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' However the term “uncertain” brings in serious negative connotation, which is worth avoiding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' As all textbooks discuss [7], this principle is a direct consequence of the superposition principle and the kinematics of canonically conjugate variables, postulates I and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The concept of the wave function is of great utility especially in visualising bound states and the electron probability density distribution in an atomic orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' But the concept seems to get confused with waves on water or on stretched membranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In Dirac notation, Ψ(x) = ⟨x|Ψ⟩, or it is nothing but the component of the state along the basis vector |x⟩ |Ψ⟩ = � x |x⟩⟨x|Ψ⟩ ≡ � x Cx|x⟩ ≡ � x Ψ(x)|x⟩ showing that it has the status of component Cx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' However the wave function can be obtained in any convenient basis such as the momentum basis Φ(p, t) ≡ ⟨p|Ψt⟩, and convenient for bound states in spherically symmetric potentials, Φ(l, m, t) ≡ ⟨l, m|Ψt⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' These can be interpreted as the corresponding “components” of the abstract state vector in that basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Indeed position space has a special status, more than any other observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' One then accords great primacy to this description, leading to mental pictures of catastrophic “collapse” of the wave function all over space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' What about the fact that we have de Broglie wavelength associated with a particle?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' A free particle is in a momentum eigenstate, with some value p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Due to the existence of 15 the fundamental constant ¯h, we can come up with a quantity of dimension of length, λ = 2π¯h/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In hindsight, once the pilot waves are abandoned, there is nothing more to the “wave picture” than this convenient re-expression of the momentum eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The argument extends to a “wave packet” which is a linear superposition of momentum eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Wave particle duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The term duality immediately suggests ambiguity of descrip- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The classic experimental situation where wave particle duality can be demon- strated is the double slit experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' If we can trace the electron as streaming through one of the two slits we call it the particle manifestation, if we forgo the knowledge of which slit it traveled through we obtain a fringe pattern characteristic of wave phenom- ena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Let us be clear that even in the “wave” avatar, the electron is going to be actually measured as a particle, by a click counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The number of clicks will now be distributed in the fringe pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Further, the wave is only a mental construct obtained by relabeling momentum eigenstates in terms of “wavelengths” λ as discussed in Wave Function topic above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' But don’t we still face the paradoxical situation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Not if we properly digest postulate V, the canonical commutation relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Effectively we are switching between position observable ( which slit) and momentum observable (free streaming to interfere) and the what we observe is easily deduced from Kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' We can summarise the situation by saying that the double slit experiment will remain a key part of pedagogical discussion as included in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 2, being the most direct manifestation of quantum phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' But it needs formalisation as Postulate V and accepted as a part of the basic tenets rather than be mystified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Collapse of the wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' We have discussed above the convenience but also the secret of the wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Due to primacy of space variable, our postulate IV gets construed as “collapse of the wave function”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' As we remarked in 3, Sec 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content='1, the enigmatic situation with respect to probabilistic outcomes will remain or get resolved with more experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' In the meantime we need to banish this term and live with this postulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Indistinguishability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' We tell students about indistinguishable quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The fallacy is that there was nothing to distinguish to begin with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The term is only a hang over of the classical ideas about billiard balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Two of them machined with great care may be impossible to distinguish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' It is with respect to this classical expectation that we dub the quanta to fall into the same descriptive category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' More carefully stated, in weakly coupled systems number is an observable quantity and these are eigenstates of that observable with value 2 or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' For photons the number is not conserved, but it is a valid observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Note that regardless of how many quanta it has, the state is normalised to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 16 We can trace two sources of the misnomer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' One is that the many-quanta states in weakly coupled systems are most conveniently obtained as tensor products of 1-particle or 1-quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This accords an unjustified primacy to the 1-quantum state, and one thinks of the purely mathematical construct as the physical one of two or more quanta being brought together, whereas it is a single state in the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Another is a more empirical source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' A state of two quanta reduces to a state of one quantum if one of them is absorbed or undergoes observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' And “which” quantum got observed and “which” remained is a meaningless question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' But after the observation of one of them we have two distinct ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' It is with respect to this putative final state that we call that system to be contain indistinguishable quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' It is common to refer to the Bose-Einstein and Fermi-Dirac prescriptions to be “statistics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This may leave the student with the idea that some randomness or incompleteness is present and we are sampling partial information, as would be done in macroscopic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The prescriptions are only providing the correct enumeration of states as tensor prod- ucts of 1-quantum states, with nothing statistical about the prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The first manifestation of these uncanny rules was deduced in the context of thermodynamics of quantum gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' It is perhaps a left over terminology from that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' A state of several quanta is merely the correctly identified basis state of the weakly coupled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' There are really no entities undergoing “tangling” with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Only if we refer to a post facto state of quanta subjected to observation that we can say there were several different entities which could have been entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Per se there are just many-quanta states as per B-E or F-D rules with nothing tangled up inside them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Correlations is a less drastic term, though still with respect to post observation states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The EPR paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' This is now a composite of several of the issues already described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The conceptual error is to think that there are “two” systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' It is actually one quantum state normalised to unity, with two quanta, the latter number being the eigenstate of the number operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' When evolved, it can have configuration space component (in the sense of point 2, Wavefunction, above) which is highly non-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Observation process breaks the quantum system, with the number operator eigenvalue having reduced to one in the remainder state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The remaining system needs to be freshly normalised to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' The uncanny situation should not be held up as mysterious or unsatisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' It only needs to be referred to postulate IV on Observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' 6 Conclusion We have summarised an essential core of the well known principles of Quantum Mechanics itemised as seven postulates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' They are not logically independent in the axiomatic sense 17 but are of primary importance to be listed independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' It is necessary that students develop direct intuition about raw quantum facts, and understand them as the sources of the mathematical laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' As such we discussed relevant basic experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' I have tried to dispel the negative perception in which a variety of quantum phenomena are held.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' After one century of success of the theory one must jettison the baggage of the past and attempt to come up with better terminology, or at least avoid using its negative contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' I have tried to argue that if one properly digests the truth of the postulates, most of the negative connotations and the sense of spooky mystery should get dispelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Finally and most importantly, I have pointed out the fallacy of instantaneous velocity which gives rise to the illusion of perfect predictive power of classical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Indeed with hindsight one may note that the continuum is a purely logical construct, a description not borne out by the material media once assumed to be continua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Functional analysis nevertheless continues to be the bulwark of Quantum Mechanics, albeit after due tweaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Perhaps there are more subtle formulations of the continuum that make contact with the reality of fundamental quantum phenomena more easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' References [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Kibble, “Classical Mechanics“, 3rd Edition, Longman 1985 [2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content='nobelprize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content='org/prizes/physics/2022/summary/ [7] See for example L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Schiff, “Quantum Mechanics”, 3rd Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=', McGraw Hill Book Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=', 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' Merzbacher, “Quantum Mechanics”, 3rd Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=' John Wiley and Sons Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=', 1999 [8] John Stachel, Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} +page_content=', “Einstein’s Miraculous Year”, Princeton University Press, 1998 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE5T4oBgHgl3EQfHA6S/content/2301.05436v1.pdf'} diff --git a/ltFJT4oBgHgl3EQfZSy-/content/2301.11531v1.pdf 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b/m9FPT4oBgHgl3EQf5DW2/content/tmp_files/2301.13195v1.pdf.txt @@ -0,0 +1,1662 @@ +Adaptive Computation with Elastic Input Sequence +Fuzhao Xue † 1 2 Valerii Likhosherstov 1 Anurag Arnab 1 Neil Houlsby 1 Mostafa Dehghani ‡ 1 Yang You 2 +Abstract +Humans have the ability to adapt the type of in- +formation they use, the procedure they employ, +and the amount of time they spend when solving +problems. However, most standard neural net- +works have a fixed function type and computation +budget regardless of the sample’s nature or dif- +ficulty. Adaptivity is a powerful paradigm as it +not only imbues practitioners with flexibility per- +taining to the downstream usage of these models +but can also serve as a powerful inductive bias for +solving certain challenging classes of problems +(Dehghani et al., 2018; Banino et al., 2021; Tay +et al., 2022). In this work, we introduce a new +approach called AdaTape, which allows for dy- +namic computation in neural networks through +adaptive tape tokens. AdaTape utilizes an elastic +input sequence by equipping an architecture with +a dynamic read-and-write tape. Specifically, we +adaptively generate input sequences using tape +tokens obtained from a tape bank which can be +either trainable or derived from input data. We ex- +amine the challenges and requirements to obtain +dynamic sequence content and length, and pro- +pose the Adaptive Tape Reading (ATR) algorithm +to achieve both goals. Through extensive experi- +ments on image recognition tasks, we show that +AdaTape can achieve better performance while +maintaining the computational cost. To facili- +tate further research, we have released code at +https://github.com/google-research/scenic. +1. Introduction +Adaptive computation is central to human intelligence. This +is clear, given that humans spend a variable amount of time +and energy on different problems depending on their com- +plexity (Meunier et al., 2009). Adaptivity in neural networks +†Work performed while at Google ‡Project lead +1Google +Brain 2National University of Singapore. Correspondence to: +Fuzhao Xue , Mostafa Dehghani +. +Preprint +is attractive for two key reasons. Firstly, adaptive computa- +tion could potentially be a powerful and essential inductive +bias for solving challenging problems that would have been +significantly harder otherwise (Dehghani et al., 2018; Ban- +ino et al., 2021). Secondly, adaptive computation could +imbue practitioners with downstream flexibility pertaining +to the usage of these models. For the most part, altering +the computation budget of a model after it has been trained +becomes almost impossible. Hence, the ability to flexibly +scale computational costs and budgets dynamically is highly +desirable. +This paper proposes AdaTape, a new general-purpose adap- +tive computation method. The key idea is to introduce +elastic input sequences via the means of a dynamic read-and- +write memory tape. Unlike all prior works that investigate +adaptivity via sparse conditional computation (Fedus et al., +2022; 2021; Lepikhin et al., 2020) or adaptivity through +recursion over architecture (Dehghani et al., 2018; Banino +et al., 2021; Graves, 2016), this work presents a new perspec- +tive that explores adaptivity with respect to input sequence +length (or read/write memory tapes from the perspective of +a Neural Turing Machine (Graves et al., 2014)). We pos- +tulate that this exploration is crucial for the development +of this class of methods and is very complementary to the +existing suite of methods developed to encourage adaptive +computation in neural networks. +AdaTape promotes adaptivity in both type and amount of +computation. Specifically, AdaTape controls (1) the con- +tents of the tape tokens (2) the number of tape tokens, that +are used for each input. To this end, AdaTape is charac- +terized by a tape bank that can be dynamically read from, +using a newly proposed dynamic halting algorithm which +we call Adaptive Tape Reading (ATR). Concretely, ATR +method adaptively and dynamically selects the content and +length of this memory tape which is appended to the inputs +of a standard Transformer (Vaswani et al., 2017). Given that +the increasing computation budget generally leads to im- +proved quality (Kaplan et al., 2020; Dehghani et al., 2021a; +Hoffmann et al., 2022; Zhai et al., 2022; Abnar et al., 2021; +Likhosherstov et al., 2021), this enables a new way for adap- +tively scaling the computation budget without adding new +parameters or applying a part of the model recursively. +To ascertain the effectiveness of the proposed AdaTape +arXiv:2301.13195v1 [cs.LG] 30 Jan 2023 + +Adaptive Computation with Elastic Input Sequence +Tokenizer +Transformer Layer #1 +Input tokens +Sample 1 +Sample 2 +Sample 1 +Sample 2 +Transformer Layer +Attention +FFN +Tape FFN +Input tokens +Tape tokens +Transformer Layer #2 +Transformer Layer #N +… +Tape Bank +Tape tokens +Query +… +Tape tokens +Adaptive +Tape +Reading +Figure 1: An overview of AdaTape. For different samples, we pick a variable number of different tokens from the tape bank. +The tape bank can be driven from input, e.g., by extracting some extra fine-grained information or it can be a set of trainable +vectors. The Adaptive Tape Reading is used to recursively select different sequences of tape tokens, with variable lengths, +for different inputs. These tokens are then simply appended to inputs and fed to the transformer encoder. +method, we first evaluate it on the challenging Parity task +(Graves, 2016; Banino et al., 2021), a standard verifica- +tion check for Adaptive Computation Time (ACT) algo- +rithms (Graves, 2016). Our results demonstrate that AdaT- +ape performs well on this problem. Meanwhile, this prob- +lem remains completely unsolvable by vanilla Transformers. +This not only verifies that the AdaTape inductive bias is +crucial in solving certain classes of problems but also as- +serts its correctness. Given that the standard Transformer +is touted as the true universal algorithm with ubiquitous +impact across all fields (Vaswani et al., 2017; Jumper et al., +2021; Dosovitskiy et al., 2020; Likhosherstov et al., 2021), +it would be unthinkable if Transformers lack the inductive +bias for a standard vector parity problem. +Finally, we conduct large-scale experiments on vision tasks +(e.g., image recognition and evaluate their few-shot accu- +racy), showing that AdaTape performs well and outperforms +vanilla Transformers when compute-matched (Dehghani +et al., 2021a) across both FLOPs and throughput. While +AdaTape does not improve efficiency during training, the +granted flexibility and adaptivity allow dynamic scaling of +the computation budget during inference. Given that the +standard practice is to train and serve n models to potentially +cater to variable computation budgets (e.g., base models for +less important workloads and large models for prioritized ex- +amples), we consider the property of having a single model +flex between multiple requirements to be highly desirable. +2. AdaTape: Adaptive Computation with +Elastic Input Sequence +Neural networks can attain adaptivity by using different +functions or variable computation budgets for different in- +puts. Consider a deep neural network as a function f(x; θ), +whose output depends on both the input x and the parameter +θ. To achieve adaptive function types, we often sparsely +activate a subset of parameters θ conditioned on x. This +type of adaptivity can also be referred to as conditional com- +putation. Studies on Mixture-of-Experts (Fedus et al., 2021; +Lepikhin et al., 2020; Xue et al., 2021; Lou et al., 2021; +Riquelme et al., 2021) introduce adaptivity on the function +type through routing and determining the computation for +each input sample. +Another line of research in adaptive computation involves +dynamic computation budget. In standard neural networks, +such as transformers, the computation budget is fixed for +different samples. However, recent studies have shown that +adaptive computation budgets can improve performance on +tasks where traditional transformers fail (Dehghani et al., +2018; Banino et al., 2021; Abnar et al., 2020). Many of +these works use dynamic depth to achieve adaptivity in +the allocation of the computation budget. For instance, +the Adaptive Computation Time (ACT) algorithm was pro- +posed by Graves (2016) for adaptive computational budget +on recurrent neural networks (Hochreiter & Schmidhuber, +1997). The Universal Transformer (UT)(Dehghani et al., +2018) extends the ACT algorithm to transformers(Vaswani +et al., 2017) by making the computation budget relying on +the number of transformer layers used for processing each +input example or token. Recent studies, such as Ponder- + +Adaptive Computation with Elastic Input Sequence +Net (Banino et al., 2021), follow a similar framework while +improving the dynamic halting mechanisms. +AdaTape not only uses different function types per input via +conditioning the adaptive tape reading mechanism on the +input representation but also adjusts the computation budget +by employing variable length memory tape for different +inputs. Figure 1 presents a high-level schema of AdaTape +in the context of image recognition. +For a given batch of input images, after tokenization (e.g., a +linear projection of non-overlapping patches from the image +in the vision transformer), AdaTape uses a vector repre- +senting each input to dynamically select a variable-sized +sequence of tape tokens from a tape bank. A tape bank +could be a set of trainable vectors or generated on-the-fly +from inputs, e.g., by using a more fine-grained tokenizer. +The selected tape tokens are appended to the original input +and fed to the Transformer layers. More precisely, AdaT- +ape defines f(x; θ), where x = xI : xT , and xI being +input tokens and xT being a sequence of tape tokens that +is generated by fATR(xI, Zbank). When using a learnable +bank, fATR in fact selects different parameters of the model +(learnable tapes) per input, which means AdaTape has adap- +tivity in terms of the function type. Also given that the +length of xt is different for each input, AdaTape allocates +different computation budgets for different inputs. Finally, a +sequence containing both input and tape tokens is passed to +a transformer encoder. For each transformer layer, the same +multi-head attention is used across all input and tape tokens. +However, two different feed-forward networks are used, one +for all tokens from the original input and the other for all +tape tokens. We observed slightly better quality by using +separate feed-forward networks for input and tape tokens. +In the next section, we will provide a brief overview of the +original ACT algorithm and explore how the assumptions +made contradict the concept of adaptive sequences length +setting. To address this issue, we present Adaptive Tape +Reading algorithm, which specifically caters to the needs of +a dynamic input sequence. We will delve into the various +components of this algorithm and explain the reasoning +behind each design decision. +2.1. Adaptive Computation Time +The ACT algorithm, as outlined in Algorithm 1, uses +a trainable linear component with sigmoid activation +sigmoid(g(·)) that computes the halting score at each step. +When the sum of the halting scores hp exceeds the halt- +ing threshold, the computation stops and future layers are +skipped through early exiting. +At each time step, the +states S are weighted by halting score p and accumulated. +When the algorithm ends, the sum of the halting scores, +psum = �T +t=1 pt, equals 1.0. +Algorithm 1 Adaptive Computation Time +1: Input: Initial states S = {s1, ..., sN}, where sn= 0; Output +states Sout = {so1, ..., so +N}, where so +n= 0; Initial halting +score hp = 0; Initial update times n = 0; Max ponder times +T; Trainable linear layer g(·); Set ϵ as a small number like +0.01; Initial ponder loss latr = 0. +2: while hp < 1.0 and n < T do +3: +Compute halting score for each token at this step p = +sigmoid(g(S)) +4: +Compute the average halting score for each sample p = +mean(p) +5: +if hp + p > 1.0 - ϵ then +6: +Update remainder r = 1 - hp +7: +Update output states Sout += S * r +8: +break +9: +end if +10: +Increase update times n += 1 +11: +Update states S = g(S) +12: +Update output states Sout += S * p +13: +Increase the accumulated halting score hp += p +14: end while +15: lact = n + r +16: return lact, So +ACT employs a loss function lact to encourage less compu- +tation. The loss lact includes two components, the number +of updates n and the remainder r. The main goal of ACT is +to control the computation by minimizing the number of up- +dates n. However, such an objective is not differentiable and +cannot be minimized directly via back-propagation. To ad- +dress that in a round-about way, ACT optimizes n together +with the remainder r. When decreasing r, the summation +of all used psum = �T +t=1 pt increases. When �T −1 +t=1 pt > +1.0 - ϵ, n would decrease by 1.0 thus ACT minimizes n +indirectly. The final loss function can usually be written as +l = lmain +λlact, where lmain represents the main training +objective, e.g., cross-entropy loss in the classification task. +Note that the stability and performance of the model when +using ACT is sensitive to the ACT loss coefficient, so, using +ACT requires careful tuning of λ. +2.2. Tape Bank +AdaTape uses a bank of tokens where all the candidate tape +tokens are stored. The adaptive tape reading mechanism +does not consider the origin of the tape tokens and only +interacts with a RB×H tensor, where B is the bank size and +H is the dimension of each token. We explore two different +methods for creating the tape bank: an input-driven bank +and a learnable bank. +Input-driven Bank +For an input-driven bank, we use in- +put data as the source to generate bank tokens on-the-fly. +The general idea is to extract a bank of tokens from the input, +while employing a different approach than what the model +tokenizer uses for mapping the raw input to a sequence of +input tokens. This enables dynamic, on-demand access to + +Adaptive Computation with Elastic Input Sequence +information from the input that is obtained using a different +point of view, e.g., a different resolution or a different level +of abstraction. For instance, for the image recognition task, +when using Vision Transformers (Dosovitskiy et al., 2020), +a simple, yet effective idea is to use different patch sizes +for generating input tokens and bank tokens. For instance, +using a smaller patch size for generating the bank can be +seen as dynamic multi-scale processing of the input where +we can select what fine-grained information from the input +is useful which is much more efficient than using all the +small patches and consuming a large amount of computa- +tion. Appendix A.3 provides more details on input-driven +bank setup for Vision Transformer. In this paper, we use the +input-driven bank in one of our image recognition experi- +ments, however, the idea can be generalized to other tasks +and setups, e.g., language tasks where more fine-grained +tokenization (Kudo & Richardson, 2018) has shown to be +effective. +Learnable bank +Using a finer-grain tokenizer for gener- +ating a tape bank is not always applicable. For instance, it +is hard to further split each node in graph transformer (Yun +et al., 2019). A different and more general approach for +generating the tape bank, is to simply use a set of train- +able vectors as tape tokens. The learnable bank can be +seen as an embedding layer with a trainable tensor of size +Zbank ∈ RB×H, from which the model can dynamically +retrieve (Zamani et al., 2022) tokens according to the com- +plexity of the example at hand. Unlike an input-driven bank, +the content of the learnable bank is fixed for different sam- +ples. However, selecting tape tokens from a learnable bank +means partially using the model’s parameters, which can be +seen as a form of sparsity in AdaTape. +2.3. Adaptive Computation Time for Elastic Input +Sequence +In order to have elasticity on the input length, we need +a mechanism that selects a variable sequence of tape to- +kens from the bank conditioning on the input representation. +Such a mechanism needs to be able to make a decision on +how many tape tokens should be appended to the input, e.g., +using a halting mechanism where tape tokens are recursively +added to the input. As the input representation, we can use +[CLS] token or the average pooling of all input tokens. +This input query vector is used for making the decision on +the number of tape tokens for each input. +One straightforward method to obtain adaptive sequence is +adapting ACT algorithm explained in Section 2.1. To this +end, we first summarize the requirements of ACT algorithm +as follows: (1) the summation of the halting score psum = +�T +t=1 pt = 1.0; (2) each halting score pt is proportional to +the importance of the current step output; (3) the computing +process of halting score pt is required to be differentiable +Algorithm 2 Adaptive Tape Reading +1: Input: Initial halting score hp = 0; A list of tokens se- +lected states = []; Initial query q ∈ R1×h; Tape token +bank Zbank ∈ RC×H; Token bank mask m = 0, where +m ∈ R1×C, where m = 0; Max ponder times T; Halting +threshold τ; Initial ponder loss latr = 0. +2: while hp < τ do +3: +Compute the inner product of query and tape tokens d = q · +Zbank[:, : h]T , where d ∈ R1×C +4: +Mask inner product d = d ⊙ (1.0 - m) +5: +Compute index of top K elements in d, set the index vector +as i ∈ R1×K, where K = T +τ +6: +Take top K elements from d by the index vector i and +denote this new vector as w, where w ∈ R1×K +7: +Normalize the elements by softmax: w = softmax( w +√ +h) +8: +Take top K tape tokens from Zbank by the index vector i, +and merge the tape tokens {s1, ..., sK} selected from bank +as one single tape token s = �K +k=1 wksk +9: +Append this token into input sequence states += s +10: +if hp + max(w) > τ then +11: +break +12: +end if +13: +Increase halting score: hp += max(w) +14: +Increase ponder loss: latr += 1.0 - �K +k=1 w2 +k or lcollect += +hp +15: +Update token mask by the one-hot vector of i: m += +sum((one-hot(i), axis=0) +16: +Update query q = 1 +2 (s + q) or q = s +17: end while +18: return latr, states +and can be well-trained. +However, unfortunately, all these requirements in ACT are +not desirable in the adaptive sequence scenario. First, in +adaptive token reading, the output state of tth step is tth tape +token we are going to use as the model input. For the steps +that we do not want to stop, we expect the weights of the +tokens to stay close to 1.0. On the other hand, for the step +we want to stop at, which means this incoming tape token is +not as important as others, we need a smaller weight than +previous tokens. Such a requirement totally contradicts to +requirements (1) and (2) in ACT. Additionally, the existence +of layer normalization LayerNorm(·) will make trainable +halting score computation layer g(·) in ACT algorithm in- +valid. The main reason is the normalization layer will ignore +the halting score pt: LayerNorm(ptzt) ≈ LayerNorm(zt). +The halting score computation layer g(·) will then cannot +be well-trained. We introduce the detailed reasoning in Ap- +pendix A.4. In summary, after analyzing the requirements +of ACT algorithm and adaptive sequence length, we found +there are some clear contradictions. Therefore, we devote +ourselves to designing a novel dynamic halting algorithm in +the following section. +2.4. Adaptive Tape Reading Mechanism +Algorithm 2 presents a pseudo code for the Adaptive Tape + +Adaptive Computation with Elastic Input Sequence +Reading. ATR is an iterative process for selecting tape +tokens from the bank. At each iteration, we select a new set +of tokens (here we select and add tape tokens one-by-one) +without replacement, conditioned on the previously selected +tokens. ATR uses a query vector q ∈ RH representing +the input at the current iteration (i.e., the sequence of all +input tokens plus already selected tape tokens) to select +the next set of tokens from a tape bank Zbank ∈ RB×H. +We compute d as the inner product of q and Zbank. Note +that in practice, to reduce the computation, we can use part +of the q and tape tokens for computing the inner product +(e.g., q[: h], and Zbank[:, : h], meaning to use the first h +elements). This can be seen as using first h elements in +q and Zbank[b, : h] as key and the remaining elements as +value, where Zbank[b, :] denotes bth tape token. To avoid the +repeated selection of tape tokens, at each iteration, we adjust +the inner product d by masking out weights of tokens that +are selected before (using the bank mask m in Algorithm 2 +that gets updated in each iteration). Then we select top- +K tape tokens according to their scores from d and create +w ∈ R1×K containing weight logits of the selected tape +tokens in form. We normalize w and apply softmax. We also +create the corresponding tape tokens vectors {s1, ..., sK} +collected from the Zbank, and compute a single vector s as +the weighted average of the selected K tokens, which is the +tape token that we add to the input at the current iteration. +To make the halting decision, we accumulate the largest +value in w into hp until it is greater or equal to a threshold +τ. Larger max(w) indicates the need for less iteration (i.e., +adding fewer tape tokens). Note that the query vector q +used for selecting the tape token is updated by averaging +the input query and current tape token or just replacing the +input query with the current tape token. +In order to incentivize shorter sequences for efficiency and +penalize the model for adding tape tokens when there is no +need, we use a similar loss term to what the original ACT +uses, i.e., l = lmain + λlatr. We observe, unlike ACT, the +adaptive tape reading is less sensitive to the value of λ. We +also empirically found that, when using a learnable bank, we +can omit the latr loss term without observing a significant +change in the performance. +ATR offers a template for elastic input sequences. To em- +phasize some of the details in the above algorithms: (1) we +observe better performance and stability when using a lower +dimension for the query; we found normalizing the inner +product by query dimension +√ +h before applying softmax +critical following the original design of attention (Vaswani +et al., 2017); (3) we found that normalizing the bank and +query using a shared LayerNorm(·) layer improves perfor- +mance and stabilizes training. +3. Experiments +We first present our experimental setup and implementation +details in Section 3.1. We report the results on the parity +task and image classification benchmarks in Section 3.2 +and 3.3. To further validate the effectiveness of our model, +we conduct an ablation study in Section 3.4. Finally, Sec- +tion 3.5 provides some insights into the behavior of adaptive +tape reading in AdaTape. +3.1. Experimental Setting +We evaluate AdaTape on both parity task and image recogni- +tion benchmarks. First, the parity task is, given a sequence +of numbers x = [x1, x2, ..., xN], where xn is 1, −1, or 0, +we are to predict whether the number of 1’s in x is even or +odd. For the parity task, we employ an input-driven bank +and use the bank as the way the model accesses the input +information, and as the input token, s, we simply use a +single trainable token vector (e.g., [CLS] token) to cre- +ate the initial query for tape token reading. We train all +models for 10000 steps with batch size 128. The learning +rate is set as 3e-5, and we use a linear warm-up for 1000 +steps. We use transformer-Tiny in this task since we can- +not see improvements when we scale it up, similar to the +observation of UT (Dehghani et al., 2018) for algorithmic +tasks. The configuration of transformer-Tiny can be found +in Appendix A.5. +For image classification benchmarks, we first conduct large- +scale pre-training on JFT-300M (Sun et al., 2017) followed +by few-shot learning on a wide range of datasets, including +ImageNet (Deng et al., 2009), Cifar100 (Krizhevsky et al., +2009) and Pets (Parkhi et al., 2012) following the protocol +of vanilla ViT (Dosovitskiy et al., 2020) and Big Trans- +fer (Kolesnikov et al., 2020). That is, we use the hidden rep- +resentation before logits computation as a feature to adapt +the upstream model, and evaluate the resulting model on the +validation/test set. We also train models on ImageNet from +scratch for easier comparison in future work. Following +existing work on ViT with adaptive computation (Yin et al., +2022), on ImageNet, we train models mainly at Tiny and +Small scales. Please see Appendix A.6 and Appendix A.7 +for details on hyper-parameters and training strategies used +for AdaTape with a learnable bank. +We compare AdaTape with standard transformers and +adaptive transformers. The standard transformers include +ViT (Dosovitskiy et al., 2020), DeiT (Touvron et al., 2021) +and PlainViT (Beyer et al., 2022). DeiT and PlainViT are +heavily-optimized models for training on ImageNet from +scratch. We also compare with adaptive transformers like +UT (Dehghani et al., 2018) and A-ViT (Yin et al., 2022). To +further enhance our baselines and achieve a more compre- +hensive comparison, we develop two improved versions of +UT as our strong baselines equipped with adaptive compu- + +Adaptive Computation with Elastic Input Sequence +22 +23 +24 +25 +26 +Input Length (Difficulty) +50 +60 +70 +80 +90 +100 +Percision (%) +Parity Task Comparison +Random +Transformer +Universal Transformer +AdaTape +Figure 2: Evaluation on the parity task. The standard transformer and universal transformer totally failed on this task, both +showing performance at the level of a random guessing baseline. +tation. We first consider UT without parameter sharing and +refer to it as Unshared Universal Transformer (U2T). We +increase the maximum number of pondering steps in U2T, +which means U2T has more layers, more computation and +more parameters. To further enhance the baselines, we also +consider stacking UT and ViT together. We found putting a +shallow UT on top of ViT totally failed due to an unstable +training process, but putting ViT on top of UT can achieve +decent performance. We refer to this model as UViT which +is our strongest baseline on image classification with adap- +tive depth. In UViT, we employ 3 UT layers on Tiny, Small, +and Base models and 6 UT layers on Large models. The +number of standard transformer layers is the same as ViT. +Adatap is implemented in JAX and is developed in +Scenic (Dehghani et al., 2022). +For reproducibil- +ity, we released the code in https://github.com/google- +research/scenic/tree/main/scenic/projects/adatape. +3.2. Evaluation on the Parity Task +We evaluate AdaTape on the parity task to study the effect of +the inductive biases that it introduces to vanilla Transformer +on a synthetic task. The Parity is the simplest non-counter- +free, or periodic regular language (Bhattamishra et al., 2020). +Simple recurrent neural networks can solve this task well +because the memory in the recurrent neural network can +record the states for finite-state automation (Abnar et al., +2021; Schwarzschild et al., 2021; Veliˇckovi´c et al., 2022; +Ibarz et al., 2022; Bansal et al., 2022). Standard transformer +totally failed in modeling such sequences (Hahn, 2020; De- +hghani et al., 2021b) as they are incapable of directly main- +taining a counter. Specifically, in AdaTape, we can use the +tape tokens to record the behavior in each recurrence step. +We first use the input sequence x to generate the bank. The +Adaptive Tape Reading algorithm plays the role to model the +parity task recurrently. Considering the target is to predict +the number of 1 in the input sequence is even or odd, there +are actually two states in the finite-state automation. We +then fix the K as 2 in Algorithm 2 to check whether there +is a state switching after each step. When K is fixed as 2, +the maximum ponder time T=2τ= N +2 , where N is the length +of input parity sequence. Such a hyper-parameter setting +is to match the period in parity and to ensure AdaTape can +collect all required tokens to make the prediction. +Figure 2 shows the results on the parity task. We can see +both the standard transformer and universal transformer +completely fail on the parity task, even if they are trained +and evaluated on a very short (simple) sequence. However, +the AdaTape performs much better than all baselines. One +reason is that AdaTape incorporates the recurrence within +the model input selection and such an inductive bias could be +crucial for solving the parity task, as it is a way to implicitly +enables AdaTape to maintain a counter, which is not possible +in the standard transformer. +3.3. Evaluation on Image Classification +We pre-train AdaTape on JFT-300M and report few-shot +results on popular image recognition benchmarks in Table 1. +We can see U2T performs worse than other models, we +suggest the reason is the unstable training loss when mak- +ing new halting decisions. The advanced baseline, UViT +performs better than ViT because of adaptivity, more com- +putation, and parameters. Our AdaTape is using less compu- +tation than UViT and performs better at all scales. AdaTape +with an input-driven bank is superior at a larger scale. We +suggest a larger scale model is better at finding informative +tokens from input and validate this reasoning in Section 3.5. +We can also observe AdaTape is not very good at throughput. +Both types of AdaTape are not as hardware friendly as base- +lines. This is the limitation of AdaTape. However, please +note this only means the training speed is slightly slower +on TPU, which is highly optimized by large-scale matrix +multiplication. Due to less real computation (GFLOPs) +and dynamic sequence length, AdaTape has the potential to +speed up inference on other hardware or smaller batch size. + +Adaptive Computation with Elastic Input Sequence +Table 1: Pre-training and transfer learning results on image classification benchmarks. We report two AdaTape models +with different bank types at each scale. AdaTape-B/32-Learn means we are using a learnable bank for AdaTape-B/32. +AdaTape-B/32-Input denotes we are using an input-driven bank. We report both GFLOPs and throughput (measured in +images / second / core). For all models with adaptive computation budget, we report the upper bound computation cost. +The pre-training is conducted on JFT-300M dataset and we report precision@1 (%) on the validation dataset. The few-shot +experiments are on ImageNet, Cifar100, and Pets datasets with Top-1 accuracy. IN25 denotes the result on ImageNet 25-shot. +Model +GFLOPs +Throughput +JFT-300M +IN10 +IN25 +Cifar10010 +Pets10 +ViT-B/32 +4.437 +793.8 +43.3 +59.4 +62.5 +72.2 +90.3 +U2T-B/32 +5.513 +343.4 +38.8 +49.5 +53.6 +64.4 +84.1 +UViT-B/32 +5.511 +478.7 +44.4 +61.9 +64.6 +74.4 +91.6 +AdaTape-B/32-Learn +5.585 +431.8 +44.4 +63.0 +65.8 +75.0 +93.2 +AdaTape-B/32-Input +6.057 +342.3 +44.7 +63.8 +65.9 +75.5 +93.0 +ViT-B/16 +17.634 +253.7 +48.6 +67.4 +70.0 +77.5 +93.0 +U2T-B/16 +22.016 +102.9 +46.6 +63.0 +66.0 +73.1 +92.5 +UViT-B/16 +22.011 +151.7 +49.0 +68.5 +71.0 +75.8 +94.0 +AdaTape-B/16-Learn +18.837 +167.1 +49.1 +70.3 +72.6 +78.7 +94.0 +AdaTape-B/16-Input +19.304 +148.8 +48.9 +69.0 +71.5 +77.4 +94.4 +ViT-L/32 +15.628 +192.3 +49.9 +69.5 +72.1 +79.4 +94.0 +UViT-L/32 +19.242 +127.1 +50.5 +70.5 +72.6 +80.1 +94.1 +AdaTape-L/32-Learn +19.497 +108.6 +50.2 +70.8 +73.3 +79.6 +95.0 +AdaTape-L/32-Input +20.141 +95.5 +50.2 +71.8 +73.8 +81.3 +95.0 +ViT-L/16 +61.724 +63.1 +54.3 +75.2 +77.0 +83.1 +95.9 +UViT-L/16 +77.114 +44.7 +54.8 +75.5 +77.4 +81.6 +95.7 +AdaTape-L/16-Learn +65.941 +44.2 +54.7 +76.5 +78.0 +82.7 +96.7 +AdaTape-L/16-Input +66.570 +40.1 +54.8 +76.7 +78.5 +84.7 +96.4 +200 +250 +300 +350 +Img/Sec/Core +70 +72 +74 +76 +78 +Top-1 Accuracy (%) + AdaTape-Ti/16 + AdaTape-S/16 + UViT-Ti/16 + UViT-S/16 + U2T-Ti/16 + U2T-S/16 + A-ViT-Ti/16 + A-ViT-S/16 + A-ViT-Ti/16(Ours) + A-ViT-S/16(Ours) +Quality-Cost Comparison with Adaptive Baselines on ImageNet +UViT +U2T +A-ViT +A-ViT(Ours) +AdaTape +Figure 3: We evaluate AdaTape by training on ImageNet from scratch. For A-ViT, we not only report their results from the +paper but also re-implement A-ViT by training from scratch, i.e., A-ViT(Ours). +We also evaluate AdaTape by training on ImageNet-1K +from scratch. For A-ViT (Yin et al., 2022), authors fine- +tuned their model to obtain adaptivity from a pre-trained ViT +checkpoint, which has a different setting with ours. For a +straightforward comparison, we not only report their results +from the paper but also re-implement A-ViT to reproduce +the results when training from scratch. The quality-cost +curve is shown in Figure.3. We can see AdaTape performs +much better than the baselines in terms of quality-cost bal- +ance. For efficiency, AdaTape-S/16 is even faster than the +Tiny-level baselines. Such results match well with the find- +ing from Dehghani et al. (2021a), which shows that the +adaptive model depth architectures are not well suited for +many accelerators like TPU. AdaTape is much more effi- +cient compared to other adaptive baselines because we are +injecting adaptivity into the input sequence instead of model +depth. For effectiveness, AdaTape can even outperform +the non-adaptive models, ViT and DeiT, with comparable +computation costs. More detailed results can be found in +Appendix A.8 +3.4. Ablation Study +We first ablate the adaptive abilities of AdaTape including +adaptive sequence content and adaptive sequence length. +For adaptive sequence content, we expect it can improve + +Adaptive Computation with Elastic Input Sequence +Table 2: Ablation study on the adaptive abilities of AdaTape. We use AdaTape with an input-driven bank as a platform. For +the model without adaptive length, we pick T tape tokens in parallel. For the model without adaptive content and adaptive +length, we remove the tape bank and use a fixed set of trainable tokens to enhance the input. We also report average/max +sequence length and sequence length variance. +Model +JFT-300M +IN 10-shot +IN 25-shot +Avg/Max SeqLen +Var SeqLen +AdaTape-B/32 +44.7 +63.8 +65.9 +57.4/59.0 +4.8 +w/o Ada Length +44.2 +63.3 +66.4 +59.0/59.0 +0.0 +w/o Ada Length&Content +43.9 +61.0 +64.1 +59.0/59.0 +0.0 +AdaTape-B/16 +48.9 +69.0 +71.5 +203.4/206.0 +7.3 +w/o Ada Length +49.2 +69.4 +71.8 +206.0/206.0 +0.0 +AdaTape-L/32 +50.2 +71.8 +73.8 +56.8/59.0 +8.5 +w/o Ada Length +50.5 +71.7 +74.2 +59.0/59.0 +0.0 +AdaTape-L/16 +54.8 +76.7 +78.5 +203.0/206.0 +10.4 +w/o Ada Length +54.5 +76.7 +78.4 +206.0/206.0 +0.0 +0 +10 +20 +X-Index +0 +5 +10 +15 +20 +25 +Y-Index +0 +10 +20 +X-Index +0 +5 +10 +15 +20 +25 +Y-Index +Figure 4: We visualize the tape token selection heatmap of AdaTape-B/32 (left) and AdaTape-B/16 (right). The hotter color +means the patch at this position is more frequently selected. +effectiveness. For the adaptive sequence length, as it has the +potential to save computation during inference, we expect +AdaTape can keep a comparable accuracy when we use the +fixed sequence length. +Adaptive sequence length is from ATR algorithm with a +recurrent token selection process. The adaptive sequence +content is mainly from a selective use of the tape bank. To +ablate the model by removing only the “adaptivity in length”, +we set the halting threshold to infinite high and always select +T tokens for different samples, which means all samples +use the maximum sequence length and the same amount +of computation costs. Note that models without adaptive +sequence length use the upper bound of computation used by +the adaptive ones. Different from adaptive sequence length, +removing only the “adaptivity of content” is not possible, +because our ATR algorithm at the end of the day uses a tape +bank (which is inherently adaptive in content). Therefore, +we remove the “adaptivity in length and content” together +by appending a fixed set of trainable tokens to every input +sample. +Results are shown in Table 2. We can see, without the adap- +tive content, there is a significant performance drop. For in- +stance, compared with AdaTape without adaptive sequence +only, AdaTape-B/32 without adaptive content degrades 2.3 +points on ImageNet 10-shot accuracy. This shows the effec- +tiveness of adaptive sequence content. When we remove the +adaptive sequence length, we can see models perform com- +parably instead of much better at all scales, which shows +the tape tokens selected are condensed and make full use of +limited input tokens. Another interesting finding is, when +we scale the model up, we can observe the increasing se- +quence length variance. This indicates larger AdaTape is +better at controlling sequence length and computation cost. +We also conduct the ablation study on the hyper-parameters, +computation cost, and the number of trainable parameters +in Appendix A.6, A.10 and A.11, respectively. +3.5. Visualization +In this section, we study the behavior of AdaTape by vi- +sualization. First, we collect the token selection results of +AdaTape with an input-driven bank on JFT-300M validation +set, and visualize them as heatmaps in Figure 4. We can see +the central patches are more frequently picked (with lighter +colors). This matches well with our prior knowledge, central +patches are usually more informative. This shows AdaTape +prefers more informative patches to improve performance. +We also visualize the token selection distribution. Please +see Appendix A.12 for details. + +Adaptive Computation with Elastic Input Sequence +4. Conclusion +We introduce AdaTape, a new approach to adaptive compu- +tation. 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Yin et al. (2022) adapted ACT algorithm to the vision transformer along the depth. However, the ACT-based +transformer training is not as stable as the vanilla transformer and it needs careful hyper-parameters tuning. To overcome +this weakness, PonderNet (Banino et al., 2021) improves the universal transformer by reformulating the halting policy as +a probabilistic model. Balagansky & Gavrilov (2022) further stabilizes this training process by removing the sampling +process in PonderNet. Another way to halt adaptively is confidence-based dynamic halting algorithms. Schuster et al. +(2021; 2022); Schwartz et al. (2020) investigated the confidence-based dynamic halting algorithms on adaptive model depth. +Our work also proposed a dynamic halting algorithm, but we are interested in pondering at input sequence length instead of +model depth. Another difference is that we have no trainable layers to compute the halting score, which is a requirement of +adaptive transformer input. Alternatively, we use the entropy of logits from the token selection layer as an indicator to make +a halting decision. +A.1.2. ADAPTIVE SEQUENCE LENGTH +Adaptive sequence length can be achieved by token pruning. For instance, Meng et al. (2022) prune the patch tokens in the +vision transformer via a lightweight decision network, and Fayyaz et al. (2022) drop the patch tokens by sampling adaptively. +Our AdaTape is different from their work as AdaTape appends a small number of tokens to the original input adaptively +instead of pruning the intermediate token representations within the model. Also, the content of the tokens in the existing +work is not adaptive. The appended tokens in AdaTape are sparsely selected from the tape bank, which is helpful to improve +the effectiveness of the transformer. +Wang et al. (2021) proposed another way to achieve adaptive sequence length for ViT. The proposed model uses a large +patch size at first and decreases the patch size adaptively for different samples. This is similar to our input-driven bank. +However, first, in our input-driven bank, instead of using all smaller patches, we use only a subset of the fine-grained patches +adaptively. In addition, instead of limiting the setup to image-only inputs, AdaTape supports a learnable bank and formulates +this problem in a more general way. +A.1.3. EXTRA TOKENS +Extra tokens are another important component of AdaTape. Song et al. (2021) use an extra special token for vision +transformer-based object detection. Burtsev et al. (2020) introduces memory tokens to augment transformer capacity. Using +learnable prompt tokens in NLP (Lester et al., 2021) can be also perceived as adding extra tokens to the input. In all these +works, the number of added tokens is fixed across different samples and often uses a deterministic set of tokens (imagine +having more than one [CLS] token). However, AdaTape selects and appends a variable number of tokens from the bank +adaptively per sample. Such an adaptive input sequence provides not only a larger capacity but also a flexible computation +budget. +A.2. Input-driven Bank Details +We introduce more details about the input-driven bank of AdaTape. We generate the input-driven bank by: +Zbank = h2(h1(Xp) + Epos) +(1) +where Xp is the input sequence with fine-grained tokenization (e.g., smaller patch size for ViT), h1 and h2 are both trainable +linear projection, Epos is position embedding. Since Zbank is conditioned on the input sample, we need to generate the +content of the bank on-the-fly. This is also the reason why an input-driven bank is slightly more computationally expensive +than a learnable bank. + +Adaptive Computation with Elastic Input Sequence +A.3. Scaling with Patch Size +100 +101 +GFLOPs +47.5 +50.0 +52.5 +55.0 +57.5 +60.0 +62.5 +ImageNet 10-shot + ViT-S/32 + ViT-S/28 + ViT-S/16 + ViT-S/14 + ViT-S/8 +Scaling ViT with Patch Size +Figure 5: We scale ViT with patch size. The model is pre-trained on JFT-300M (Sun et al., 2017). We report the few-shot +performance on ImageNet (Deng et al., 2009) +We fix all other parameters and scale ViT with a smaller patch size to check the effectiveness of smaller patches. We +experiment with ViT-S which will not become expensive as the sequence length grows. For example, training ViT-B/8 with +data parallelism on 512 TPUv3 cores has the OOM issue. In Figure 5, when adding more computation, although there is a +significant accuracy improvement, more patches introduce very expensive computation costs. Such a trade-off inspires us to +use smaller patches as the source to generate bank for a efficient fine-grained data modeling. +A.4. LayerNorm is making ACT invaild in adaptive sequence +We argue that naively applying layer normalization in the transformer invalidates the ACT algorithm in AdaTape. To justify +this, we first assume we are using a transformer with the following standard transformer architecture: +X′ = MSA(LayerNorm(X)) + X +(2) +X′′ = FFN(LayerNorm(X′)) + X′ +(3) +where MSA(·) is multi-head attention layer and FFN(·) is feed-forward network. We are going to replace X by ptzt, +where pt is the halting score of tth tape token zt. Then, the output of first LayerNorm(·) should be: +LayerNorm(ptzt) = +ptzt − 1 +H +�H +h=1 ptzt,h +� +1 +H +�H +h=1(ptzt,h − �H +j=1 ptzt,j)2 + ϵ +∗ γ + β +(4) +since ϵ is a very small constant, it is reasonable to write the equation above as: +LayerNorm(ptzt) ≈ +zt − 1 +H +�H +h=1 zt,h +� +1 +H +�H +h=1(zt,h − �H +j=1 zt,j)2 + ϵ +∗ γ + β = LayerNorm(zt) +(5) +We can see pt is ignored during the normalization, which means the value of pt cannot change the output of the first +LayerNorm(·) in practice. Since pt is the output of the trainable layer g(·) in ACT, the g(·) cannot be really trained well by +back-propagation. To alleviate this issue, one possible solution is applying the weight pt to tape tokens after normalization +layers. This is a simple and straightforward solution that may make sense. However, empirically, we observe the weights of +all tokens will increase to 1.0 very fast in practice and the model will then always only append one token even if we do not +use any loss function to penalize longer sequence. We suggest the reason is that uniformed token mean and variance are +highly desired by attention layers. Anyhow, this result is not expected because there is no adaptive ability observed. In +addition, we observed the training is also extremely unstable under this design. +In summary, we cannot apply scalar weight to the tape token, so we cannot have a trainable linear layer to compute the +halting score p. That makes the ACT-based dynamic halting mechanism, including PonderNet, not applicable to AdaTape. + +Adaptive Computation with Elastic Input Sequence +We, therefore, are required to design a new adaptive computation algorithm for elastic input sequence, i.e., Adaptive Tape +Reading. Such a reasoning process can also be extended to other future conditional and adaptive computation work. For +instance, if you want to feed the layer norm with one token after applying weight on the token, you need to be careful +about where is the weight from. If this weight is from a trainable layer like ACT, we must check whether this layer can be +well-trained via reasoning. +A.5. Configuration for Transformer-Tiny +Table 3: Transformer configuration for all tiny-level models. +Transformer-Tiny +Depth +12 +Hidden Dimension +196 +MLP Dimension +768 +#Attention Heads +3 +We use the tiny configuration for all models on the parity task. As shown in Table 3, the Tiny level transformer still uses 12 +layers, which is the same as the transformer base. The difference is that tiny configuration has smaller hidden dimensions +and fewer attention heads. +A.6. Hyper-parameters +Table 4: Hyper-parameters for AdaTape on image classification +AdaTape-Learn +AdaTape-Input +Max ponder times T +10 +10 +Halting threshold τ +2.0 +2.0 (B) /1.0 (L) +Loss weight λ +0 +0.01 +Bank Size C +10000 +784 +On JFT-300M pre-training, we follow Dosovitskiy et al. (2020) and train all models for 7 epochs. We use the same learning +rate, batch size, and learning schedule. Customized hyper-parameters for AdaTape are summarized in Table 4. We employed +the fixed max ponder times for all models. Smaller τ on AdaTape-L with an input-driven bank. The bank size is 10000 for +AdaTape-learn. We use bank size 784 for AdaTape Input as we set patch size as 8 to generate tokens from images with +224×224 resolution. AdaTape with a learnable bank can be trained without halting loss. Also, note that we append tape +tokens after first transformer encoder layer for better query quality and tape selection. +Table 5: Hyper-parameters when training on ImageNet-1K only. Ti, S and B denote Tiny, Small, and Base scales. +Name +Value +Learning Rate +0.001 +Linear Warmup Steps +10000 +Learning Rate Decay +Cosine Decay +Optimizer +AdamW +(β1, β2) +(0.9, 0.999) +Weight Decay +1e-4(Ti), 8e-5(S&B) +Epoch +300 +Batch Size +1024 +Mixup +0.2(Ti), 0.5(S&B) +Label Smoothing +0(Ti), 0.1(S&B) +RandAug +(2,10)(Ti), (2,15)(S&B) +For ImageNet training from scratch, we summarized the data augmentation and corresponding hyper-parameters in Table 5. +Similar with existing work (Beyer et al., 2022), we used Mixup (Zhang et al., 2017), RandAug (Cubuk et al., 2020) and + +Adaptive Computation with Elastic Input Sequence +label smoothing (Szegedy et al., 2016) to improve the robustness. +A.7. Tricks of Learnable Bank Training +We found the training of AdaTape with a learnable bank is relatively unstable. To alleviate this, we propose two tricks to +improve the training process. The core idea of these two tricks is to improve the diversity of tape tokens and encourage +the model to explore the tape bank. We first add noise to query q = q + λ ϵ, where ϵ is sampled from standard normal +distribution and λ is the weight of noise. We set λ as 0.01 during training. We also mask a subset of the tape tokens in the +bank randomly. To implement this, we initialize m = 0 + b for ATR algorithm, where b ∈ R1×C and bc ∼ Bernoulli(p). +We set p as 0.1 by default. We also observed that larger p can further improve the training stability. +A.8. More results on ImageNet +Table 6: Results of training on ImageNet-1K only. We use the input-dirven bank for AdaTape. ∗ denotes that we approximate +the throughput by the models with similar architecture and input pipeline. For A-ViT, we not only report their results from +the paper but also re-implement A-ViT by training from scratch, i.e., A-ViT(Ours). +Model +Adaptive +#Param +Throughput +ImageNet Top-1 (%) +ViT-Ti/16 +5.7 +387.5 +58.7 +DeiT-Ti/16 +5.7 +381.8∗ +71.3 +PlainViT-Ti/16 +5.7 +381.8 +73.0 +U2T-Ti/16 +✓ +6.1 +364.3 +70.2 +A-ViT-Ti/16 +✓ +5.7 +226.6∗ +71.0 +A-ViT-Ti/16(Ours) +✓ +5.7 +226.6 +73.2 +AdaTape-Ti/16 +✓ +6.3 +380.9 +73.6 +ViT-S/16 +22.0 +385.7 +75.2 +DeiT-S/16 +22.0 +365.8∗ +78.9 +PlainViT-S/16 +22.0 +365.8 +79.2 +U2T-S/16 +✓ +23.8 +163.2 +74.5 +A-ViT-S/16 +✓ +22.0 +166.6∗ +78.6 +A-ViT-S/16(Ours) +✓ +22.0 +166.6 +77.0 +AdaTape-S/16 +✓ +24.3 +366.5 +79.5 +PlainViT-B/16 +87.1 +159.9 +79.5 +AdaTape-B/16 +✓ +94.9 +130.5 +80.8 +We train with ImageNet-1K only and summarized the results in Table 6. We can see AdaTape outperforms all adaptive +baselines by a large margin. Even compared to highly-optimized baselines without adaptivity like PlainViT (Beyer et al., +2022) and DeiT (Touvron et al., 2021), AdaTape can still surpass them with a comparable computation budget. For instance, +AdaTape-S/16 uses only 0.4× training cost and 0.3× parameters but achieves almost comparable results with PlainViT-B/16. +We may also note that Tiny scale models cannot achieve much higher throughput than Small scale models on ImageNet. +We suggest the reason is that the efficiency bottlenecks are mainly from data loading and preprocessing instead of the +computation budget within neural networks. +A.9. Ablation on Hyper-Parameters +Number of tape tokens +We investigate the effect of the number of tape tokens in this section. In the model with adaptive +sequence length, the number of tape tokens is controlled by the model adaptively. To control the real length directly, we +use the AdaTape without adaptive length as a platform. We sweep the number of tape tokens T over {5, 10, 20, 40} and +summarize the results in Figure 6. We can see an obvious improvement when we increase the T from 5 to 10. However, the +model is saturated after that. Since using a longer sequence means more computation budget, we select to use 10 tape tokens +as the default choice. +Bank Size +We also sweep the bank size over {1e+2, 1e+3, 1e+4, 1e+5} for AdaTape with a learnable bank. As shown in +Figure 7, the AdaTape performs best when we have 1e + 4 tape tokens in the bank. As we fix the number of recurrences as +10 in ATR algorithm by default, a larger bank will only increase the computation cost linearly. + +Adaptive Computation with Elastic Input Sequence +23 +24 +25 +Number of Tape Tokens +62.8 +62.9 +63.0 +63.1 +63.2 +63.3 +63.4 +ImageNet 10-Shot +Figure 6: We sweep the number of tape tokens over {5, 10, 20, 40}. Considering more tape tokens mean much more +computation cost, we set 10 as our default choice. +102 +103 +104 +105 +Bank Size +62.0 +62.2 +62.4 +62.6 +62.8 +63.0 +ImageNet 10-Shot +Figure 7: We sweep the bank size over {1e+2, 1e+3, 1e+4, 1e+5} for AdaTape with a learnable bank, and found 1e+4 is the +sweep point. +A.10. Ablation on Computation Cost +Although AdaTape only increases the computation cost slightly, to further verify the improvement is from more reasonable +design and adaptive abilities instead of more computation, we conduct ablation experiments on computation cost. First, as +shown in Figure 8, even if a smaller patch size can improve ViT, AdaTape is still outperforming ViT with less computation. +In addition, we report the quality-cost comparison in terms of throughput in Figure 9. We can see AdaTape can outperform +ViT significantly with smaller latency. +A.11. Ablation on Number of Trainable Parameters +Table 7: Ablation study on the number of trainable parameters. +Model +GFLOPs +Throughput +#Param +IN 10-shot +ViT-B/28 +5.730 +661.1 +100.9 +61.9 +ViT-B/28-3FFN +5.744 +517.0 +200.1 +62.4 +AdaTape-B/32 +5.585 +431.8 +185.6 +63.0 +ViT-B/14 +23.254 +155.1 +99.7 +68.4 +ViT-B/14-3FFN +23.239 +148.8 +198.9 +69.0 +AdaTape-B/16 +18.837 +167.1 +192.5 +70.3 +Since we have two FFNs in every transformer block, AdaTape has more trainable parameters than ViT. To validate that the +improvement is not just caused by having more parameters, we increase the trainable parameters in ViT by adding 2 more +FFN layers to process different tokens. Then, ViT has 3 FFNs in total. The first FFN is used to handle [CLS] token. The +second FFN and the third FFN are fed by half of the patch tokens, respectively. The results are summarized in Table 7. We + +Adaptive Computation with Elastic Input Sequence +5 +10 +15 +20 +GFLOPs +60 +62 +64 +66 +68 +70 +ImageNet 10-shot (%) + ViT-B/32 + ViT-B/28 + ViT-B/16 + ViT-B/14 + AdaTape-B/32 + AdaTape-B/16 +Quality-Cost of Base models +ViT-B +AdaTape-B +20 +40 +60 +80 +GFLOPs +70 +72 +74 +76 +ImageNet 10-shot + ViT-L/32 + ViT-L/16 + ViT-L/14 + AdaTape-L/32 + AdaTape-L/16 +Quality-Cost of Large models +ViT-L +AdaTape-L +Figure 8: Ablation Study on the quality-cost trade-off. We scale the standard transformer (i.e., ViT) to a smaller patch size, +e.g., ViT-B/14 and ViT-L/14, and compare it with AdaTape. +50 +100 +150 +200 +250 +Throughput(img/sec/core) +68 +70 +72 +74 +76 +ImageNet 10-shot + ViT-B/16 + ViT-B/14 + ViT-L/16 + ViT-L/14 + AdaTaoe-B/16 + AdaTape-L/16 +Quality-Cost Comparison +ViT +AdaTape +Figure 9: we report the quality-cost comparison in terms of throughput to verify that AdaTape is better than baselines with +comparable computation cost. +can see AdaTape outperforms ViT with 3 FFNs using less computation and fewer parameters. For instance, AdaTape-B/16 +surpasses ViT-B/14-3FFN by 1.3% in terms of top-1 accuracy on ImageNet 10-shot. +A.12. Visualization of Token Selection Distribution +Similar to Section 3.5, we collect the token selection results on JFT-300M validation set. We sort the tape token index by the +frequency and visualize the token selection distribution in Figure 10. We can observe the token selection decision obey +long-tail distribution. That shows our AdaTape prefers the tape tokens at some specific positions, which is similar to our +observation in Figure 4, i.e., central patches are frequently selected. + +Adaptive Computation with Elastic Input Sequence +Figure 10: We visualize the tape token selection distribution in AdaTape-B/32 (left) and AdaTape-B/16 (right). The index id +is sorted by the value of the selected frequency + diff --git a/m9FPT4oBgHgl3EQf5DW2/content/tmp_files/load_file.txt b/m9FPT4oBgHgl3EQf5DW2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9a4e2fd33f6c5c7009ca303b55a0fac520f3a869 --- /dev/null +++ b/m9FPT4oBgHgl3EQf5DW2/content/tmp_files/load_file.txt @@ -0,0 +1,1434 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf,len=1433 +page_content='Adaptive Computation with Elastic Input Sequence Fuzhao Xue † 1 2 Valerii Likhosherstov 1 Anurag Arnab 1 Neil Houlsby 1 Mostafa Dehghani ‡ 1 Yang You 2 Abstract Humans have the ability to adapt the type of in- formation they use, the procedure they employ, and the amount of time they spend when solving problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf'} +page_content=' However, most standard neural net- works have a fixed function type and computation budget regardless of the sample’s nature or dif- ficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf'} +page_content=' Adaptivity is a powerful paradigm as it not only imbues practitioners with flexibility per- taining to the downstream usage of these models but can also serve as a powerful inductive bias for solving certain challenging classes of problems (Dehghani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf'} +page_content=' Banino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf'} +page_content=' Tay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf'} +page_content=' In this work, we introduce a new approach called AdaTape, which allows for dy- namic computation in neural networks through adaptive tape tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf'} +page_content=' AdaTape utilizes an elastic input sequence by equipping an architecture with a dynamic read-and-write tape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf'} +page_content=' Specifically, we adaptively generate input sequences using tape tokens obtained from a tape bank which can be either trainable or derived from input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf'} +page_content=' We ex- amine the challenges and requirements to obtain dynamic sequence content and length, and pro- pose the Adaptive Tape Reading (ATR) algorithm to achieve both goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf'} +page_content=' Through extensive experi- ments on image recognition tasks, we show that AdaTape can achieve better performance while maintaining the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf'} +page_content=' To facili- tate further research, we have released code at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf'} +page_content='com/google-research/scenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf'} +page_content=' Introduction Adaptive computation is central to human intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf'} +page_content=' This is clear, given that humans spend a variable amount of time and energy on different problems depending on their com- plexity (Meunier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf'} +page_content=' Adaptivity in neural networks †Work performed while at Google ‡Project lead 1Google Brain 2National University of Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9FPT4oBgHgl3EQf5DW2/content/2301.13195v1.pdf'} +page_content=' Correspondence to: Fuzhao Xue G[object][confidence] then +8: +Update G[object] with the direction informa- +tion between object and agent, one-hot label +vector and confidence. +9: +end if +10: +end for +11: +end for +12: +Learn the representation Zt of ERG using GCN by +equation 1. +13: +Using Zt as attention map to the label embedding ex- +pression D. +14: +Output environment representation Ot as equation 2. +15: end for +embeds each input node with a matrix Wg ∈ RD×U, where D +is the dimension of the node features. Afterwards, our GCN +embeds all nodes through the adjacency matrix E ∈ R|U|×U, +and outputs a new encoding Z ∈ R|U|×U, which expresses +the environment, including the objects in the environment, the +relationship between objects, and the relationship between +objects and the agent. Executing line 12 of Algorithm 1, our +proposed ERG is expressed as: +Z = f(E · X · Wg) +(1) +where f(·) represents the ReLU activation function. Note that +our model learns both the node embedding Wg and the adja- +cency matrix E. This flexible representation of edges con- +tains rich spatial relationships between category entities. The +learning process of ERG is shown in Figure 3. Because our +ERG is learned, it can be adjusted during navigation accord- +ing to the actual needs of the agent and can also be adapted to +different environments and tasks. +Attention for Environment Representation +In order to make the agent more focus on the object of in- +terest and further enhance the matching ability of environ- +ment and instructions, we use the attention mechanism to +generate our final environment representation. First, Tiny- +BERT is used to unify the language embedding of detection +labels and navigation instructions into the same space, and +obtain the label embedding expression D ∈ R|U|×m. Then +we use Z as the attention map to label embedding. Corre- +sponding to the last two steps in Algorithm 1, our graph at- +tention is expressed as: +Ot = f(Zt · D) +(2) +and there are no learnable parameters here. +Ot is the +semantic-level environment representation. + +3.3 +Overview VLN-CE Framework +Figure 4 depicts the framework of our complete cross- +modal attention navigator. +In this paper, we refer to +[Hong et al., 2022], which predicts candidate waypoints in +advance to narrow the action space and bridge the gap be- +tween discrete and continuous environments. First, the candi- +date waypoint predictor uses only RGB-D visual information +to estimate k navigable positions for the agent in a continu- +ous environment at each time step. The process is similar to +construct a local navigable graph centered on the agent. +For vision, we use two ResNet50s [He et al., 2016] to en- +code RGB and depth observations separately, one pre-trained +on ImageNet [Russakovsky et al., 2015] for classification and +one pre-trained on Gibson [Xia et al., 2018] for point-goal +navigation. We denote the encoded visual observations as +{Vrgb +t,j }k +j=1 ∈ R2048 for RGB and {Vdepth +t,j +}k +j=1 ∈ R128 for +depth. Then, at each time step, we fuse the RGB, depth and +heading corresponding to each candidate waypoint to get the +final visual information as +Vj = [Vrgb +j +Wrgb; Vdepth +j +Wdepth; dj]Wfuse +(3) +where W are learnable linear projections and dj is a vector +encoding representing the relative orientation of the jth can- +didate viewpoint. +For environment representation, previous works only +consider RGB and depth, which cannot allow the agent to +understand the environment in detail and makes it difficult to +achieve a high degree of matching between vision and lan- +guage. So we propose a richer environment representation +based on ERG. At each time step, we construct the ERG Gt +and output the graph encoding Zt using GCN. Then we use +{Zt,u}U +u=1 as the attention map to the detection label to ob- +tain the final environment representation {Ot,u}U +u=1, where +U is the number of nodes. +For language, we use Tinybert to get embeddings L = +{l1, l2, . . . , lM} for each instruction of length M. A bidirec- +tional LSTM [Hochreiter and Schmidhuber, 1997] is used to +encode L as +C = {c1, c2, . . . , cM} = Bi-SLTM(l1, l2, . . . , lM) +(4) +For navigation, we consider CMA [Wang et al., 2019] as +our policy network. As the navigation progresses, the agent’s +visual perception changes accordingly. Attention-based his- +torical trajectory encoder GRU encodes the historical state of +the agent as +ht = GRU([ ¯ +Ot, ¯Vt, at−1Waction], ht−1) +(5) +where Waction is a learnable linear projection, at−1 is the ac- +tion taken at the last time step, ht−1 is the historical state +context of the last time step, ¯Vt = �k +j=1 αt,jVt,j is the +weighted sum of the visual features after attention, and ¯ +Ot = +�U +u=1 βt,uOt,u is the weighted sum of environmentrepresen- +tation after attention, Specifically, we utilize the dot product +attention to obtain ¯Vt and ¯ +Ot, +¯Vt = Attn(ht−1, {Vt,j}k +j=1) += +� +j +softmax(ht−1W (1) +h (Vt,jWV)T)Vt,j. +(6) +Waypoint Predictor +Turn around and exit the room. +Turn right and go past the +stairs. Keep walking straight +and stop in the middle once you +past the bar. +Object Detection +Environment +Representation Module +f +Bi-LSTM +� +Attn +ResNet50 +ResNet50 +f +GRU +1 +th - +th +Attn +Attn +Action +Selection +ta +TinyBert +1 +ta - +Heading +Attn +Attn +Figure 4: Overview of our cross-modal attention VLN-CE frame- +work. Our framework consists of a waypoint predictor, an environ- +ment representation, a visual representation, a language understand- +ing and an action decision module. +¯ +Ot = Attn(ht−1, {Ot,u}U +u=1) += +� +u +softmax(ht−1W (2) +h (Ot,uWO)T)Ot,u. +(7) +where W are learnable linear projections. +We condition on the visual historical state to locate the lan- +guage instruction of current interest. The vision-conditioned +textual feature is computed at each time step based on the +historical state ht as +�Ct = Attn(ht, {ci}M +i=1). +(8) +Then we look for visual and semantic emphasis based on +language. The text-conditioned visual feature is computed at +each time step based on the textual feature �Ct as +�Vt = Attn( �Ct, {Vj}k +j=1). +(9) +Similarly, The text-conditioned environment feature is +computed as +� +Ot = Attn( �Ct, {Ou}U +u=1). +(10) +Finally, we perform action selection based on historical +state ht, textual feature �Ct, visual feature �Vt and environment +feature � +Ot to decide which candidate waypoint to choose. +The probability of each waypoint is calculated as +pt,j = softmax([ht, �Ct, �Vt, � +Ot]Wa(Vt,j, Wv)T ). +(11) +The agent chooses the candidate direction with the highest +probability and navigates to that location. +3.4 +Training and Loss +Since the environment is static, object-to-object relationships +should not change as the agent’s perspective changes. We +hope that, at adjacent moments, the learned representation of +relationship between the same pair of objects is as consistent +as possible. We enhance the similarity of relational expres- +sions by optimizing a consistency loss: +LG = −log exp(CosSim(e, e+)/τ) +� exp(CosSim(e, e∗)/τ) +(12) + +where e is the feature representation of the edge in the ERG +at the current moment, e+ is the representation of the same +relationship at adjacent moments, each e∗ is the relationship +expression between all objects, τ is the scaling parameter, +and CosSim(·) is L2 Dot product between normalized fea- +tures. The closer the relationship is expressed, the higher the +CosSim value. By optimizing this objective, we encourage +the relationship expression of the same pair of objects in dif- +ferent perspectives to be consistent and different from other +relationships. +The agent’s navigation ability is trained by imitation learn- +ing with a cross-entropy loss as +LIL = − +� +t +a∗ +t log(pt) +(13) +where pt is the action probability and a∗ +t is the oracle action. +The overall loss function of our model can be written as +L = LIL + LG. +(14) +4 +Experiments +4.1 +Experimental Setup +VLN-CE Dataset +VLN-CE dataset [Krantz et al., 2020] is employed to eval- +uate our cross-modal attention with graph (Graph-CMA) +framework. +It contains 4475 trajectories. +Each trajec- +tory provides ego-centered images from Habitat simulator +[Savva et al., 2019], three natural language instructions and a +pre-computed shortest path via low-level actions. The dataset +is split into train, validation-seen and validation-unseen sets. +Validation-seen dataset shares the same scenes with training +set. Environments in validation-unseen dataset are not ex- +posed to the agent. +Metrics +Six metrics are established to verify the effectiveness of our +VLN framework: +• TL (Trajectory Length) measures the average length of +the predicted trajectories in navigation. +• NE (Navigation Error) measures the average distance (in +meter) between the agent’s stopping position in the predicted +trajectory and the goal in the reference trajectory. +• nDTW (normalized Dynamic Time Wraping) measures +the normalized cumulative distance between reference path +and agent position. +• OSR (Oracle Success Rate) is the proportion of the clos- +est point in the predicted trajectory to the target in the refer- +ence trajectory within a threshold distance. +• SR (Success Rate) is the proportion of the agent stopping +in the predicted route within a threshold distance of the goal +in the reference route. +• SPL (Success weighted by inverse Path Length) is a com- +prehensive metric method integrating SR and TL that takes +both effectiveness and efficiency into account. +In our discussion, we will primarily verify the performance +of our method through SR and SPL. +Implementation Details +We implement our agent on the Habitat simulator. +Af- +ter each step, the agent will obtain the images of sur- +roundings. +The semantic information is generated re- +ferring to Faster RCNN pretrained with Visual Genome ++ Res101 and PyTorch [Ren et al., 2015]. +The instruc- +tions and semantic information are embedded by pretrained +TinyBert [Jiao et al., 2020]. +The visual information is ex- +tracted from RGB and Depth images through Resnet50 (pre- +trained on ImageNet [Russakovsky et al., 2015] and Gibson +[Xia et al., 2018] separately). The waypoint predictor will +predict at most 6 candidate waypoint views in each step. Then +an ERG is established from the candidate waypoint views’ se- +mantic information. The GCN for updating ERG is composed +of 5 hidden layers. +State-of-Art Baselines +To elaborate the effectiveness of our proposed environment +representation and our new cross-modal attention navigation +framework, we compare our method with other state-of-art +VLN-CE methods. Specifically, methods are as followed. +CMA [Wang et al., 2019]:This method proposes a se- +quence to sequence model which enables the agent to have +cross-modal attention and spatial visual reasoning ability. +SASRA [Irshad et al., 2021]: This method designs for the +agent a hybrid transformer recurrent cross-modal model fo- +cusing on aligning top-down local ego-centric semantic map- +ping with language. +LAW[Raychaudhuri et al., 2021]: This method guides the +agent to learn policy with a language-aligned supervision +scheme and a metric which measures the sub-instructions the +agent has completed during navigation. +Waypoint Model [Krantz et al., 2021]: This method pro- +vides a language-conditioned waypoint prediction network +for the agent. Then the agent picks waypoint referring to its +hidden states and environmental features. +BG-CMA [Hong et al., 2022]: This method discretizes the +continuous environment and provide candidate waypoints for +the agent. Agent focuses on choosing candidate direction re- +ferring to a reinforced Cross-Modal Matching approach. +4.2 +Main Results +We validate the agent with our proposed Graph-CMA on +VLN-CE dataset. As can be seen in Val-seen, our method +achieves best performance in almost all metrics. The results +are in Table 1, which is divided into two parts. The upper part +shows the claimed performance of several more classic meth- +ods in the field of VLN-CE. In the comparison among results, +Waypoint models outperforms other three methods in almost +all metrics. In val-unseen dataset, the improvements of our +Graph-CMA in SR and SPL are 3% absolute (8.3% relative) +and 1% absolute (3% relative) compared with Waypoint mod- +els. In lower part of the table, we focus on comparing the lat- +est research (BG-CMA) results with our method. BG-CMA +[Hong et al., 2022] is claimed to outperform Waypoint mod- +els in all metrics except TL metric. It is worth noting that we +do not directly show the results mentioned in the BG-CMA +paper, but the results trained by us. This is because that our + +Methods +Val-seen +Val-unseen +TL↓ +NE↓ +nDTW↑ +OSR↑ +SR↑ +SPL↑ +TL↓ +NE↓ +nDTW↑ +OSR↑ +SR↑ +SPL↑ +CMA +9.26 +7.12 +54 +46 +37 +35 +8.64 +7.37 +51 +40 +32 +30 +LAW +- +- +58 +- +40 +37 +- +- +54 +- +35 +31 +SASRA +8.89 +7.17 +53 +- +36 +34 +7.89 +8.32 +47 +- +24 +22 +Waypoint models +8.54 +5.48 +- +53 +46 +43 +7.62 +6.31 +- +40 +36 +34 +*BG-CMA +13.9 +5.16 +56 +59 +43 +38 +9.21 +6.55 +55 +42 +36 +32 +*Our (Graph-CMA) +11.8 +5.04 +59 +61 +46 +42 +9.96 +6.20 +56 +48 +39 +35 +Table 1: Experimental results showing the performance of baselines and our method. (* represents that the results are trained in our compu- +tation environment). +Table 2: Quantitative Comparison showing the effect of our ERG. +Val-unseen +Methods +TL↓ +NE↓ +OSR↑ +SR↑ +SPL↑ +*CMA +8.64 +7.37 +40 +32 +30 +*CMA+ERG +8.16 +6.98 +43 +35 +33 +*LAW +9.87 +7.26 +58 +35 +31 +*LAW+ERG +9.95 +7.18 +59 +37 +32 +*BG-CMA +9.21 +6.55 +42 +36 +32 +*BG-CMA+ERG +9.96 +6.20 +48 +39 +35 +code is modified based on BG-CMA, but the computation en- +vironment difference cause that we achieve the different re- +sults of BG-CMA with claimed. Therefore, we choose to +show validation results under our unified computation envi- +ronment, denoted by * here. In comparison results, the im- +provements of our Graph-CMA in SR are 3 % absolute (7% +relative ) in val-seen and 3 % absolute (8% relative) in val- +unseen compared with BG-CMA. The improvements of our +Graph-CMA in SPL are 4 % absolute (10.5% relative ) in val- +seen and 3 % absolute (9.4% relative) in val-seen compared +with BG-CMA. The comparison shows that our method en- +hances the navigation performance and generalization ability +of the agent. +4.3 +Quantitative Comparison with ERG +To reveal the effectiveness of ERG, we design an quantitative +comparison for the ERG. To be specific, we do comparison +between the baselines and baselines with our proposed ERG. +Table 2 shows that our proposed ERG significantly improves +the baselines performances. In the comparison about CMA +baseline in Val-unseen, CMA+ERG improves 3% absolute +(9.4% relative) in SR and 3% absolute (10% relative) in SPL. +Notice CMA method focuses on cross-modal attention spa- +tial visual reasoning ability. The improvement reveals ERG +has better cross-modal matching and visual reasoning ability +than CMA. In the comparison about LAW baseline in Val- +unseen, LAW+ERG improves 2% absolute (5.7% relative) +in SR and 1% absolute (3.2% relative) in SPL. Notice LAW +method focuses on language understanding ability during the +trajectory. The improvement reveals ERG has better language +understanding than LAW. In the comparison about BG-CMA +baseline in Val-unseen, apart from improvements in SR and +SPL, BG-CMA+ERG also improves 6% absolute (14.3% rel- +ative) in OSR. Notice OSR measures success for each agent +under oracle stopping rule, i.e whether agent stops at closest +room +stair +door +hallway +doorway +bedroom +table +bathroom +chair +step +kitchen +bed +top +couch +wall +glass +sink +staircase +fireplace +pool +head +house +rug +painting +desk +patio +window +counter +floor +mirror +bar +foot +plant +archway +picture +building +carpet +rail +bench +sofa +balcony +wood +sign +cabinet +light +refrigerator +vase +toilet +shelf +pillar +walkway +flower +lamp +stool +statue +bathtub +stove +back +television +art +stone +column +tree +tile +switch +base +center +dresser +mat +case +deck +hand +water +screen +rock +oven +bookshelf +chandelier +rope +arch +washer +sculpture +bookcase +frame +circle +brick +seat +clock +gate +book +pot +ceiling +curtain +man +towel +bike +animal +planter +arm +poster +Figure 5: The agent’s understanding of the environment during nav- +igation. +point to the goal on its trajectory. The huge improvement in +this metric shows that ERG can stop more precisely when it +is close to destination. This further demonstrates the ERG’s +strong environment understanding capability. +4.4 +Qualitative Analysis +For a more intuitive view of how our method works for the +VLN-CE task, we visualize an qualitative example of agent’s +trajectory in Figure 5. The bottom left part of the figure is the +platform of agent’s trajectory. The upper left part of the figure +is the candidate waypoint views. The right part of the figure +is the expression of ERG generated by GCN as the attention +map for interest objects. Objects are represented by column +vectors. In the trajectory, the agent successfully reaches the +target destination, with a comprehensive understanding of the +environment. It first picks 6 waypoint views from images and +builds ERG from semantic information. Then from the graph +updated, the agent gains the attention of the objects in the +environment. From the attention map we can see that ERG +shows strong cross-modal matching ability. +5 +Conclusion +In this paper, we focus on the agent’s ability to understand +the environment for VLN-CE and propose a new environ- +ment representation. First, we introduce semantic informa- +tion to construct an ERG based on object detection results. +Then, GCN is used to learn the relational representation of +object-object and object-agent in ERG. Finally, the environ- +ment representation is obtained by combining the ERG with +object label embeddings. In order to embed the ERG into the + +00.0 +O'OJ +0'05 +E0.0 +0'04 +0'02navigation, a novel cross-modal attention navigation frame- +work is proposed with a loss dedicated for ERG. 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In Proceedings of the IEEE/CVF Conference on +Computer Vision and Pattern Recognition, pages 12689– +12699, 2021. + diff --git a/mNE3T4oBgHgl3EQfKQmi/content/tmp_files/load_file.txt b/mNE3T4oBgHgl3EQfKQmi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..94e6d3621b810ca61c8342a84a7c9a5b31b00863 --- /dev/null +++ b/mNE3T4oBgHgl3EQfKQmi/content/tmp_files/load_file.txt @@ -0,0 +1,718 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf,len=717 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='04352v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='CV] 11 Jan 2023 Graph based Environment Representation for Vision-and-Language Navigation in Continuous Environments Ting Wang1,2 , Zongkai Wu2,∗ , Feiyu Yao and Donglin Wang2,∗ 1Zhejiang University 2Westlake University {wangting, wuzongkai}@westlake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='cn, feiyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='yao@columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='edu, wangdonglin@westlake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='cn Abstract Vision-and-Language Navigation in Continuous Environments (VLN-CE) is a navigation task that requires an agent to follow a language instruction in a realistic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The understanding of envi- ronments is a crucial part of the VLN-CE task, but existing methods are relatively simple and direct in understanding the environment, without delving into the relationship between language instructions and visual environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Therefore, we propose a new environment representation in order to solve the above problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' First, we propose an Environ- ment Representation Graph (ERG) through object detection to express the environment in semantic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' This operation enhances the relationship be- tween language and environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Then, the rela- tional representations of object-object, object-agent in ERG are learned through GCN, so as to obtain a continuous expression about ERG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Sequentially, we combine the ERG expression with object label embeddings to obtain the environment representa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Finally, a new cross-modal attention naviga- tion framework is proposed, incorporating our en- vironment representation and a special loss func- tion dedicated to training ERG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Experimental result shows that our method achieves satisfactory perfor- mance in terms of success rate on VLN-CE tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Further analysis explains that our method attains better cross-modal matching and strong generaliza- tion ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 1 Introduction Vision-and-language navigation (VLN) [Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2018] is a task that requires an agent to navigate from an arbitrary initial position to a described destination by understanding real-time image information and language instructions in a photo-realistic 3D environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' In recent years, many studies [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Fried et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2019a] have done model improvement and method innovation on this basis to achieve satisfactory results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' However, some perfect assumptions of VLN are far from real robots, including hop- ping navigation, known environment topology, and precise Walk to the table and through the open door and go into the first room on your left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Wait there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' bl d Figure 1: For VLN-CE, the agent needs to complete an instruction in a real environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Without sufficient understanding of the environ- ment, the agent can easily become confused about the instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' When the agent has a good understanding of the environment and is aware of the objects and their relationships, the navigation task is easy to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Therefore, the Vision-and-Language Navigation in Continuous Environments (VLN-CE) [Krantz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2020] task is proposed to provide a more realistic platform for VLN robot by performing a set of low-level actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' This means that the agent in continuous environments needs to have better environment understanding capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' At present, the work on the VLN-CE task is still rela- tively lacking, and there is a large space for improvement in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The VLN-CE dataset [Krantz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2020] is obtained by converting the navigation trajectories of the Room-to-Room dataset [Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2018] into the Habitat Simulator [Savva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Visual informa- tion in the dataset is real image captured by the cam- era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' [Irshad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2021] propose a cross-modal Semantic- Linguistic Attention Map Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' [Krantz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2021], [Raychaudhuri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2021] and [Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2022] convert low-level action predictions into waypoint predictions in dif- ferent ways, achieving the best current navigation perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' But they do not realize the importance of understand- ing the environment and the intrinsic connection between the visual environment and language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' These problems make it impossible for the agent to achieve a high-level matching of vision and language, thus limiting the navigation perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' For example, as shown in Figure 1, the language instruc- tion “Walk to the table and through the open door and go into door mrron chair table0 20 100 120 00 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 112 J20 TS2 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 12 20 52 0the first room on your left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Wait there.” requires the agent to understand the language and interact with the environment to complete the navigation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Most of previous works di- rectly extract the visual feature from image to make naviga- tion decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' However, referring to humans, the objects such as “table”, “door”, and “room” should be first identified in sequence, and then the navigation decision is made by under- standing the relationship between object-object and object- human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Such operations require the agent to be able to detect the objects in vision and establish an environment represen- tation in semantic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Finally, the agent should make navi- gation decision by matching this representation and language instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' In order to achieve the above idea, we propose a new envi- ronment representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' First, we analyze all object descrip- tions in language instructions and summarize a vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Then, we introduce object detection technology, determine the target to be detected according to the summarized vocab- ulary, and identify the objects in the environment within the agent’s field of vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' To express the environment, we pro- pose an Environment Representation Graph (ERG) with the detection results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Each node information of ERG includes direction information between object and agent, one-hot la- bel vector and confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Such operation can improve the agent’s environment understanding capacity under language conditions, and can relatively reduce the gap caused by the texture difference among different buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' However, to express the environment, it is not enough to simply introduce semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' It is also necessary to know the relationship of object-object and object-agent, that is, the expression of edges in ERG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Whether manu- ally defining discrete relationship symbols or estimating dis- tances, there are problems such as high labor costs or miss- ing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Therefore, we consider Graph Convolution Network (GCN) [Satorras and Estrach, 2018] to learn the re- lationship between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' That is, our ERG neither requires a pre-defined adjacency matrix nor relies on additional ex- ternal knowledge, but learns relational representations from the training dataset with the help of object detection knowl- edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Sequentially, we obtain a continuous expression about ERG, which can imply more information and can be flexibly adjusted according to the needs of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Finally, we mul- tiply the ERG expression as the attention map by the object label embedding to obtain the final environment representa- tion at the semantic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' It is worth noting that in order to further alleviate the expression differences between naviga- tion instructions and object detection labels, we use TinyBert to transform them into the same embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' After that, we design a cross-modal attention navigation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Our framework gets three inputs about the en- vironment: our environment representation, RGB and depth, and utilizes the GRU to carry the overall information flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The historical context encoded by the GRU, in addition to the previous RGB-D, will also include the environment rep- resentation processed by soft-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' In the action se- lection module, our environment representation features are also added to make decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Our cross-modal attention framework not only improves the connections between multi- modal information, but also amplifies the influence of our environment representations on navigation decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' In the training phase, in addition to considering the navigation cross-entropy loss, we also propose a new loss function that considering the spatial consistency of the relationship be- tween the same pair of objects when the perspective changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Overall, the main contributions of this article are summarized as follows: We propose an environment representation for VLN-CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' We introduce semantic information and construct the ERG in real time through object detection, where the relational representation is learned by GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' We propose a new cross-modal attention VLN-CE framework combined with an environment representa- tion module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' In the training phase, We consider a com- bination of the two losses, the navigation cross-entropy loss and the spatial consistency loss of the relationship between the same pair of objects when the perspective changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The experimental results demonstrate that our graph based environment representation indeed helps to im- prove the navigation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Better cross-modal matching is attained and the generalization of the navi- gation model is enhanced in unseen environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 2 Related Work Vision-and-Language navigation The Vision- and-Language Navigation (VLN) task proposed by [Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2018] has attracted extensive research attention in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Many works make improvements and innovations on this basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2018] first combine model-free and model-based deep reinforcement learning for VLN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Speaker-Follower [Fried et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2018] performs data augmentation by generating language instructions for efficient navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' To address the cross-modal grounding and information feedback issues, [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2019] propose a novel Reinforced Cross- Modal Matching (RCM) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Similarly, the Regretful Agent [Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2019b] and Tactical Rewind backtrack [Ke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2019] introduce a self-monitoring mechanism and global signals, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Referring to previous methods , [Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2020] introduce four self-supervised auxiliary reasoning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' [Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2021] propose a new transformer-based multimodal framework for navigator and speaker, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2021a] propose a VLN model based on curriculum learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' ENVEDIT [Jialu Li, 2022] and REM [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2021] perform data augmentation by editing environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' In addition, there are many works [Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Majumdar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Guhur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2021] to improve VLN performance through various pre-training methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' [Jain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2019] propose a new dataset Room-for-Room (R4R) and a new metric Coverage weighted by Length Score (CLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' [Ku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2020] introduce a multilingual Room-Across-Room (RxR) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' [Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2020] transfer the VLN task from the simulation environment to the physical robotic platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' [Krantz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2020] eliminate unrealistic assumptions ERG TinyBert bed, window, door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Environment Representation Attn Attn 1 th - Text state GRU Action selection Waypoint Prediction + Object Detection Heading Angle Confidence Embedded Label Heading Angle Confidence Embedded Label door bed window hallway mirror bed window hallway mirror door bed window hallway mirror door bed window hallway mirror Figure 2: Illustration of environment representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The waypoint predictor predicts k candidate directions for the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Then ERG is generated by extracting object one-hot label features, confidence and orientation through object detection from K candidate waypoint views as node information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' To further facilitate the matching of environment and language, ORG is used as the attention map to encode the TinyBERT embedding of the label to obtain the final environment expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' about sparse navigation graphs and propose the VLN- CE task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' [Krantz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Raychaudhuri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2022] propose waypoint-based VLN-CE nav- igation model in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2021b] introduce a meta-learning-based visual perception general- ization strategy that enables agents to adapt to new sensor configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' [Irshad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2021] propose a cross-modal Semantic-Linguistic Attention Map Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Graph for navigation In the field of intelligent navigation, there are some works that have been combined with graph and achieved good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' There are the following researches in the field of object navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' [Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2020] propose three complementary techniques: an object relational graph, a trial-driven imitation learning and a memory-augmented tentative policy network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Similarly, [Xiaobo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2021] in- troduce an Agent-Centric Relation Graph (ACRG) for learn- ing visual representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' [Gadre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2022] propose scene representations suitable for different downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' An DOA graph [Dang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2022] explicitly learns attention re- lationships between objects and allocates more reasonable attention resources to object features and global image fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' A novel two-layer hierarchical reinforcement learning method with a Goals Relational Graph [Ye and Yang, 2021] is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2021b] propose an online learn- ing mechanism based on a hierarchical object-to-zone (HOZ) graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' As for VLN-related work, a Structured Scene Mem- ory (SSM) [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2021a] is proposed to capture vi- sual and geometric cues in the environment, and is equipped with a collect-read controller for information gathering and long-term navigation reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' [Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2020] propose a Language and Visual Entity Relationship Graph and a message-passing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' [Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2021] introduce a Scenario Oriented Object Navigation (SOON) task and pro- pose a graph-based exploration method, which models navi- gation states as graphs and stabilizes training by learning sub- optimal trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Unlike them, we will realize the environ- ment representation at the semantic level through building a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 3 Methodology In this paper, we aim to create our environment representa- tion for introducing rich semantic and contextual information, which enables understanding and reasoning of environments and facilitates cross-modal matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' We start by describ- ing the VLN-CE task and the specific learning process of the ERG-centered environment representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Then we show overall VLN-CE framework incorporating our environment representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Finally, we introduce the training and loss of the whole model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='1 Task Definition At each time step t, the VLN-CE agent receives the instruc- tion L, the RGB observation Vrgb t and the depth observa- tion Vdepth t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The language instruction L is a sentence com- posed of M words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Visual observations Vrgb t and Vdepth t re- ceived in real time are egocentric RGBD images from the simulator with a resolution of 256 × 256 and a horizontal field-of-view of 90 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Then the agent predicts the next action at based on the current state and the above informa- tion (vision and language).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' In this paper, we add our ERG- centric environment representation for decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The entire navigation can be viewed as a Partially Observable Markov Decision Process (POMDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The action space A of all agents consists of four simple, low-level actions - Move forward 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='25m, Turn left or Turn right 15 degrees, and Stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The agent repeats this process and performs a series of ac- tions a = {a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' , at} until the “Stop” action is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The success of the VLN-CE task can be declared only if the agent’s position is close enough to the instruction target when it makes the “Stop” action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='2 Environment Representation In this section, we explain how an ERG can be constructed in real-time to obtain richer environment representations for navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' We first discuss building local node representa- tions by object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Then we discuss how to obtain rela- tional representations in ERG with GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Finally, we take the output of the ERG as an attention map, and combine it with the label embedding to get the final environment representa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' A general illustration of the environment representation is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Detection for Node Representation In order to obtain a more contextual environment rep- resentation, we first need to detect all objects from the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' We achieve this by using the Faster RCNN [Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2015] pretrained on Visual Genome [Krishna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2017] as our object detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' As shown in Figure 2, at each time step, we localize all objects of inter- est through Faster RCNN for the egocentric RGB images of k candidate waypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' In this paper, we analyze all object descriptions in language instructions and summarize a vocab- ulary, and pre-set U = 100 object categories of interest based on instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Then we define a graph G = (N, E) to con- struct our ERG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Each node n ∈ N represents an entity cate- gory in the environment, and each edge ε ∈ E represents a re- lationship between two entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' According to each detection result, the heading feature dj, the confidence q and a one-hot encoded label vector r will serve as a local node represen- tation of our ERG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' dj is a heading feature vector consisting of [sinφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' cosφ], where φ is the heading angle of the candi- date waypoint relative to the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Each time, we aggregate all detections in k directions and construct only one ERG, as described in Algorithm 1, lines 4 to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' These operations con- struct the ERG to make it easier for an agent to understand the environment, and introduce semantic information that can al- leviate the domain gap caused by style and texture differences among different buildings to a large extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Learning for Our ERG Regarding the expression of edges, we have considered many possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' For example, artificially pre-defined dis- crete relationship symbols, such as “besides”, “above”, “in- side”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' On the one hand, this method is time-consuming and labor-intensive, and on the other hand, it may lose a lot of important information such as the specific performance of relative positions or metric relationships, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Alterna- tively, using a distance metric as a relational expression also produces errors while losing information, because our per- spective changes in real time and our navigation instructions are not guided by distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Therefore, we consider apply- ing GCN to acquire continuous representations of edges in a learned manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Our GCN takes all nodes X ∈ R|U|×D as input, and then ReLU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Hidden layer Input Hidden layer ReLU Output Figure 3: The learning process of ERG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Algorithm 1 Establish environment representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 1: for Each t-th epochs do 2: Obtain k candidate RGB images through waypoint predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 3: Build ERG G with empty node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 4: for j = 1 : k do 5: Detect objects by Faster RCNN from j-th candidate image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 6: for j = 1 : k do 7: if q > G[object][confidence] then 8: Update G[object] with the direction informa- tion between object and agent, one-hot label vector and confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 9: end if 10: end for 11: end for 12: Learn the representation Zt of ERG using GCN by equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 13: Using Zt as attention map to the label embedding ex- pression D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 14: Output environment representation Ot as equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 15: end for embeds each input node with a matrix Wg ∈ RD×U, where D is the dimension of the node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Afterwards, our GCN embeds all nodes through the adjacency matrix E ∈ R|U|×U, and outputs a new encoding Z ∈ R|U|×U, which expresses the environment, including the objects in the environment, the relationship between objects, and the relationship between objects and the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Executing line 12 of Algorithm 1, our proposed ERG is expressed as: Z = f(E · X · Wg) (1) where f(·) represents the ReLU activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Note that our model learns both the node embedding Wg and the adja- cency matrix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' This flexible representation of edges con- tains rich spatial relationships between category entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The learning process of ERG is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Because our ERG is learned, it can be adjusted during navigation accord- ing to the actual needs of the agent and can also be adapted to different environments and tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Attention for Environment Representation In order to make the agent more focus on the object of in- terest and further enhance the matching ability of environ- ment and instructions, we use the attention mechanism to generate our final environment representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' First, Tiny- BERT is used to unify the language embedding of detection labels and navigation instructions into the same space, and obtain the label embedding expression D ∈ R|U|×m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Then we use Z as the attention map to label embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Corre- sponding to the last two steps in Algorithm 1, our graph at- tention is expressed as: Ot = f(Zt · D) (2) and there are no learnable parameters here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Ot is the semantic-level environment representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='3 Overview VLN-CE Framework Figure 4 depicts the framework of our complete cross- modal attention navigator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' In this paper, we refer to [Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2022], which predicts candidate waypoints in advance to narrow the action space and bridge the gap be- tween discrete and continuous environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' First, the candi- date waypoint predictor uses only RGB-D visual information to estimate k navigable positions for the agent in a continu- ous environment at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The process is similar to construct a local navigable graph centered on the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' For vision, we use two ResNet50s [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2016] to en- code RGB and depth observations separately, one pre-trained on ImageNet [Russakovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2015] for classification and one pre-trained on Gibson [Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2018] for point-goal navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' We denote the encoded visual observations as {Vrgb t,j }k j=1 ∈ R2048 for RGB and {Vdepth t,j }k j=1 ∈ R128 for depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Then, at each time step, we fuse the RGB, depth and heading corresponding to each candidate waypoint to get the final visual information as Vj = [Vrgb j Wrgb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Vdepth j Wdepth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' dj]Wfuse (3) where W are learnable linear projections and dj is a vector encoding representing the relative orientation of the jth can- didate viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' For environment representation, previous works only consider RGB and depth, which cannot allow the agent to understand the environment in detail and makes it difficult to achieve a high degree of matching between vision and lan- guage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' So we propose a richer environment representation based on ERG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' At each time step, we construct the ERG Gt and output the graph encoding Zt using GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Then we use {Zt,u}U u=1 as the attention map to the detection label to ob- tain the final environment representation {Ot,u}U u=1, where U is the number of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' For language, we use Tinybert to get embeddings L = {l1, l2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' , lM} for each instruction of length M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' A bidirec- tional LSTM [Hochreiter and Schmidhuber, 1997] is used to encode L as C = {c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' , cM} = Bi-SLTM(l1, l2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' , lM) (4) For navigation, we consider CMA [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2019] as our policy network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' As the navigation progresses, the agent’s visual perception changes accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Attention-based his- torical trajectory encoder GRU encodes the historical state of the agent as ht = GRU([ ¯ Ot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' ¯Vt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' at−1Waction],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' ht−1) (5) where Waction is a learnable linear projection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' at−1 is the ac- tion taken at the last time step,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' ht−1 is the historical state context of the last time step,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' ¯Vt = �k j=1 αt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='jVt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='j is the weighted sum of the visual features after attention,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' and ¯ Ot = �U u=1 βt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='uOt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='u is the weighted sum of environmentrepresen- tation after attention,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' we utilize the dot product attention to obtain ¯Vt and ¯ Ot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' ¯Vt = Attn(ht−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' {Vt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='j}k j=1) = � j softmax(ht−1W (1) h (Vt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='jWV)T)Vt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' (6) Waypoint Predictor Turn around and exit the room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Turn right and go past the stairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Keep walking straight and stop in the middle once you past the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Object Detection Environment Representation Module f Bi-LSTM � Attn ResNet50 ResNet50 f GRU 1 th - th Attn Attn Action Selection ta TinyBert 1 ta - Heading Attn Attn Figure 4: Overview of our cross-modal attention VLN-CE frame- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Our framework consists of a waypoint predictor, an environ- ment representation, a visual representation, a language understand- ing and an action decision module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' ¯ Ot = Attn(ht−1, {Ot,u}U u=1) = � u softmax(ht−1W (2) h (Ot,uWO)T)Ot,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' (7) where W are learnable linear projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' We condition on the visual historical state to locate the lan- guage instruction of current interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The vision-conditioned textual feature is computed at each time step based on the historical state ht as �Ct = Attn(ht, {ci}M i=1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' (8) Then we look for visual and semantic emphasis based on language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The text-conditioned visual feature is computed at each time step based on the textual feature �Ct as �Vt = Attn( �Ct, {Vj}k j=1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' (9) Similarly, The text-conditioned environment feature is computed as � Ot = Attn( �Ct, {Ou}U u=1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' (10) Finally, we perform action selection based on historical state ht, textual feature �Ct, visual feature �Vt and environment feature � Ot to decide which candidate waypoint to choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The probability of each waypoint is calculated as pt,j = softmax([ht, �Ct, �Vt, � Ot]Wa(Vt,j, Wv)T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' (11) The agent chooses the candidate direction with the highest probability and navigates to that location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='4 Training and Loss Since the environment is static, object-to-object relationships should not change as the agent’s perspective changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' We hope that, at adjacent moments, the learned representation of relationship between the same pair of objects is as consistent as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' We enhance the similarity of relational expres- sions by optimizing a consistency loss: LG = −log exp(CosSim(e, e+)/τ) � exp(CosSim(e, e∗)/τ) (12) where e is the feature representation of the edge in the ERG at the current moment, e+ is the representation of the same relationship at adjacent moments, each e∗ is the relationship expression between all objects, τ is the scaling parameter, and CosSim(·) is L2 Dot product between normalized fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The closer the relationship is expressed, the higher the CosSim value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' By optimizing this objective, we encourage the relationship expression of the same pair of objects in dif- ferent perspectives to be consistent and different from other relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The agent’s navigation ability is trained by imitation learn- ing with a cross-entropy loss as LIL = − � t a∗ t log(pt) (13) where pt is the action probability and a∗ t is the oracle action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The overall loss function of our model can be written as L = LIL + LG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' (14) 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='1 Experimental Setup VLN-CE Dataset VLN-CE dataset [Krantz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2020] is employed to eval- uate our cross-modal attention with graph (Graph-CMA) framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' It contains 4475 trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Each trajec- tory provides ego-centered images from Habitat simulator [Savva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2019], three natural language instructions and a pre-computed shortest path via low-level actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The dataset is split into train, validation-seen and validation-unseen sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Validation-seen dataset shares the same scenes with training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Environments in validation-unseen dataset are not ex- posed to the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Metrics Six metrics are established to verify the effectiveness of our VLN framework: TL (Trajectory Length) measures the average length of the predicted trajectories in navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' NE (Navigation Error) measures the average distance (in meter) between the agent’s stopping position in the predicted trajectory and the goal in the reference trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' nDTW (normalized Dynamic Time Wraping) measures the normalized cumulative distance between reference path and agent position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' OSR (Oracle Success Rate) is the proportion of the clos- est point in the predicted trajectory to the target in the refer- ence trajectory within a threshold distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' SR (Success Rate) is the proportion of the agent stopping in the predicted route within a threshold distance of the goal in the reference route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' SPL (Success weighted by inverse Path Length) is a com- prehensive metric method integrating SR and TL that takes both effectiveness and efficiency into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' In our discussion, we will primarily verify the performance of our method through SR and SPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Implementation Details We implement our agent on the Habitat simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Af- ter each step, the agent will obtain the images of sur- roundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The semantic information is generated re- ferring to Faster RCNN pretrained with Visual Genome + Res101 and PyTorch [Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The instruc- tions and semantic information are embedded by pretrained TinyBert [Jiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The visual information is ex- tracted from RGB and Depth images through Resnet50 (pre- trained on ImageNet [Russakovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2015] and Gibson [Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2018] separately).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The waypoint predictor will predict at most 6 candidate waypoint views in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Then an ERG is established from the candidate waypoint views’ se- mantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The GCN for updating ERG is composed of 5 hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' State-of-Art Baselines To elaborate the effectiveness of our proposed environment representation and our new cross-modal attention navigation framework, we compare our method with other state-of-art VLN-CE methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Specifically, methods are as followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' CMA [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2019]:This method proposes a se- quence to sequence model which enables the agent to have cross-modal attention and spatial visual reasoning ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' SASRA [Irshad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2021]: This method designs for the agent a hybrid transformer recurrent cross-modal model fo- cusing on aligning top-down local ego-centric semantic map- ping with language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' LAW[Raychaudhuri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2021]: This method guides the agent to learn policy with a language-aligned supervision scheme and a metric which measures the sub-instructions the agent has completed during navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Waypoint Model [Krantz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2021]: This method pro- vides a language-conditioned waypoint prediction network for the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Then the agent picks waypoint referring to its hidden states and environmental features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' BG-CMA [Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2022]: This method discretizes the continuous environment and provide candidate waypoints for the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Agent focuses on choosing candidate direction re- ferring to a reinforced Cross-Modal Matching approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='2 Main Results We validate the agent with our proposed Graph-CMA on VLN-CE dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' As can be seen in Val-seen, our method achieves best performance in almost all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The results are in Table 1, which is divided into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The upper part shows the claimed performance of several more classic meth- ods in the field of VLN-CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' In the comparison among results, Waypoint models outperforms other three methods in almost all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' In val-unseen dataset, the improvements of our Graph-CMA in SR and SPL are 3% absolute (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='3% relative) and 1% absolute (3% relative) compared with Waypoint mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' In lower part of the table, we focus on comparing the lat- est research (BG-CMA) results with our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' BG-CMA [Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2022] is claimed to outperform Waypoint mod- els in all metrics except TL metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' It is worth noting that we do not directly show the results mentioned in the BG-CMA paper, but the results trained by us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' This is because that our Methods Val-seen Val-unseen TL↓ NE↓ nDTW↑ OSR↑ SR↑ SPL↑ TL↓ NE↓ nDTW↑ OSR↑ SR↑ SPL↑ CMA 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='26 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='12 54 46 37 35 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='64 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='37 51 40 32 30 LAW 58 40 37 54 35 31 SASRA 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='89 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='17 53 36 34 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='89 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='32 47 24 22 Waypoint models 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='54 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='48 53 46 43 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='62 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='31 40 36 34 BG-CMA 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='16 56 59 43 38 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='21 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='55 55 42 36 32 Our (Graph-CMA) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='04 59 61 46 42 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='96 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='20 56 48 39 35 Table 1: Experimental results showing the performance of baselines and our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' (* represents that the results are trained in our compu- tation environment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Table 2: Quantitative Comparison showing the effect of our ERG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Val-unseen Methods TL↓ NE↓ OSR↑ SR↑ SPL↑ CMA 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='64 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='37 40 32 30 CMA+ERG 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='98 43 35 33 LAW 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='87 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='26 58 35 31 LAW+ERG 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='95 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='18 59 37 32 BG-CMA 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='21 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='55 42 36 32 BG-CMA+ERG 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='96 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='20 48 39 35 code is modified based on BG-CMA, but the computation en- vironment difference cause that we achieve the different re- sults of BG-CMA with claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Therefore, we choose to show validation results under our unified computation envi- ronment, denoted by * here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' In comparison results, the im- provements of our Graph-CMA in SR are 3 % absolute (7% relative ) in val-seen and 3 % absolute (8% relative) in val- unseen compared with BG-CMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The improvements of our Graph-CMA in SPL are 4 % absolute (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='5% relative ) in val- seen and 3 % absolute (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='4% relative) in val-seen compared with BG-CMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The comparison shows that our method en- hances the navigation performance and generalization ability of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='3 Quantitative Comparison with ERG To reveal the effectiveness of ERG, we design an quantitative comparison for the ERG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' To be specific, we do comparison between the baselines and baselines with our proposed ERG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Table 2 shows that our proposed ERG significantly improves the baselines performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' In the comparison about CMA baseline in Val-unseen, CMA+ERG improves 3% absolute (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='4% relative) in SR and 3% absolute (10% relative) in SPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Notice CMA method focuses on cross-modal attention spa- tial visual reasoning ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The improvement reveals ERG has better cross-modal matching and visual reasoning ability than CMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' In the comparison about LAW baseline in Val- unseen, LAW+ERG improves 2% absolute (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='7% relative) in SR and 1% absolute (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='2% relative) in SPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Notice LAW method focuses on language understanding ability during the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The improvement reveals ERG has better language understanding than LAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' In the comparison about BG-CMA baseline in Val-unseen, apart from improvements in SR and SPL, BG-CMA+ERG also improves 6% absolute (14.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='towel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='bike ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='animal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='planter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='arm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='poster ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='Figure 5: The agent’s understanding of the environment during nav- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='igation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' point to the goal on its trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The huge improvement in this metric shows that ERG can stop more precisely when it is close to destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' This further demonstrates the ERG’s strong environment understanding capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content='4 Qualitative Analysis For a more intuitive view of how our method works for the VLN-CE task, we visualize an qualitative example of agent’s trajectory in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The bottom left part of the figure is the platform of agent’s trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The upper left part of the figure is the candidate waypoint views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' The right part of the figure is the expression of ERG generated by GCN as the attention map for interest objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Objects are represented by column vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' In the trajectory, the agent successfully reaches the target destination, with a comprehensive understanding of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' It first picks 6 waypoint views from images and builds ERG from semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Then from the graph updated, the agent gains the attention of the objects in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' From the attention map we can see that ERG shows strong cross-modal matching ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' 5 Conclusion In this paper, we focus on the agent’s ability to understand the environment for VLN-CE and propose a new environ- ment representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' First, we introduce semantic informa- tion to construct an ERG based on object detection results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Then, GCN is used to learn the relational representation of object-object and object-agent in ERG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Finally, the environ- ment representation is obtained by combining the ERG with object label embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' In order to embed the ERG into the 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content="0 O'OJ 0'05 E0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content="0 0'04 0'02navigation, a novel cross-modal attention navigation frame- work is proposed with a loss dedicated for ERG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Experiments along with further analysis validate the cross-modal matching and strong generalization ability of our proposed environment representation and the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' References [Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=', 2018] Peter Anderson, Qi Wu, Damien Teney, Jake Bruce, Mark Johnson, Niko S¨underhauf, Ian Reid, Stephen Gould, and Anton van den Hengel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' Vision- and-language navigation: Interpreting visually-grounded navigation instructions in real environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfKQmi/content/2301.04352v1.pdf'} +page_content=' In Proceed- ings of the IEEE Conference on Computer Vision and Pat- tern Recognition (CVPR), pages 3674–3683, 2018.' metadata={'source': 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0000000000000000000000000000000000000000..e2b7fa3ea8f5d59930b24cb40aa1d9c6b9f748d6 --- /dev/null +++ b/oNE1T4oBgHgl3EQfOwM_/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c4b5d962a7dd152c82a0ac2bf38eebf5bd57ee783760a4d25358b8e40d8c0dfd +size 120025 diff --git a/oNFMT4oBgHgl3EQf6zEt/content/tmp_files/2301.12461v1.pdf.txt b/oNFMT4oBgHgl3EQf6zEt/content/tmp_files/2301.12461v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7e5c326a80cfc93b3f84b846eaddf9d173ce7d3a --- /dev/null +++ b/oNFMT4oBgHgl3EQf6zEt/content/tmp_files/2301.12461v1.pdf.txt @@ -0,0 +1,1223 @@ +arXiv:2301.12461v1 [eess.SY] 29 Jan 2023 +Stochastic Wasserstein Gradient Flows +using Streaming Data with an Application +in Predictive Maintenance +Nicolas Lanzetti ∗ Efe C. Balta ∗ Dominic Liao-McPherson ∗ +Florian D¨orfler ∗ +∗ Automatic Control Laboratory, ETH Z¨urich, Physikstrasse 3, 8092 +Z¨urich, Switzerland. +Abstract: We study estimation problems in safety-critical applications with streaming data. +Since estimation problems can be posed as optimization problems in the probability space, +we devise a stochastic projected Wasserstein gradient flow that keeps track of the belief of +the estimated quantity and can consume samples from online data. We show the convergence +properties of our algorithm. Our analysis combines recent advances in the Wasserstein space +and its differential structure with more classical stochastic gradient descent. We apply our +methodology for predictive maintenance of safety-critical processes: Our approach is shown to +lead to superior performance when compared to classical least squares, enabling, among others, +improved robustness for decision-making. +Keywords: Wasserstein gradient flows, streaming data, predictive maintenance +1. INTRODUCTION +Providing performance guarantees for parameter estima- +tion algorithms operating with streaming data is a key +challenge when developing methods for safety-critical ap- +plications across various domains of engineering and data +science. Ideally, one should be able to (i) efficiently handle +streaming data in real time, without resorting to com- +putationally expensive one-shot numerical routines, and +(ii) rigorously quantify the uncertainty related to the es- +timated quantity. In this paper, we focus on probabilistic +approaches to uncertainty quantification rather than set- +based ones (Combettes, 1993). +A prominent approach for parameter estimation with +streaming data is Recursive Least Squares (RLS). In RLS, +the online solution is obtained by “updating” the previous +solution with the latest measurement. RLS avoids the +need to store and invert large data matrices and provides +probabilistic guarantees on its estimate when the process +is linear and all distributions are Gaussian. RLS is a +special case of Bayes filter (see e.g., S¨arkk¨a (2013); Sullivan +(2015)), whose many variants (e.g., particle filters, ex- +tended Kalman filters, etc.) are the dominant approaches +for inference using non-Gaussian distributions. Bayes filter +is powerful but inflexible, it can be challenging to integrate +side information into the filter and it can be difficult to +implement due to the need to compute high-dimensional +integrals. +In this work, we propose a different approach based +on the theory of optimization in Wasserstein probability +spaces (Jordan et al., 1998; Ambrosio et al., 2005; Lanzetti +⋆ e-mails:{lnicolas,ebalta,dliaomc,dorfler}@ethz.ch. +This re- +search is supported by the Swiss National Science Foundation +through NCCR Automation (Grant Number 180545). +et al., 2022). We pose the parameter estimation problem +as an optimization problem in the probability space and +devise a stochastic projected gradient flow to iteratively +compute its optimal solution using samples obtained from +streaming data. Our approach maintains and iteratively +improves an estimate of the probability measure of the esti- +mated quantities and does not require a-priori assumptions +on the probability measures (e.g., Gaussianity), but rather +works in the space of all probability measures with finite +second moment. Our proposed framework is more flexible +than the Bayes filter in the sense that its intuitive to add +side information e.g., constraints on the support of the +final distribution or on the variance (indeed, it can be used +to recover maximum likelihood estimator for stochastic +least squares problems (Rigollet and Weed, 2018)). +A motivating application of interest is Predictive Main- +tenance (PdM), where the goal is to efficiently maintain +a safety-critical process (e.g., with minimal interruption) +before an unsafe event occurs (see Pech et al. (2021) for a +recent survey of results). The PdM problem is challenging +from an online algorithmic perspective since, in practice, +there is often only historical data on the nominal operation +and little or no data on the unsafe operation. Moreover, +the problem calls for careful risk analysis: Too conservative +decisions impact performance and efficiency, while unsafe +events, if they occur, might lead to catastrophic failures. +The current state-of-the-art consists of rule-based meth- +ods and estimation strategies that rely on predetermined +distribution models (Hu and Chen, 2020; Kanso et al., +2022). With our work, we learn the model of the underlying +process without a-priori assumptions on its probability +measure to improve the overall performance by reducing +conservativeness. + +Our contributions are twofold. First, we propose a novel +stochastic projected gradient flow for optimization in the +probability space that operates on streaming data and +study its convergence properties. Our analysis combines +tools from optimal transport and differential calculus +in the probability space with more classical projected +stochastic gradient descent. We prove that similar to the +Euclidean setting, our scheme yields convergence to a +ball around the optimal solution. Second, we apply our +scheme to the predictive maintenance of the damping +ratio of a second-order system and demonstrate improved +performance relative to a classical least-squares approach. +2. BACKGROUND +In this section, we briefly review our notation, basics of +measure theory and optimal transport, geodesic convexity, +and Wasserstein gradients. For more details, we refer +the reader to Villani (2009); Ambrosio et al. (2005); +Santambrogio (2015); Lanzetti et al. (2022). +Notation +We consider the Euclidean space Rd, with +the usual Euclidean norm ∥·∥. For a matrix A ∈ Rm×n, +we denote by σmin(A) and by σmax(A) its minimum and +maximum singular value, respectively. If m = n, we use +the notation λmin(A) and by λmax(A) for the minimum +and maximum eigenvalue of A. +Basics in Measure Theory +We denote by P(Rd) the +space of (Borel) probability measures over Rd and by +P2(Rd) := {µ ∈ P(Rd) : +� +Rd ∥x∥2dµ(x) < +∞} the space +of probability measures with finite second moment. We +denote the Dirac measures at x ∈ Rd by δx, defined by +δx(A) = 1 if and only if x ∈ A. We denote the support +of a probability measure µ ∈ P2(Rd) by supp(µ) ⊂ Rd. +The pushforward of a measure µ ∈ P(Rd) via a (Borel) +map T : Rd → Rd is denoted by T#µ and defined by +(T#µ)(B) = µ(T −1(B)) for all B ⊂ Rd Borel. For any +f : Rd → R, T#µ-integrable it holds +� +Rd f(x)d(T#µ)(x) = +� +Rd f(T (x))dµ(x). +Moreover, a sequence of probability measures (µn)n∈N ⊂ +P(Rd) converges narrowly to µ ∈ P(Rd) if +� +Rd φ(x)dµ(x) → +� +Rd φ(x)dµ(x) for all continuous and bounded functions +φ : Rd → R. +Wasserstein distance +The (type 2) Wasserstein distance +between two probability measures µ ∈ P(Rd) and ν ∈ +P(Rd) is defined by +W2(µ, ν) := +� +min +γ∈Γ(µ,ν) +� +Rd×Rd ∥x − y∥2dγ(x, y) +� 1 +2 +, +where Γ(µ, ν) is the set of transport plans, that is, of +probability measures on Rd × Rd whose first marginal is µ +and second marginal is ν; i.e., Γ(µ, ν) = {γ ∈ P(Rd ×Rd) : +(proj1)#γ = µ, (proj2)#γ = ν} where proj1 and proj2 are +projection operators (e.g., proj1(x, y) = x). We denote by +Γo(µ, ν) the (non-empty) set of optimal couplings between +µ and ν. It is well-known that the Wasserstein distance is +a distance on P2(Rd). +Geodesic convexity +Given µ0 ∈ P(Rd) and µ1 ∈ P(Rd), +we define the geodesic between them by µt = ((1 − +t) proj1 +t proj2)#γ, where γ ∈ Γo(µ0, µ1) is an optimal +transport plan between µ0 and µ1. Since optimal transport +plans are generally not unique, there might exist multiple +geodesics between µ0 and µ1. Accordingly, a functional +J : P2(Rd) → R is α-geodesically convex if for all µ0, µ1 ∈ +P2(Rd) there exists a geodesic µt so that J(µt) ≤ (1 − +t)J(µ0)+tJ(µ1)− α +2 t(1−t)W2(µ0, µ1)2. For instance, µ �→ +Eµ [V ] is α-geodesically convex if and only if V : Rd → R +is α-convex (i.e., convex with convexity parameter α) and +µ �→ Varµ [xi] is geodesically convex (with α = 0), where +Varµ [xi] denotes the variance of xi. +Wasserstein gradient +A function ∇µJ(µ) ∈ L2(Rd, Rd; µ) +is a Wasserstein gradient of a real-valued functional over +the probability space J : P2(Rd) → R if it approximates J +“linearly”; i.e., for all γ ∈ Γo(µ, ν) +J(ν)−J(µ) = +� +Rd ∇µJ(µ)(x)⊤(y−x)dγ(x, y)+o(W2(µ, ν)), +where o(W2(µ, ν)) denotes high-order term. Wasserstein +gradients are well-defined for many functionals of practical +interest. In particular, we have ∇µEµ [V ] = ∇V for any +smooth V : Rd → R with at most quadratic growth (i.e., +the Wasserstein gradient of an expected value is simply +the gradient of the function in the expected value) and +∇ Varµ [xi] = 2(xi−Eµ [xi]). For the Wasserstein gradients +of more functionals, we refer to Lanzetti et al. (2022). +3. STOCHASTIC PROJECTED GRADIENT +DESCENT IN PROBABILITY SPACES +We construct our estimation method by encoding our +objectives in an optimization problem and adapting a +gradient descent algorithm to operate using samples from +the system obtained with streaming data. Consider the +optimization problem +inf +µ∈P2(Rd)J(µ) +s.t. +supp(µ) ⊂ Θ, +(1) +we seek to minimize a real-valued lower semi-continuous 1 +function J : P2(Rd) → R over the probability space sub- +ject to a support constraint. The functional J can encode +standard expected values of real-valued quantities, but also +other costs such as the variance, the Wasserstein distance +from a reference probability measure, or the Kullback- +Leibler divergence. We impose the following assumption +on (1): +Assumption 3.1. The set Θ ⊂ Rd is closed and convex. +Since we only have access to streaming data, we cannot +evaluate J and its Wasserstein gradient ∇µJ exactly. +Thus, we solve (1) via a stochastic projected gradient +descent, where at each iteration k ∈ N we have access +to an unbiased noisy estimate of the Wasserstein gradient +of J and we leverage projections to enforce the support +constraint. More specifically, our scheme reads +µ(k + 1) = projsupp⊂Θ [(Id −τξk)#µ(k)] +µ(0) = µ0 ∈ P2(Rd), +(2) +where Id is the identity map on Rd, ξk is an unbiased +estimate of the Wasserstein gradient, i.e., +1 Here, lower semi-continuity is intended with respect to the conver- +gence induced by the Wasserstein distance. + +E [ξk] = ∇µJ(µ(k)), +τ ∈ R>0 is a step size, and projsupp⊂Θ [·] denotes the +projection (w.r.t. to the Wasserstein distance) onto the +set of probability measures with support contained in Θ. +Later, we demonstrate how we construct our gradient +estimate ξk using streaming data. +We impose the following assumption on our noisy gradi- +ents: +Assumption 3.2. (Finite second moment). The estimate of +the gradient has bounded variance. In particular, there +exists σ > 0 and C > 0 so that +E +� +∥ξ∥2 +L2(Rd,Rd;µ) +� +≤ σ2 + C(J(µ) − J(µ∗)). +This assumption is mild: It stipulates that the second +moment of the norm of the gradient at µ is controlled +by the suboptimality of µ. Whenever it is uniformly (in µ) +upper bounded, Assumption 3.2 holds trivially. +3.1 Projections in the Wasserstein Space +Our proposed algorithm includes a projection onto the set +of probability measures with support in Θ, denoted by +projsupp⊂Θ [·], which is defined by +projsupp⊂Θ[µ] = arg min +¯µ∈P2(R) +W2(µ, ¯µ) +s.t. supp(¯µ) ⊂ Θ. +Our next result states the projection of a probability +measure onto the set of probability measures with support +contained in Θ is (i) well-defined and (ii) results from +pushforward of µ via the projection operator projΘ : Rd → +Θ on Rd. Intuitively, we can thus compute projections by +“projecting every (infinitesimal) particle of µ to Θ”: +Proposition 3.1. (Projections). Let Assumption 3.1 hold. +Then, projΘ [·] is well-defined and for all µ ∈ P2(Rd) +projsupp⊂Θ [µ] = (projΘ[·])# µ. +(3) +Since every point x ∈ Rd can be embedded to a probability +measure δx ∈ P2(Rd), Assumption 3.1 is necessary for the +existence of a unique projection. Indeed, if it fails to hold, +then the projection operator is ill-defined even on Rd. In +Proposition 3.1, we show that it is also sufficient. +3.2 Convergence Analysis +We now study the convergence properties of the itera- +tion (2). Similarly to Euclidean settings, the stochastic +projected Wasserstein gradient descent (2) converges to a +(Wasserstein) ball centered at the optimal solution of (1): +Theorem 3.2. (Convergence). Let J : P2(Rd) → R be +Wasserstein differentiable and α-geodesically convex with +convexity parameter α > 0, let Assumptions 3.1 and 3.2 +hold, let µ∗ ∈ P2(Rd) be the optimal solution of (1), and +let τ ∈ (0, min{1/α, 2/C}). Then, for all k ∈ N +E{ξj}k +j=0 � +W2(µk+1, µ∗)2� +≤ (1 − τα)k +� +W2(µ0, µ∗)2 − τσ2 +α +� ++ τσ2 +α , +(4) +In particular, +1) we have +lim sup +k→∞ +E{ξj}k +j=1 [W2(µk+1, µ∗)] ≤ +� +τσ2 +α , +(5) +2) +lim sup +k→∞ +E +�� +∥mk+1 − m∗∥2 + +���S1/2 +k+1 − (S∗)1/2 +��� +2 +F +� +≤ +� +τσ2 +α , +(6) +where the expectation is taken w.r.t. {ξj}k +j=0, ∥A∥F +denotes of the Frobenius norm of A ∈ Rd×d, mk and +m∗ are the mean of µk and µ∗, respectively, and Sk +and S∗ are their covariance matrices, +3) for any L-Lipschitz continuous function ϕ : Rd → R, +we have +lim sup +k→∞ +E{ξj}k +j=0 +����∥ϕ∥L2(Rd,R,µ)−∥ϕ∥L2(Rd,R,µ∗) +��� +� +≤ L +� +τσ2 +α . +(7) +where the L2 norm w.r.t. a probability measure ν is +defined as +∥ϕ∥2 +L2(Rd,R,ν) = +� +Rd ϕ(x)2dν(x). +We can specialize our results to the noise-free case, simply +setting σ = 0. This way, we recover the convergence +properties of Wasserstein gradient flows, studied, e.g., +in Ambrosio et al. (2005): +Corollary 3.3. (Noise-free case). Let σ = 0. Then, +lim +k→∞ W2(µk, µ∗) = 0. +Our results predicate convergence in expectation to a +Wasserstein ball. This conclusion is in line with stan- +dard stochastic gradient descent; e.g., see Bottou et al. +(2018). Furthermore, the iterates not only convergence to +a Wasserstein ball but also provide practically relevant in- +formation if the generated solution (µk)k∈N is subsequently +used for prediction or estimation purposes. In particular, +one can deploy recent results in uncertainty propagation +(Aolaritei et al., 2022) to study how Wasserstein balls +propagate through prediction processes or leverage the +framework of distributionally robust optimization to eval- +uate the worst-case risk over Wasserstein balls (Moha- +jerin Esfahani and Kuhn, 2018; Blanchet and Murthy, +2019; Gao and Kleywegt, 2022). +4. ESTIMATION WITH STREAMING DATA +We next specialize our scheme (2) to a meaningful special +case and illustrate how it can be applied to problems with +streaming data. +We assume access to a stream of data {yk} generated by +the process +yk = Wθ∗ + wk +(8) +where W ∈ Rd×d is the known process matrix, θ∗ ∈ Θ is +the parameter we would like to estimate, and wk is zero- +mean uncorrelated noise with finite variance, probability +measure ν, and support W ⊂ Rd. We pose the following +parameter estimation problem + +inf +µ∈P2(Θ) J(µ) :=1 +2 +� +W +� +Θ +∥Wθ∗ + w − Wθ∥2dµ(θ)dν(w) ++ ρ +2 Varµ [θ1 + . . . + θd] , +s.t. +supp(µ) ⊂ Θ, +(9) +where ρ > 0. In words, µ is a probability measure over +estimators θ: We penalize the expected estimation error +and a regularization term accounting for high variance, +and we impose that the estimator lies in a set Θ. If we +could solve this problem (i.e., we had access to all data in a +batch), then we would obtain a Dirac probability measure +at the least squares estimator (provided that it lies in Θ). +Nonetheless, since we only have access to online streaming +data, we need to compute the solution iteratively. We +impose mild assumptions on the noise as well as some +structure in the linear model W: +Assumption 4.1. (Noise). The noise w is zero-mean and +has finite variance σ2 +w > 0. +Assumption 4.2. (Invertible linear model). The matrix W +is invertible. +Intuitively, Assumption 4.1 allows us to show that the +second moment of the stochastic gradient is well-behaved +(cf. Assumption 3.2), which allows us to deploy Theo- +rem 3.2. Assumption 4.2, instead, is required to ensure +strong (geodesic) convexity of the objective function. +Our estimation problem (9) involves the unknown true +parameters θ∗ and cannot be solved directly. Instead, we +derive a data-driven algorithm using {yk}. To start, we +show that the Wasserstein gradient of J is well-defined, +derive an expression for computing it, and show that +Assumption 3.2, required for Theorem 3.2, holds true: +Lemma 4.1. (Wasserstein gradients). Let Assumption 4.1 +hold. The Wasserstein gradient of J reads +∇µJ(µ)(θ) = W ⊤W (θ − θ∗) + ρ (θ − Eµ [θ]) +Moreover, +ξ(θ, ˆy) = W ⊤ (Wθ − ˆy) + ρ (θ − Eµ [θ]) , +(10) +is an unbiased estimate of ∇µJ(µ) so that +E [ξ] = ∇µJ(µ) +(11) +and +E +� +∥ξ∥L2(Rd,Rd;µ) +� +≤4 max{σmax(W)2, ρ} (J(µ) − J(µ∗)) ++ 4 max{σmax(W)2, ρ}σ2. +(12) +Remark 4.1. We can perturb ξ via any function f of θ and +of a random parameter ζ which satisfies E [f(ζ, θ)] = 0 +for all θ ∈ supp(µ), and still obtain an unbiased estimate +of the Wasserstein gradient. Of course, this will increase +its second moment, which imposes a re-evaluation of the +upper bound (12). +We solve (9) using {yk} via the following stochastic gradi- +ent descent iteration: +µ(k + 1) = projsupp⊂Θ [(Id −τξk)#µ(k)] += (projΘ[Id +τξk])# µ(k). +(13) +where ξk = ξ(θ, yk) is our streaming data based estimate +of ∇µJ. +The convergence analysis for (13) is a corollary of Theo- +rem 3.2. +Corollary 4.2. (Convergence). Let Assumptions 3.1, 4.1, +and 4.2 hold. Let τ ∈ (0, 1/(2 max{σmax(W)2, ρ})), and +let (µk)k∈N ⊂ P2(Rd) be the sequence generated by (13) +and µ∗ be the optimal solution. Then, +E{ξj}k +j=0 � +W2(µk+1, µ∗)2� +≤ (1−σmin(W)2τ)k � +W2(µ0, µ∗)2−τησ2 +w +� ++τησ2 +w, +(14) +where η := 4 max{σmax(W)2, ρ}/σmin(W)2. In particular: +1) lim supk→∞ E{ξj}k +j=0 [W2(µk+1, µ∗)] ≤ σw√ητ, +2) with mk and m∗ being the mean of µk and µ∗, +respectively, and Sk and S∗ being their covariance +matrices, we have +lim sup +k→∞ +E{ξj}k +j=0 +�� +∥mk+1 − m∗∥2+ +���S1/2 +k+1−(S∗)1/2 +��� +2 +F +� +≤ σw +√ητ, +where ∥A∥F denotes of the Frobenius norm of A ∈ +Rd×d, +3) for any L-Lipschitz continuous function ϕ : Rd → R. +lim sup +k→∞ +E{ξj}k +j=0 +����∥ϕ∥L2(Rd,R,µ) − ∥ϕ∥L2(Rd,R,µ∗) +��� +� +≤ Lσw +√ητ. +5. PREDICTIVE MAINTENANCE OF THE +DAMPING RATIO +Consider the second-order system +¨z + a ˙z + b(z − r + ε) = 0, +(15) +where a, b ∈ R are parameters, ε is measurement noise +with bounded variance and r ∈ R is a reference signal. +Our goal is to monitor the damping ratio +ζ := +a +2 +√ +b +(16) +and ensure that it does not fall below a safe lower bound +ζmin ∈ R. This leads to the following safe set of values for +(a, b): +Qsafe = +� +(a, b) ∈ R2 : +a +2 +√ +b +≥ ζmin +� +⊂ R2. +The parameters a and b vary slowly with time according +to the equation +y(t) = +� +a(t) +b(t) +� += +� +a0 − λ1t +b0 + λ2t +� +, +(17) +where t ≥ 0 is the amount of time that has passed +since the system was last maintained, θ = (λ1, λ2) ∈ +R2 are unknown decay parameters, and a0, b0 ∈ R are +known constants. At time t = 0, i.e., immediately after +maintenance, the parameters a and b belong to the safe set; +i.e., (a0, b0) ∈ Qsafe. The coefficients λ1 and λ2 are positive +(i.e., Θ = R2 +≥0) so that ζ(t) decreases with time and the +system will eventually exit the safe set. The behavior of +the system as it decays is illustrated in Figure 1. +We are interested in deciding when to perform mainte- +nance on the system (15), which resets t = 0 and the +parameters a and b. Performing maintenance is expensive +and it is desirable to do it as infrequently as possible while +still ensuring that a(t) and b(t) remain in Qsafe. Ideally we +would always maintain the system at time +t⋆ = sup{t ≥ 0 : y(t) ∈ Qsafe}. +(18) + +0 +20 +40 +60 +80 +100 +−2 +−1 +0 +1 +2 +3 +Time [s] +Position +Day 1, ζ ≈ 1.25 +Day 30, ζ ≈ 0.43 +Day 60, ζ ≈ 0.12 +0 +20 +40 +60 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +Days since +maintenance +Damping ratio [–] +Fig. 1. As the damping ratio decays over time (right), the +response of the system becomes increasingly oscilla- +tory (left). +However, in practice θ is unknown and we cannot directly +measure y, hence we must infer θ from noisy data, use +it to estimate t⋆, and account for the uncertainty in our +estimation in our decision-making process. +5.1 Estimation +The parameters y cannot be directly measured but must +be estimated based on trajectories of the system (15). +For a fixed value of t (remember that a(t) and b(t) vary +slowly relative to the dynamics of (15)) applying Euler +discretization with a sampling period ∆t > 0 to (15) yields +the discrete-time system +xk+1 = +� +1 +∆t +−∆tb 1 − ∆ta +� +xk + +� +0 +∆tb +� +(r + εk), +(19) +where the state is x = (z, ˙z), which we can rewrite as +xk+1 = A(y)xk + B(y)(rk + εk). +For sufficiently small ∆t, if a, b > 0 then (15) is ro- +bustly stable about z = r. We then measure trajectories +{ˆxk, rk}N +k=0 of (19) and estimate y(t) using the least- +squares estimator +ˆy(t) = arg min +y +N−1 +� +k=0 +∥ˆxk+1 − A(y)ˆxk − B(y)rk∥2. +(20) +The presence of the noise term ε in (19) introduces noise +in the estimator (20). Thus, in practice, we obtain noisy +measurements +ˆy(t) = +� +a(t) +b(t) +� += +� +a0 − λ1t +b0 + λ2t +� ++ w(t) +where the noise term w(t) ∈ R2 is uncorrelated in time and +is assumed to have bounded variance. Note that, in this +case, the noise w might not be not zero-mean, as it results +from the propagation of ε through the argmin in (20). In +our numerical case study, we generate trajectories of 100 +seconds with sampling time ∆t = 0.001s and suppose εk +is uniform between −3 and +3. +5.2 Probabilistic Predictive Maintenance +To ensure robustness and careful decision-making, we +adopt a probabilistic approach and encapsulate our belief +about θ = (λ1, λ2) in a probability distribution µ ∈ P2(R2) +that will enable us to quantify our uncertainty about θ. +This opens the floor to stochastic and (distributionally) +robust decision-making; e.g., with νt denoting the proba- +bility distribution of y(t), we can use a chance constraint +t⋆ = sup {t ≥ 0 : Pνt[Qsafe] = Px∼νt[x ∈ Qsafe] ≥ 1 − α} +for some confidence level α ∈ (0, 1) or rely on the mean +prediction +t⋆ = sup {t ≥ 0 : Eνt [x] = Ex∼νt [x] ∈ Qsafe} . +Each day k, we obtain degradation data {ˆyk, tk} from +the system. To put our PDM problem (17) in the form +of (9) we consider the difference between two consecutive +measurements, happening every tk+1 − tk = T > 0 time +units, and obtain a new measurement function +˜y = y(t + T ) − y(t) = +� +−λ1T +λ2T +� ++ +� +˜w1 +˜w2 +� +, +for which we have data {ˆyk+1 − ˆyk}. Thus we have W = +diag(−T, T ). The noise ˜w is zero-mean, since E [ ˜wi] = +E [wi(t + T ) − wi(t)] = 0, and has variance +Var [ ˜wi] = Var [wi(t + T )] + Var [wi(t)] =: σ2 +w, +and the parameters are known to lie in the set Θ = R2 +≥0 +which defines the support or µ. +To obtain a practical implementation, we implement (13) +using particles, i.e., in the setting of probability mea- +sures that have finitely many samples of form µ(k) = +1 +N +�N +i=1 δxi with {xi}N +i=1 ⊂ Rd. In this work, N = 1000. +In this case, the update equation for probability mea- +sures (13) simplifies to +µ(k + 1) = 1 +N +N +� +i=1 +δprojΘ[xi−τξk(xi,yk)]. +(21) +where τ > 0 is a step size and ξ is the unbiased estimator +of the Wasserstein gradient from Lemma 4.1. That is, +one can simply pushforward all “particles” of µ(k) and +then project them individually to Θ. It suffices therefore +to keep track of the location xi ∈ Rd of every particle +i ∈ {1, . . . , N}. In particular, the update rule is then +xi �→ projΘ[xi − τξk(xi, yk)], which can be evaluated in +parallel for each particle i. +5.3 Numerical results +We use the true values a0 = 2.5, b0 = 1 (known) and +λ1 = 2/60, λ2 = 5/60 (unknown). We initialize the par- +ticles’ position via uniform sampling in [0, 8/60]2 ⊂ R2. +We weigh the variance with ρ = 0.1 and consider T = 5 +days. At each time step of the algorithm, (i) we collect an +estimate of y as described in Section 5.1, (ii) we run one +iterate of our gradient descent scheme, with τ sufficiently +small and an additional zero-mean noise term (Gaussian +with standard deviation 0.02) in the Wasserstein gradient +(cf. Remark 4.1). Thereafter, we use the probability mea- +sure to construct a confidence interval for the damping, +which can be used to schedule maintenance. For instance, +Figure 2 shows the confidence interval constructed at day +15, compared against a classical least-squares estimate. +This way, we can predict maintenance. We collect the +predictive maintenance time at each iteration in Figure 3. +As can be seen, our approach has superior performance +than the classic least squares, as it readily enables robust +decision-making, which consistently leads to safe estimates +of the maintenance time. + +10 +15 +20 +25 +30 +35 +40 +45 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Time [day] +Predicted damping ratio [–] +True +Observed +Our mean prediction +Our estimate +LS prediction +Fig. 2. Prediction of the evolution of the damping ratio +constructed at day 15, alongside with its standard +least squares prediction and the true evolution of the +damping ratio. For our estimate, we construct the +confidence interval via the 10% percentile and the 90% +percentile. The vertical lines highlight the intersection +of the curves with the threshold ζmin = 0.4. The green +area denotes safe operation (ζ ≥ ζmin) and the red +area denotes unsafe operation (ζ < ζmin). +10 +15 +20 +25 +30 +35 +40 +45 +15 +20 +25 +30 +35 +40 +Time [day] +Suggested maintenance time [day] +True +Our estimate +LS estimate +Fig. 3. Predicted maintenance time at each time step. +Here, classical least squares fails to predict the main- +tenance time. Our algorithm, instead, robustly sug- +gests to schedule maintenance a few days in advance. +The green area denotes safe operation (ζ ≥ ζmin) and +the red area denotes unsafe operation (ζ < ζmin). +6. CONCLUSIONS +In this work, we present a novel stochastic Wasserstein +gradient flow method to efficiently perform estimation in +probability spaces with streaming data. Our formal results +provide a convergence analysis of our online stochastic +optimization method, which provides convergence to a +ball around the optimal solution, similar to the standard +Euclidean setting. We illustrate the utility of the proposed +method in an application of predictive maintenance to +show the benefit over classical approaches such as simple +least-squares with Gaussianity assumptions. Overall, our +method provides a flexible online estimation tool to esti- +mate a rich set of processes without any assumptions on +the model of the underlying distribution. Future work will +consider providing further results under relaxed settings +such as non-strong convexity and applications to real- +world physical examples. +REFERENCES +Ambrosio, L., Gigli, N., and Savar´e, G. (2005). Gradient +flows: in metric spaces and in the space of probability +measures. Springer Science & Business Media. +Aolaritei, L., Lanzetti, N., Chen, H., and D¨orfler, F. +(2022). Uncertainty propagation via optimal transport +ambiguity sets. arXiv preprint arXiv:2205.00343. +Blanchet, J. and Murthy, K. (2019). Quantifying distribu- +tional model risk via optimal transport. 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Entropic optimal trans- +port is maximum-likelihood deconvolution. +Comptes +Rendus Mathematique, 356(11-12), 1228–1235. +Santambrogio, F. (2015). Optimal transport for applied +mathematicians. Birk¨auser, NY, 55(58-63), 94. +S¨arkk¨a, S. (2013). Bayesian filtering and smoothing. 3. +Cambridge university press. +Sullivan, T.J. (2015). Introduction to uncertainty quantifi- +cation, volume 63. Springer. +Villani, C. (2009). Optimal transport: old and new, volume +338. Springer. + +Appendix A. PROOFS +A.1 Proof of Proposition 3.1 +Proof. We split the proof in three parts. +Existence: Since the Wasserstein is lower semi-continuous +(w.r.t. narrow convergence), and has compact (w.r.t. nar- +row convergence) level sets, it suffices to prove that the +set {ν ∈ P(R) : supp(ν) ⊂ Θ} ⊂ P(Rd) is closed +(w.r.t. narrow convergence). Let (νn)n∈N ⊂ P(Rd) so that +supp(νn) ⊂ Θ for all n ∈ N, and assume that (νn)n∈N +converges narrowly to ν ∈ P(Rd). We seek to prove that +supp(ν) ⊂ Θ. For m ∈ N, let fm(x) be continuous and +bounded so that (i) fm(x) = 1 if x ∈ Θ and (ii) converges +pointwise to 1Θ(x) as m → ∞. As Θ is closed, such fm +can always be constructed. Since supp(νn) ⊂ Θ for every +m ∈ N and every n ∈ N +� +Rd fm(x)dνn(x) = +� +Θ +fm(x)dνn(x) = νn(Θ) = 1. +Moreover, as fm is continuous and bounded, for any fixed +m ∈ N the definition of narrow convergence gives +1 = lim +n→∞ +� +Rd fm(x)dνn(x) = +� +Rd fm(x)dν(x). +We can now deploy dominated convergence (|fm| is uni- +formly dominated by an ν-integrable function) to conclude +1 = lim +m→∞ +� +Rd fm(x)dν(x) = +� +Rd lim +m→∞ fm(x)dν(x) += +� +Rd 1Θ(x)dν(x) = ν(Θ). +Thus, supp(ν) ⊂ Θ, and the set {ν ∈ P(Rd) : supp(ν) ⊂ +Θ} is closed (w.r.t. narrow convergence), as desired. +Uniqueness: Assume two non-equal projections ¯µ1, ¯µ2 ∈ +P2(Rd) exist. Let γ1 ∈ Γo(¯µ, ¯µ1) and γ2 ∈ Γo(¯µ, ¯µ2), +and let µ1/2 be the generalized geodesics with µ1/2 = +� 1 +2 proj2 + 1 +2 proj3 +� +# γ, where γ ∈ Γ(¯µ, ¯µ1, ¯µ2) ⊂ P(Rd × +Rd × Rd) results from gluing γ1 and γ2 (via the Gluing +Lemma, e.g., Lemma 5.3.2 in Ambrosio et al. (2005); more +generally, see Chapter 9.2 in Ambrosio et al. (2005) for an +introduction to generalized geodesics. Since γ1 and γ2 are +both concentrated on Θ × Θ, γ must also have support in +Θ × Θ × Θ. Thus, +supp(µ1/2) ⊂ +�1 +2 proj2 +1 +2 proj3 +� +(Θ × Θ × Θ) += 1 +2Θ ⊕ 1 +2Θ = Θ, +where the last equality follows from the convexity of Θ and +the definition of Minkovsky sum of sets. This shows that +µ1/2 is feasible. The squared Wasserstein distance from µ +is known to be 2-convex along this geodesic. Thus, +W2(µ1/2, ¯µ)2 ≤ 1 +2W2(¯µ0, ¯µ)2+ 1 +2W2(µ1, ¯µ)− 1 +4W2(¯µ0, ¯µ1)2 += W2(¯µ0, ¯µ)2 − 1 +4W2(¯µ0, ¯µ1)2. +Since ¯µ0 ̸= ¯µ1, W2(¯µ0, ¯µ1) > 0. However, this implies +W2(µ1/2, ¯µ) < W2(¯µ0, ¯µ), which contradicts optimality of +¯µ0 and ¯µ1. +Equation (3): We will use Kantorovich duality. Let ν ∈ +P2(Rd) with supp(ν) ⊂ Θ and let 1Θ(x) be 0 in Ω and +∞ +outside. Clearly, 1Θ is zero ν-a.e., and so � +Rd 1Θ(x)dν(x) = +0 for any ν ∈ A. Thus, Kantorovich duality (e.g., Chapter +5 in Villani (2009)), with f c(y) = supx∈Rd f(x)−∥x − y∥2, +gives +W2(ν, µ)2 ≥ +� +Rd 1Θ(x)dν(x) − +� +Rd(1Θ)c(y)dµ(y) += − +� +Rd sup +x∈Rd 1Θ(x) − ∥x − y∥2dµ(x) += +� +Rd inf +x∈Θ ∥x − y∥2dµ(x). +(A.1) +Moreover, projΘ[·] is trivially a (possibly suboptimal) +transport map from µ to (projΘ[·])#µ. Thus, +W2((projΘ[·])#µ, µ)2 ≤ +� +Rd ∥x − projΘ[x]∥2dµ(x) += +� +Rd inf +x∈Θ ∥x − y∥2dµ(x), +(A.2) +where the last equality follows from the definition of +projection. We can now combine (A.1) and (A.2) to +conclude that for all ν ∈ P2(Rd) with supp(ν) ⊂ Θ +W2(ν, µ) ≥ W2((projΘ[·])#µ, µ). +Since supp((projΘ[·])#µ) ⊂ Θ and projections are unique, +we establish (3). This concludes the proof. +A.2 Proof of Theorem 3.2 +Proof. We start with the proof of (4). The other state- +ments then follow. Let γ ∈ Γo(µk, µ∗), where µ∗ is well- +defined; indeed, if J is α-geodesically convex with α > 0 +and lower semi-continuous w.r.t. the convergence induced +by the Wasserstein distance, a unique minimizer exists +(e.g., see Section 11.2 in Ambrosio et al. (2005)). Then, +Eξk � +W2(µk+1, µ∗)2� +♣= Eξk � +W2((projΘ) ◦ (Id −τξk)#µk, µ∗)2� +♥ +≤ Eξk +�� +Rd ∥θk − θ∗∥2 +d +� +(projΘ ◦(Id −τξk)) × Id)# γ +� +(θk, θ∗) +� +□= Eξk +�� +Rd ∥projΘ[θk − τξk(θk)] − θ∗∥2dγ(θk, θ∗) +� +△ +≤ Eξk +�� +Rd ∥θk − τξk(θk) − θ∗∥2dγ(θk, θ∗) +� += Eξk +�� +Rd ∥θk − θ∗∥2dγ(θk, θ∗) +� ++ τ 2Eξk +�� +Rd ∥ξk(θk)∥2dγ(θk, θ∗) +� +− 2τEξk +�� +Rd ⟨θk − θ∗, ξk(θk)⟩ dγ(θk, θ∗) +� +♠ +≤ W2(µk, µ∗)2 + τ2Eξk +�� +Rd ∥ξk(θk)∥2dµk(θk) +� +− 2τEξk +�� +Rd ⟨θk − θ∗, ξk(θk)⟩ dγ(θk, θ∗) +� +≤ W2(µk, µ∗)2 + τ2Eξk +�� +Rd ∥ξk(θk)∥2dµk(θk) +� +− 2τ +� +Rd +� +θk − θ∗, Eξk [ξk(θk)] +� +dγ(θk, θ∗) + +≤ W2(µk, µ∗)2 + τ 2Eξk +�� +Rd ∥ξk(θk)∥2dµk(θk) +� +− 2τ +� +Rd ⟨θk − θ∗, ∇µJ(µ)(θk)⟩ dγ(θk, θ∗) +♦ +≤ W2(µk, µ∗)2 + τ 2(σ2 + C(J(µk) − J(µ∗))) ++ 2τ +� +J(µ∗) − J(µk) − α +2 W2(µk, µ∗)2� +≤ (1 − ατ)W2(µk, µ∗)2 + τ 2σ2 ++ (J(µk) − J(µ∗))(Cτ 2 − 2τ), +where in ♣ we used the definition of µk+1; in ♥ we used +that (projΘ ◦(Id −τξk)×Id)#γ is a (possibly sub-optimal) +transport plan between (projΘ ◦(Id −τξk))#µk and µ∗ (by +Lemma 3.3 Aolaritei et al. (2022)), i.e., +(projΘ ◦(Id−τξk) × Id)#γ ∈ Γ((projΘ◦(Id−τξk))#µk, µ∗) +is candidate (but generally suboptimal) plan for the +Wasserstein distance between (projΘ ◦(Id −τξk))#µk and +µ∗; in □ we used +� +gdf#µ = +� +g ◦ fdµ; in △ we used +non-expansivness of the projection operator (together with +projΘ[θ∗] = θ∗ for all θ∗ ∈ supp(µ∗) ⊂ Θ); ♠ follows +from the definition of γ; and in ♦ we used properties of +Wasserstein gradients of α-convex functionals (e.g., see +Proposition 2.8 in Lanzetti et al. (2022)), together with +Assumption 3.2. Since by assumption τ ≤ 2/C, Cτ 2 − +2τ ≤ 0, and +Eξk � +W2(µk+1, µ∗)2� +≤ (1 − ατ)W2(µk, µ∗)2 + τ 2σ2. +We can now proceed iteratively to obtain +E{ξj}k +j=0 � +W2(µk+1, µ∗)2� +≤ (1 − τα)k +� +W2(µ0, µ∗)2 − σ2τ +α +� ++ σ2τ +α , +This establishes (4). +We now prove (5). By assumption 0 < τ < 1/α, 1 − +ατ ∈ (0, 1), and so the limit k → ∞, (4) gives +lim sup +k→∞ +E{ξj}k +j=0 � +W2(µk+1, µ∗)2� +≤ τσ2 +α . +By Jensen inequality, together with continuity and mono- +tonicity of x �→ x2, we have +� +lim sup +k→∞ +E{ξj}k +j=0 [W2(µk+1, µ∗)] +�2 += lim sup +k→∞ +� +E{ξj}k +j=0 [W2(µk+1, µ∗)] +�2 +≤ lim sup +k→∞ +E{ξj}k +j=0 � +W2(µk+1, µ∗)2� +≤ τσ2 +α . +Monotonicity x �→ √x establishes (5). +We now prove (6), it suffices to observe that, in virtue of +Gelbrich’s bound (Gelbrich (1990)), we have +W2(µ, ν) ≥ +� +∥mµ − mν∥2 + +���S1/2 +µ +− S1/2 +ν +��� +2 +with mµ and and Sµ (mν and Sν) being the mean and +covariance matrices of µ (ν). Thus, (6) follows from (5). +Finally, (7). follows from Villani (2009, Proposition 7.29), +observing that Rd is locally compact and replacing ϕ via +|ϕ|. This concludes the proof. +A.3 Proof of Corollary 3.3 +Proof. The proof follows directly from Theorem 3.2, +together with the well-known fact convergence in the +Wasserstein distance is equivalent to weak convergence in +P2(Rd); see Chapter 6 in Villani (2009). +A.4 Proof of Lemma 4.1 +Proof. We prove the statements separately. +The proof of (10) follows from Section 2 in Lanzetti et al. +(2022). To prove (11) observe +E [ξk] = E +� +W ⊤ (Wθ − ˆyi) + ρ (θ − Eµ [θ]) +� += W ⊤ (Wθ − E [ˆyi]) + ρ (θ − Eµ [θ]) += W ⊤ (Wθ − Wθ∗) + ρ (θ − Eµ [θ]) = ∇µJ(µ). +For the proof of (12), observe that +E +� +∥ξ∥2 +L2(Rd,Rd;µ) +� += E +���W ⊤(Wθ − Wθ∗ − w) + ρ (θ − Eµ [θ]) +��2 +L2(Rd,Rd;µ) +� +≤ 2E +�� +Θ +��W ⊤ (Wθ − Wθ∗ − w) +��2dµ(θ) +� ++ 2ρ2 +� +Θ +∥θ − Eµ [θ]∥2dµ(θ) +≤ 4σmax(W)2 1 +2E +�� +Θ +∥Wθ − Wθ∗ − w∥2dµ(θ) +� ++ 4ρρ +2 +� +Θ +∥θ − Eµ [θ]∥2dµ(θ) +≤ 4 max{σmax(W)2, ρ}J(µ), +where we used the definition of J for the last inequality. +Thus, +E +� +∥ξ∥2 +L2(Rd,Rd;µ) +� +≤ 4 max{σmax(W)2, ρ} (J(µ) − J(µ∗) + J(µ∗)) . +≤ 4 max{σmax(W)2, ρ} +� +J(µ) − J(µ∗) + +� +W +∥w∥2dν(w) +� += 4 max{σmax(W)2, ρ} (J(µ) − J(µ∗)) ++ 4 max{σmax(W)2, ρ}σ2. +This concludes the proof. +A.5 Proof of Corollary 4.2 +Proof. We just need to evaluate the convexity parameters +of J and σ2, C, as defined in Assumption 3.2. Since +µ �→ Eµ [f] is α-geodesically convex if and only if f +is α-convex, we have that J is geodesically convex with +α = λmin(W ⊤W) = σmin(W)2. Moreover, by Lemma 4.1, +we have +σ2 = 4 max{σmax(W)2, ρ}σ2 +w +C = 4 max{σmax(W)2, ρ}. +Then, the result follows from Theorem 3.2, as +σ2τ +α += 4 max{σmax(W)2, ρ}σ2 +wτ +σmin(W)2 += ησ2 +wτ. +Also, α ≤ 2C. Thus, the condition on τ simplifies to τ ∈ +(0, 1/(2 max{σmax(W)2, ρ})). This concludes the proof. + diff --git a/oNFMT4oBgHgl3EQf6zEt/content/tmp_files/load_file.txt b/oNFMT4oBgHgl3EQf6zEt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a52d29c319cb7a4dfbe73f5b319c34beefbcbba4 --- /dev/null +++ b/oNFMT4oBgHgl3EQf6zEt/content/tmp_files/load_file.txt @@ -0,0 +1,599 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf,len=598 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='12461v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='SY] 29 Jan 2023 Stochastic Wasserstein Gradient Flows using Streaming Data with an Application in Predictive Maintenance Nicolas Lanzetti ∗ Efe C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Balta ∗ Dominic Liao-McPherson ∗ Florian D¨orfler ∗ ∗ Automatic Control Laboratory, ETH Z¨urich, Physikstrasse 3, 8092 Z¨urich, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Abstract: We study estimation problems in safety-critical applications with streaming data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Since estimation problems can be posed as optimization problems in the probability space, we devise a stochastic projected Wasserstein gradient flow that keeps track of the belief of the estimated quantity and can consume samples from online data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We show the convergence properties of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Our analysis combines recent advances in the Wasserstein space and its differential structure with more classical stochastic gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We apply our methodology for predictive maintenance of safety-critical processes: Our approach is shown to lead to superior performance when compared to classical least squares, enabling, among others, improved robustness for decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Keywords: Wasserstein gradient flows, streaming data, predictive maintenance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' INTRODUCTION Providing performance guarantees for parameter estima- tion algorithms operating with streaming data is a key challenge when developing methods for safety-critical ap- plications across various domains of engineering and data science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Ideally, one should be able to (i) efficiently handle streaming data in real time, without resorting to com- putationally expensive one-shot numerical routines, and (ii) rigorously quantify the uncertainty related to the es- timated quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' In this paper, we focus on probabilistic approaches to uncertainty quantification rather than set- based ones (Combettes, 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' A prominent approach for parameter estimation with streaming data is Recursive Least Squares (RLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' In RLS, the online solution is obtained by “updating” the previous solution with the latest measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' RLS avoids the need to store and invert large data matrices and provides probabilistic guarantees on its estimate when the process is linear and all distributions are Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' RLS is a special case of Bayes filter (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', S¨arkk¨a (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Sullivan (2015)), whose many variants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', particle filters, ex- tended Kalman filters, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=') are the dominant approaches for inference using non-Gaussian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Bayes filter is powerful but inflexible, it can be challenging to integrate side information into the filter and it can be difficult to implement due to the need to compute high-dimensional integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' In this work, we propose a different approach based on the theory of optimization in Wasserstein probability spaces (Jordan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Ambrosio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Lanzetti ⋆ e-mails:{lnicolas,ebalta,dliaomc,dorfler}@ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' This re- search is supported by the Swiss National Science Foundation through NCCR Automation (Grant Number 180545).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We pose the parameter estimation problem as an optimization problem in the probability space and devise a stochastic projected gradient flow to iteratively compute its optimal solution using samples obtained from streaming data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Our approach maintains and iteratively improves an estimate of the probability measure of the esti- mated quantities and does not require a-priori assumptions on the probability measures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', Gaussianity), but rather works in the space of all probability measures with finite second moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Our proposed framework is more flexible than the Bayes filter in the sense that its intuitive to add side information e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', constraints on the support of the final distribution or on the variance (indeed, it can be used to recover maximum likelihood estimator for stochastic least squares problems (Rigollet and Weed, 2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' A motivating application of interest is Predictive Main- tenance (PdM), where the goal is to efficiently maintain a safety-critical process (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', with minimal interruption) before an unsafe event occurs (see Pech et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (2021) for a recent survey of results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The PdM problem is challenging from an online algorithmic perspective since, in practice, there is often only historical data on the nominal operation and little or no data on the unsafe operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Moreover, the problem calls for careful risk analysis: Too conservative decisions impact performance and efficiency, while unsafe events, if they occur, might lead to catastrophic failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The current state-of-the-art consists of rule-based meth- ods and estimation strategies that rely on predetermined distribution models (Hu and Chen, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Kanso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' With our work, we learn the model of the underlying process without a-priori assumptions on its probability measure to improve the overall performance by reducing conservativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Our contributions are twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' First, we propose a novel stochastic projected gradient flow for optimization in the probability space that operates on streaming data and study its convergence properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Our analysis combines tools from optimal transport and differential calculus in the probability space with more classical projected stochastic gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We prove that similar to the Euclidean setting, our scheme yields convergence to a ball around the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Second, we apply our scheme to the predictive maintenance of the damping ratio of a second-order system and demonstrate improved performance relative to a classical least-squares approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' BACKGROUND In this section, we briefly review our notation, basics of measure theory and optimal transport, geodesic convexity, and Wasserstein gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' For more details, we refer the reader to Villani (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Ambrosio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Santambrogio (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Lanzetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Notation We consider the Euclidean space Rd, with the usual Euclidean norm ∥·∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' For a matrix A ∈ Rm×n, we denote by σmin(A) and by σmax(A) its minimum and maximum singular value, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' If m = n, we use the notation λmin(A) and by λmax(A) for the minimum and maximum eigenvalue of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Basics in Measure Theory We denote by P(Rd) the space of (Borel) probability measures over Rd and by P2(Rd) := {µ ∈ P(Rd) : � Rd ∥x∥2dµ(x) < +∞} the space of probability measures with finite second moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We denote the Dirac measures at x ∈ Rd by δx, defined by δx(A) = 1 if and only if x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We denote the support of a probability measure µ ∈ P2(Rd) by supp(µ) ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The pushforward of a measure µ ∈ P(Rd) via a (Borel) map T : Rd → Rd is denoted by T#µ and defined by (T#µ)(B) = µ(T −1(B)) for all B ⊂ Rd Borel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' For any f : Rd → R, T#µ-integrable it holds � Rd f(x)d(T#µ)(x) = � Rd f(T (x))dµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Moreover, a sequence of probability measures (µn)n∈N ⊂ P(Rd) converges narrowly to µ ∈ P(Rd) if � Rd φ(x)dµ(x) → � Rd φ(x)dµ(x) for all continuous and bounded functions φ : Rd → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Wasserstein distance The (type 2) Wasserstein distance between two probability measures µ ∈ P(Rd) and ν ∈ P(Rd) is defined by W2(µ, ν) := � min γ∈Γ(µ,ν) � Rd×Rd ∥x − y∥2dγ(x, y) � 1 2 , where Γ(µ, ν) is the set of transport plans, that is, of probability measures on Rd × Rd whose first marginal is µ and second marginal is ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', Γ(µ, ν) = {γ ∈ P(Rd ×Rd) : (proj1)#γ = µ, (proj2)#γ = ν} where proj1 and proj2 are projection operators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', proj1(x, y) = x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We denote by Γo(µ, ν) the (non-empty) set of optimal couplings between µ and ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' It is well-known that the Wasserstein distance is a distance on P2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Geodesic convexity Given µ0 ∈ P(Rd) and µ1 ∈ P(Rd), we define the geodesic between them by µt = ((1 − t) proj1 +t proj2)#γ, where γ ∈ Γo(µ0, µ1) is an optimal transport plan between µ0 and µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Since optimal transport plans are generally not unique, there might exist multiple geodesics between µ0 and µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Accordingly, a functional J : P2(Rd) → R is α-geodesically convex if for all µ0, µ1 ∈ P2(Rd) there exists a geodesic µt so that J(µt) ≤ (1 − t)J(µ0)+tJ(µ1)− α 2 t(1−t)W2(µ0, µ1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' For instance, µ �→ Eµ [V ] is α-geodesically convex if and only if V : Rd → R is α-convex (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', convex with convexity parameter α) and µ �→ Varµ [xi] is geodesically convex (with α = 0), where Varµ [xi] denotes the variance of xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Wasserstein gradient A function ∇µJ(µ) ∈ L2(Rd, Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' µ) is a Wasserstein gradient of a real-valued functional over the probability space J : P2(Rd) → R if it approximates J “linearly”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', for all γ ∈ Γo(µ, ν) J(ν)−J(µ) = � Rd ∇µJ(µ)(x)⊤(y−x)dγ(x, y)+o(W2(µ, ν)), where o(W2(µ, ν)) denotes high-order term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Wasserstein gradients are well-defined for many functionals of practical interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' In particular, we have ∇µEµ [V ] = ∇V for any smooth V : Rd → R with at most quadratic growth (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', the Wasserstein gradient of an expected value is simply the gradient of the function in the expected value) and ∇ Varµ [xi] = 2(xi−Eµ [xi]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' For the Wasserstein gradients of more functionals, we refer to Lanzetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' STOCHASTIC PROJECTED GRADIENT DESCENT IN PROBABILITY SPACES We construct our estimation method by encoding our objectives in an optimization problem and adapting a gradient descent algorithm to operate using samples from the system obtained with streaming data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Consider the optimization problem inf µ∈P2(Rd)J(µ) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' supp(µ) ⊂ Θ, (1) we seek to minimize a real-valued lower semi-continuous 1 function J : P2(Rd) → R over the probability space sub- ject to a support constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The functional J can encode standard expected values of real-valued quantities, but also other costs such as the variance, the Wasserstein distance from a reference probability measure, or the Kullback- Leibler divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We impose the following assumption on (1): Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The set Θ ⊂ Rd is closed and convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Since we only have access to streaming data, we cannot evaluate J and its Wasserstein gradient ∇µJ exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Thus, we solve (1) via a stochastic projected gradient descent, where at each iteration k ∈ N we have access to an unbiased noisy estimate of the Wasserstein gradient of J and we leverage projections to enforce the support constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' More specifically, our scheme reads µ(k + 1) = projsupp⊂Θ [(Id −τξk)#µ(k)] µ(0) = µ0 ∈ P2(Rd), (2) where Id is the identity map on Rd, ξk is an unbiased estimate of the Wasserstein gradient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', 1 Here, lower semi-continuity is intended with respect to the conver- gence induced by the Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' E [ξk] = ∇µJ(µ(k)), τ ∈ R>0 is a step size, and projsupp⊂Θ [·] denotes the projection (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' to the Wasserstein distance) onto the set of probability measures with support contained in Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Later, we demonstrate how we construct our gradient estimate ξk using streaming data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We impose the following assumption on our noisy gradi- ents: Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (Finite second moment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The estimate of the gradient has bounded variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' In particular, there exists σ > 0 and C > 0 so that E � ∥ξ∥2 L2(Rd,Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='µ) � ≤ σ2 + C(J(µ) − J(µ∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' This assumption is mild: It stipulates that the second moment of the norm of the gradient at µ is controlled by the suboptimality of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Whenever it is uniformly (in µ) upper bounded, Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2 holds trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1 Projections in the Wasserstein Space Our proposed algorithm includes a projection onto the set of probability measures with support in Θ, denoted by projsupp⊂Θ [·], which is defined by projsupp⊂Θ[µ] = arg min ¯µ∈P2(R) W2(µ, ¯µ) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' supp(¯µ) ⊂ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Our next result states the projection of a probability measure onto the set of probability measures with support contained in Θ is (i) well-defined and (ii) results from pushforward of µ via the projection operator projΘ : Rd → Θ on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Intuitively, we can thus compute projections by “projecting every (infinitesimal) particle of µ to Θ”: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (Projections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Let Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Then, projΘ [·] is well-defined and for all µ ∈ P2(Rd) projsupp⊂Θ [µ] = (projΘ[·])# µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (3) Since every point x ∈ Rd can be embedded to a probability measure δx ∈ P2(Rd), Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1 is necessary for the existence of a unique projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Indeed, if it fails to hold, then the projection operator is ill-defined even on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' In Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1, we show that it is also sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2 Convergence Analysis We now study the convergence properties of the itera- tion (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Similarly to Euclidean settings, the stochastic projected Wasserstein gradient descent (2) converges to a (Wasserstein) ball centered at the optimal solution of (1): Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (Convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Let J : P2(Rd) → R be Wasserstein differentiable and α-geodesically convex with convexity parameter α > 0, let Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2 hold, let µ∗ ∈ P2(Rd) be the optimal solution of (1), and let τ ∈ (0, min{1/α, 2/C}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Then, for all k ∈ N E{ξj}k j=0 � W2(µk+1, µ∗)2� ≤ (1 − τα)k � W2(µ0, µ∗)2 − τσ2 α � + τσ2 α , (4) In particular, 1) we have lim sup k→∞ E{ξj}k j=1 [W2(µk+1, µ∗)] ≤ � τσ2 α , (5) 2) lim sup k→∞ E �� ∥mk+1 − m∗∥2 + ���S1/2 k+1 − (S∗)1/2 ��� 2 F � ≤ � τσ2 α , (6) where the expectation is taken w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' {ξj}k j=0, ∥A∥F denotes of the Frobenius norm of A ∈ Rd×d, mk and m∗ are the mean of µk and µ∗, respectively, and Sk and S∗ are their covariance matrices, 3) for any L-Lipschitz continuous function ϕ : Rd → R, we have lim sup k→∞ E{ξj}k j=0 ����∥ϕ∥L2(Rd,R,µ)−∥ϕ∥L2(Rd,R,µ∗) ��� � ≤ L � τσ2 α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (7) where the L2 norm w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' a probability measure ν is defined as ∥ϕ∥2 L2(Rd,R,ν) = � Rd ϕ(x)2dν(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We can specialize our results to the noise-free case, simply setting σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' This way, we recover the convergence properties of Wasserstein gradient flows, studied, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', in Ambrosio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (2005): Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (Noise-free case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Let σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Then, lim k→∞ W2(µk, µ∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Our results predicate convergence in expectation to a Wasserstein ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' This conclusion is in line with stan- dard stochastic gradient descent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', see Bottou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Furthermore, the iterates not only convergence to a Wasserstein ball but also provide practically relevant in- formation if the generated solution (µk)k∈N is subsequently used for prediction or estimation purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' In particular, one can deploy recent results in uncertainty propagation (Aolaritei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', 2022) to study how Wasserstein balls propagate through prediction processes or leverage the framework of distributionally robust optimization to eval- uate the worst-case risk over Wasserstein balls (Moha- jerin Esfahani and Kuhn, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Blanchet and Murthy, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Gao and Kleywegt, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' ESTIMATION WITH STREAMING DATA We next specialize our scheme (2) to a meaningful special case and illustrate how it can be applied to problems with streaming data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We assume access to a stream of data {yk} generated by the process yk = Wθ∗ + wk (8) where W ∈ Rd×d is the known process matrix, θ∗ ∈ Θ is the parameter we would like to estimate, and wk is zero- mean uncorrelated noise with finite variance, probability measure ν, and support W ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We pose the following parameter estimation problem inf µ∈P2(Θ) J(µ) :=1 2 � W � Θ ∥Wθ∗ + w − Wθ∥2dµ(θ)dν(w) + ρ 2 Varµ [θ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' + θd] , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' supp(µ) ⊂ Θ, (9) where ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' In words, µ is a probability measure over estimators θ: We penalize the expected estimation error and a regularization term accounting for high variance, and we impose that the estimator lies in a set Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' If we could solve this problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', we had access to all data in a batch), then we would obtain a Dirac probability measure at the least squares estimator (provided that it lies in Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Nonetheless, since we only have access to online streaming data, we need to compute the solution iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We impose mild assumptions on the noise as well as some structure in the linear model W: Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (Noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The noise w is zero-mean and has finite variance σ2 w > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (Invertible linear model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The matrix W is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Intuitively, Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1 allows us to show that the second moment of the stochastic gradient is well-behaved (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2), which allows us to deploy Theo- rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2, instead, is required to ensure strong (geodesic) convexity of the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Our estimation problem (9) involves the unknown true parameters θ∗ and cannot be solved directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Instead, we derive a data-driven algorithm using {yk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' To start, we show that the Wasserstein gradient of J is well-defined, derive an expression for computing it, and show that Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2, required for Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2, holds true: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (Wasserstein gradients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Let Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The Wasserstein gradient of J reads ∇µJ(µ)(θ) = W ⊤W (θ − θ∗) + ρ (θ − Eµ [θ]) Moreover, ξ(θ, ˆy) = W ⊤ (Wθ − ˆy) + ρ (θ − Eµ [θ]) , (10) is an unbiased estimate of ∇µJ(µ) so that E [ξ] = ∇µJ(µ) (11) and E � ∥ξ∥L2(Rd,Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='µ) � ≤4 max{σmax(W)2, ρ} (J(µ) − J(µ∗)) + 4 max{σmax(W)2, ρ}σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (12) Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We can perturb ξ via any function f of θ and of a random parameter ζ which satisfies E [f(ζ, θ)] = 0 for all θ ∈ supp(µ), and still obtain an unbiased estimate of the Wasserstein gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Of course, this will increase its second moment, which imposes a re-evaluation of the upper bound (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We solve (9) using {yk} via the following stochastic gradi- ent descent iteration: µ(k + 1) = projsupp⊂Θ [(Id −τξk)#µ(k)] = (projΘ[Id +τξk])# µ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (13) where ξk = ξ(θ, yk) is our streaming data based estimate of ∇µJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The convergence analysis for (13) is a corollary of Theo- rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (Convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Let Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Let τ ∈ (0, 1/(2 max{σmax(W)2, ρ})), and let (µk)k∈N ⊂ P2(Rd) be the sequence generated by (13) and µ∗ be the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Then, E{ξj}k j=0 � W2(µk+1, µ∗)2� ≤ (1−σmin(W)2τ)k � W2(µ0, µ∗)2−τησ2 w � +τησ2 w, (14) where η := 4 max{σmax(W)2, ρ}/σmin(W)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' In particular: 1) lim supk→∞ E{ξj}k j=0 [W2(µk+1, µ∗)] ≤ σw√ητ, 2) with mk and m∗ being the mean of µk and µ∗, respectively, and Sk and S∗ being their covariance matrices, we have lim sup k→∞ E{ξj}k j=0 �� ∥mk+1 − m∗∥2+ ���S1/2 k+1−(S∗)1/2 ��� 2 F � ≤ σw √ητ, where ∥A∥F denotes of the Frobenius norm of A ∈ Rd×d, 3) for any L-Lipschitz continuous function ϕ : Rd → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' lim sup k→∞ E{ξj}k j=0 ����∥ϕ∥L2(Rd,R,µ) − ∥ϕ∥L2(Rd,R,µ∗) ��� � ≤ Lσw √ητ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' PREDICTIVE MAINTENANCE OF THE DAMPING RATIO Consider the second-order system ¨z + a ˙z + b(z − r + ε) = 0, (15) where a, b ∈ R are parameters, ε is measurement noise with bounded variance and r ∈ R is a reference signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Our goal is to monitor the damping ratio ζ := a 2 √ b (16) and ensure that it does not fall below a safe lower bound ζmin ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' This leads to the following safe set of values for (a, b): Qsafe = � (a, b) ∈ R2 : a 2 √ b ≥ ζmin � ⊂ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The parameters a and b vary slowly with time according to the equation y(t) = � a(t) b(t) � = � a0 − λ1t b0 + λ2t � , (17) where t ≥ 0 is the amount of time that has passed since the system was last maintained, θ = (λ1, λ2) ∈ R2 are unknown decay parameters, and a0, b0 ∈ R are known constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' At time t = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', immediately after maintenance, the parameters a and b belong to the safe set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', (a0, b0) ∈ Qsafe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The coefficients λ1 and λ2 are positive (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', Θ = R2 ≥0) so that ζ(t) decreases with time and the system will eventually exit the safe set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The behavior of the system as it decays is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We are interested in deciding when to perform mainte- nance on the system (15), which resets t = 0 and the parameters a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Performing maintenance is expensive and it is desirable to do it as infrequently as possible while still ensuring that a(t) and b(t) remain in Qsafe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Ideally we would always maintain the system at time t⋆ = sup{t ≥ 0 : y(t) ∈ Qsafe}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (18) 0 20 40 60 80 100 −2 −1 0 1 2 3 Time [s] Position Day 1, ζ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='25 Day 30, ζ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='43 Day 60, ζ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='12 0 20 40 60 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2 Days since maintenance Damping ratio [–] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' As the damping ratio decays over time (right), the response of the system becomes increasingly oscilla- tory (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' However, in practice θ is unknown and we cannot directly measure y, hence we must infer θ from noisy data, use it to estimate t⋆, and account for the uncertainty in our estimation in our decision-making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1 Estimation The parameters y cannot be directly measured but must be estimated based on trajectories of the system (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' For a fixed value of t (remember that a(t) and b(t) vary slowly relative to the dynamics of (15)) applying Euler discretization with a sampling period ∆t > 0 to (15) yields the discrete-time system xk+1 = � 1 ∆t −∆tb 1 − ∆ta � xk + � 0 ∆tb � (r + εk), (19) where the state is x = (z, ˙z), which we can rewrite as xk+1 = A(y)xk + B(y)(rk + εk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' For sufficiently small ∆t, if a, b > 0 then (15) is ro- bustly stable about z = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We then measure trajectories {ˆxk, rk}N k=0 of (19) and estimate y(t) using the least- squares estimator ˆy(t) = arg min y N−1 � k=0 ∥ˆxk+1 − A(y)ˆxk − B(y)rk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (20) The presence of the noise term ε in (19) introduces noise in the estimator (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Thus, in practice, we obtain noisy measurements ˆy(t) = � a(t) b(t) � = � a0 − λ1t b0 + λ2t � + w(t) where the noise term w(t) ∈ R2 is uncorrelated in time and is assumed to have bounded variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Note that, in this case, the noise w might not be not zero-mean, as it results from the propagation of ε through the argmin in (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' In our numerical case study, we generate trajectories of 100 seconds with sampling time ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='001s and suppose εk is uniform between −3 and +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2 Probabilistic Predictive Maintenance To ensure robustness and careful decision-making, we adopt a probabilistic approach and encapsulate our belief about θ = (λ1, λ2) in a probability distribution µ ∈ P2(R2) that will enable us to quantify our uncertainty about θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' This opens the floor to stochastic and (distributionally) robust decision-making;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', with νt denoting the proba- bility distribution of y(t), we can use a chance constraint t⋆ = sup {t ≥ 0 : Pνt[Qsafe] = Px∼νt[x ∈ Qsafe] ≥ 1 − α} for some confidence level α ∈ (0, 1) or rely on the mean prediction t⋆ = sup {t ≥ 0 : Eνt [x] = Ex∼νt [x] ∈ Qsafe} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Each day k, we obtain degradation data {ˆyk, tk} from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' To put our PDM problem (17) in the form of (9) we consider the difference between two consecutive measurements, happening every tk+1 − tk = T > 0 time units, and obtain a new measurement function ˜y = y(t + T ) − y(t) = � −λ1T λ2T � + � ˜w1 ˜w2 � , for which we have data {ˆyk+1 − ˆyk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Thus we have W = diag(−T, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The noise ˜w is zero-mean, since E [ ˜wi] = E [wi(t + T ) − wi(t)] = 0, and has variance Var [ ˜wi] = Var [wi(t + T )] + Var [wi(t)] =: σ2 w, and the parameters are known to lie in the set Θ = R2 ≥0 which defines the support or µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' To obtain a practical implementation, we implement (13) using particles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', in the setting of probability mea- sures that have finitely many samples of form µ(k) = 1 N �N i=1 δxi with {xi}N i=1 ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' In this work, N = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' In this case, the update equation for probability mea- sures (13) simplifies to µ(k + 1) = 1 N N � i=1 δprojΘ[xi−τξk(xi,yk)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (21) where τ > 0 is a step size and ξ is the unbiased estimator of the Wasserstein gradient from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' That is, one can simply pushforward all “particles” of µ(k) and then project them individually to Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' It suffices therefore to keep track of the location xi ∈ Rd of every particle i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' In particular, the update rule is then xi �→ projΘ[xi − τξk(xi, yk)], which can be evaluated in parallel for each particle i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='3 Numerical results We use the true values a0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='5, b0 = 1 (known) and λ1 = 2/60, λ2 = 5/60 (unknown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We initialize the par- ticles’ position via uniform sampling in [0, 8/60]2 ⊂ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We weigh the variance with ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1 and consider T = 5 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' At each time step of the algorithm, (i) we collect an estimate of y as described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1, (ii) we run one iterate of our gradient descent scheme, with τ sufficiently small and an additional zero-mean noise term (Gaussian with standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='02) in the Wasserstein gradient (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Thereafter, we use the probability mea- sure to construct a confidence interval for the damping, which can be used to schedule maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' For instance, Figure 2 shows the confidence interval constructed at day 15, compared against a classical least-squares estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' This way, we can predict maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We collect the predictive maintenance time at each iteration in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' As can be seen, our approach has superior performance than the classic least squares, as it readily enables robust decision-making, which consistently leads to safe estimates of the maintenance time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' 10 15 20 25 30 35 40 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='8 Time [day] Predicted damping ratio [–] True Observed Our mean prediction Our estimate LS prediction Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Prediction of the evolution of the damping ratio constructed at day 15, alongside with its standard least squares prediction and the true evolution of the damping ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' For our estimate, we construct the confidence interval via the 10% percentile and the 90% percentile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The vertical lines highlight the intersection of the curves with the threshold ζmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The green area denotes safe operation (ζ ≥ ζmin) and the red area denotes unsafe operation (ζ < ζmin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' 10 15 20 25 30 35 40 45 15 20 25 30 35 40 Time [day] Suggested maintenance time [day] True Our estimate LS estimate Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Predicted maintenance time at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Here, classical least squares fails to predict the main- tenance time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Our algorithm, instead, robustly sug- gests to schedule maintenance a few days in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The green area denotes safe operation (ζ ≥ ζmin) and the red area denotes unsafe operation (ζ < ζmin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' CONCLUSIONS In this work, we present a novel stochastic Wasserstein gradient flow method to efficiently perform estimation in probability spaces with streaming data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Our formal results provide a convergence analysis of our online stochastic optimization method, which provides convergence to a ball around the optimal solution, similar to the standard Euclidean setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We illustrate the utility of the proposed method in an application of predictive maintenance to show the benefit over classical approaches such as simple least-squares with Gaussianity assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Overall, our method provides a flexible online estimation tool to esti- mate a rich set of processes without any assumptions on the model of the underlying distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Future work will consider providing further results under relaxed settings such as non-strong convexity and applications to real- world physical examples.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Optimal transport for applied mathematicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Birk¨auser, NY, 55(58-63), 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' S¨arkk¨a, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Bayesian filtering and smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Cambridge university press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Sullivan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Introduction to uncertainty quantifi- cation, volume 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Villani, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Optimal transport: old and new, volume 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' PROOFS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We split the proof in three parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Existence: Since the Wasserstein is lower semi-continuous (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' narrow convergence), and has compact (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' nar- row convergence) level sets, it suffices to prove that the set {ν ∈ P(R) : supp(ν) ⊂ Θ} ⊂ P(Rd) is closed (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' narrow convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Let (νn)n∈N ⊂ P(Rd) so that supp(νn) ⊂ Θ for all n ∈ N, and assume that (νn)n∈N converges narrowly to ν ∈ P(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We seek to prove that supp(ν) ⊂ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' For m ∈ N, let fm(x) be continuous and bounded so that (i) fm(x) = 1 if x ∈ Θ and (ii) converges pointwise to 1Θ(x) as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' As Θ is closed, such fm can always be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Since supp(νn) ⊂ Θ for every m ∈ N and every n ∈ N � Rd fm(x)dνn(x) = � Θ fm(x)dνn(x) = νn(Θ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Moreover, as fm is continuous and bounded, for any fixed m ∈ N the definition of narrow convergence gives 1 = lim n→∞ � Rd fm(x)dνn(x) = � Rd fm(x)dν(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We can now deploy dominated convergence (|fm| is uni- formly dominated by an ν-integrable function) to conclude 1 = lim m→∞ � Rd fm(x)dν(x) = � Rd lim m→∞ fm(x)dν(x) = � Rd 1Θ(x)dν(x) = ν(Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Thus, supp(ν) ⊂ Θ, and the set {ν ∈ P(Rd) : supp(ν) ⊂ Θ} is closed (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' narrow convergence), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Uniqueness: Assume two non-equal projections ¯µ1, ¯µ2 ∈ P2(Rd) exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Let γ1 ∈ Γo(¯µ, ¯µ1) and γ2 ∈ Γo(¯µ, ¯µ2), and let µ1/2 be the generalized geodesics with µ1/2 = � 1 2 proj2 + 1 2 proj3 � # γ, where γ ∈ Γ(¯µ, ¯µ1, ¯µ2) ⊂ P(Rd × Rd × Rd) results from gluing γ1 and γ2 (via the Gluing Lemma, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2 in Ambrosio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' more generally, see Chapter 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2 in Ambrosio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (2005) for an introduction to generalized geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Since γ1 and γ2 are both concentrated on Θ × Θ, γ must also have support in Θ × Θ × Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Thus, supp(µ1/2) ⊂ �1 2 proj2 +1 2 proj3 � (Θ × Θ × Θ) = 1 2Θ ⊕ 1 2Θ = Θ, where the last equality follows from the convexity of Θ and the definition of Minkovsky sum of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' This shows that µ1/2 is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The squared Wasserstein distance from µ is known to be 2-convex along this geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Thus, W2(µ1/2, ¯µ)2 ≤ 1 2W2(¯µ0, ¯µ)2+ 1 2W2(µ1, ¯µ)− 1 4W2(¯µ0, ¯µ1)2 = W2(¯µ0, ¯µ)2 − 1 4W2(¯µ0, ¯µ1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Since ¯µ0 ̸= ¯µ1, W2(¯µ0, ¯µ1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' However, this implies W2(µ1/2, ¯µ) < W2(¯µ0, ¯µ), which contradicts optimality of ¯µ0 and ¯µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Equation (3): We will use Kantorovich duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Let ν ∈ P2(Rd) with supp(ν) ⊂ Θ and let 1Θ(x) be 0 in Ω and +∞ outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Clearly, 1Θ is zero ν-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', and so � Rd 1Θ(x)dν(x) = 0 for any ν ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Thus, Kantorovich duality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', Chapter 5 in Villani (2009)), with f c(y) = supx∈Rd f(x)−∥x − y∥2, gives W2(ν, µ)2 ≥ � Rd 1Θ(x)dν(x) − � Rd(1Θ)c(y)dµ(y) = − � Rd sup x∈Rd 1Θ(x) − ∥x − y∥2dµ(x) = � Rd inf x∈Θ ∥x − y∥2dµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1) Moreover, projΘ[·] is trivially a (possibly suboptimal) transport map from µ to (projΘ[·])#µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Thus, W2((projΘ[·])#µ, µ)2 ≤ � Rd ∥x − projΘ[x]∥2dµ(x) = � Rd inf x∈Θ ∥x − y∥2dµ(x), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2) where the last equality follows from the definition of projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We can now combine (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2) to conclude that for all ν ∈ P2(Rd) with supp(ν) ⊂ Θ W2(ν, µ) ≥ W2((projΘ[·])#µ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Since supp((projΘ[·])#µ) ⊂ Θ and projections are unique, we establish (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We start with the proof of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The other state- ments then follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Let γ ∈ Γo(µk, µ∗), where µ∗ is well- defined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' indeed, if J is α-geodesically convex with α > 0 and lower semi-continuous w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' the convergence induced by the Wasserstein distance, a unique minimizer exists (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', see Section 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2 in Ambrosio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (2005)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Eξk � W2(µk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' µ∗)2� ♣= Eξk � W2((projΘ) ◦ (Id −τξk)#µk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' µ∗)2� ♥ ≤ Eξk �� Rd ∥θk − θ∗∥2 d � (projΘ ◦(Id −τξk)) × Id)# γ � (θk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' θ∗) � □= Eξk �� Rd ∥projΘ[θk − τξk(θk)] − θ∗∥2dγ(θk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' θ∗) � △ ≤ Eξk �� Rd ∥θk − τξk(θk) − θ∗∥2dγ(θk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' θ∗) � = Eξk �� Rd ∥θk − θ∗∥2dγ(θk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' θ∗) � + τ 2Eξk �� Rd ∥ξk(θk)∥2dγ(θk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' θ∗) � − 2τEξk �� Rd ⟨θk − θ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' ξk(θk)⟩ dγ(θk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' θ∗) � ♠ ≤ W2(µk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' µ∗)2 + τ2Eξk �� Rd ∥ξk(θk)∥2dµk(θk) � − 2τEξk �� Rd ⟨θk − θ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' ξk(θk)⟩ dγ(θk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' θ∗) � ≤ W2(µk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' µ∗)2 + τ2Eξk �� Rd ∥ξk(θk)∥2dµk(θk) � − 2τ � Rd � θk − θ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Eξk [ξk(θk)] � dγ(θk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' θ∗) ≤ W2(µk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' µ∗)2 + τ 2Eξk �� Rd ∥ξk(θk)∥2dµk(θk) � − 2τ � Rd ⟨θk − θ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' ∇µJ(µ)(θk)⟩ dγ(θk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' θ∗) ♦ ≤ W2(µk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' µ∗)2 + τ 2(σ2 + C(J(µk) − J(µ∗))) + 2τ � J(µ∗) − J(µk) − α 2 W2(µk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' µ∗)2� ≤ (1 − ατ)W2(µk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' µ∗)2 + τ 2σ2 + (J(µk) − J(µ∗))(Cτ 2 − 2τ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' where in ♣ we used the definition of µk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' in ♥ we used that (projΘ ◦(Id −τξk)×Id)#γ is a (possibly sub-optimal) transport plan between (projΘ ◦(Id −τξk))#µk and µ∗ (by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='3 Aolaritei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (2022)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', (projΘ ◦(Id−τξk) × Id)#γ ∈ Γ((projΘ◦(Id−τξk))#µk, µ∗) is candidate (but generally suboptimal) plan for the Wasserstein distance between (projΘ ◦(Id −τξk))#µk and µ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' in □ we used � gdf#µ = � g ◦ fdµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' in △ we used non-expansivness of the projection operator (together with projΘ[θ∗] = θ∗ for all θ∗ ∈ supp(µ∗) ⊂ Θ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' ♠ follows from the definition of γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' and in ♦ we used properties of Wasserstein gradients of α-convex functionals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=', see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='8 in Lanzetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (2022)), together with Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Since by assumption τ ≤ 2/C, Cτ 2 − 2τ ≤ 0, and Eξk � W2(µk+1, µ∗)2� ≤ (1 − ατ)W2(µk, µ∗)2 + τ 2σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We can now proceed iteratively to obtain E{ξj}k j=0 � W2(µk+1, µ∗)2� ≤ (1 − τα)k � W2(µ0, µ∗)2 − σ2τ α � + σ2τ α , This establishes (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We now prove (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' By assumption 0 < τ < 1/α, 1 − ατ ∈ (0, 1), and so the limit k → ∞, (4) gives lim sup k→∞ E{ξj}k j=0 � W2(µk+1, µ∗)2� ≤ τσ2 α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' By Jensen inequality, together with continuity and mono- tonicity of x �→ x2, we have � lim sup k→∞ E{ξj}k j=0 [W2(µk+1, µ∗)] �2 = lim sup k→∞ � E{ξj}k j=0 [W2(µk+1, µ∗)] �2 ≤ lim sup k→∞ E{ξj}k j=0 � W2(µk+1, µ∗)2� ≤ τσ2 α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Monotonicity x �→ √x establishes (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We now prove (6), it suffices to observe that, in virtue of Gelbrich’s bound (Gelbrich (1990)), we have W2(µ, ν) ≥ � ∥mµ − mν∥2 + ���S1/2 µ − S1/2 ν ��� 2 with mµ and and Sµ (mν and Sν) being the mean and covariance matrices of µ (ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Thus, (6) follows from (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Finally, (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' follows from Villani (2009, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='29), observing that Rd is locally compact and replacing ϕ via |ϕ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='3 Proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The proof follows directly from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2, together with the well-known fact convergence in the Wasserstein distance is equivalent to weak convergence in P2(Rd);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' see Chapter 6 in Villani (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='4 Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We prove the statements separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' The proof of (10) follows from Section 2 in Lanzetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' To prove (11) observe E [ξk] = E � W ⊤ (Wθ − ˆyi) + ρ (θ − Eµ [θ]) � = W ⊤ (Wθ − E [ˆyi]) + ρ (θ − Eµ [θ]) = W ⊤ (Wθ − Wθ∗) + ρ (θ − Eµ [θ]) = ∇µJ(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' For the proof of (12), observe that E � ∥ξ∥2 L2(Rd,Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='µ) � = E ���W ⊤(Wθ − Wθ∗ − w) + ρ (θ − Eµ [θ]) ��2 L2(Rd,Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='µ) � ≤ 2E �� Θ ��W ⊤ (Wθ − Wθ∗ − w) ��2dµ(θ) � + 2ρ2 � Θ ∥θ − Eµ [θ]∥2dµ(θ) ≤ 4σmax(W)2 1 2E �� Θ ∥Wθ − Wθ∗ − w∥2dµ(θ) � + 4ρρ 2 � Θ ∥θ − Eµ [θ]∥2dµ(θ) ≤ 4 max{σmax(W)2, ρ}J(µ), where we used the definition of J for the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Thus, E � ∥ξ∥2 L2(Rd,Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='µ) � ≤ 4 max{σmax(W)2, ρ} (J(µ) − J(µ∗) + J(µ∗)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' ≤ 4 max{σmax(W)2, ρ} � J(µ) − J(µ∗) + � W ∥w∥2dν(w) � = 4 max{σmax(W)2, ρ} (J(µ) − J(µ∗)) + 4 max{σmax(W)2, ρ}σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='5 Proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' We just need to evaluate the convexity parameters of J and σ2, C, as defined in Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Since µ �→ Eµ [f] is α-geodesically convex if and only if f is α-convex, we have that J is geodesically convex with α = λmin(W ⊤W) = σmin(W)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Moreover, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='1, we have σ2 = 4 max{σmax(W)2, ρ}σ2 w C = 4 max{σmax(W)2, ρ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Then, the result follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content='2, as σ2τ α = 4 max{σmax(W)2, ρ}σ2 wτ σmin(W)2 = ησ2 wτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Also, α ≤ 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' Thus, the condition on τ simplifies to τ ∈ (0, 1/(2 max{σmax(W)2, ρ})).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/oNFMT4oBgHgl3EQf6zEt/content/2301.12461v1.pdf'} diff --git a/odE1T4oBgHgl3EQf1wVk/content/2301.03471v1.pdf b/odE1T4oBgHgl3EQf1wVk/content/2301.03471v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..cbb813f7d4d4cdf421091cef865da12210539361 --- /dev/null +++ b/odE1T4oBgHgl3EQf1wVk/content/2301.03471v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bf86d34dd8327cf8e3c219aaa560b3f448137aa8413e2076765ae9442561a831 +size 1174112 diff --git a/odE1T4oBgHgl3EQf1wVk/vector_store/index.faiss b/odE1T4oBgHgl3EQf1wVk/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..786e887adbab35948681c631489fdb3fb913b1e1 --- /dev/null +++ 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of a linear numerical scheme +approximating nonlinear fluid–structure interactions ∗ +Sebastian Schwarzacher† +Bangwei She‡,† +Karel T˚uma† +January 13, 2023 +Abstract +In this paper, we propose a linear and monolithic finite element method for the approximation +of an incompressible viscous fluid interacting with an elastic and deforming plate. We use the +arbitrary Lagrangian–Eulerian (ALE) approach that works in the reference domain, meaning that +no re-meshing is needed during the numerical simulation. For time discretization, we employ the +backward Euler method. For space discretization, we respectively use P1-bubble, P1, and P1 finite +elements for the approximation of the fluid velocity, pressure, and structure displacement. We +show that our method fulfills the geometrical conservation law and dissipates the total energy on +the discrete level. Moreover, we prove the (optimal) linear convergence with respect to the sizes +of the time step τ and the mesh h. We present numerical experiments involving a substantially +deforming fluid domain that do validate our theoretical results. A comparison with a fully implicit +(thus nonlinear) scheme indicates that our semi-implicit linear scheme is faster and as accurate as +the fully implicit one, at least in stable configurations. +Keywords: fluid-structure interaction, Navier–Stokes equations, stability, error estimates, fi- +nite element method, divergence-free projection +MSC(2010): 35Q30, 76N99, 74F10, 65M12, 65M60 +Contents +1 +Introduction +2 +1.1 +Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2 +Weak formulation and stability +5 +2.1 +Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2.2 +Weak formulation on the reference domain . . . . . . . . . . . . . . . . . . . . . . . . . +7 +2.3 +Energy stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +3 +Numerical method +8 +3.1 +Time discretization. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +3.2 +Discrete Reynolds transport theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +3.3 +Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +3.4 +The numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +4 +Stability +11 +∗All authors thank for the support of the ERC-CZ Grant LL2105 CONTACT, the program GJ19-11707Y of the Czech +national grant agency (GAˇCR) and the Charles University Research program No. UNCE/SCI/023. S. S. and B. S. also +thank the Primus research program PRIMUS/19/SCI/01. +†Department of Analysis, Faculty of Mathematics and Physics, Charles University (schwarz@karlin.mff.cuni.cz, +ktuma@karlin.mff.cuni.cz) +‡Academy for Multidisciplinary Studies, Capital Normal University; Institute of Mathematics of the Czech Academy +of Sciences (she@math.cas.cz). +1 +arXiv:2301.05014v1 [math.NA] 12 Jan 2023 + +5 +Interpolation operators +14 +5.1 +Interpolation operator for the solid deformation . . . . . . . . . . . . . . . . . . . . . . +14 +5.2 +Interpolation operator for the fluid velocity +. . . . . . . . . . . . . . . . . . . . . . . . +15 +6 +Error estimates +19 +6.1 +The time projection +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +6.2 +Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +7 +Numerical experiments +23 +7.1 +Problem description +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +7.2 +Convergence rates +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +7.3 +Comparison between the semi-implicit Scheme-R and fully implicit scheme . . . . . . . +24 +8 +Conclusion +26 +A Appendix: Interpolation operators +29 +B Appendix: Useful equalities and estimates +30 +B.1 +Proof of the error equation (6.5) +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +30 +B.2 +Preliminary estimates +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +31 +B.3 +Secondary estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +33 +B.4 +Proof of estimates (6.10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +C Numerical Implementation +40 +C.1 Implementation of semi-implicit Scheme-R . . . . . . . . . . . . . . . . . . . . . . . . . +40 +C.2 Implementation of fully implicit scheme +. . . . . . . . . . . . . . . . . . . . . . . . . . +41 +1 +Introduction +Fluid–structure interaction (FSI) problems occur in many engineering applications, from aero-elasticity +to civil engineering and bio-mechanical problems, such as the design of aircraft wings, wind turbines +and heat exchangers, the response of bridges and skyscrapers to wind force, blood flow in arteries, see +[5, 6, 29] among others. +Numerical simulation of FSI problems has been largely studied and great progress has been achieved +during the past decades; see, for examples, [4, 8, 18, 20, 24] and references therein. Concerning the +numerical stability analysis, we would like to mention the nice results of Luk´aˇcov´a-Medvid’ov´a et al. +[23], Bukaˇc and Muha [10], Lozovskiy et al. [21, 22], Hecht and Pironneau [17], and Wang et al. [30] +as examples. However, in terms of convergence analysis, there are certainly many more efforts to be +made. To our best knowledge, only a few results are available on this topic; see Bukaˇc and Muha [10], +Burman et al. [11, 12], Fern´andez and Mullaert [15], and Seboldt and Bukaˇc [28]. In this direction, +all available literature results are not only under the assumption that the displacement of the solid +structure is infinitesimal but also based on the ignorance of the convection of the fluid motion. The +main target of this paper is to show the convergence of a numerical approximation without these +restrictions. +For that reason, we study an archetypical setting of fluid-structure interaction. In our setting +a one-dimensional plate is situated on the top of a two-dimensional container filled with a viscous +incompressible liquid governed by the Navier–Stokes equations. The plate is governed by a hyperbolic +equation driven by fluid traction. It may deform largely and therefore the Eulerian fluid domain is +time-changing. This implies a severe nonlinear coupling between the structure and fluid equation. +In order to solve the FSI problem numerically, we introduce a linear, implicit-explicit (semi- +implicit), and monolithic finite element method. For time discretization, we take the backward Euler +method. For space discretization, we start with the so-called arbitrary Lagrangian–Eulerian (ALE) +mapping and transfer the time-dependent domain to a fixed reference grid. Then, we use an inf-sup +stable finite element pair (P1-bubble/P1) on the reference domain for the fluid, and P1 elements for +2 + +the structure. Our aims of the paper are to design an energy stable scheme and to show the (optimal) +convergence rate of the numerical solution. +The key point in the construction of the stability of our linear and semi-implicit scheme is that +we keep the scheme implicit with respect to the velocities. In particular, the velocities of the solid +structure and fluid are coupled implicitly in time, see also a similar construction of Lozovskiy et al. +[21]. Nevertheless, the scheme is linear as we take the fluid domain explicitly, which means it is given +by the deformation of the plate of the previous time-step. Further, the convective term of the fluid is +linearized in a stable manner. +To some extent the current paper can be viewed as a numerical counterpart of Schwarzacher and +Sroczinski [27], where the authors investigated the distance between a weak solution and a strong +solution, while the aim of this paper is to investigate the distance between a numerical solution +obtained by a finite element method and a smooth solution. +In order to adapt this result to a +discrete numerical scheme, rather sophisticated analytic tools have to be invented. +In particular, +good projection operators have to be invented for a smooth solution. The challenge comes from the +change of the fluid domain in time, which results in several non-trivial analytic difficulties on all levels +when studying the convergence rate. Roughly, there are three different sources of errors that have to +be estimated: i) the mismatch between the continuous geometry and the discrete geometry; ii) the +respective different divergence-free constraints; iii) the projector of the fluid-velocity which has to fit a +rather particular choice of a projector according to the structure equation. The first point is overcome +by a change of variables. The second point is already very technical. For that, we introduce a Fortin +operator for variable geometries in order to inherit the discrete solenoidality from the continuous +one. Then the divergence-free condition destroyed by a change of variable is resolved by a Bogovskij +correction recently developed by Kampschulte et al. [19]. The last point, the mismatch between the +interpolation operator of the fluid at the boundary turns out to be the hardest to overcome. The +reason is that the structure equation is of the fourth order in space. Hence, a discrete bi-Laplacian +naturally appears. In order to gain suitable estimates for the structure part, a very particular choice of +projector, the so-called Riesz projection operator, has to be used. Further, we have to solve a discrete +Stokes problem in order to find a suitable projector of the fluid velocity that possesses these particular +boundary values. +The main result of the paper is that there is a monolithic, linear, and fully-discrete scheme of +the FSI problem (1.1)–(1.4), which satisfies under suitable conditions: +• Total energy stability (see Theorem 4.1) +• Linear convergence w.r.t. the space and time discretization parameters (see Theorem 6.2) +The highlight of the paper reads: +• The proposed semi-implicit scheme is linear; the absence of nonlinear iterations makes the +approach computationally cheaper than the fully implicit scheme, yet provides equally +good results. +• We show the energy stability of the method. +• We show the linear convergence rate of the method with respect to the computational +parameters τ (the time step) and h (mesh size). Such a result has not been achieved in +literature for any structure interacting with fluids described by the Navier–Stokes equa- +tions. +• The discrete structure displacement is defining the real-time geometry of the Eulerian +fluid domain. +• The convergence rate is optimal as demonstrated by numerical experiments, see Section 7. +3 + +1.1 +Problem formulation +In this paper, we are interested in the interaction between an incompressible viscous fluid and a thin +elastic structure, which is part of the fluid boundary. More precisely, we consider the motion of an +incompressible and viscous fluid flow in a time-dependent domain +Ωη(t) = {x = (x1, x2) ∈ Σ × (0, η(t, x1))} ⊂ R2, +where Σ = (0, L1), L1 is the length of the domain, η = η(t, x1) > 0 represents the height of the +upper boundary ΓS(t) of the fluid domain Ωη. For the sake of simplicity, we assume that i) the flow +is periodic in the x1-direction; ii) the upper boundary is formed by an elastic structure that can move +in the x2-direction; iii) the bottom boundary ΓD is a solid wall; iv) initially Ωη(0) = Σ × [0, 1]. +In this paper, we shall use the ALE method and directly work on a time-independent reference +domain �Ω = Ωη(0) instead of the time-dependent domain Ωη(t). To this end, we introduce an ALE +mapping Aη that maps the reference domain �Ω to the time dependent domain Ωη, i.e. +Aη : �Ω �→ Ωη, +(�x1, �x2) �→ (x1, x2) = Aη(t, �x) = (�x1, η�x2) , +see Figure 1 for a graphical illustration of the domain and ALE mapping. +�Ω +ΓD +�ΓS = Σ +0 +�x1 +L1 +1 +�x2 +Aη +Ω(t) +ΓD +ΓS(t) +η +0 +x1 +L1 +x2 +Figure 1: Time dependent domain and the ALE mapping +Fluid model. +The motion of the incompressible viscous fluid is described by the Navier–Stokes +equations +� +� +� +� +� +ϱf (∂tu + (u · ∇)u) − div T(u, p) = 0, +in (0, T) × Ωη +div u = 0, +in (0, T) × Ωη, +u = 0, +on (0, T) × ΓD, +(1.1) +where ϱf, u = u(t, x), and p = p(t, x) are the fluid density (given constant), velocity field, and +pressure, respectively. The Cauchy stress T reads1 +T = 2µ(∇u)S − pI +with the constant viscosity coefficient µ > 0, +and the superscript S denotes the symmetric operator for a matrix-valued function A, meaning that +AS = A + AT. +Structure model. +The motion of the elastic structure is given by +� +ϱs∂tξ + L(η) = f, +on (0, T) × Σ, +∂2 +x1η = 0, +on (0, T) × ∂Σ, +(1.2) +where ϱs > 0 is the density of the structure, ξ = ∂tη is the velocity of the structure, f is the interaction +force acting on the structure due to fluid motion, and +L(η) = −γ1∂2 +x1η − γ2∂2 +x1ζ − γ3∂2 +x1∂tη, +ζ = −∂2 +x1η, +where γ1 > 0, γ2 > 0, γ3 ≥ 0 are given constants. +Further, the initial data of the problem read +u(0) = u0 in Ωη(0) +and +η(0, ·) = η0, ξ(0, ·) = ξ0 in Σ. +(1.3) +1We adopt the following notations: (∇u)ij = ∂jui, (v · (∇u))i = �d +j=1 vj∂iuj and +� +v · ∇u +� +i = +� +(v · ∇)u +� +i = +�d +j=1 vj∂jui. Note that v · ∇u ̸= v · (∇u) but v · ∇u = (v · ∇)u = ∇u · v. +4 + +Coupling conditions. +Finally, to close the system, we require coupling conditions at the fluid- +structure interface, which are the so-called kinematic and dynamic boundary conditions: +• the kinematic coupling condition +u(x) = ξ(x1)e2, +∀ x = (x1, η) ∈ ΓS. +(1.4a) +• the dynamic coupling condition +f = −e2 · +� +JT(u, p) ◦ AηF−T � +· e2, +(1.4b) +where F = F(η) is the Jacobian of the mapping Aη and J = J(η) is the corresponding determi- +nant. In the current setting, we have +F(η) = ∇�xAη = +� +1 +0 +�x2∂x1η +η +� +and +J(η) = det(F(η)) = η. +(1.5) +The plan of the paper is the following. +In Section 2 we discuss the weak formulation and +stability of our FSI problem on the continuous level. In Section 3 we introduce the numerical method. +In Section 4 we prove the stability of the numerical solution on the discrete level. In Section 5 we +introduce interpolation operators that are specially designed to fit both the divergence-free velocity +field and the kinematic coupling condition. These operators are essential in Section 6, where we show +the convergence rate of the numerical solution towards a strong solution. In Section 7 we present the +numerical experiments. Finally, in Section 8 we give a short conclusion of the achievements in the +paper. +2 +Weak formulation and stability +In this section, we introduce a weak formulation of the FSI problem (1.1)–(1.4) and prove that a +solution to the weak formulation is energy stable. +To begin, we introduce the standard notations W k,p(D) and Lp(D) on a generic domain D for +the Sobolev space and Lebesgue space, respectively. Further, we denote by W k,p +0 +the functions with +zero traces on the boundary. In order to specify functions on the reference domain, we shall use the +superscript “�”. For example, for a generic function v = v(x) defined on Ωη we write on the reference +domain that �v = v(Aη(�x)) = v ◦ Aη. Next, we recall the Piola transformation [13] for the mapping +Aη: +dx = η d�x, +dS(x) = |ηF−T �n| dS(�x), +n = ηF−T �n +|ηF−T �n|, +div �x(ηF−1) = 0, +div xq ◦ Aη = 1 +ηdiv �x +� +ηF−1�q +� += ∇�x�q : F−T , +∇q ◦ Aη = ∇�x�q F−1, +(2.1) +where dx (resp. dS(x)) is the volume (resp. face) integral in the time-dependent domain, d�x (resp. +dS(�x)) is the volume integral in the reference domain, q and q are generic vector-valued and scalar +functions, respectively. Note that we have emphasized here the dependence of the differential operators +∇ and div with respect to x and �x. Hereinafter, if no confusion occurs, we shall simply write ∇ (resp. +div ) instead of both ∇x and ∇�x (resp. div x and div �x). +Now, we define a new velocity field w that describes the change of the fluid domain (ALE mapping) +in time. It reads +�w(�x) := ∂tAη = (0, ∂tη �x2) +and +w(x) := �w(�x) ◦ A−1 +η (x) = (0, x2 ∂tη/η) . +(2.2) +Then, it is easy to observe the so-called Euler expansion +div w = ∂tη/η. +(2.3) +5 + +According to the chain rule, we have +∂t�v(�x) = d +dtv(Aη(�x)) = ∂tv(x) + ∂tAη(�x) · ∇v(x) += ∂tv(x) + w(x) · ∇v(x) =: ∂M +t v(x), +(2.4) +where ∂M +t +represents a material-type time derivative. +With the above notations, it is easy to check the Reynolds transport theory +∂t +� +Ωη +v dx = +� +Ωη +∂tv dx + +� +∂Ωη +vw · n dS(x) += +� +Ωη +(∂tv + div (vw)) dx = +� +Ωη +� +∂M +t v + vdiv w +� +dx. +(2.5) +Further, we denote v as the relative velocity of the fluid with respect to the fluid domain. It reads +v = u − w +satisfying the boundary condition +v|∂Ωη = 0. +Thanks to the above boundary condition and the incompressibility condition (1.1)2, we observe for +any differentiable test function ϕ that +� +Ωη +� +(v · ∇)u +� +· ϕ dx = +� +Ωη +� +ϕ · (∇u) · v + 1 +2(div v + div w +� +�� +� +=div u=0 +)u · ϕ +� +dx += +� +Ωη +div w u +2 · ϕ dx + 1 +2 +� +Ωη +� +ϕ · (∇u) − u · (∇ϕ) +� +· v dx. +Accordingly, we may reformulate the time derivative and convective terms in the following way +� +Ωη +� +∂tu + (u · ∇)u +� +· ϕ dx = +� +Ωη +� +∂tu + (w · ∇)u +� +· ϕ dx + +� +Ωη +� +(v · ∇)u +� +· ϕ dx += +� +Ωη +� +∂M +t u + div w u +2 +� +· ϕ dx + 1 +2 +� +Ωη +� +ϕ · (∇u) − u · (∇ϕ) +� +· v dx. +(2.6) +Finally, we introduce the following abbreviation for the sake of simplicity +as(η, ζ, ξ, ψ) = +� +Σ +(γ1∂x1η∂x1ψ + γ2∂x1ζ∂x1ψ + γ3∂x1ξ∂x1ψ) dx1 +with ζ = −∂2 +x1η. +(2.7) +2.1 +Weak formulation +Before introducing the weak formulation, we introduce the space of coupled test functions to accom- +modate the no-slip boundary condition (1.4a). +Wη = +� +(ϕ, ψ) ∈ W 1,2(Ωη) × L2(Σ) : ψ(x)e2 = ϕ(x, η(x)), ϕ = 0 on ΓD +� +. +Now we are ready to present the weak formulation of the FSI problem (1.1)–(1.4). +Definition 2.1 (Weak formulation of the FSI problem on Ωη). Let (p, u, ξ, η) be a “solution” to the +FSI problem (1.1)–(1.4). We say the following formula is a weak formulation of the FSI problem. +� +Ωη +qdiv u dx = 0 +for all +q ∈ L2(Ωη); +(2.8a) +ϱf +� +Ωη +� +∂M +t u + div w u +2 +� +· ϕ dx + 1 +2 +� +Ωη +� +ϕ · (∇u) − u · (∇ϕ) +� +· v dx ++ +� +Ωη +T(u, p) : ∇ϕ dx + ϱs +� +Σ +∂tξψ dx1 + as(η, ζ, ξ, ψ) = 0, +(2.8b) +for all (ϕ, ψ) ∈ Wη, where as is given in (2.7), and ζ = −∂2 +x1η. +6 + +Note that (2.8a) is directly obtained by +� +Ωη (1.1)2 × q dx for q ∈ L2(Ωη) and (2.8b) is obtained by +calculating +� +Ωη (1.1)1 × ϕ dx + +� +Σ (1.2)1 × ψ dx1 for the coupled test functions (ϕ, ψ) ∈ Wη, where we +have used the following identity due to the Piola transformation (2.1), the fact that the test functions +ϕ and ψ are coupled, and the coupling condition (1.4b). +� +ΓS +ϕ · T · n dS(x) = +� +Σ +ϕ ◦ Aη · (T ◦ Aη) · (ηF−T e2) dx1 += +� +Σ +ψe2 · (ηT ◦ AηF−T ) · e2 dx1 = − +� +Σ +fψ dx1. +2.2 +Weak formulation on the reference domain +By means of a change of variables, we may reformulate the weak formulation (2.8) from the current +domain Ωη onto the reference domain �Ω. +Lemma 2.2 (Weak formulation of the FSI problem on �Ω). Let (p, u, ξ, η) satisfy the weak formulation +(2.8) of the FSI problem on Ωη with the test functions (q, ϕ, ψ) ∈ L2(Ωη) × Wη. Let (�p, �q, �u, �ϕ) = +(p, q, u, ϕ) ◦ Aη. Then there hold +� +�Ω +�q∇�u : M d�x = 0; +(2.9a) +ϱf +� +�Ω +� +η∂t�u + ∂tη �u +2 +� +· �ϕ d�x + 1 +2ϱf +� +�Ω +� �ϕ · (∇�u) − �u · (∇ �ϕ) +� +· F−1 · �vη d�x ++ +� +�Ω +�T(�u, �p) : +� +∇ �ϕF−1� +η d�x + ϱs +� +Σ +∂tξψ dx1 + as(η, ζ, ξ, ψ) = 0, +(2.9b) +where as is given in (2.7), ζ = −∂2 +x1η, and +�T(�u, �p) = T(u, p) ◦ Aη = 2µ(∇�uF−1)S − �pI, +M := M(η) := ηF−T = +� η +−�x2∂x1η +0 +1 +� +. +(2.10) +Proof. First, recalling (2.4) we know that +∂M +t u ◦ Aη = ∂t�u, +which together with the Euler expansion (2.3) yield +ϱf +� +Ωη +� +∂M +t u + div w u +2 +� +· ϕ dx = ϱf +� +�Ω +� +η∂t�u + ∂tη �u +2 +� +· �ϕ d�x +Next, using the Piola transformation (2.1) we have +� +Ωη +ϕ · (∇u) · v dx = +� +�Ω +�ϕ · (∇�uF−1) · �vη d�x, +which indicates +1 +2ϱf +� +Ωη +� +ϕ · (∇u) − u · (∇ϕ) +� +· v dx = 1 +2ϱf +� +�Ω +� �ϕ · (∇�u) − �u · (∇ �ϕ) +� +· F−1 · �vη d�x. +Analogously, we find +� +Ωη +T : ∇ϕ dx = +� +Ωη +� +2µ(∇u)S − pI +� +: ∇ϕ dx += +� +�Ω +� +2µ(∇�uF−1)S − �pI +� +: +� +∇ �ϕF−1� +η d�x = +� +�Ω +�T : +� +∇ �ϕF−1� +η d�x, +and +� +Ωη +qdiv u dx = +� +�Ω +�q∇�u : F−T η d�x = +� +�Ω +�q∇�u : M d�x. +Consequently, collecting the above equalities we derive (2.9) from (2.8), which completes the proof. +7 + +2.3 +Energy stability +Finally, we are ready to show the stability of the FSI problem (1.1)–(1.4). Indeed, any solution to its +weak formulation (2.8) (or equivalently (2.9)) satisfies the following energy stability. +Lemma 2.3 (Stability of the continuous problem). +Let (p, u, ξ, η) be a solution to (2.8) (or (�p, �u, ξ, η) be a solution to (2.9)). Then the following +energy estimates hold. +∂t +� +Ωη +1 +2ϱf|u|2 dx + ∂t +� +Σ +1 +2 +� +ϱs|ξ|2 + γ1|∂x1η|2 + γ2|∂2 +x1η|2� +dx1 ++ 2µ +� +Ωη +|(∇u)S|2 dx + +� +Σ +γ3|∂x1ξ|2 dx1 = 0. +(2.11) +Proof. First, we derive by setting (q, ϕ, ψ) = (p, u, ξ) in the weak formulation (2.8) that +ϱf +� +Ωη +� +∂M +t +|u|2 +2 ++ |u|2 +2 div w +� +dx + 2µ +� +Ωη +|(∇u)S|2 dx ++ ϱs +� +Σ +∂t +|ξ|2 +2 dx1 + +� +Σ +� +γ1∂x1η∂x1ξ + γ2∂2 +x1η∂2 +x1ξ + γ3|∂x1ξ|2� +dx1 = 0. +Then we finish the proof after noticing ξ = ∂tη and employing the Reynolds transport theory (2.5) +with v = |u|2 +2 . +3 +Numerical method +In this section, we discretize the weak formulation introduced in the last section by a suitable finite +element method. +3.1 +Time discretization. +We start with time discretization. Let τ be the time increment and tk = kτ for k = 0, 1, . . . , N(≡ T/τ). +Then we denote by (pk +h, uk +h, ξk +h, ηk +h) the numerical approximation of the FSI problem at time tk. Further, +for any set of pointwise in time approximation {vk +h}N +k=0 we extend it to the whole time interval [0, T] +in the following way +vh(t) = vk +h in [tk, tk+1). +(3.1) +The discrete Eulerian domain Ωk +ηh = Ωηk +h at time tk is determined by the discrete ALE mapping: +Ak +ηh : +�Ω �→ Ωk +ηh, +Ak +ηh(�x) = (�x1, ηk +h�x2). +Again Aηh is a piecewise constant in time function in the sense of (3.1). For a generic function vh +(including test functions) defined on Ωηh we have �vh = vh ◦Aηh on �Ω. Here we emphasize that �ηh = ηh +and �ξh = ξh as their domain of definition Σ is time independent. +To approximate the time derivatives ∂tv and ∂M +t v we introduce +Dtvk +h = vk +h − vk−1 +h +τ +≈ +∂tvk, +DM +t vk +h = vk +h − vk−1 +h +◦ Xk−1 +k +τ +≈ +∂M +t vk, +(3.2) +where Xj +i = Aj +ηh ◦ (Ai +ηh)−1 denotes the mapping from Ωi +ηh to Ωj +ηh. +8 + +3.2 +Discrete Reynolds transport theory. +Analogous to the continuous definitions (1.5) and (2.2) and the identity (2.3) we have +Fh = F(ηh) = ∇�xAηh = +� +1 +0 +�x2∂x1ηh +ηh +� +, +det(Fh) = ηh. +(3.3) +�wh = DtAηh = (0, Dtηh�x2), wh = �wh ◦ (Aηh)−1 = +� +0, Dtηh +ηh +x2 +� +, +(3.4) +and +div wh = Dtηh +ηh +. +(3.5) +Next, realizing the equality +Dt�vk +h = (DM +t vk +h) ◦ Ak +ηh +we observe the discrete analogue of the Reynolds transport theorem (2.5), see also [26, Lemma 1]. +Lemma 3.1 (Discrete Reynolds transport). +Let Dt and DM +t +be given in (3.2), then for any k = 1, · · · , NT we have +Dt +� +Ωkηh +vk +h dx = +� +Ωkηh +� +DM +t vk +h + div wk +hvk−1 +h +◦ Xk−1 +k +� +dx. +(3.6) +Remark 3.2. Choosing vh = 1Dh in Lemma 3.1 for any Dh ⊂ Ωηh, we have the geometric conservation +law, i.e., +Dt|Dh| = Dt +� +Ωηh +1 dx = +� +Ωηh +div wh dx = +� +∂Dh +wh · n dS(x). +Further, as we keep in our numerical scheme that wh = uh on the boundary, where the velocity field +is weakly divergence-free, we have +Dt|Ωηh| = +� +∂Ωηh +wh · n dS(x) = +� +∂Ωηh +uh · n dS(x) = +� +Ωηh +div uh dx = 0. +Indeed, our method does fulfill the above equality, as our boundary condition is uk−1 +h += wk +h and we use +a flat reference geometry. Therefore +Dt|Ωk +ηh| = +� +∂Ωkηh +wk +h · n dS(x) = +� +Σ +Dtηk +h dx1 = +� +Σ +ξk−1 +h +dx1 = +� +Ωk−1 +ηh +div uk−1 +h += 0, +where the last equality is due to the weakly divergence-free condition (3.13a) with the choice of test +function q = 1. +3.3 +Spatial discretization +Let Th be a shape regular and quasi-uniform triangulation of the reference domain �Ω, where h stands +for the maximum diameter of all elements of Th. +Let Σh be the surface mesh of Th on the top +boundary �ΓS. We denote by K ∈ Th a generic element in Th and by σ ∈ Σh a generic face element in +Σh. Moreover, we introduce the following function spaces on �Ω +�V f +h = +� +�ϕ ∈ W 1,2(�Ω; Rd) +��� �ϕ ∈ P1(K) ⊕ B1(K), ∀K ∈ Th, �ϕ|ΓD = 0 +� +, +(3.7) +B1(K) = +� +φ ∈ P3(K) +��φ(ai) = 0, +where ai, i = 1, 2, 3, are vertices of K ∈ Th +� +, +(3.8) +�Qf +h = +� +�q ∈ C0(�Ω) +����q ∈ P1(K), ∀K ∈ Th +� +, +(3.9) +V s +h = +� +ψ ∈ W 1,2 +0 (Σ) +���ψ ∈ P1(σ), ∀σ ∈ Σh +� +, +(3.10) +9 + +where Pn(K) (resp. Pn(σ)) denote polynomials of degree not greater than n on K (resp. on σ). +Further, we denote +� +Wηh = +� +( �ϕ, ψ) ∈ �V f +h × V s +h +��� �ϕ(�x1, 1) = ψ(�x1)e2 +� +. +Finally, we denote �V fsi +h += �Qf +h × � +Wηh and V fsi +h +(t) = �V fsi +h +◦ A−1 +ηh (t). +Let us point out that by using the linear finite element space V s +h for the structure displacement +η, we cannot directly discretize the bi-Laplacian term. Therefore, we decide to approximate the bi- +Laplacian via duality, which still requires a discrete Laplace operator. To this end, we introduce the +following discrete Laplace operator for ηh ∈ V s +h by seeking ∂2 +x1,hηh ∈ V s +0,h := V s +h ∩ L2 +0(Σ) such that +� +Σ +∂2 +x1,hηh ψ dx1 + +� +Σ +∂x1ηh∂x1ψ dx1 = 0 +for all ψ ∈ V s +h . +(3.11) +Here, we would like to point out that V s +0,h is a finite dimensional space and in view of the assumptions +on the grid the stiffness matrix of (3.11) is invertible. Thus, it admits a unique solution. +3.4 +The numerical method +With the notations introduced above, we propose a monolithic finite element method for the dis- +cretization of the weak formulation (2.9). +Scheme-R(A monolithic finite element method on the reference domain �Ω). +For k = 1, . . . , N we +seek (�pk +h, �uk +h, ξk +h, ηk+1 +h +) ∈ �V fsi +h +× V s +h with ξk +h = Dtηk+1 +h +such that for all (�q, �ϕ, ψ) ∈ �V fsi +h +there hold +� +�Ω +�q∇�uk +h : Mk +h d�x = 0, +(3.12a) +ϱf +� +�Ω +Dt�uk +h · �ϕηk +h d�x + 1 +2ϱf +� +�Ω +Dtηk +h�uk∗ +h · �ϕ d�x ++ 1 +2ϱf +� +�Ω +� �ϕ · (∇�uk +h) − �uk +h · (∇ �ϕ) +� +· (Fk +h)−1 · �vk−1 +h +ηk +h d�x ++ +� +�Ω +�T(�uk +h, �pk +h) : +� +∇ �ϕ(Fk +h)−1� +ηk +h d�x ++ ϱs +� +Σ +Dtξk +hψ dx1 + as(ηk+1 +h +, ζk+1 +h +, ξk +h, ψ) = 0, +(3.12b) +where �uk∗ +h += 2�uk−1 +h +− �uk +h, �vk−1 +h += �uk−1 +h +− �wk +h, Mk +h = M(ηk +h) is given in (2.10), as is given in (2.7), +ζh = −∂2 +x1,hηh is the (minus) discrete Laplace uniquely defined by (3.11), �T is given in (2.10), and +the discrete initial data are given by u0 +h = Pf +hu0, ξ0 +h = Rs +hξ0, η0 +h = Rs +hη0, and η1 +h = η0 +h + τξ0 +h. Here +Pf +h : W 1,2(�Ω) �→ �V f +h is a suitable projection operator and Rs +h : W 1,2(�Ω) �→ V s +h is a Riesz projection +operator to be clarified in the next section. +Note that Scheme-R approximates the FSI problem (1.1)–(1.4) based on the weak formula- +tion (2.9) in the reference domain �Ω. It is linear and belongs to the monolithic approach. Practically, +it is more convenient to work with the reference domain as it is time-independent and no need for +re-meshing. Nevertheless, many researchers appreciate working with the current domain Ωηh (approx- +imation of Ωη). To this end, we present the following equivalent formulation of Scheme-R on the +current domain. +Scheme-C(A monolithic finite element method on the current (push-forward) domain Ωηh). +Given +the initial data (1.3) we set u0 +h = Pf +hu0, ξ0 +h = Rs +hξ0, η0 +h = Rs +hη0, and η1 +h = η0 +h + τξ0 +h. +Then for +k = 1, . . . , N we seek (pk +h, uk +h, ξk +h, ηk+1 +h +) ∈ V fsi +h +×V s +h with ξk +h = Dtηk+1 +h +such that for all (q, ϕ, ψ) ∈ V fsi +h +there hold +� +Ωηk +qdiv uk +h dx = 0, +(3.13a) +10 + +ϱf +� +Ωkηh +� +DM +t uk +h + div wk +h +uk∗ +h +2 +� +· ϕ dx ++ 1 +2ϱf +� +Ωkηh +� +ϕ · (∇uk +h) − uk +h · (∇ϕ) +� +· vk−1 +h +◦ Xk−1 +k +dx ++ +� +Ωkηh +T(uk +h, pk +h) : ∇ϕ dx + ϱs +� +Σ +Dtξk +hψ dx1 + as(ηk+1 +h +, ζk+1 +h +, ξk +h, ψ) = 0, +(3.13b) +where vk−1 +h += uk−1 +h +− wk +h ◦ Xk +k−1, uk∗ +h = 2uk−1 +h +◦ Xk−1 +k +− uk +h, as is given in (2.7), and ζh = −∂2 +x1,hηh is +the (minus) discrete Laplace given by (3.11). +Remark 3.3. +1. We omit the proof on how to identify the equivalence of Scheme-R and Scheme-C as it is +similar to the proof of Lemma 2.2. +2. In Scheme-C (or equivalently Scheme-R) we solve for each time step tk, k ∈ {1, . . . , N}, the +fluid variables (uk +h, pk +h) in an explicit domain Ωk +ηh and solve the structure variable ηk+1 +h +, which +determines the fluid domain Ωk+1 +ηh +of the next time step tk+1. This differs from many monolithic +schemes defined in an implicit domain Ωk +ηh (or their equivalent form in the reference domain) +when ηk +h instead of ηk+1 +h +is unknown at time step tk. Such a kind of solver “time splitting” +helps us to define a linear scheme without destroying the stability of the numerical solutions, see +Theorem 4.1. +Remark 3.4 (On the extension to 3D/2D). Many parts of our analysis are also valid when con- +sidering a three-dimensional fluid domain with a two-dimensional plate attached to it. However, the +regularity of the (approximated) fluid domain is essentially weaker a priori. Observe that if the plate +is two-dimensional, the discrete domain in space can not even be assumed to be uniformly Lipschitz +continuous, as in two dimensions W 2,2 does not embed into Lipschitz functions. +Remark 3.5. Note that we approximate the boundary deformation η with a piecewise linear finite +element space, resulting in a linear ALE mapping and a linear deformation of the fluid domain. +Therefore, the geometry of the fluid domain is automatically captured at every time step, as every +element K ∈ Th is preserved as a triangle. +Let us point out that the fourth order derivative in the structure (due to the bi-Laplacian term) +is avoided by the introduction of a discrete Laplace operator, which maps a piecewise linear function +space into the same space, see (3.11). +4 +Stability +In this section, we show the stability of the Scheme-C (or equivalently Scheme-R). We start with +the following observation by recalling the discrete Laplace operator (3.11). +� +Σ +∂x1ξk +h∂x1(−∂2 +x1,hηk+1 +h +) dx1 = +� +Σ +∂2 +x1,hξk +h∂2 +x1,hηk+1 +h +dx1 = +� +Σ +Dt∂2 +x1,hηk+1 +h +∂2 +x1,hηk+1 +h +dx1 += 1 +2 +� +Σ +� +Dt|∂2 +x1,hηk+1 +h +|2 + τ|Dt∂2 +x1,hηk+1 +h +|2� +dx1 += Dt +� +Σ +1 +2|∂2 +x1,hηk+1 +h +|2 dx1 + τ +2 +� +Σ +|∂2 +x1,hξk +h|2 dx1, +(4.1) +where we have used the algebraic equality +(a − b)a = 1 +2(a2 − b2) + 1 +2(a − b)2. +(4.2) +11 + +Then, recalling (2.7) with the test function ψ = ξh and thanks to (4.1), we find +as(ηk+1 +h +, −∂2 +x1,hηk+1 +h +, ξk +h, ξk +h) += Dt +�γ1 +2 +���∂x1ηk+1 +h +��� +2 +L2(Σ) + γ2 +2 +���∂2 +x1,hηk+1 +h +��� +2 +L2(Σ) +� ++ γ3 +���∂x1ξk +h +��� +2 +L2(Σ) + Dk +s, +where Dk +s = γ1τ +2 +� +Σ +|∂x1ξk +h|2 dx1 + γ2τ +2 +� +Σ +|∂2 +x1,hξk +h|2 dx1 ≥ 0. +(4.3) +Now we are ready to show the energy estimates. +Theorem 4.1 (Energy estimates). Let {(pk +h, uk +h, ξk +h, ηk+1 +h +)}N +k=1 be the solution of Scheme-C +(or equivalently, let {(�pk +h, �uk +h, ξk +h, ηk+1 +h +)}N +k=1 be the solution of Scheme-R). Then we have the +following stability result for all m = 1, . . . , N +Em +h + τ +m +� +k=1 +� +2µ +���(∇uk +h)S��� +2 +L2(Ωkηh) + γ3 +���∂x1ξk +h +��� +2 +L2(Σ) + Dk +num +� += E0 +h +(4.4) +where for any k = 0, . . . , N the total energy Ek +h and the numerical dissipation Dk +num read +Ek +h = ϱf +2 +���uk +h +��� +2 +L2(Ωkηh) + ϱs +2 +���ξk +h +��� +2 +L2(Σ) + γ1 +2 +���∂x1ηk+1 +h +��� +2 +L2(Σ) + γ2 +2 +���∂2 +x1,hηk+1 +h +��� +2 +L2(Σ) +Dk +num = ϱfτ +2 +� +Ωkηh +ηk−1 +h +ηk +h +|DM +t uk +h|2 dx + ϱsτ +2 +���Dtηk+1 +h +��� +2 +L2(Σ) + γ1τ +2 +���∂x1ξk +h +��� +2 +L2(Σ) + γ2τ +2 +���∂2 +x1,hξk +h +��� +2 +L2(Σ) +Proof. We test the numerical method (3.13) by (q, ϕ, ψ) = (pk +h, uk +h, ξk +h) to get +ϱf +� +Ωkηh +� +DM +t uk +h · uk +h + div wk +h +� +uk−1 +h +◦ Xk−1 +k +− uk +h +2 +� +· uk +h +� +dx ++2µ +���(∇uk +h)S��� +2 +L2(Ωkηh) + ϱs +� +Σ +Dtξk +hξk +h dx1 + as(ηk+1 +h +, ζk+1 +h +, ξk +h, ξk +h) = 0. +(4.5) +Next, using the equality (4.2) we get +� +Σ +Dtξk +hξk +h dx1 = 1 +2 +� +Σ +Dt|ξk +h|2 dx1 + τ +2 +� +Σ +|Dtξk +h|2 dx1, +(4.6) +and +� +Ωkηh +� +DM +t uk +h · uk +h + div wk +h +� +uk−1 +h +◦ Xk−1 +k +− uk +h +2 +� +· uk +h +� +dx += +� +Ωkηh +� +DM +t +|uk +h|2 +2 ++ τ +2|DM +t uk +h|2 + div wk +h(uk−1 +h +◦ Xk−1 +k +· uk +h − |uk +h|2 +2 +) +� +dx += Dt +� +Ωkηh +1 +2|uk +h|2 dx + I0, +(4.7) +where we have also used the discrete Reynolds transport formula (3.6). Here, the term I0 reads +I0 = +� +Ωkηh +� +τ +2|DM +t uk +h|2 + div wk +h +� +uk−1 +h +◦ Xk−1 +k +· uk +h − |uk +h|2 +2 +− |uk−1 +h +◦ Xk−1 +k +|2 +2 +�� +dx += +� +Ωkηh +�τ +2 − τ 2 +2 div wk +h +� +|DM +t uk +h|2 dx = τ +2 +� +Ωkηh +ηk−1 +h +ηk +h +|DM +t uk +h|2 dx, +where we have used the (3.5). +12 + +Substituting (4.6) and (4.7) into (4.5) and owing to (4.3), we derive +DtEk +h + 2µ +���(∇uk +h)S��� +2 +L2(Ωkηh) + γ3 +���∂x1ξk +h +��� +2 +L2(Σ) + Dk +num = 0. +(4.8) +Finally, computing τ �m +k=1 (4.8) yields (4.4), which completes the proof. +The above stability estimate can be rewritten in the reference domain as +ϱf +� +�Ω +1 +2ηm +h |�um +h |2 d�x + ϱs +� +Σ +1 +2|ξm +h |2 dx1 + γ1 +2 +��∂x1ηm+1 +h +��2 +L2(Σ) + γ2 +2 +��∂2 +x1,hηm+1 +h +��2 +L2(Σ) ++ 2µτ +m +� +k=1 +� +�Ω +ηk +h|(∇�uk +h(Fk +h)−1)S|2 d�x + γ3τ +m +� +k=1 +���∂x1ξk +h +��� +2 +L2(Σ) ++ τ 2 +2 +m +� +k=1 +ϱf +� +�Ω +ηk−1 +h +|Dt�uk +h|2 d�x + τ 2 +2 +m +� +k=1 +ϱs +� +Σ +|Dtξk +h|2 dx1 ++ γ1τ 2 +2 +m +� +k=1 +� +Σ +|∂x1ξk +h|2 dx1 + γ2τ 2 +2 +m +� +k=1 +� +Σ +|∂2 +x1,hξk +h|2 dx1 += ϱf +� +�Ω +1 +2η0 +h|u0 +h|2 d�x + ϱs +� +Σ +1 +2|ξ0 +h|2 dx1 + γ1 +2 +��∂x1η1 +h +��2 +L2(Σ) + γ2 +2 +��∂2 +x1,hη1 +h +��2 +L2(Σ) , +(4.9) +Note that ηh appears on the left-hand-side (LHS) of the energy estimates (4.4) (see also (4.9)) and +determines if all terms on the LHS of the energy balance are non-negative or not. Therefore, it is +important to preserve the positivity of ηh in order to get a priori estimates. Actually, there exists a +T0 > 0 such that for all T ≤ T0 we have no contact between the upper surface and the bottom surface +of Ωη, see [26, Lemma 5]. More precisely, if ηh(0) > η, for every c there exists a T, such that +ηh ≥ η − c +∀ t ∈ [0, T]. +(4.10) +From Theorem 4.1 and the above assumption, we have the following uniform estimates. +Corollary 4.2. Let the initial data satisfy u0 ∈ W 1,2(Ωη(0); R2), η0 ∈ W 1,2(Σ), and ξ0 ∈ W 1,2(Σ). +Let (ph, uh, ξh, ηh) = {(pk +h, uk +h, ξk +h, ηk+1 +h +)}N +k=1 be a solution to Scheme-C (or equivalently (�ph, �uh, ξh, ηh) +be a solution to Scheme-R) with (τ, h) ∈ (0, 1)2 and let (4.10) hold. Then we have the following +uniform bounds. +∥ξh∥L∞(0,T;L2(Σ)) + ∥∂x1ηh∥L∞(0,T;L2(Σ)) + +��∂2 +x1,hηh +�� +L∞(0,T;L2(Σ)) +<∼ 1, +��η−1 +h +�� +L∞((0,T)×Σ) + ∥ηh∥L∞((0,T)×Σ) + ∥∂x1ηh∥L∞((0,T)×Σ) +<∼ 1, +∥Fh∥L∞((0,T)×Σ;R2×2) + +��F−1 +h +�� +L∞((0,T)×Σ;R2×2) +<∼ 1, +∥�uh∥L∞(0,T;L2(�Ω)) + +��(∇�uh(Fh)−1)S�� +L2((0,T)×�Ω) +<∼ 1, +∥∇�uh∥L2((0,T)×�Ω) +<∼ 1, +∥�uh∥L2(0,T;Lq1(�Ω)) +<∼ 1, +∥ξh∥L2(0,T;L∞(Σ)) +<∼ ∥∇�uh∥L2((0,T)×�Ω) +<∼ 1, +∥�wh∥L∞(0,T;L2(�Ω)) + ∥�wh∥L2(0,T;L∞(�Ω)) +<∼ 1, +∥�vh∥L2(0,T;Lq1(�Ω)) + ∥�vh∥L∞(0,T;L2(�Ω)) +<∼ 1, +∥�vh∥Lq2(0,T;Lq1(�Ω;R2)) +<∼ 1. +(4.11) +for any q1 ∈ [1, ∞) and q2 ∈ [1, ∞). +Proof. Noticing that η1 +h = η0 +h + τξ0 +h, τ < 1, and the algebraic inequality (a + b)2 ≤ 2(a2 + b2), we know +that +γ1 +2 +��∂x1η1 +h +��2 +L2(Σ) + γ2 +2 +��∂2 +x1,hη1 +h +��2 +L2(Σ) +13 + +≤ γ1 +���∂x1η0 +h +��2 +L2(Σ) + τ 2 ��∂x1ξ0 +h +��2 +L2(Σ) +� ++ γ2 +���∂2 +x1,hη0 +h +��2 +L2(Σ) + τ 2 ��∂2 +x1,hξ0 +h +��2 +L2(Σ) +� +< γ1 +���∂x1η0 +h +��2 +L2(Σ) + +��∂x1ξ0 +h +��2 +L2(Σ) +� ++ γ2 +���∂2 +x1,hη0 +h +��2 +L2(Σ) + +��∂2 +x1,hξ0 +h +��2 +L2(Σ) +� +≤ c(γ1, γ2, ∥η0∥W 1,2(Σ) , ∥ξ0∥W 1,2(Σ)), +where we have used the stability of the Riesz projection operator in the last step, see (5.5). Therefore, +the right-hand side of the energy estimate (4.4) is uniformly bounded by a positive constant. Then, +we have (4.11)1 and (4.11)4 after noticing ηh ≥ η > 0. +Further, by Korn’s inequality, Sobolev’s +inequalities (5.1), the assumption (4.10), and triangular inequality, we get all the rest estimates. +5 +Interpolation operators +A critical difficulty in convergence analysis is the appropriate choice of interpolation operators for the +couple (u, ξ), that inherit not only the kinematic coupling condition at the fluid-structure interface +but also the divergence-free property of the velocity field. Before digging into this issue, we recall +some analytic estimates that we will use frequently. First, the discrete Sobolev inequalities +∥�v∥Lq1(�Ω) +<∼ ∥�v∥W 1,2(�Ω) for �v ∈ W 1,2(�Ω), +q1 ∈ [1, ∞), +(5.1a) +∥v∥L∞(Σ) +<∼ ∥v∥W 1,2(Σ) for v ∈ W 1,2 +0 (Σ). +(5.1b) +Next, we recall the standard projection error, see e.g. Boffi et. al [7] +���ΠQ +h p − p +��� +W k,s +<∼ h ∥p∥W k+1,s , k = 1, 2, s ∈ [1, ∞]. +(5.2) +where ΠQ +h : L2(�Ω) �→ Qf +h is any suitable projection operator satisfying the above (see for example [7, +Section 2.2]), which we will use for the pressure. The projection operators for the velocities (both +for the fluid and the solid) are much more complicated in this framework. It starts already with the +necessity of a careful choice of the interpolation operator for solid deformation. +5.1 +Interpolation operator for the solid deformation +For the solid we will use the Riesz projection as the interpolation operator. Let η ∈ W 1,2(Σ), our +Riesz projection operator Rs +h reads +� +Σ +∂x1(Rs +hη − η)∂x1ψ dx1 = 0 +∀ ψ ∈ V s +h with +� +Σ +Rs +hη dx1 = +� +Σ +η dx1. +(5.3) +Recalling the discrete Laplace (3.11) we know that for any η ∈ W 2,2(Σ) it holds +� +Σ +(∂2 +x1,hRs +hη − ∂2 +x1η) ψ dx1 = 0 +∀ ψ ∈ V s +h . +(5.4) +Setting ψ as Rs +hη and ∂2 +x1,hRs +hη respectively in (5.3) and (5.4), we obtain by H¨older’s inequality that +∥∂x1Rs +hη∥L2(Σ) +<∼ ∥∂x1η∥L2(Σ) +and +��∂2 +x1,hRs +hη +�� +L2(Σ) +<∼ +��∂2 +x1η +�� +L2(Σ) , +(5.5) +where by a <∼ b we mean a ≤ cb for a positive constant c that is independent of the computational +parameters τ and h. Further, if η ∈ W 3,2(Σ), we have +��∂2 +x1,hRs +hη − ∂2 +x1η +�� +L2(Σ) +<∼ h ∥η∥W 3,2(Σ) , +(5.6) +see the detailed proof in Lemma A.1 for not only one-dimensional Σ but also a multi-dimensional +domain. +This concludes all necessary estimates that we need for the approximation of η as well as the +structure velocity ξ. Building on these properties we have to extend this projection into the fluid- +reference domain. +14 + +5.2 +Interpolation operator for the fluid velocity +Given the divergence-free velocity field u with the boundary condition u|ΓS = ξe2, our aim here is to +construct an interpolation operator Πf +h : W 1,2(�Ω)) → �V f +h , such that for q ∈ [2, ∞) +h1− 2 +q +����u − Πf +h�u +��� +Lγ(�Ω) + h +���∇�u − ∇Πf +h�u +��� +L2(�Ω) ≤ �Ch2 ∥∆�u∥L2(�Ω) , +which has to satisfy also the following two restrictions: +• Kinematic condition Pf +hu|ΓS = Rs +hξe2. +• Weakly divergence-free condition. Here, one may naively consider the form +� +�Ω �q∇Pf +h �u : M d�x = +0. However, it is necessary to have +� +�Ω �q∇Pf +h �u : Mh d�x = 0 for the convenience of convergence +analysis, see Remark B.4. +The next theorem takes care of the first bullet. For it we need the following lemma of analytical +extensions, see [19, Proposition 3.5]. +Lemma 5.1. Assume that Ωηh is a given subgraph, with ηh > δ. Let φ ∈ C∞ +0 ((Σ × [0, δ/2); [0, ∞)) +such that +� +Σ×[0,δ/2) φ dx = 1. Then there is an extension operator Eηh : W k,p(Σ) → W k,p(Ωη), for +k ∈ {0, 1, 2, ...} and 1 < p < ∞, such that the following hold: +1. Eηh(ξ)(x1, x2) = (0, ξ(x1)) for x2 ∈ [δ, ∞). +2. div Eηh(ξ) = ∂2φ +� +Σ ξ dx1. +3. +��∇kEηh(ξ) +�� +Lp(Ωη) ≤ c ∥ξ∥W k,p(Σ), +where c depends on p, k and δ only. +Note that div u = 0, implying that 0 = +� +ΓS(t) v · ν dx1 = +� +Σ ∂tη dx1. It implies that +div Eηh(∂tη) = ∂2φ +� +Σ +∂tη dx1 = 0 for any ηh ≥ δ as above. +(5.7) +Our construction follows tightly [7, Section 2.2 and Section 8.4], where more details on the notation +and arguments can be found. Following the argumentation there it seems more natural to work on the +computed Eulerian grid, that is the grid pushed forward by Aηh. For that reason, we first introduce +the auxiliary operator Pf +h on Ωηh that eventually becomes the basis for the desired operator Πf +h. +Theorem 5.2. Let the grids Th and Σh respectively defined on the domains �Ω and Σ be shape regular +and quasi-uniform. Moreover, let ΓS = {(x1, ηh(x1)) : x1 ∈ Σ} satisfy minΣ ηh ≥ δ and ∥∂x1ηh∥L∞ ≤ L +for some positive constants δ and L. Then there exists an interpolation operator +Pf +h : +Wηh → Wηh, +that satisfies for (ξ, �ϕ) ∈ � +Wηh, γ < ∞ and ϕ := �ϕ ◦ A−1 +ηh +���ϕ − Pf +hϕ +��� +Lγ(Ωηh) + h +���∇(ϕ − Pf +hϕ) +��� +L2(Ωηh) +<∼ h2 ∥ �ϕ∥W 2,2(�Ω) + h2 ∥ξ∥W 2,2(Σ) , +���Pf +hϕ +��� +Lγ(Ωηh) + +���∇Pf +hϕ +��� +L2(Ωηh) +<∼ ∥ �ϕ∥ W 1,2(�Ω) + ∥ξ∥W 1,2(Σ) , +where the bounds depend linearly on 1 +δ, L, L +δ . Moreover, we find Pf +hϕ(x1, ηh(x1)) = (0, Rs +hξ(x1)) on Σ +and +� +Ωηh +qdiv Pf +hϕ dx = +� +Ωηh +qdiv ϕ dx +∀ q ∈ Qf +h. +The above construction on the variable domain Ωηh implies the following corollary for the reference +domain. +15 + +Corollary 5.3. Under the assumption of the Theorem 5.2, there exists +�Pf +h : +� +Wηh → � +Wηh +satisfying for (ξ, �ϕ) ∈ � +Wηh and γ < ∞ that +��� �ϕ − �Pf +h �ϕ +��� +Lγ(�Ω) + h +���∇( �ϕ − �Pf +h �ϕ) +��� +L2(�Ω) +<∼ h2 ∥ �ϕ∥W 2,2(�Ω) + h2 ∥ξ∥W 2,2(Σ) , +��� �Pf +h �ϕ +��� +Lγ(�Ω) + +���∇ �Pf +h �ϕ +��� +L2(�Ω) +<∼ ∥ �ϕ∥W 1,2(�Ω) + ∥ξ∥W 1,2(Σ) , +where the bounds depend linearly on 1 +δ, L, L +δ . Moreover, we find �Pf +hϕ(x1, 1) = (0, Rs +hξ(x1)) on Σ and +� +�Ω +�q∇ �Pf +h �ϕ : M(ηh) d�x = +� +�Ω +�q∇ �ϕ : M(ηh) d�x +∀ �q ∈ �Qf +h. +Proof of Theorem 5.2. The proof is split into two parts. +Part I: construction of a Fortin operator on Ωηh. +We start with the operator �Π1 which is the piecewise affine interpolation operator on the reference +grid constructed in [7, Section 2.2], that naturally preserves zero boundary values component wisely. +In particular, we may define Ps +hξ as +�Π1( �ϕ)(x1, 1) = +� +�Π1(�ϕ1)(x1, 1), �Π1(�ϕ2)(x1, 1) +� +=: (0, Ps +hξ(x1)), +which is by the construction of a function in V s +h . Accordingly, we define +Π1ϕ := �Π1( �ϕ) ◦ Aηh. +In order to show the necessary bounds, we realize by [7, equation (2.2.20)] and by the uniform Lipschitz +bounds of ηh that +∥∇(Π1ϕ − ϕ)∥L2(Ωηh) ≤ cγ,L +����Π1 �ϕ − �ϕ +��� +W 1,2(�Ω) ≲ min +� +∥ �ϕ∥W 1,2(�Ω) , h ∥ �ϕ∥W 2,2(�Ω) +� +and +∥Π1ϕ − ϕ∥L2(Ωηh) ≲ min +� +∥ �ϕ∥L2(�Ω) , h ∥ �ϕ∥W 1,2(�Ω) , h2 ∥ �ϕ∥W 2,2(�Ω) +� +. +This finishes the construction of Π1. +Next, we construct Π2. We start by recalling that �V f +h is piecewise affine. Hence composed with +Aηh these objects are not any more piecewise affine. But as by our assumptions Aηh is bi-Lipschitz, all +necessary bounds for Π1 are directly inherited from the bounds of �Π1 with an additional dependence on +δ, L. Let us focus on a generic reference cell K ∈ Th with its bubble function bK ∈ P3(K) ∩ W 1,2 +0 (K). +It is obvious that +∥bK∥Lp(K) ∼ h +2 +p and ∥∇bK∥Lp(K) ∼ h +2 +p −1. +Analog estimates with dependence on δ and L do also hold for bK ◦ Aηh as ηh is uniformly bounded. +Next, we show how to map the bubble function onto the current domain Ωηh according to the +change of geometry. Let +Bηh = +� � +K +aKbK ◦ Aηh : aK ∈ R2� +be the set of the potential bubble functions pushed forward by Aηh. Our aim is to find a projector +Π2 : W 1,2(Ωηh; R2) → Bd +ηh = +� � +K aKbK ◦ Aηh : aK ∈ R2� +that satisfies +� +K +� +Aηh(K) +(Π2ϕ − ϕ) · ∇q dx = 0, +16 + +for all q = �q ◦ Aηh, �q ∈ �V f +h . Let �q = c + a′x1 + ax2 on K for some constants a′, c, a ∈ R. Then (here +we take ∇ as a column vector) +∇q(x1, x2) = +� +a′ − +ax2 +η2 +h(x1)∂x1ηh(x1) +a +ηh(x1) +� += +� +1 +− +x2 +η2 +h(x1)∂x1ηh(x1) +0 +1 +ηh(x1) +� �a′ +a +� +=: Aηh(x1) +�a′ +a +� +and thus +ϕ · ∇q = (ϕ1, ϕ2)Aηh +�a′ +a +� +. +This allows us to define Π2(ϕ)|Aηh(K) = � +K∈Th βKbK ◦ Aηh, where βK ∈ R2 is determined by the +equation +βT +K +� +Aηh(K) +bK ◦ AηhAηh dx = +� +Aηh(K) +ϕTAηh dx. +It is easy to check that Aηh is bounded from above and below by positive constants and +c1h2 +∥ηh∥∞ +≤ +� +Aηh(K) +bK ◦ Aηh(x1, x2) +1 +ηh(x1) dx ≤ c2h2 +δ +Consequently the matrix MK = +� +Aηh(K) bK ◦ AηhAηh dx is invertible with +|M−1 +K | ≤ +1 +| det( +� +Aηh(K) bK ◦ AηhAηh dx)| +� +Aηh(K) +bK ◦ Aηh|Aηh| dx ≤ ch−2, +where c depends linearly on L and L +δ . Further, we find +|βK| ≤ |M−1 +K | ∥Aηh∥∞ ∥ϕ∥L1(Aηh(K)) ≤ ch−2 ∥ϕ∥L1(Aηh(K)) ≤ ch− 2 +p ∥ϕ∥Lp(Aηh(K)) +where in the last step we have used Jensen’s inequality. Using the above estimate, we have +∥∇Π2ϕ∥Lp(Aηh(K)) ≤ |βK| ∥∇(bK ◦ Aηh)∥Lp(Aηh(K)) ≤ ch−1 ∥ϕ∥Lp(Aηh(K)) , +and +∥Π2ϕ∥Lp(Aηh(K)) ≤ |βK| ∥(bK ◦ Aηh)∥Lp(Aηh(K)) ≤ c ∥ϕ∥Lp(Aηh(K)) , +which allows us to follow the arguments at the end of [7, Section 8.4] to gain the expected estimates +and bounds for the operator: +Πf +h(ϕ) := Π1(ϕ) + Π2(ϕ − Π1(ϕ)). +Part II: a Fortin operator with appropriate boundary values +By construction Πf +hϕ(x1, ηh(x1)) = Π1ϕ(x1, ηh(x1)) =: (0, Ps +hξ(x1)), with Ps +h being an interpola- +tion operator for V s +h with natural stability properties and error bounds. The problem is that, unlike +Rs +h, the operator Ps +h does not have the required second-order estimates (in particular Lemma A.1 does +not hold). Nevertheless by the orthogonality of the error for Rs +h and the estimates of first order for +Ps +hξ(x1), we find that +∥∂x1(Rs +hξ − Ps +hξ)∥L2(Σ) ≤ ∥∂x1(ξ − Ps +hξ)∥L2(Σ) ≤ ch +��∂2 +x1ξ +�� +L2(Σ) . +(5.8) +The desired projector turns out to be the solution to a discrete Stokes problem: We derive it for ηh, +ϕ and ξh fixed by minimizing +� +Ωηh +|∇(ψh − ϕ)|2 dx +17 + +over the class of all ψh ∈ V f +h , with ψh(x1, ηh(x1)) = (0, ξh(x1)) on Σ, which satisfy the discrete +divergence-free property: +� +Ωηh div (ψh)·q dx for all q ∈ Qf +h. The minimizer is then defined as Pf +hϕ. The +respective Euler-Lagrange equation becomes the discrete solution to an approximate Stokes problem +� +Ωηh +∇(Pf +hϕ − ϕ) · ∇ψh dx = 0, +for all ψh ∈ V f +h with zero boundary values and which are discretely divergence-free. The error of +the projector is of two kinds. The first is the error stemming from the prescribed boundary values, +and the second is the discretization error. For the first, we take the linear divergence-free extension +Eηh(Rs +hξ − ξ) given by Lemma 5.1. Now we can take ψh = Πf +h(ϕ + Eηh(Rs +hξ − ξ)) as competitor in the +minimization. Indeed, as (0, ξ(x1)) = ϕ(x1, ηh(x1)), we find that Πf +h(ϕ + Eηh(Rs +hξ − ξ))(x1, ηh(x1)) = +Πf +h(Eηh(ξ))(x1, ηh(x1)) = (0, Ps +h(ξ)(x1) in Σ. This implies (as the projector is the minimizer) that +���∇(Pf +hϕ − ϕ) +��� +2 +L2(Ωηh) ≤ +���∇(Πf +h(ϕ + Eηh(Rs +hξ − ξ)) − ϕ) +��� +2 +L2(Ωηh) +≤ c +���∇ϕ − ∇Πf +hϕ +��� +2 +L2(Ωηh) + c +���∇(Πf +hEηh(Rs +hξ − ξ)) +��� +2 +L2(Ωηh) . +The first term is estimated directly by the properties of the Fortin operator. The second one is by the +stability of the Fortin operator, Lemma 5.1 and (5.8). +Remark 5.4 (On the importance of the proper choice of an interpolation operator). The deep rea- +son why the interpolation has to be solved as a discrete PDE, is that the solid matter and the fluid +matter have totally different properties, even so they are coupled. Our scheme follows the direct path +that is also used in the existence theory, where already in the approximation the coupling and the +different matters are simultaneously (monolithically) solved. The fact that this uniform (and linear) +approximation does indeed converge properly can only be revealed by imitating the coupling between +two solutions of independent PDEs. This imitation is exactly performed by solving a discrete boundary +value problem. +The last step for the interpolation of u is the correction of the divergence due to the change of +variables. For that, we use another analytic tool developed in [19, Theorem 3.3]. It is the so-called +universal Bogovskij operator. Universal it is, because it is independent of the particular (Lipschitz) +geometry. We cite the important estimate in the following lemma. +Lemma 5.5. There is an operator B : {f ∈ Lp(Ωηh) : +� +Ωηh f dx = 0} → W 1,p +0 (Ωηh) for any Ωηh for +1 < p < ∞ that is a given subgraph, with minΣ ηh > δ and ∥∂xηh∥L∞ ≤ L, such that the following +hold: +1. div B(f) = f. +2. ∥B(f)∥W 1,p(Ω) ≤ ∥f∥Lp. +The above lemma and Corollary 5.3 lead to the final statement of this section. +Theorem 5.6. Let Ωηh ⊂ R2 be a subgraph and let the assumptions of Theorem 5.2 hold. Then there +exists +Πf +h : +� +Wη → � +Wηh, +satisfying for (ξ, u) ∈ Wη and γ < ∞ that +����u − Πf +h�u +��� +Lγ(�Ω) + h +���∇(�u − Πf +h�u) +��� +L2(�Ω) +<∼ h2 ∥�u∥W 2,2(�Ω) + h2 ∥ξ∥W 2,2(Σ) + h ∥η − ηh∥W 1,2(Σ) , +���Πf +h�u +��� +Lγ(�Ω) + +���∇Πf +h�u +��� +L2(�Ω) +<∼ ∥�u∥W 1,2(�Ω) + ∥ξ∥W 1,2(Σ) + ∥η − ηh∥W 2,2(Σ) , +(5.9) +18 + +where the bounds depend linearly on 1 +δ, L, L +δ . Moreover, we find Πf +hu(x1, 1) = (0, Rs +hξ(x1)) on Σ and +� +�Ω +�q∇Πf +hu : M(ηh) dx = 0 +∀ �q ∈ �Qf +h. +(5.10) +Proof. The proof takes the function ϕ := u◦Aηh ◦A−1 +η −B(div (u◦Aηh ◦A−1 +η )). By the Gauss theorem +we note that +� +Ωηh +div (u ◦ Aηh ◦ A−1 +η ) dx = 0, +hence B is well defined and so div ϕ = 0 on Ωηh. Further +div (u ◦ Aηh ◦ A−1 +η ) = +� η +ηh +− 1 +� +∂2u2 ◦ Aηh ◦ A−1 +η ++ ∂1 +� η +ηh +� +x2∂2u1 ◦ Aηh ◦ A−1 +η , +which implies as W 1,∞(Σ) ⊂ W 2,2(Σ) that +��div u ◦ Aηh ◦ A−1 +η +�� +L2(Ωηh) ≤ c ∥η − ηh∥W 2,2(Σ) ∥∇�u∥L2(�Ω) . +Hence by Lemma 5.5 and a change of variable we find +∥ �ϕ − �u∥W 1,2(�Ω) ≤ c ∥η − ηh∥W 2,2(Σ) ∥∇�u∥L2(�Ω) and �ϕ(x1, 1) = ξ(x1). +Then, we define Πf +h�u = �Pf +h �ϕ for which now the result follows from the previous estimates and Corol- +lary 5.3. +6 +Error estimates +In this section, we study the error between the numerical solution (�ph, �uh, ξh, ηh) of Scheme-R and +its target smooth solution (�p, �u, ξ, η). Here we assume the existence of a smooth solution of (1.1)–(1.4) +in the following class +� +� +� +� +� +� +� +� +� +� +� +η > η, η ∈ L2(0, T; W 3,2(Σ)) ∩ W 2,2(0, T; W 2,2(Σ)), +�u ∈ L∞(0, T; W 1,2(�Ω; R2)) ∩ L2(0, T; W 2,2(�Ω; R2)) +∂t�u ∈ L2(0, T; W 1,2(�Ω; R2)), +�p ∈ L∞(0, T; L2(�Ω)), ∇p ∈ L2((0, T) × �Ω). +(6.1) +6.1 +The time projection +Very relevant in this highly nonlinear coupled system is to choose a set of appropriate time-value tτ +k, +k = 1, . . . , NT , at which we will compare the continuous equation with its numerical approximation. +For a given τ and k, we denote +(�pk, �uk, ξk, ηk) := (�p, �u, ξ, η)(tτ +k). +Then, according to our smoothness assumption (6.1), we may choose the value tτ +k ∈ [kτ, (k + 1)τ) in +such a way that +τ +� +∥�u(tτ +k)∥2 +W 2,2(�Ω) + ∥∂t�u(tτ +k)∥2 +W 1,2(�Ω) + +��∂2 +t �u(tτ +k) +��2 +L2(�Ω) + +∥η(tτ +k)∥2 +W 2,3(Σ) + ∥ξ(tτ +k)∥2 +W 2,4(Σ) + ∥∂tξ(tτ +k)∥2 +W 2,2(Σ) +� +≤ +� (k+1)τ +kτ +� +∥�u(t)∥2 +W 2,2(�Ω) + ∥∂t�u(t)∥2 +W 1,2(�Ω) + +��∂2 +t �u(t) +��2 +L2(�Ω) ++ ∥η(t)∥2 +W 2,3(Σ) + ∥ξ(t)∥2 +W 2,2(Σ) ∥∂tξ(t))∥2 +W 2,2(Σ) +� +dt, +19 + +which is possible to find by the continuity of the integral, if the right-hand side is bounded. +In +particular, we find that +τ +NT +� +k=1 +� ����uk��� +2 +W 2,2(�Ω) + +���∂t�uk��� +2 +W 1,2(�Ω) + +���∂2 +t �uk��� +2 +L2(�Ω) + +���ηk��� +2 +W 2,2(Σ) + +���∂tξk��� +2 +W 2,2(Σ) +� +≤ +� T +0 +� +∥�u∥2 +W 2,2(�Ω) + ∥∂t�u∥2 +W 1,2(�Ω) + +��∂2 +t �u +��2 +L2(�Ω) + ∥η∥2 +W 2,2(Σ) + ∥∂tξ)∥2 +W 2,2(Σ) +� +dt +(6.2) +Actually, this right-hand side summarizes our regularity assumptions on the solution. All the above +regularity requirements do follow from these assumptions. +Remark 6.1 (On the regularity assumptions). When comparing the assumptions on the smooth so- +lution with the theory for the heat/wave equation (or the 2D/Navier-Stokes equation), one realizes +that we have the same regularity assumptions for the fluid as in the non-moving case. For the plate, +which also deduces the domain essentially one more time-derivative has to be assumed, as nonlinear +equations of a similar type can be expected. +6.2 +Main result +Before introducing the main result, let us denote the following error terms for each time step k ∈ +{1, . . . , NT }. +ek +p = �pk +h − �pk = (�pk +h − ΠQ +h �pk) + (ΠQ +h �pk − �pk) =: δk +p + Ik +p , +ek +u = �uk +h − �uk = (�uk +h − Πf +h�uk) + (Πf +h�uk − �uk) =: δk +u + Ik +u, +ek +ξ = ξk +h − ξk = (ξk +h − Rs +hξk) + (Rs +hξk − ξk) =: δk +ξ + Ik +ξ , +ek +η = ηk +h − ηk = (ηk +h − Rs +hηk) + (Rs +hηk − ηk) =: δk +η + Ik +η , +ek +ζ = ζk +h − ζk = (ζk +h + ∂2 +x1,hRs +hηk) + (−∂2 +x1,hRs +hηk − ζk) =: δk +ζ + Ik +ζ , +(6.3) +where ζh = −∂2 +x1,hηh and ζ = −∂2 +x1η. Now we are ready to present the main result of the paper. +Theorem 6.2 (Convergence rate). Let {(�pk +h, �uk +h, ξk +h, ηk+1 +h +)}NT +k=1 be the solution of Scheme-R +(3.12), and let (�u, �p, ξ, η)(t), t ∈ (0, T), be a strong solution of (1.1)–(1.4) belonging to the class +(6.1). Then for any m ∈ {1, · · · , NT } it holds +1 +2ϱf +� +�Ω +|em +u |2ηm +h d�x + 1 +2 +� +Σ +� +ϱs|em +ξ |2 + γ1|∂x1em+1 +η +|2 + γ2|em+1 +ζ +|2� +dx1 ++ 2µτ +m +� +k=1 +� +�Ω +���∇ek +u(Fk +h)−1��� +2 +d�x + γ3ϱsτ +m +� +k=1 +� +Σ +|∂x1δk +ξ |2 dx1 +<∼ τ 2 + h2. +In particular, we have the following convergence rates +∥eu∥L∞(0,T;L2(�Ω;R2)) + ∥eξ∥L∞(0,T;L2(Σ)) + ∥∂x1eη∥L∞(0,T;L2(Σ)) + ∥eζ∥L∞(0,T;L2(Σ)) ++ ∥∇eu∥L2((0,T)×�Ω;R2×2) + γ3 ∥∂x1eξ∥L2((0,T)×Σ;R2) +<∼ τ + h. +(6.4) +Proof. First, we subtract the weak formulation (2.9b) from the numerical scheme (3.12b) and get +� +�Ω +ϱf(ηk +hDtek +u + 1 +2Dtηk +hek∗ +u ) · �ϕ d�x + 2µ +� +�Ω +� +∇ek +u(Fk +h)−1�S : +� +∇ �ϕ(Fk +h)−1� +ηk +h d�x ++ ϱs +� +Σ +Dtek +ξψ dx1 + as(ek+1 +η +, ek+1 +ζ +, ek +ξ, ψ) = − +7 +� +i=1 +Rk +i ( �ϕ, ψ), +(6.5) +20 + +where +Rk +1( �ϕ, ψ) = ϱf +� +�Ω +� +ek +η∂t�uk + ηk +h(Dt�uk − ∂t�uk) +� +· �ϕ d�x, +Rk +2( �ϕ, ψ) = 1 +2ϱf +� +�Ω +� +(ek−1 +ξ +− τDtξk)�uk∗ − τ∂tηkDt�uk� +· �ϕ d�x, +Rk +3( �ϕ, ψ) = 1 +2ϱf +� +�Ω +� +�ϕ · (∇ek +u) − ek +u · (∇ �ϕ) +� +· (Fk +h)−1�vk−1 +h +ηk +h d�x ++ 1 +2ϱf +� +�Ω +� +�ϕ · (∇�uk) − �uk · (∇ �ϕ) +� +· +� +(Fk +h)−1�vk−1 +h +ηk +h − (Fk)−1�vkηk� +d�x, +Rk +4( �ϕ, ψ) = +� +�Ω +ek +p∇ �ϕ : Mk +h d�x + +� +�Ω +�pk∇ �ϕ : +� +Mk +h − Mk� +d�x, +Rk +5( �ϕ, ψ) = 2µ +� +�Ω +�� +∇�u(Fk +h)−1�S : (∇ �ϕ(Fk +h)−1)ηk +h − +� +∇�u(F)−1�S : (∇ �ϕ(Fk)−1)ηk� +d�x, +Rk +6( �ϕ, ψ) = ϱs +� +Σ +(Dtξk − ∂tξk)ψ dx1, +Rk +7( �ϕ, ψ) = −γ1 +� +Σ +∂2 +x1(ηk+1 − ηk) ψ dx1 − γ2 +� +Σ +∂2 +x1(ζk+1 − ζk) ψ dx1. +(6.6) +The precise justification of (6.5) is given in Appendix B.1. By setting ( �ϕ, ψ) = (δk +u, δk +ξ ) in (6.5) and +sum up from k = 1 to m we derive +−τ +m +� +k=1 +7 +� +i=1 +Rk +i (δk +u, δk +ξ ) = τ +m +� +k=1 +� +�Ω +ϱf(ηk +hDtek +u + 1 +2Dtηk +hek∗ +u ) · δk +u d�x ++ 2µτ +m +� +k=1 +� +�Ω +� +∇ek +u(Fk +h)−1�S : +� +∇δk +u(Fk +h)−1� +ηk +h d�x ++ τ +m +� +k=1 +ϱs +� +Σ +Dtek +ξδk +ξ dx1 + τ +m +� +k=1 +as(ek+1 +η +, ek+1 +ζ +, ek +ξ, δk +ξ ). +(6.7) +Further, applying the algebraic equalities (4.2) and (B.10) to the above right-hand-side, we get (simi- +larly as was performed for the stability estimate) +− τ +m +� +k=1 +7 +� +i=1 +Rk +i (δk +u, δk +ξ ) = τ +m +� +k=1 +� +�Ω +ϱf +� +ηk +hDt(δk +u + Ik +u) + 1 +2Dtηk +h(δk∗ +u + Ik∗ +u ) +� +· δk +u d�x ++ 2µτ +m +� +k=1 +� +�Ω +� +∇(δk +u + Ik +u)(Fk +h)−1�S : (∇δk +u(Fk +h)−1)ηk +h d�x ++ τ +m +� +k=1 +ϱs +� +Σ +Dt(δk +ξ + Ik +ξ ) δk +ξ dx1 + τ +m +� +k=1 +as(ek+1 +η +, ek+1 +ζ +, ek +ξ, δk +ξ ) += δm +E − δ0 +E + τ +m +� +k=1 +Dk +phys + τ +m +� +k=1 +Dk +num + Gf + Gs, +(6.8) +where +δk +E = +� +�Ω +1 +2ϱfηk +h|δk +u|2 d�x + 1 +2 +� +Σ +� +ϱs|δk +ξ |2 + γ1|∂x1δk+1 +η +|2 + γ2|δk+1 +ζ +|2� +dx1, +δk +D =2µ +� +�Ω +ηk +h| +� +∇δk +u(Fk +h)−1�S|2 d�x + γ3 +� +Σ +|∂x1δk +ξ |2 dx1, +Dk +num =τ +2ϱf +� +�Ω +ηk−1 +h +|Dtδk +u|2 d�x + τ +2 +� +Σ +� +ϱs|Dtδk +ξ |2 + γ1|Dt∂x1δk+1 +η +|2 + γ2|Dtδk+1 +ζ +|2� +dx1 ≥ 0, +Gf =τ +m +� +k=1 +� +�Ω +ϱf +� +ηk +hDtIk +u + 1 +2Dtηk +hIk∗ +u +� +· δk +u d�x +21 + ++ 2µτ +m +� +k=1 +� +�Ω +� +∇Ik +u(Fk +h)−1�S : (∇δk +u(Fk +h)−1)ηk +h d�x, +Gs =γ1τ +m +� +k=1 +� +Σ +∂x1δk+1 +η +∂x1(Dtηk+1 − ∂tηk) dx1 − γ2τ +m +� +k=1 +� +Σ +δk+1 +ζ +∂2 +x1(Dtηk+1 − ∂tηk) dx1 ++ τ +m +� +k=1 +� +Σ +ϱsDtIξδk +ξ dx1. +Next, we reformulate (6.8) in the following form. +δm +E − δ0 +E + τ +m +� +k=1 +δk +D + τ +m +� +k=1 +Dk +num = −τ +m +� +k=1 +7 +� +i=1 +Rk +i − Gf − Gs. +(6.9) +Then, by Young’s inequality, H¨older’s inequality, the interpolation error in Theorem 5.6, and the +uniform bounds (4.11), we estimate the right-hand-side of the above equation as +�����τ +m +� +k=1 +7 +� +i=1 +Rk +i + Gf + Gs +����� +<∼ τ 2 + h2 + cτ +m +� +k=1 +δm +E + 2αµτ +m +� +k=1 +� +�Ω +���∇δk +u(Fk +h)−1��� +2 +ηk +h d�x, +(6.10) +see Appendix B.4. Further, substituting the above estimate into (6.9) and noticing the initial error +δ0 +E = 0, we get (using also the lower bound of η, ηh) that +δm +E + (1 − α)2µτ +m +� +k=1 +� +�Ω +���∇δk +u(Fk +h)−1��� +2 +ηk +h d�x + γ3ϱsτ +m +� +k=1 +� +Σ +|∂x1δξ|2 dx1 +<∼ τ 2 + h2 + τ +m +� +k=1 +δk +E. +By choosing any α ∈ (0, 1) and using Gr¨onwall’s inequality, we get +δm +E + τ +m +� +k=1 +δk +D +<∼ τ 2 + h2. +Recalling the interpolation errors (Theorem 5.6 and (5.6)) and the regularity of the strong solution +(6.1) we get +1 +2ϱf +� +�Ω +|Im +u |2ηm +h d�x + 1 +2 +� +Σ +� +ϱs|Im +ξ |2 + γ1|∂x1Im+1 +η +|2 + γ2|Im+1 +ζ +|2� +dx1 ++ τ +m +� +k=1 +� +2µ +� +�Ω +���∇Ik +u(Fk +h)−1��� +2 +ηk +h d�x + γ3ϱs +� +Σ +|∂x1Ik +ξ |2 dx1 +� +<∼ h2. +Finally, due to the triangular inequality, we sum up the previous two estimates and get +1 +2ϱf +� +�Ω +|em +u |2ηm +h d�x + 1 +2 +� +Σ +� +ϱs|eξ|2 + γ1|∂x1em+1 +η +|2 + γ2|em+1 +ζ +|2� +dx1 ++ τ +m +� +k=1 +� +2µ +� +�Ω +���∇ek +u(Fk +h)−1��� +2 +ηk +h d�x + γ3ϱs +� +Σ +|∂x1δk +ξ |2 dx1 +� +<∼ τ 2 + h2, +(6.11) +which provides the proof for small T ≤ T0 such that (4.10) is valid. +Next, we show that T can be arbitrarily large if η ≥ η on [0, T]. We start with a fixed T0 such +that ηh ≥ +η +2, this can be found by [26, Lemma 5]. Then, by the above estimate (6.11), we know that +∥eη(T0)∥L∞ ≤ c(τ + h), +where the constant c depends on the lower bound +η +2. Recalling η(t) ≥ η we know that +ηh(T0) ≥ η − c(τ + h), +22 + +which actually is much larger than +η +2, if τ, h are small enough. Hence by [26, Lemma 5], there is a +T1 > T0, such that +ηh(T1) ≥ 2η +3 − c(τ + h) ≥ η +2 +for τ and h small enough, where T1 depends only on the initial energy of the problem but is independent +of τ and h. Hence we can repeat the above argument with the same lower bound +η +2. It implies for +τ, h → 0 that this procedure can be repeated arbitrarily many times, thus (4.10) hold for any large +T. +7 +Numerical experiments +In this section, we define a problem that we use to study the convergence rate of the linear semi-implicit +Scheme-R (3.12) on a reference domain �Ω. +This semi-implicit scheme is then compared with the +nonlinear fully implicit scheme corresponding to the weak form (2.9). Both numerical implementations +are described in detail in Appendices C.1 and C.2. +7.1 +Problem description +In our experiments, the domain �Ω is a rectangle of dimensions 2×1 with periodic boundary conditions +in the x1−direction, i.e. the solution on the left boundary coincides with the solution on the right +boundary. On the bottom we have no-slip boundary conditions. At t = 0, we prescribe zero initial +conditions for all unknowns. Moreover, we set ϱf = ϱs = 1, γ1 = γ2 = 0.1, and γ3 = 0 since we +wish to solve a problem with a non-dissipating elastic shell. The flow is driven by the external force g +periodic in x1 direction. The force is applied up to t = 0.2 such that a big amplitude of the structure +deformation is produced. Next, the force is turned off and the system is left to relax. The force g +reads +g = +�200t sin(2πx) +t ≤ 0.2, +0 +t > 0.2. +Snapshots of the simulation are given in Figure 2. +(a) +(b) +(c) +(d) +(e) +(f) +Figure 2: Snapshots of the simulation: (a) t = 0.2 end of loading, (b) t = 0.26 maximum of amplitude, +(c) t = 0.35, (d) t = 0.41, (e) t = 0.45, (f) t = 0.53 another maximum. The color scale depicts +pressure, arrows show the direction of the velocity field. +23 + +7.2 +Convergence rates +The simulation is computed for t ∈ [0, T], T = 1.0 for six different time steps τ = 5 × 10−3, 2.5 × +10−3, 1.25 × 10−3, 6.25 × 10−4, 3.125 × 10−4 and τmin = 1 × 10−4 on six different meshes with the mesh +sizes h = 2.83 × 10−1, 1.41 × 10−1, 7.07 × 10−2, 3.54 × 10−2, 1.77 × 10−2 and hmin = 8.84 × 10−3. The +solution with the finest mesh (corresponding to 410 880 degrees of freedom (dofs) in Step 1 of the +implementation of Scheme-R, see Appendix C.1) and the smallest time step is used as the reference +solution. +The solutions for different mesh refinements and the smallest time step are compared to the +reference solution, specifically, we record all summands of the right-hand-side of Theorem 6.2, these +are: ∥eu∥L∞(L2), ∥eξ∥L∞(L2), ∥eη∥L∞(L2), ∥∇eη∥L∞(L2), ∥eζ∥L∞(L2) and ∥∇eu∥L2(L2). The convergence +with respect to the mesh size h is given in the Table 1. The graphs depicting the convergence rate +with respect to the mesh size h are shown in Figure 3. The convergence with respect to the time step +τ is provided in Table 2 and Figure 4. +h +∥eu∥L∞(L2) +∥eξ∥L∞(L2) +∥eη∥L∞(L2) +∥∇eη∥L∞(L2) +∥eζ∥L∞(L2) +∥∇eu∥L2(L2) +2.83 × 10−1 +1.20 × 100 +2.84 × 100 +2.22 × 10−1 +1.41 × 100 +9.22 × 100 +1.23 × 101 +1.41 × 10−1 +3.19 × 10−1 +5.80 × 10−1 +5.99 × 10−2 +3.79 × 10−1 +2.42 × 100 +7.51 × 100 +7.07 × 10−2 +1.05 × 10−1 +1.39 × 10−1 +1.52 × 10−2 +1.34 × 10−1 +6.02 × 10−1 +4.11 × 100 +3.54 × 10−2 +2.78 × 10−2 +3.31 × 10−2 +3.65 × 10−3 +6.57 × 10−2 +1.44 × 10−1 +2.12 × 100 +1.77 × 10−2 +5.91 × 10−3 +6.64 × 10−3 +7.32 × 10−4 +2.94 × 10−2 +2.89 × 10−2 +1.04 × 100 +Table 1: Convergence of errors with mesh refinement (using fixed time step τ = τmin); reference +solution: hmin = 8.84 × 10−3, τmin = 10−4. +τ +∥eu∥L∞(L2) +∥eξ∥L∞(L2) +∥eη∥L∞(L2) +∥∇eη∥L∞(L2) +∥eζ∥L∞(L2) +∥∇eu∥L2(L2) +5.00 × 10−3 +2.55 × 10−1 +5.50 × 10−1 +4.23 × 10−2 +2.66 × 10−1 +1.67 × 100 +1.61 × 100 +2.50 × 10−3 +1.36 × 10−1 +2.87 × 10−1 +2.21 × 10−2 +1.39 × 10−1 +8.74 × 10−1 +8.52 × 10−1 +1.25 × 10−3 +6.87 × 10−2 +1.43 × 10−1 +1.10 × 10−2 +6.91 × 10−2 +4.35 × 10−1 +4.28 × 10−1 +6.25 × 10−4 +3.25 × 10−2 +6.73 × 10−2 +5.17 × 10−3 +3.25 × 10−2 +2.05 × 10−1 +2.02 × 10−1 +3.12 × 10−4 +1.37 × 10−2 +2.83 × 10−2 +2.17 × 10−3 +1.36 × 10−2 +8.60 × 10−2 +8.45 × 10−2 +Table 2: Convergence of errors with time step refinement (using fixed mesh size h = hmin); reference +solution: hmin = 8.84 × 10−3, τmin = 10−4. +In Theorem 6.2 we proved that the convergence rate is linear both in h (space) and τ (time) for the +sum of all errors mentioned. This is justified by the experiments. In time the convergence is indeed +linear for all summands (see Figure 4), in space we observe a quadratic convergence for ∥eu∥L∞(L2), +∥eξ∥L∞(L2), ∥eη∥L∞(L2), ∥eζ∥L∞(L2), but a linear convergence for ∥∇eη∥L∞(L2) and ∥∇eu∥L2(L2) (see +Figure 3). +7.3 +Comparison between the semi-implicit Scheme-R and fully implicit scheme +Since our proposed Scheme-R performs in accordance with the (optimal) predictions, we decided to +test how well it behaves with respect to the fully implicit scheme, as many researchers believe that a +monolithic scheme should be implemented fully implicitly. In the fully implicit scheme we solve a fully +implicit nonlinear problem based on the weak form (2.9), the details of the implementation are given +in Appendix C.2. The main difference is the following. In the semi-implicit Scheme-R, as described +in Appendix C.1, every time step is solved in two steps. First, we solve a linear problem for velocity +u, the second order derivative of the mesh displacement ζ, and the pressure p. This is followed by +a second step in which we update the mesh displacement η. In the fully implicit scheme we solve +everything at once, which, however, requires to solve a more expensive nonlinear problem. It turns +out that both schemes produce the same solution. +Since the main difference between the two schemes is in the time splitting, we compare the numeri- +cal errors ∥∇eη∥L∞(L2) and ∥∇eη∥L∞(L2) for several different time steps τ. Similarly as in the previous +24 + +10−1.8 10−1.6 10−1.4 10−1.2 +10−1 +10−0.8 10−0.6 +10−2 +10−1 +100 +1 +order 2 +1 +order 1 +h +Errors to reference solution +∥eu∥L∞(L2) +∥eξ∥L∞(L2) +∥eη∥L∞(L2) +∥∇eη∥L∞(L2) +∥eζ∥L∞(L2) +∥∇eu∥L2(L2) +Figure 3: +Mesh convergence comparison for ∥eu∥L∞(L2), ∥eξ∥L∞(L2), ∥eη∥L∞(L2), ∥∇eη∥L∞(L2), +∥eζ∥L∞(L2) and ∥∇eu∥L2(L2). For a better comparison, the plots of the errors are shifted to start +from the same point. +10−3.6 10−3.4 10−3.2 +10−3 +10−2.8 10−2.6 10−2.4 10−2.2 +10−1 +100 +1 +order 1 +τ +Errors to reference solution +∥eu∥L∞(L2) +∥eξ∥L∞(L2) +∥eη∥L∞(L2) +∥∇eη∥L∞(L2) +∥eζ∥L∞(L2) +∥∇eu∥L2(L2) +Figure 4: +Timestep convergence comparison ∥eu∥L∞(L2), ∥eξ∥L∞(L2), ∥eη∥L∞(L2), ∥∇eη∥L∞(L2), +∥eζ∥L∞(L2) and ∥∇eu∥L2(L2). For a better comparison, the errors start at the same point. +25 + +10−3.6 10−3.4 10−3.2 +10−3 +10−2.8 10−2.6 10−2.4 10−2.2 +10−1 +100 +τ +Errors to reference solution +∥∇eu∥L2(L2) semi-implicit +∥∇eu∥L2(L2) fully implicit +∥∇eη∥L∞(L2) semi-implicit +∥∇eη∥L∞(L2) fully implicit +Figure 5: Timestep convergence comparison semi-implicit vs. +fully implicit for ∥∇eu∥L2(L2) and +∥∇eη∥L∞(L2). +subsection, we compute the errors with respect to the same reference solution obtained by the semi- +implicit Scheme-R with the finest mesh hmin = 8.84 × 10−3 and smallest time step τ = 1 × 10−4 for +the numerical solutions of these two schemes. The graph of convergence in time is shown in Figure 5. +Note that the two discrete problems are computed on the same mesh h = hmin for both schemes. +In case of the semi-implicit Scheme-R we solve a linear problem of size 410 880 dofs in Step 1 and +a linear problem of size 153 920 dofs in Step 2 every time step. In case of the fully implicit scheme +we solve a nonlinear problem of size 564 800 dofs every time step. All problems are computed on a +server equipped with Intel Xeon Gold 6240 CPU, and (although the code works in parallel) for the +purpose of comparison we run them in serial. We have recorded the CPU time of the computations +for the largest and smallest time step τ, see Table 3. For the largest time step τ = 5 × 10−3 the semi- +implicit scheme is 5.53 times faster than the fully implicit scheme that needs to solve three Newton +iterations in average in every time step and solves a slightly larger problem. For the smallest time +step τ = 3.12 × 10−4 the semi-implicit scheme is 3.88 times faster because the fully implicit scheme +needs in average only two Newton iterations per time step. +Scheme +τ +Avg Newton its +CPU time [min] +Fully implicit +5.00 × 10−3 +3 +135.5 +Semi-implicit +5.00 × 10−3 +– +24.5 +Fully implicit +3.12 × 10−4 +2 +1 310.7 +Semi-implicit +3.12 × 10−4 +– +338.0 +Table 3: Comparison of CPU times (in minutes) for semi-implicit and fully implicit schemes. +8 +Conclusion +We have introduced a novel semi-implicit and linear scheme for the approximation of the interaction +between an incompressible fluid and elastic shell allowing for large deformation. By this we mean +that the domain of definition for the fluid is time changing and the changes of the domain can be +arbitrarily large as long as no topological change appears. We have proved that the scheme is energy +stable and it converges to the smooth solution linearly with respect to the mesh size h and time step +τ. We have implemented the scheme in FEniCS and observed that the convergence rates are optimal. +Possibly, the rates can be improved for some terms, where we observed quadratic growth which paves +26 + +the way for further research. We have compared our semi-implicit scheme with a fully implicit scheme +that not only provides the same convergence rates as our scheme but does produce almost exactly the +same solution. Moreover, our scheme overperforms the fully nonlinear scheme several times in terms +of consumed CPU time. +The analysis presented here, in particular the development of the interpolation operators does +form the basis for new theoretical numerical investigations. It is shown that suitable interpolation +operators for nonlinear equations coupled via their geometry can be constructed. Hence they motivate +a respective convergence analysis for the plethora of applications involving such couplings. 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Fluids and +Structures 98: paper No. 103117, 2020. +28 + +A +Appendix: Interpolation operators +In this part, we present some useful estimate/equality for the interpolation operators used in our +paper. First, we show the approximation error of the discrete Laplace ∂2 +x1,h given by (3.11) of a Riesz +projection operator Rs +h defined by (5.3). +Lemma A.1. Let η ∈ W 3,2 ∩ W 1,2 +0 (Σ), Σ ⊂ Rn, n = 2, 3, and V s +h ⊂ W 1,2 +0 (Σ) is a closed subspace. +For any ψ ∈ V s +h , let ∆hφ ∈ V s +h satisfy +− +� +Σ +∆hφψdz = +� +Σ +∇φ · ∇ψdz +and Rs +h satisfy (5.3) in n dimensions, i.e., +� +Σ +(∇η − ∇Rs +hη) · ∇ψ dz = 0 +Moreover, we assume there exists a projection Ph : L2(Σ) → V s +h satisfying +∥η − Phη∥2 ≤ ch ∥∇η∥2 ∀ η ∈ W 1,2(Σ). +Then +∥∆η − ∆hRs +hη∥2 ≤ ch ∥∇∆η∥2 . +Proof. By the definition of the Riesz projection, and the discrete Laplace, we find +∥∆η − ∆hRs +hη∥2 +2 = +� +Σ +(∆η − ∆hRs +hη) (∆η − ∆hRs +hη) dz += +� +Σ +(∆η − ∆hRs +hη) (Ph∆η − ∆hRs +hη) dz + +� +Σ +(∆η − ∆hRs +hη) (∆η − Ph∆η) dz += − +� +Σ +(∇η − ∇Rs +hη) · (∇(Ph∆η − ∆hRs +hη)) dz + +� +Σ +(∆η − Ph∆η) (∆η − ∆hRs +hη) dz +≤ ∥∆η − Ph∆η∥2 ∥∆η − ∆hRs +hη∥ . +This implies the wanted estimate by the assumed property of Ph: +∥∆η − Ph∆η∥2 ≤ ch ∥∇∆η∥2 . +Remark A.2. In our setting one possibility is to choose as Ph the L2-Projection into V s +h defined by +� +(η − Phη) φh dz = 0 for all φ ∈ V s +h , which is known to satisfy in our setting the needed estimate +∥η − Phη∥2 ≤ ch ∥∇η∥2. Note that as Phη ∈ V s +h by definition, it is in particular a Lipschitz function +and possesses a weak gradient. +Lemma A.3. Let �uh and �u be respectively the solution to (3.12a) and (2.9a) with the test function +�q ∈ �Qf +h. Then it holds +0 = +� +�Ω +�q∇δu : Mh d�x. +(A.1) +Proof. Using (3.12a) and the property of the fine constructed projection (5.10), we derive +� +�Ω +�q∇δu : Mh d�x = +� +�Ω +�q∇�uh : Mh d�x − +� +�Ω +�q∇Πf +h�u : Mh d�x = 0. +29 + +B +Appendix: Useful equalities and estimates +B.1 +Proof of the error equation (6.5) +In this part, we show the details how to obtain the equation (6.5) satisfied by the errors. First, for +any k = 1, . . . , NT we subtract the weak formulation (2.9b) from the numerical scheme (3.12b) and +get +7 +� +i=1 +T k +i = 0, +(B.1) +where T k +i reads (keeping in mind that Rk +i , i = 1, . . . , 7, are given in (6.6)) +T k +1 =ϱf +� +�Ω +(ηk +hDt�uk +h − ηk∂t�uk) · �ϕ d�x +=ϱf +� +�Ω +� +ηk +hDt(�uk +h − �uk) + ηk +h(Dt�uk − ∂t�uk) + (ηk +h − ηk)∂t�uk� +· �ϕ d�x +=ϱf +� +�Ω +ηk +hDtek +u · �ϕ d�x + Rk +1, +T k +2 =1 +2ϱf +� +�Ω +(Dtηk +h�uk∗ +h − ∂tηk�uk) · �ϕ d�x +=1 +2ϱf +� +�Ω +� +Dtηk +h(�uk∗ +h − �uk∗) + (Dtηk +h − ∂tηk)�uk∗ + ∂tηk(�uk∗ − �uk) +� +· �ϕ d�x +=1 +2ϱf +� +�Ω +Dtηk +hek∗ +u · �ϕ d�x + Rk +2, +T k +3 =1 +2ϱf +� +�Ω +� �ϕ · (∇�uk +h) − �uk +h · (∇ �ϕ) +� +· (Fk +h)−1 · �vk−1 +h +ηk +h d�x +− 1 +2ϱf +� +�Ω +� �ϕ · (∇�uk) − �uk · (∇ �ϕ) +� +· (Fk)−1 · �vkηk d�x +=1 +2ϱf +� +�Ω +� +�ϕ · (∇ek +u) − ek +u · (∇ �ϕ) +� +· (Fk +h)−1�vk−1 +h +ηk +h d�x ++ 1 +2ϱf +� +�Ω +� +�ϕ · (∇�uk) − �uk · (∇ �ϕ) +� +· +� +(Fk +h)−1�vk−1 +h +ηk +h − (Fk)−1�vkηk� +d�x +=Rk +3, +T k +4 = +� +�Ω +� +�pk +h∇ �ϕ : Mk +h − �pk∇ �ϕ : Mk� +d�x += +� +�Ω +ek +p∇ �ϕ : Mk +h d�x + +� +�Ω +�pk∇ �ϕ : +� +Mk +h − Mk� +d�x = Rk +4, +T k +5 =2µ +� +�Ω +�� +∇�uk +h(Fk +h)−1�S : +� +∇ �ϕ(Fk +h)−1� +ηk +h − +� +∇�uk(Fk)−1�S : +� +∇ �ϕ(Fk)−1� +ηk� +d�x +=2µ +� +�Ω +� +∇ek +u(Fk +h)−1�S : +� +∇ �ϕ(Fk +h)−1� +ηk +h d�x ++ 2µ +� +�Ω +�� +∇�uk(Fk +h)−1�S : (∇ �ϕ(Fk +h)−1)ηk +h − +� +∇�uk(Fk)−1�S : (∇ �ϕ(Fk)−1)ηk� +d�x +=2µ +� +�Ω +� +∇ek +u(Fk +h)−1�S : +� +∇ �ϕ(Fk +h)−1� +ηk +h d�x + Rk +5, +T k +6 =ϱs +� +Σ +(Dtξk +h − ∂tξk)ψ dx1 = ϱs +� +Σ +Dtek +ξψ dx1 + Rk +6, +T k +7 =as(ηk+1 +h +, ζk +h, ξk +h, ψ) − as(ηk, ζk, ξk, ψ) +=as(ek+1 +η +, ek+1 +ζ +, ek +ξ, ψ) + γ1 +� +Σ +∂x1(ηk+1 − ηk)∂x1ψ dx1 + γ2 +� +Σ +∂x1(ζk+1 − ζk)∂x1ψ dx1 +=as(ek+1 +η +, ek+1 +ζ +, ek +ξ, ψ) + Rk +7. +30 + +Consequently, substituting the above expansions of the Ti-terms into (B.1) and shifting the Ri-terms +to the right-hand-side, we derive (6.5). +B.2 +Preliminary estimates +In this part we show some preliminary estimates and equalities. First, we show the estimates related +to the time discretization operator Dt given by (3.2). +Lemma B.1. Let φ ∈ L2((0, T) × D) for D ∈ {Σ, �Ω}. Then we have +τ +N +� +k=1 +���Dtφk��� +2 +L2(D) +<∼ ∥∂tφ∥2 +L2((0,T)×D) , +(B.2a) +τ +N +� +k=1 +���Dtφk − ∂tφk��� +2 +L2(D) +<∼ τ 2 ��∂2 +t φ +��2 +L2((0,T)×D) , +(B.2b) +τ +N +� +k=1 +���Dtφk+1 − ∂tφk��� +2 +L2(D) +<∼ τ 2 ��∂2 +t φ +��2 +L2((0,T)×D) . +(B.2c) +Proof. We only need to prove (B.2c), the others follow analogously.2 To begin, we recall the product +rule (uv)′ = u′v + uv′ and apply it for u = (tk+1 − t)/τ and v = ∂tφ(t), which yields +� tk+1 +tk +tk+1 − t +τ +∂2 +t φ(t) dt = +�tk+1 − t +τ +∂tφ(t) +�tk+1 +tk ++ 1 +τ +� tk+1 +tk +∂tφ(t) dt = Dtφk+1 − ∂tφ(tk). +Thanks to the above equality and H¨older’s inequality, we obtain +τ +N +� +k=1 +���Dtφk+1 − ∂tφk��� +2 +L2(D) = τ +N +� +k=1 +� +D +����� +� tk+1 +tk +tk+1 − t +τ +∂2 +t φ(t) dt +����� +2 +dx +≤ τ +N +� +k=1 +� +D +�� tk+1 +tk +���� +tk+1 − t +τ +���� +2 +dt +� �� tk+1 +tk +��∂2 +t φ(t) +��2 dt +� +dx +≤ 1 +3τ 2 +� +D +N +� +k=1 +� tk+1 +tk +��∂2 +t φ(t) +��2 dtdx = 1 +3τ 2 ��∂2 +t φ(t) +��2 +L2((0,T)×D) , +which proves (B.2c). +Lemma B.2. Let η ∈ W 2,2(Σ), ξ = ∂tη, ζ = −∂2 +x1η, ηh ∈ V s +h , ξk +h = Dtηk+1 +h +, k = 1, . . . , NT , +ζh = −∂2 +x1,hηh, and ψ ∈ V s +h . Let as be given by (2.7) and the notation of the errors be given by (6.3). +Then +δk +ξ = Dtδk+1 +η ++ Rs +h(Dtηk+1 − ∂tηk), +(B.3) +δζ = −∂2 +x1,hδη, +� +Σ +ψδζ dx1 = − +� +Σ +∂2 +x1,hψ δη dx1. +(B.4) +� +Σ +∂x1Iη∂x1ψ dx1 = 0, +� +Σ +∂x1Iξ∂x1ψ dx1 = 0, +(B.5) +� +Σ +Iζψ dx1 = 0, +� +Σ +∂x1Iζ∂x1ψ dx1 = 0, +(B.6) +� +Σ +∂x1δζ∂x1ψ dx1 = +� +Σ +∂2 +x1,hδη∂2 +x1,hψ dx1, +(B.7) +2For (B.2a) and (B.2b) and p = 2 = q a proof can be found in Lemma 5.4 and Lemma 5.3 of [10], respectively. +31 + +� +Σ +∂x1δk+1 +η +∂x1δk +ξ dx1 = +� +Σ +� +Dt +|∂x1δk+1 +η +|2 +2 ++ τ +2|Dt∂x1δk+1 +η +|2 +� +dx1 ++ +� +Σ +∂x1δk+1 +η +∂x1(Dtηk+1 − ∂tηk) dx1, +(B.8) +� +Σ +∂x1δk+1 +ζ +∂x1δk +ξ dx1 = +� +Σ +� +Dt +|δk+1 +ζ +|2 +2 ++ τ +2|Dtδk+1 +ζ +|2 +� +dx1 +− +� +Σ +δk+1 +ζ +∂2 +x1(Dtηk+1 − ∂tηk) dx1, +(B.9) +as(ek+1 +η +, ek+1 +ζ +, ek +ξ, δk +ξ ) = Dt +� +Σ +�γ1 +2 |∂x1δk+1 +η +|2 + γ2 +2 |δk+1 +ζ +|2� +dx1 + γ3 +� +Σ +|δk +ξ |2 dx1 ++ τ +2 +� +Σ +� +γ1|Dt∂x1δk+1 +η +|2 + γ2|Dtδk+1 +ζ +|2� +dx1 ++ +� +Σ +� +γ1∂x1δk+1 +η +∂x1(Dtηk+1 − ∂tηk) − γ2δk+1 +ζ +∂2 +x1(Dtηk+1 − ∂tηk) +� +dx1. +(B.10) +Proof. Recalling equalities +ξk = ∂tηk, ξk +h = Dtηk+1 +h +and the errors defined in (6.3) we get (B.3) +δk +ξ − Dtδk+1 +η += (ξk +h − Rs +hξk) − (Dtηk+1 +h +− DtRs +hηk+1) += DtRs +hηk+1 − Rs +hξk = Rs +h(Dtηk+1 − ∂tηk). +Moreover, it is easy to check +δζ = ζh + ∂2 +x1,hRs +hη = ∂2 +x1,hRs +hη − ∂2 +x1,hηh = −∂2 +x1,hδη. +Further, recalling the discrete Laplace operator, we complete the proof of (B.4), i.e., +δζ = ζh + ∂2 +x1,hRs +hη = ∂2 +x1,hRs +hη − ∂2 +x1,hηh = −∂2 +x1,hδη, +� +Σ +ψδζ dx1 = − +� +Σ +ψ∂2 +x1,hδη dx1 = +� +Σ +∂x1ψ∂x1δη dx1 = − +� +Σ +∂2 +x1,hψ δη dx1. +Next, recalling the Riesz projection (5.3) we immediate get (B.5) +� +Σ +∂x1Iη∂x1ψ dx1 = +� +Σ +∂x1(Rs +hη − η)∂x1ψ dx1 = 0, +� +Σ +∂x1Iξ∂x1ψ dx1 = +� +Σ +∂x1(Rs +hξ − ξ)∂x1ψ dx1 = 0. +Analogously, we find +� +Σ +Iζψ dx1 = +� +Σ +(∂2 +x1η − ∂2 +x1,hRs +hη)ψ dx1 = +� +Σ +(∂x1Rs +hη − ∂x1η)∂x1ψ dx1 = 0. +Then, setting ∂2 +x1,hψ in the above equality, we get (B.6) +� +Σ +∂x1Iζ∂x1ψ dx1 = − +� +Σ +Iζ∂2 +x1,hψ dx1 = 0. +Further, recalling (3.11) and (5.4) we get (B.7) +� +Σ +∂x1δζ∂x1ψ dx1 = − +� +Σ +δζ∂2 +x1,hψ dx1 = − +� +Σ +(ζh + ∂2 +x1,hRs +hη)∂2 +x1,hψ dx1 += +� +Σ +(∂2 +x1,hηh − ∂2 +x1,hRs +hη)∂2 +x1,hψ dx1 = +� +Σ +∂2 +x1,hδη∂2 +x1,hψ dx1 +32 + +Using the above equalities (B.3) and (B.7), the algebraic inequality (4.2), and the Riesz projection +(5.3) we get (B.8) +� +Σ +∂x1δk+1 +η +∂x1δk +ξ dx1 = +� +Σ +∂x1δk+1 +η +� +∂x1Dtδk+1 +η ++ ∂x1Rs +h(Dtηk+1 − ∂tηk) +� +dx1 += +� +Σ +� +Dt +|∂x1δk+1 +η +|2 +2 ++ τ +2|Dt∂x1δk+1 +η +|2 +� +dx1 + +� +Σ +∂x1δk+1 +η +∂x1(Dtηk+1 − ∂tηk) dx1 +Analogously, using the above equalities (B.3), (B.4), and (B.7), the algebraic inequality (4.2), and the +Riesz projection (5.4), we get (B.9) +� +Σ +∂x1δk+1 +ζ +∂x1δk +ξ dx1 = − +� +Σ +δk+1 +ζ +∂2 +x1,hδk +ξ dx1 = +� +Σ +∂2 +x1,hδk+1 +η +∂2 +x1,hδk +ξ dx1 += +� +Σ +∂2 +x1,hδk+1 +η +∂2 +x1,hDtδk+1 +η +dx1 + +� +Σ +∂2 +x1,hδk+1 +η +∂2 +x1,hRs +h(Dtηk+1 − ∂tηk) dx1 += +� +Σ +� +Dt +|∂2 +x1,hδk+1 +η +|2 +2 ++ τ +2|Dt∂2 +x1,hδk+1 +η +|2 +� +dx1 + +� +Σ +∂2 +x1,hδk+1 +η +∂2 +x1(Dtηk+1 − ∂tηk) dx1 += +� +Σ +� +Dt +|δk+1 +ζ +|2 +2 ++ τ +2|Dtδk+1 +ζ +|2 +� +dx1 − +� +Σ +δk+1 +ζ +∂2 +x1(Dtηk+1 − ∂tηk) dx1. +Using the equalities (B.5) and (B.6) we know that +as(Ik+1 +η +, Ik+1 +ζ +, Ik +ξ , ψ) = 0. +Consequently, collecting the above equality together with (B.8) and (B.9), we get +as(ek+1 +η +, ek+1 +ζ +, ek +ξ, δk +ξ ) = as(δk+1 +η +, δk+1 +ζ +, δk +ξ , δk +ξ ) += γ1 +� +Σ +∂x1δk+1 +η +∂x1δk +ξ dx1 + γ2 +� +Σ +∂x1δk+1 +ζ +∂x1δk +ξ dx1 + γ3 +� +Σ +∂x1δk +ξ ∂x1δk +ξ dx1 += γ1 +� +Σ +� +Dt +|∂x1δk+1 +η +|2 +2 ++ τ +2|Dt∂x1δk+1 +η +|2 +� +dx1 + γ1 +� +Σ +∂x1δk+1 +η +∂x1(Dtηk+1 − ∂tηk) dx1 ++ γ2 +� +Σ +� +Dt +|δk+1 +ζ +|2 +2 ++ τ +2|Dtδk+1 +ζ +|2 +� +dx1 − γ2 +� +Σ +δk+1 +ζ +∂2 +x1(Dtηk+1 − ∂tηk) dx1 + γ3 +� +Σ +|δk +ξ |2 dx1 += Dt +� +Σ +�γ1 +2 |∂x1δk+1 +η +|2 + γ2 +2 |δk+1 +ζ +|2� +dx1 + γ3 +� +Σ +|δk +ξ |2 dx1 ++ τ +2 +� +Σ +� +γ1|Dt∂x1δk+1 +η +|2 + γ2|Dtδk+1 +ζ +|2� +dx1 ++ +� +Σ +� +γ1∂x1δk+1 +η +∂x1(Dtηk+1 − ∂tηk) − γ2δk+1 +ζ +∂2 +x1(Dtηk+1 − ∂tηk) +� +dx1, +which proves (B.10) and completes the proof. +B.3 +Secondary estimates +Lemma B.3. Let (p, u, ξ, η) be a target smooth solution of the FSI problem (1.1)–(1.4) belonging to +the class (6.1). Let (ph, �uh, ξh, ηh) be a solution to the numerical scheme (3.13). Then, +∥F∥L∞L∞ + +��F−1�� +L∞L∞ + ∥M∥L∞L∞ + +��M−1�� +L∞L∞ +<∼ 1, +(B.11) +∥eη∥LγL∞ ≤ ∥∂x1δη∥LγL2 + h +��∂2 +x1η +�� +LγL2 , 1 ≤ γ ≤ ∞, +(B.12) +∥∂x1eη∥L2L∞ +<∼ ∥δζ∥L2L2 + h +��∂3 +x1η +�� +L2L2 +(B.13) +33 + +and +∥Mh − M∥L2L∞ +<∼ ∥∂x1δη∥L2L2 + ∥δζ∥L2L2 + h +���∂2 +x1η +�� +L2L2 + +��∂3 +x1η +�� +L2L2 +� +. +(B.14) +Further +∥Fh − F∥L2L∞ +<∼ ∥∂x1δη∥L2L2 + ∥δζ∥L2L2 + h +���∂2 +x1η +�� +L2L2 + +��∂3 +x1η +�� +L2L2 +� +(B.15) +and +��F−1 +h +− F−1�� +L2L∞ +<∼1 + ∥∂x1η∥L∞L∞ +η2 +∥∂x1δη∥L2L2 + 1 +η ∥δζ∥L2L2 ++ h +�1 + ∥∂x1η∥L∞L∞ +η2 +��∂2 +x1η +�� +L2L2 + 1 +η +��∂3 +x1η +�� +L2L2 +� +. +(B.16) +For p ∈ [1, ∞) we find +���ξk +h +��� +Lp ≤ +���∇uk +h +��� +L2 , +(B.17) +and +∥Dtδη∥L2Lp +<∼ ∥δξ∥L2Lp + τ +��∂2 +t ∂xη +�� +L2L2 . +(B.18) +For p ∈ [1, ∞) we find +∥�uh − �u∥L2Lp +<∼ ∥δu∥L2Lp + h ∥�u∥L2W 2,2 , +(B.19) +∥�wh − �w∥L2Lp +<∼ ∥δξ∥L2Lp + τ +��∂2 +t ∂xη +�� +L2L2 + h ∥η∥L2W 2,2 , +(B.20) +and finally +ϱfτ +m +� +k=1 +� � +�Ω +|�vk−1 +h +ηk +h − Fk +h(Fk)−1�vkηk|2 d�x +� 2 +p <∼ τ +m +� +k=1 +α +� +�Ω +|∇δk +u|2ηk +h d�x + 1 +α +� +�Ω +|δk +u|2ηk +h d�x ++ ∥δξ∥2 +L2Lp + c1 ∥δη∥2 +L2Lp + c2 ∥∂x1δη∥2 +L2Lp + c3τ 2 + c4h2, +(B.21) +where +c1 = ϱf/η ∥�v∥2 +L∞L∞ +� +1 + ∥M∥2 +L∞L∞ +� +, +c2 = ϱf/η ∥M∥2 +L∞L∞ ∥�v∥2 +L∞L∞ , +c3 = ϱfη +� +∥∂t�u∥2 +L2Lp + +��∂2 +t ∂xη +��2 +L2L2 +� +, +c4 = c1 ∥∂x1η∥L2Lp + c2 +��∂2 +x1η +�� +L2L2 + ϱfη ∥∇�u∥2 +L2Lp . +(B.22) +Proof. Here we shall frequently recall the estimates (4.11). First, (B.11) is obvious as ηh and η are +bounded from above and below by positive constants, as well as ∂x1η and ∂x1ηh are bounded from +above. +By the triangular inequality and the Sobolev inequality we get (B.12) +∥ηh − η∥LγL∞ ≤ ∥δη∥LγL∞ + ∥Iη∥LγL∞ +<∼ ∥∂x1δη∥LγL2 + h ∥∂x1η∥LγL∞ +<∼ ∥∂x1δη∥LγL2 + h +��∂2 +x1η +�� +LγL2 . +Analogously, we have (B.13) +∥∂x1ηh − ∂x1η∥L2L∞ ≤ ∥∂x1δη∥L2L∞ + ∥∂x1Iη∥L2L∞ +<∼ +��∂2 +x1,hδη +�� +L2L2 + h +��∂2 +x1η +�� +L2L∞ += ∥δζ∥L2L2 + h +��∂2 +x1η +�� +L2L∞ +<∼ ∥δζ∥L2L2 + h +��∂3 +x1η +�� +L2L2 . +Recalling the definition of M and Mh, using triangular inequality and the estimates (B.12) and (B.13) +we obtain (B.14) +∥Mh − M∥L2L∞ = +���� +� eη +−�x2∂x1eη +0 +0 +����� +L2L∞ +<∼ ∥∂x1δη∥L2L2 + ∥δζ∥L2L2 + h +���∂2 +x1η +�� +L2L2 + +��∂3 +x1η +�� +L2L2 +� +. +34 + +Analogously, we get (B.15) +∥Fh − F∥L2L∞ = +���� +� +0 +0 +−�x2∂x1eη +eη +����� +L2L∞ +<∼ ∥∂x1δη∥L2L2 + ∥δζ∥L2L2 + h +���∂2 +x1η +�� +L2L2 + +��∂3 +x1η +�� +L2L2 +� +, +and (B.16) +��F−1 +h +− F−1�� +L2L∞ = +����� +� +0 +0 +−�x2 +∂x1ηh +ηh ++ �x2 +∂x1η +η +1 +ηh − 1 +η +������ +L2L∞ +≤ +���� +ηh∂x1η − η∂x1ηh +ηηh +���� +L2L∞ + +���� +η − ηh +ηηh +���� +L2L∞ = +���� +eη∂x1η + η∂x1eη +ηηh +���� +L2L∞ + +���� +eη +ηηh +���� +L2L∞ +<∼ 1 + ∥∂x1η∥L∞L∞ +η2 +∥eη∥L2L∞ + 1 +η ∥∂x1eη∥L2L∞ +<∼ 1 + ∥∂x1η∥L∞L∞ +η2 +� +∥∂x1δη∥L2L2 + h ∥∂x1η∥L2L∞ +� ++ 1 +η +� +∥δζ∥L2L2 + h +��∂2 +x1η +�� +L2L∞ +� +<∼ 1 + ∥∂x1η∥L∞L∞ +η2 +∥∂x1δη∥L2L2 + 1 +η ∥δζ∥L2L2 ++ h +�1 + ∥∂x1η∥L∞L∞ +η2 +∥∂x1η∥L2L∞ + 1 +η +��∂2 +x1η +�� +L2L∞ +� +. +The estimate (B.17) is a consequence of the trace estimate and Sobolev embedding in 1-D. For (B.18) +we first recall (B.3) and the triangular inequality to get +���Dtδk+1 +η +��� +Lp ≤ +���δk +ξ +��� +Lp + +���Rs +h(Dtηk+1 − ∂tηk) +��� +Lp . +Then the estimate follows from the continuity of Rs +h and Lemma B.1 as +���Rs +h(Dtηk+1 − ∂tηk) +��� +L2Lp +<∼ +���∂x1(Dtηk+1 − ∂tηk) +��� +L2L2 +<∼ τ +��∂2 +t ∂x1η +�� +L2L2 . +The proof of (B.19) is by Sobolev embedding and the interpolation estimate, +∥�uh − �u∥L2Lp ≤ ∥δu∥L2Lp + ∥Iu∥L2Lp +<∼ ∥δu∥L2Lp + ∥∇Iu∥L2L2 +<∼ ∥δu∥L2Lp + h ∥�u∥L2W 2,2 . +Recalling the definition of �wh and �w, using again Sobolev embedding, Lemma B.1, the estimate (B.18) +and the interpolation inequality we get (B.20): +∥�wh − �w∥L2Lp +<∼ ∥Dtηh − ∂tη∥L2Lp +≤ ∥Dt(ηh − η)∥L2Lp + ∥Dtη − ∂tη∥L2Lp +<∼ ∥Dtδη∥L2Lp + ∥DtIη∥L2Lp + ∥Dt∂x1η − ∂t∂x1η∥L2L2 +<∼ ∥Dtδη∥L2Lp + ∥∇DtIη∥L2L2 + τ +��∂2 +t ∂x1η +�� +L2L2 +<∼ ∥δξ∥L2Lp + h ∥η∥L2W 2,2 + τ +��∂2 +t ∂x1η +�� +L2L2 . +By the triangular inequality, (B.12), (B.19), and (B.20) we get (B.21) +ϱfτ +m +� +k=1 +� � +�Ω +|�vk−1 +h +ηk +h − Fk +h(Fk)−1�vkηk|p d�x +� 2 +p += ϱfτ +m +� +k=1 +� � +�Ω +|(�vk−1 +h +− �vk)ηk +h + �vk(ηk +h − ηk) + (I − Fk +h(Fk)−1)�vkηk|p d�x +� 2 +p +≤ ϱfτ +m +� +k=1 +� � +�Ω +| − τDt�uk +h + ek +u + �wk +h − �wk|pηk +h d�x +� 2 +p ++ ϱfτ +m +� +k=1 +� � +�Ω +|�vk(ηk +h − ηk)|p/ηk +h d�x +� 2 +p +35 + ++ ϱfτ +m +� +k=1 +� � +�Ω +|(I − Fk +h(Fk)−1)�vkηk|p/ηk +h d�x +� 2 +p +<∼ τ +m +� +k=1 +� � +�Ω +ϱf|δk +u|pηk +h d�x +� 2 +p ++ ∥δξ∥2 +L2Lp + c1 ∥δη∥2 +L2Lp + c2 ∥∂x1δη∥2 +L2Lp + c3τ 2 + c4h2, +where ci, i = 1, 2, 3, 4 are given above in (B.22), interpolation implies for 2 +p = θ, p < ∞ that +� � +�Ω +ϱf|δk +u|pηk +h d�x +� 2 +p <∼ +���∇δk +u +��� +(1−θ)2 +L2 +���δk +u +��� +2θ +L2 +<∼ α +���∇δk +u +��� +2 +L2 + 1 +α +���δk +u +��� +2 +L2 , +which closes the proof. +B.4 +Proof of estimates (6.10) +Proof. First, by Young’s inequality, the interpolation error, Theorem 5.6, and the uniform bounds +(4.11), we find +|Gf| = +���τ +m +� +k=1 +� +�Ω +ϱf +� +ηk +hDtIk +u + 1 +2Dtηk +hIk∗ +u +� +· δk +u d�x ++ 2µτ +m +� +k=1 +� +�Ω +� +∇Ik +u(Fk +h)−1�S : (∇δk +u(Fk +h)−1)ηk +h d�x +���, +<∼ +m +� +k=1 +τ +� +�Ω +ϱfηk +h|δk +u|2 d�x + 2αµ +m +� +k=1 +τ +� +�Ω +|∇δk +u(Fk +h)−1|2ηk +h d�x + ch2 +(B.23) +for any fixed α > 0, where +c = 1 +4ϱf +� +η ∥∂t∇�u∥2 +L2L2 + 1 +4η ∥ξh∥2 +L2L∞ ∥∇�u∥2 +L∞L2 +� ++ 2µη +4α ∥Fh∥L∞L∞ ∥�u∥L2W 2,2 . +Note that this constant is indeed bounded by the stability of the discrete solution, as ∥ξh∥2 +L2L∞ can +be bounded by (B.17). Second, by Young’s inequality, the time discretization error (B.2c), we can +control Gs in the following way. +|Gs| = +���γ1τ +m +� +k=1 +� +Σ +∂x1δk+1 +η +∂x1(Dtηk+1 − ∂tηk) dx1 − γ2τ +m +� +k=1 +� +Σ +δk+1 +ζ +∂2 +x1(Dtηk+1 − ∂tηk) dx1 ++ τ +m +� +k=1 +� +Σ +ϱsDtIξδk +ξ dx1 +��� +<∼ +m +� +k=1 +τ +� +Σ +� +γ1|∂x1δk+1 +η +|2 + γ2|δk+1 +ζ +|2 + ϱs|δk +ξ |2� +dx1 + 1 +4ϱs ∥∂tIξ∥2 +L2L2 ++ 1 +4 +m +� +k=1 +τ +� +Σ +� +γ1|Dt∂x1ηk+1 − ∂t∂x1ηk|2 + γ2|Dt∂2 +x1ηk+1 − ∂t∂2 +x1ηk|2� +dx1 +<∼ +m +� +k=1 +τ +� +Σ +� +γ1|∂x1δk+1 +η +|2 + γ2|δk+1 +ζ +|2 + ϱs|δk +ξ |2� +dx1 + c1τ 2 + c2h2, +(B.24) +where +c1 = 1 +4γ1 +��∂2 +t ∂x1η +��2 +L2((0,T)×Σ) + 1 +4γ2 +��∂2 +t ∂2 +x1η +��2 +L2((0,T)×Σ) , +c2 = 1 +4ϱs ∥∂t∂x1ξ∥2 +L2((0,T)×Σ) = 1 +4ϱs +��∂2 +t ∂x1η +��2 +L2((0,T)×Σ) . +Next, we analyze the Ri terms. +36 + +Rk +1-term +�����τ +m +� +k=1 +Rk +1 +����� = +�����τ +m +� +k=1 +ϱf +� +�Ω +� +ek +η∂t�uk + ηk +h(Dt�uk − ∂t�uk) +� +· δk +u d�x +����� +≤ τ +m +� +k=1 +ϱf +� +�Ω +ηk +h|δk +u|2 d�x + 1 +2τ +m +� +k=1 +ϱf +� +�Ω +|ek +η|2/ηk +h|∂t�uk|2 d�x ++ 1 +2τ +m +� +k=1 +ϱf +� +�Ω +ηk +h|Dt�uk − ∂t�uk|2 d�x +≤ τ +m +� +k=1 +ϱf +� +�Ω +ηk +h|δk +u|2 d�x + c1τ +m +� +k=1 +� +Σ +|∂x1δk +η|2 dx1 + c2h2 + c3τ 2, +where we have used (B.12) and the constants read +c1 = ϱf +η ∥∂t�u∥2 +L∞(0,T;L2(�Ω;Rd)) , +c2 = ϱf +η ∥∂x1η∥2 +L2(0,T;L∞(Σ)) ∥∂t�u∥2 +L∞(0,T;L2(�Ω;Rd)) , +c3 = ϱfη +2 +��∂2 +t �u +��2 +L2((0,T)×�Ω;Rd) . +Rk +2-term +�����τ +m +� +k=1 +Rk +2 +����� = +���� +1 +2ϱf +� +�Ω +� +(ek−1 +ξ +− τDtξk)�uk∗ − τ∂tηkDt�uk� +· δk +u d�x +���� +≤ τ +m +� +k=1 +ϱf +� +�Ω +ηk +h|δk +u|2 d�x + τ 2 ϱf +2η ∥∂t�u∥2 +L∞(0,T;L2(�Ω;Rd)) ∥∂tη∥2 +L2(0,T;L∞(Σ)) ++ ϱf +η +� +τ 2 ∥∂tξ∥2 +L2((0,T)×Σ) + τ +m +� +k=1 +� +Σ +|δk−1 +ξ ++ Ik−1 +ξ +|2 dx1 +� +∥�u∥L∞((0,T)×�Ω;Rd) +<∼ τ +m +� +k=1 +ϱf +� +�Ω +ηk +h|δk +u|2 d�x + τ +m +� +k=1 +� +Σ +|δk +ξ |2 dx1 + c1τ 2 + c2h2, +where +c1 = ϱf +2η ∥∂t�u∥2 +L∞(0,T;L2(�Ω;Rd)) ∥∂tη∥2 +L2(0,T;L∞(Σ)) + ϱf +η +��∂2 +t η +��2 +L2((0,T)×Σ) ∥�u∥L∞((0,T)×�Ω;Rd) , +c2 = ϱf +η ∥∂t∂x1η∥L2((0,T)×Σ) ∥�u∥L∞((0,T)×�Ω;Rd) . +Rk +3-term +For this term we recall Ladyzenskaja’s estimate in 2D +∥f∥2 +L4 ≤ ∥∇f∥L2 ∥f∥L2 . +(B.25) +We use it to find by the previous arguments that +����τ +m +� +k=1 +Rk +3 +���� ≤ +����τ +m +� +k=1 +1 +2ϱf +� +�Ω +� +δk +u · (∇Ik +u) − Ik +u · (∇δk +u) +� +· (Fk +h)−1�vk−1 +h +ηk +h d�x ++ τ +m +� +k=1 +1 +2ϱf +� +�Ω +� +δk +u · (∇�uk) − �uk · (∇δk +u) +� +· (Fk +h)−1 � +�vk−1 +h +ηk +h − Fk +h(Fk)−1�vkηk� +d�x +���� +<∼ τ +m +� +k=1 +� +�Ω +|δk +u||∇Ik +u||�vk−1 +h +| + |δk +u||∇uk||�vk−1 +h +ηk +h − Fk +h(Fk)−1�vk| d�x +37 + ++ τ +m +� +k=1 +� +�Ω +|∇δk +u||Ik +u||�vk−1 +h +| + |∇δk +u||uk||�vk−1 +h +ηk +h − Fk +h(Fk)−1�vk| d�x +<∼ τ +m +� +k=1 +����vk−1 +h +��� +L4 +� ���δk +u +��� +L4 +���∇Ik +u +��� +L2 + +���∇δk +u +��� +L2 +���Ik +u +��� +L4 +� ++ τ +m +� +k=1 +����vk−1 +h +ηk +h − Fk +h(Fk)−1�vk��� +L4 +� ���δk +u +��� +L2 +���∇uk��� +L4 + +���∇δk +u +��� +L2 +���uk��� +L4 +� +<∼ c5h2τ +α +m +� +k=1 +����vk +h +��� +2 +W 1,2 + ατ +m +� +k=1 +���∇δk +u +��� +2 +L2 ++ τ +m +� +k=1 +����vk−1 +h +ηk +h − Fk +h(Fk)−1�vk��� +L4 +� ���δk +u +��� +L2 +���uk��� +W 2,2 + +���∇δk +u +��� +L2 +� +, +with +c5 = +��∇2u +��2 +L2(L2) sup +k +����vk��� +2 +L2 . +Now (B.21) and Young’s inequality implies that +����τ +m +� +k=1 +Rk +3 +���� +<∼ ατ +m +� +k=1 +���∇δk +u +��� +2 +L2 + 1 +α +� +�Ω +|δk +u|2ηk +h d�x + ∥δξ∥2 +L2L4 ++ c1 ∥δη∥2 +L2L4 + c2 ∥∂x1δη∥2 +L2L4 + c3τ 2 + ˜c4h2 +for c1, c2, c3, c4 given in (B.22) with +˜c4 = c4 + c5h2τ +α +m +� +k=1 +����vk +h +��� +2 +W 1,2 + ∥u∥2 +L2W 2,2 . +Finally by using (B.17) we estimate +���δk +ξ +��� +2 +L4 +<∼ +���δk +ξ +��� +L2 +���δk +ξ +��� +L∞ +<∼ +���δk +ξ +��� +L2 +���∇δk +u +��� +L2 +<∼ α +���∇δk +u +��� +2 +L2 + 1 +α +���δk +ξ +��� +2 +L2 , +With that and Sobolev embedding we conclude that +����τ +m +� +k=1 +Rk +3 +���� +<∼ ατ +m +� +k=1 +���∇δk +u +��� +2 +L2 + 1 +α +� +�Ω +|δk +u|2ηk +h d�x + 1 +α +���δk +ξ +��� +2 +L2L2 ++ c1 ∥∂x1δη∥2 +L2L2 + c2 +��∂2 +x1δη +��2 +L2L2 + c3τ 2 + ˜c4h2. +Rk +4-term +By using the identity (A.1), Young’s inequality, H¨older’s inequality, and the interpolation +error (5.2), we obtain +�����τ +m +� +k=1 +Rk +4 +����� = +�����τ +m +� +k=1 +� +�Ω +ek +p∇δk +u : Mk +h d�x + +� +�Ω +�pk∇δk +u : +� +Mk +h − Mk� +d�x +����� += +�����τ +m +� +k=1 +� +�Ω +� +Ik +p ∇δk +u : Mk +h + �pk∇δk +u : +� +Mk +h − Mk�� +d�x +����� +<∼ 2αµτ +m +� +k=1 +� +�Ω +|∇δk +u(Fk +h)−1|2ηk +h d�x + +η +4αµ ∥Ip∥2 +L2((0,T)×�Ω) + ++ +1 +4αµη ∥�p∥2 +L∞L∞ ∥Fh∥2 +L∞L∞ ∥Mh − M∥2 +L2(0,T;L∞(�Ω)) +38 + +<∼ 2αµτ +m +� +k=1 +� +�Ω +|∇δk +u(Fk +h)−1|2ηk +h d�x + c1 +� +∥∂x1δη∥2 +L2((0,T)×Σ) + +��∂2 +x1δη +��2 +L2((0,T)×Σ) +� ++ c2h2, +where we have used (B.14) and the constants read +c1 = +1 +4αµη ∥�p∥2 +L∞L2 ∥Fh∥2 +L∞((0,T)×�Ω;Rd×d) , +c2 = +1 +4αµη ∥∇�p∥2 +L2L2 + c1 +� +∥∂x1η∥2 +L2(0,T;L∞(Σ)) + +��∂2 +x1η +��2 +L2(0,T;L∞(Σ)) +� +Remark B.4. Thanks to the nice interpolation operator which produces the divergence-free condition +(5.10) with the covariance Mh instead of M. Otherwise, we would lose the equality (A.1) and has to +estimate τ �m +k=1 +� +�Ω δk +p∇δk +u : Mk +h d�x, in which the pressure error δp is not available in our setting. +Rk +5-term +Applying Young’s inequality, H¨older’s inequality, (B.14), and (B.16), we obtain +�����τ +m +� +k=1 +Rk +5 +����� = +�����τ +m +� +k=1 +� +�Ω +�� +∇�uk(Fk +h)−1�S : +� +∇δk +u(Fk +h)−1ηk +h +� +− +� +∇�uk(Fk)−1�S : +� +∇δk +u(Fk)−1ηk�� +d�x +����� +≤ +�����τ +m +� +k=1 +� +�Ω +� +∇�uk� +(Fk +h)−1 − (Fk)−1��S +: +� +∇δk +u(Fk +h)−1ηk +h +� +d�x +����� ++ +�����τ +m +� +k=1 +� +�Ω +� +∇�uk(Fk)−1�S : +� +∇δk +u(Fk +h)−1(ηk +h − Fk +h(Fk)−1ηk� +d�x +����� +<∼ ατ +m +� +k=1 +� +�Ω +���� +� +∇δk +u(Fk +h)−1�S���� +2 +ηk +h d�x + η +2α ∥∇�u∥2 +L∞L2 +��(Fh)−1 − (F)−1��2 +L2L∞ ++ η +2α ∥∇�u∥2 +L∞L2 +��F−1��2 +L∞L∞ ∥Fh∥2 +L∞L∞ +��MT +h − MT��2 +L2L∞ +<∼ ατ +m +� +k=1 +� +�Ω +���� +� +∇δk +u(Fk +h)−1�S���� +2 +ηk +h d�x + c1 ∥∂x1δη∥2 +L2((0,T)×Σ) + c2 ∥δζ∥2 +L2((0,T)×Σ) + c3h2 +where +c1 = η +2α ∥∇�u∥2 +L∞L2 +���F−1��2 +L∞L∞ ∥Fh∥2 +L∞L∞ + 1 + ∥∂x1η∥L∞L∞ +η2 +� +, +c2 = η +2α ∥∇�u∥2 +L∞L2 +���F−1��2 +L∞L∞ ∥Fh∥2 +L∞L∞ + 1 +η +� +, +c3 =c1 +��∂2 +x1η +��2 +L2((0,T)×Σ) + c2 +��∂3 +x1η +��2 +L2((0,T)×Σ) . +Rk +6-term +By Young’s inequality and (B.2b) we obtain +|τ +m +� +k=1 +Rk +6| = +�����τ +m +� +k=1 +ϱs +� +Σ +(Dtξk − ∂tξk)δk +ξ dx1 +����� +<∼ τ 2 +4ϱs +��∂2 +t ξ +��2 +L2((0,T)×Σ) + τ +m +� +k=1 +� +Σ +ϱs|δξ|2 dx1. +Rk +7-term. +�����τ +m +� +k=1 +Rk +7 +����� = +�����τ +m +� +k=1 +γ1 +� +Σ +∂x1(ηk+1 − ηk)∂x1δk +ξ dx1 + τ +m +� +k=1 +γ2 +� +Σ +∂x1(ζk+1 − ζk)∂x1δk +ξ dx1 +����� +<∼ τ +m +� +k=1 +τ +� ���∂2 +x1Dtηk��� + +���∂2 +x1Dtζk��� +L2 +� ���δk +ξ +��� +L2 +39 + +<∼ τ +m +� +k=1 +���δk +ξ +��� +2 +L2 + τ 2( +��∂4 +x1ξ +��2 +L2L2 + +��∂2 +x1ξ +��2 +L2L2) +Consequently, collecting all the above estimates we get +�����τ +m +� +k=1 +7 +� +i=1 +Rk +i + Gf + Gs +����� +<∼ τ 2 + h2 + cτ +m +� +k=1 +δk +E + 2αµτ +m +� +k=1 +� +�Ω +���∇δk +u(Fk +h)−1��� +2 +ηk +h d�x, +which proves (6.10). +C +Numerical Implementation +In this appendix we provide details of the numerical implementations of semi-implicit Scheme-R (3.12) +and monolithic fully implicit (2.9) both computed on the reference domain �Ω and both implemented +using FEniCS finite element method [1]. Here, let us point out that, instead of implementing the +height of the structure η, we take a shift η = η − 1 (independent of �x2) and then linearly extend it +to the whole domain via η = η�x2. Moreover, the structure velocity ξ on Γ is directly replaced by the +second component of the fluid velocity ξ = u2. Further, instead of ζ we shall use z as the second order +derivative of the new η. Hereinafter, we shall frequently drop the superscript “�” for simplicity of the +notation. +C.1 +Implementation of semi-implicit Scheme-R +We implemented Scheme-R (3.12), the monolithic method on the reference domain �Ω. The domain �Ω +is approximated by regular triangles K ∈ Th with the typical mesh size h. The problem comprises four +global unknowns: velocity u, pressure p, mesh displacement in x2−direction η and its second order +derivative z. The velocity-pressure pair is approximated with the inf-sup compatible MINI element [3], +where the velocity is approximated by the piecewise linear continuous elements enlarged with the cubic +bubbles, mesh displacement by the same elements as the velocity and the second order derivative of +the mesh displacement is approximated by piecewise linear elements, for the definitions of the discrete +function spaces see (3.7). +For the time stepping we use a backward Euler method with a fixed time step τ, we denote by +uk, zk, pk and ηk the unknowns at the kth time step, i.e. at time t = kτ and u0, z0, p0, η0 are prescribed +initial conditions (in our case equal to zero). Since in case of zero initial conditions, it holds η1 = η0, +we may shift the time index k (resp. k + 1) to k − 1 (resp. k) for the structure variables. This +semi-implicit scheme is linear and the corresponding system of linear equations is solved with the +direct solver MUMPS [2]. Components of the velocity u are denoted by (u1, u2). +Displacement ηk is computed explicitly using the y-component of the velocity uk +2 on the top bound- +ary Γ, i.e. +ηk = ηk−1 + τ uk +2 +on Γ. +The following quantities are used in the discretized weak form. The deformation gradient F is obtained +from the displacement, J is its determinant and ˙J its time derivative. All are evaluated at the (k−1)st +time level +Fk−1 = I + ∇(0, ηk−1)T, +Jk−1 = det(Fk−1), +˙Jk−1 = DtJk−1 = Jk−1 − Jk−2 +τ +. +Finally, v is the relative velocity of the fluid and �T is the Cauchy stress tensor after the ALE trans- +formation +vk−1 = uk−1 − (0, ξk−1)T, +�Tk = −pkI + 2µ +� +∇uk(Fk−1)−1�S +, +where ξk−1 = Dtηk−1 is the mesh velocity that is computed after Step 2 when the displacement η is +prolongated into the whole domain �Ω. +40 + +The whole simulation consists of two steps. In Step 1 we solve for velocity u, its Laplace z and +pressure p, explicitly compute the value of η on the top boundary Γ and in Step 2 we linearly expand +it to the whole domain �Ω. +Step 1 We solve for u, z and p. +� +�Ω +Jk−1 tr +� +∇uk � +Fk−1�−1� +q d�x = 0, +� +�Ω +� +zb + +� +∂x1ηk−1 + τ ∂x1uk +2 +� +∂x1b +� +d�x = 0, +ρf +� +�Ω +Jk−1Dtuk · ϕϕϕ d�x + 1 +2ρf +� +�Ω +˙Jk−1(2uk − uk−1) · ϕϕϕ d�x + 1 +2ρf +� +�Ω +Jk−1∇uk � +Fk−1�−1 +vk−1 · ϕϕϕ d�x +− 1 +2ρf +� +�Ω +Jk−1∇ϕϕϕ +� +Fk−1�−1 +vk−1 · uk d�x + +� +�Ω +Jk−1 tr +� +�Tk∇ϕϕϕ(Fk−1)−1) +� +d�x ++ +� +Γ +ρsDtuk +2 ϕ2 dS(x) + γ1 +� +Γ +� +∂x1ηk−1 + τ ∂x1uk +2 +� +∂x1ϕ2 dS(x) +− γ2 +� +Γ +∂x1z ∂x1ϕ2 dS(x) + γ3 +� +Γ +∂x1u2 ∂x1ϕ2 dS(x) + +� +Γ +fϕ2 dS(x) = 0, +where ϕϕϕ = (ϕ1, ϕ2) is the test function corresponding to the velocity u and ϕ1, ϕ2 its components, +q is the test function for the pressure p and b the test function for z. Finally, f denotes the +y-component of the force acting on the boundary Γ. +Step 2 We linearly prolongate the displacement η to the whole domain �Ω by solving +� +�Ω +∂x2ηk ∂x2ψ d�x = 0, +for all ψ ∈ B. Here, ηk = ηk−1 + τ uk +2, where uk +2 is obtained in Step 1. +C.2 +Implementation of fully implicit scheme +In Section 7.3 we compare our Scheme-R to the fully implicit method based on the weak form (2.9). +As in the case of implementation of Scheme-R, the domain �Ω is approximated by regular triangles +T ∈ Th and the problem comprises four global unknowns: velocity u, pressure p, mesh displacement +in x2−direction η and its second derivative z. The velocity-pressure pair (u, p) is approximated with +the MINI element, mesh displacement η is from the same space as velocity, and the second derivative +of the displacement z is approximated by piecewise linear elements. +The time derivatives are approximated by the backward Euler time scheme, the nonlinearities are +treated with the Newton solver, and the consequent set of linear equations by direct solver MUMPS. +Knowing the solution uk−1, ηk, zk, pk on the previous time level, we are solving the fully implicit +nonlinear problem. Thus, we solve for uk, ηk, zk and pk satisfying the continuity equation +� +�Ω +Jk tr +� +∇uk (Fk)−1� +q d�x = 0, +the coupled momentum equation +ρf +� +�Ω +JkDtuk · ϕϕϕ d�x + 1 +2ρf +� +�Ω +DtJk uk · ϕϕϕ d�x + 1 +2ρf +� +�Ω +Jk∇uk(Fk)−1vk · ϕϕϕ d�x +− 1 +2ρf +� +�Ω +Jk∇ϕϕϕ(Fk)−1vk · uk d�x + +� +�Ω +Jk tr +� +�Tk∇ϕϕϕ(Fk)−1) +� +d�x ++ +� +Γ +ρsDtuk +2 ϕ2 dS(x) + γ1 +� +Γ +∂x1ηk ∂x1ϕ2 dS(x) +41 + +− γ2 +� +Γ +∂x1z ∂x1ϕ2 dS(x) + γ3 +� +Γ +∂x1u2 ∂x1ϕ2 dS(x) + +� +Γ +fϕ2 dS(x) = 0. +Further, the discrete Laplace equation for zk and the harmonic extension of ηk +� +�Ω +� +zkb + ∂x1ηk ∂x1b +� +d�x = 0, +� +�Ω +∂x2ηk ∂x2ψ d�x = 0, +for all test functions q, b,ϕϕϕ and ψ. Here, we used the same notion as above +Fk = I + ∇(0, ηk)T, +Jk = det Fk, +vk = uk − (0, Dtηk)T, +�Tk = −pkI + 2µ +� +∇uk(Fk)−1�S +and the components of the velocity u are denoted by (u1, u2). The problem is periodic in x1 direction, +with a homogeneous Dirichlet boundary conditions for u and η on the bottom boundary, and Dtηk = uk +2 +on the top boundary. +42 + diff --git a/ptE4T4oBgHgl3EQfUQxX/content/tmp_files/load_file.txt b/ptE4T4oBgHgl3EQfUQxX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dae84aec9ff99be04b68557234ad26829312511a --- /dev/null +++ b/ptE4T4oBgHgl3EQfUQxX/content/tmp_files/load_file.txt @@ -0,0 +1,2762 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf,len=2761 +page_content='Stability and error estimates of a linear numerical scheme approximating nonlinear fluid–structure interactions ∗ Sebastian Schwarzacher† Bangwei She‡,† Karel T˚uma† January 13, 2023 Abstract In this paper, we propose a linear and monolithic finite element method for the approximation of an incompressible viscous fluid interacting with an elastic and deforming plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' We use the arbitrary Lagrangian–Eulerian (ALE) approach that works in the reference domain, meaning that no re-meshing is needed during the numerical simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For time discretization, we employ the backward Euler method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For space discretization, we respectively use P1-bubble, P1, and P1 finite elements for the approximation of the fluid velocity, pressure, and structure displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' We show that our method fulfills the geometrical conservation law and dissipates the total energy on the discrete level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Moreover, we prove the (optimal) linear convergence with respect to the sizes of the time step τ and the mesh h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' We present numerical experiments involving a substantially deforming fluid domain that do validate our theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' A comparison with a fully implicit (thus nonlinear) scheme indicates that our semi-implicit linear scheme is faster and as accurate as the fully implicit one, at least in stable configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Keywords: fluid-structure interaction, Navier–Stokes equations, stability, error estimates, fi- nite element method, divergence-free projection MSC(2010): 35Q30, 76N99, 74F10, 65M12, 65M60 Contents 1 Introduction 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 Problem formulation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 4 2 Weak formulation and stability 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 Weak formulation .' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 15 6 Error estimates 19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 The time projection .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 Main result .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': 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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 24 8 Conclusion 26 A Appendix: Interpolation operators 29 B Appendix: Useful equalities and estimates 30 B.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 40 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 Implementation of fully implicit scheme .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 41 1 Introduction Fluid–structure interaction (FSI) problems occur in many engineering applications, from aero-elasticity to civil engineering and bio-mechanical problems, such as the design of aircraft wings, wind turbines and heat exchangers, the response of bridges and skyscrapers to wind force, blood flow in arteries, see [5, 6, 29] among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Numerical simulation of FSI problems has been largely studied and great progress has been achieved during the past decades;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' see, for examples, [4, 8, 18, 20, 24] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Concerning the numerical stability analysis, we would like to mention the nice results of Luk´aˇcov´a-Medvid’ov´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' [23], Bukaˇc and Muha [10], Lozovskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' [21, 22], Hecht and Pironneau [17], and Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' [30] as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' However, in terms of convergence analysis, there are certainly many more efforts to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' To our best knowledge, only a few results are available on this topic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' see Bukaˇc and Muha [10], Burman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' [11, 12], Fern´andez and Mullaert [15], and Seboldt and Bukaˇc [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In this direction, all available literature results are not only under the assumption that the displacement of the solid structure is infinitesimal but also based on the ignorance of the convection of the fluid motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The main target of this paper is to show the convergence of a numerical approximation without these restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For that reason, we study an archetypical setting of fluid-structure interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In our setting a one-dimensional plate is situated on the top of a two-dimensional container filled with a viscous incompressible liquid governed by the Navier–Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The plate is governed by a hyperbolic equation driven by fluid traction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' It may deform largely and therefore the Eulerian fluid domain is time-changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' This implies a severe nonlinear coupling between the structure and fluid equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In order to solve the FSI problem numerically, we introduce a linear, implicit-explicit (semi- implicit), and monolithic finite element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For time discretization, we take the backward Euler method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For space discretization, we start with the so-called arbitrary Lagrangian–Eulerian (ALE) mapping and transfer the time-dependent domain to a fixed reference grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then, we use an inf-sup stable finite element pair (P1-bubble/P1) on the reference domain for the fluid, and P1 elements for 2 the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Our aims of the paper are to design an energy stable scheme and to show the (optimal) convergence rate of the numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The key point in the construction of the stability of our linear and semi-implicit scheme is that we keep the scheme implicit with respect to the velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In particular, the velocities of the solid structure and fluid are coupled implicitly in time, see also a similar construction of Lozovskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Nevertheless, the scheme is linear as we take the fluid domain explicitly, which means it is given by the deformation of the plate of the previous time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Further, the convective term of the fluid is linearized in a stable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' To some extent the current paper can be viewed as a numerical counterpart of Schwarzacher and Sroczinski [27], where the authors investigated the distance between a weak solution and a strong solution, while the aim of this paper is to investigate the distance between a numerical solution obtained by a finite element method and a smooth solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In order to adapt this result to a discrete numerical scheme, rather sophisticated analytic tools have to be invented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In particular, good projection operators have to be invented for a smooth solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The challenge comes from the change of the fluid domain in time, which results in several non-trivial analytic difficulties on all levels when studying the convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Roughly, there are three different sources of errors that have to be estimated: i) the mismatch between the continuous geometry and the discrete geometry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ii) the respective different divergence-free constraints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' iii) the projector of the fluid-velocity which has to fit a rather particular choice of a projector according to the structure equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The first point is overcome by a change of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The second point is already very technical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For that, we introduce a Fortin operator for variable geometries in order to inherit the discrete solenoidality from the continuous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then the divergence-free condition destroyed by a change of variable is resolved by a Bogovskij correction recently developed by Kampschulte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The last point, the mismatch between the interpolation operator of the fluid at the boundary turns out to be the hardest to overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The reason is that the structure equation is of the fourth order in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Hence, a discrete bi-Laplacian naturally appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In order to gain suitable estimates for the structure part, a very particular choice of projector, the so-called Riesz projection operator, has to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Further, we have to solve a discrete Stokes problem in order to find a suitable projector of the fluid velocity that possesses these particular boundary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The main result of the paper is that there is a monolithic, linear, and fully-discrete scheme of the FSI problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4), which satisfies under suitable conditions: Total energy stability (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1) Linear convergence w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' the space and time discretization parameters (see Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2) The highlight of the paper reads: The proposed semi-implicit scheme is linear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' the absence of nonlinear iterations makes the approach computationally cheaper than the fully implicit scheme, yet provides equally good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' We show the energy stability of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' We show the linear convergence rate of the method with respect to the computational parameters τ (the time step) and h (mesh size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Such a result has not been achieved in literature for any structure interacting with fluids described by the Navier–Stokes equa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The discrete structure displacement is defining the real-time geometry of the Eulerian fluid domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The convergence rate is optimal as demonstrated by numerical experiments, see Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 Problem formulation In this paper, we are interested in the interaction between an incompressible viscous fluid and a thin elastic structure, which is part of the fluid boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' More precisely, we consider the motion of an incompressible and viscous fluid flow in a time-dependent domain Ωη(t) = {x = (x1, x2) ∈ Σ × (0, η(t, x1))} ⊂ R2, where Σ = (0, L1), L1 is the length of the domain, η = η(t, x1) > 0 represents the height of the upper boundary ΓS(t) of the fluid domain Ωη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For the sake of simplicity, we assume that i) the flow is periodic in the x1-direction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ii) the upper boundary is formed by an elastic structure that can move in the x2-direction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' iii) the bottom boundary ΓD is a solid wall;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' iv) initially Ωη(0) = Σ × [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In this paper, we shall use the ALE method and directly work on a time-independent reference domain �Ω = Ωη(0) instead of the time-dependent domain Ωη(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' To this end, we introduce an ALE mapping Aη that maps the reference domain �Ω to the time dependent domain Ωη, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Aη : �Ω �→ Ωη, (�x1, �x2) �→ (x1, x2) = Aη(t, �x) = (�x1, η�x2) , see Figure 1 for a graphical illustration of the domain and ALE mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' �Ω ΓD �ΓS = Σ 0 �x1 L1 1 �x2 Aη Ω(t) ΓD ΓS(t) η 0 x1 L1 x2 Figure 1: Time dependent domain and the ALE mapping Fluid model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The motion of the incompressible viscous fluid is described by the Navier–Stokes equations � � � � � ϱf (∂tu + (u · ∇)u) − div T(u, p) = 0, in (0, T) × Ωη div u = 0, in (0, T) × Ωη, u = 0, on (0, T) × ΓD, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1) where ϱf, u = u(t, x), and p = p(t, x) are the fluid density (given constant), velocity field, and pressure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The Cauchy stress T reads1 T = 2µ(∇u)S − pI with the constant viscosity coefficient µ > 0, and the superscript S denotes the symmetric operator for a matrix-valued function A, meaning that AS = A + AT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Structure model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The motion of the elastic structure is given by � ϱs∂tξ + L(η) = f, on (0, T) × Σ, ∂2 x1η = 0, on (0, T) × ∂Σ, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2) where ϱs > 0 is the density of the structure, ξ = ∂tη is the velocity of the structure, f is the interaction force acting on the structure due to fluid motion, and L(η) = −γ1∂2 x1η − γ2∂2 x1ζ − γ3∂2 x1∂tη, ζ = −∂2 x1η, where γ1 > 0, γ2 > 0, γ3 ≥ 0 are given constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Further, the initial data of the problem read u(0) = u0 in Ωη(0) and η(0, ·) = η0, ξ(0, ·) = ξ0 in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3) 1We adopt the following notations: (∇u)ij = ∂jui, (v · (∇u))i = �d j=1 vj∂iuj and � v · ∇u � i = � (v · ∇)u � i = �d j=1 vj∂jui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Note that v · ∇u ̸= v · (∇u) but v · ∇u = (v · ∇)u = ∇u · v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 4 Coupling conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Finally, to close the system, we require coupling conditions at the fluid- structure interface, which are the so-called kinematic and dynamic boundary conditions: the kinematic coupling condition u(x) = ξ(x1)e2, ∀ x = (x1, η) ∈ ΓS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4a) the dynamic coupling condition f = −e2 · � JT(u, p) ◦ AηF−T � e2, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4b) where F = F(η) is the Jacobian of the mapping Aη and J = J(η) is the corresponding determi- nant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In the current setting, we have F(η) = ∇�xAη = � 1 0 �x2∂x1η η � and J(η) = det(F(η)) = η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5) The plan of the paper is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In Section 2 we discuss the weak formulation and stability of our FSI problem on the continuous level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In Section 3 we introduce the numerical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In Section 4 we prove the stability of the numerical solution on the discrete level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In Section 5 we introduce interpolation operators that are specially designed to fit both the divergence-free velocity field and the kinematic coupling condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' These operators are essential in Section 6, where we show the convergence rate of the numerical solution towards a strong solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In Section 7 we present the numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Finally, in Section 8 we give a short conclusion of the achievements in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 2 Weak formulation and stability In this section, we introduce a weak formulation of the FSI problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4) and prove that a solution to the weak formulation is energy stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' To begin, we introduce the standard notations W k,p(D) and Lp(D) on a generic domain D for the Sobolev space and Lebesgue space, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Further, we denote by W k,p 0 the functions with zero traces on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In order to specify functions on the reference domain, we shall use the superscript “�”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For example, for a generic function v = v(x) defined on Ωη we write on the reference domain that �v = v(Aη(�x)) = v ◦ Aη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Next, we recall the Piola transformation [13] for the mapping Aη: dx = η d�x, dS(x) = |ηF−T �n| dS(�x), n = ηF−T �n |ηF−T �n|, div �x(ηF−1) = 0, div xq ◦ Aη = 1 ηdiv �x � ηF−1�q � = ∇�x�q : F−T , ∇q ◦ Aη = ∇�x�q F−1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1) where dx (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' dS(x)) is the volume (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' face) integral in the time-dependent domain, d�x (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' dS(�x)) is the volume integral in the reference domain, q and q are generic vector-valued and scalar functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Note that we have emphasized here the dependence of the differential operators ∇ and div with respect to x and �x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Hereinafter, if no confusion occurs, we shall simply write ∇ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' div ) instead of both ∇x and ∇�x (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' div x and div �x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Now, we define a new velocity field w that describes the change of the fluid domain (ALE mapping) in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' It reads �w(�x) := ∂tAη = (0, ∂tη �x2) and w(x) := �w(�x) ◦ A−1 η (x) = (0, x2 ∂tη/η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2) Then, it is easy to observe the so-called Euler expansion div w = ∂tη/η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3) 5 According to the chain rule, we have ∂t�v(�x) = d dtv(Aη(�x)) = ∂tv(x) + ∂tAη(�x) · ∇v(x) = ∂tv(x) + w(x) · ∇v(x) =: ∂M t v(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4) where ∂M t represents a material-type time derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' With the above notations, it is easy to check the Reynolds transport theory ∂t � Ωη v dx = � Ωη ∂tv dx + � ∂Ωη vw · n dS(x) = � Ωη (∂tv + div (vw)) dx = � Ωη � ∂M t v + vdiv w � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5) Further, we denote v as the relative velocity of the fluid with respect to the fluid domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' It reads v = u − w satisfying the boundary condition v|∂Ωη = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Thanks to the above boundary condition and the incompressibility condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1)2, we observe for any differentiable test function ϕ that � Ωη � (v · ∇)u � ϕ dx = � Ωη � ϕ · (∇u) · v + 1 2(div v + div w � �� � =div u=0 )u · ϕ � dx = � Ωη div w u 2 · ϕ dx + 1 2 � Ωη � ϕ · (∇u) − u · (∇ϕ) � v dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Accordingly, we may reformulate the time derivative and convective terms in the following way � Ωη � ∂tu + (u · ∇)u � ϕ dx = � Ωη � ∂tu + (w · ∇)u � ϕ dx + � Ωη � (v · ∇)u � ϕ dx = � Ωη � ∂M t u + div w u 2 � ϕ dx + 1 2 � Ωη � ϕ · (∇u) − u · (∇ϕ) � v dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6) Finally, we introduce the following abbreviation for the sake of simplicity as(η, ζ, ξ, ψ) = � Σ (γ1∂x1η∂x1ψ + γ2∂x1ζ∂x1ψ + γ3∂x1ξ∂x1ψ) dx1 with ζ = −∂2 x1η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='7) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 Weak formulation Before introducing the weak formulation, we introduce the space of coupled test functions to accom- modate the no-slip boundary condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Wη = � (ϕ, ψ) ∈ W 1,2(Ωη) × L2(Σ) : ψ(x)e2 = ϕ(x, η(x)), ϕ = 0 on ΓD � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Now we are ready to present the weak formulation of the FSI problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 (Weak formulation of the FSI problem on Ωη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let (p, u, ξ, η) be a “solution” to the FSI problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' We say the following formula is a weak formulation of the FSI problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' � Ωη qdiv u dx = 0 for all q ∈ L2(Ωη);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8a) ϱf � Ωη � ∂M t u + div w u 2 � ϕ dx + 1 2 � Ωη � ϕ · (∇u) − u · (∇ϕ) � v dx + � Ωη T(u, p) : ∇ϕ dx + ϱs � Σ ∂tξψ dx1 + as(η, ζ, ξ, ψ) = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8b) for all (ϕ, ψ) ∈ Wη, where as is given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='7), and ζ = −∂2 x1η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 6 Note that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8a) is directly obtained by � Ωη (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1)2 × q dx for q ∈ L2(Ωη) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8b) is obtained by calculating � Ωη (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1)1 × ϕ dx + � Σ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2)1 × ψ dx1 for the coupled test functions (ϕ, ψ) ∈ Wη, where we have used the following identity due to the Piola transformation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1), the fact that the test functions ϕ and ψ are coupled, and the coupling condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' � ΓS ϕ · T · n dS(x) = � Σ ϕ ◦ Aη · (T ◦ Aη) · (ηF−T e2) dx1 = � Σ ψe2 · (ηT ◦ AηF−T ) · e2 dx1 = − � Σ fψ dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 Weak formulation on the reference domain By means of a change of variables, we may reformulate the weak formulation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8) from the current domain Ωη onto the reference domain �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 (Weak formulation of the FSI problem on �Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let (p, u, ξ, η) satisfy the weak formulation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8) of the FSI problem on Ωη with the test functions (q, ϕ, ψ) ∈ L2(Ωη) × Wη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let (�p, �q, �u, �ϕ) = (p, q, u, ϕ) ◦ Aη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then there hold � �Ω �q∇�u : M d�x = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9a) ϱf � �Ω � η∂t�u + ∂tη �u 2 � �ϕ d�x + 1 2ϱf � �Ω � �ϕ · (∇�u) − �u · (∇ �ϕ) � F−1 · �vη d�x + � �Ω �T(�u, �p) : � ∇ �ϕF−1� η d�x + ϱs � Σ ∂tξψ dx1 + as(η, ζ, ξ, ψ) = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9b) where as is given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='7), ζ = −∂2 x1η, and �T(�u, �p) = T(u, p) ◦ Aη = 2µ(∇�uF−1)S − �pI, M := M(η) := ηF−T = � η −�x2∂x1η 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' First, recalling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4) we know that ∂M t u ◦ Aη = ∂t�u, which together with the Euler expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3) yield ϱf � Ωη � ∂M t u + div w u 2 � ϕ dx = ϱf � �Ω � η∂t�u + ∂tη �u 2 � �ϕ d�x Next, using the Piola transformation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1) we have � Ωη ϕ · (∇u) · v dx = � �Ω �ϕ · (∇�uF−1) · �vη d�x, which indicates 1 2ϱf � Ωη � ϕ · (∇u) − u · (∇ϕ) � v dx = 1 2ϱf � �Ω � �ϕ · (∇�u) − �u · (∇ �ϕ) � F−1 · �vη d�x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Analogously, we find � Ωη T : ∇ϕ dx = � Ωη � 2µ(∇u)S − pI � : ∇ϕ dx = � �Ω � 2µ(∇�uF−1)S − �pI � : � ∇ �ϕF−1� η d�x = � �Ω �T : � ∇ �ϕF−1� η d�x, and � Ωη qdiv u dx = � �Ω �q∇�u : F−T η d�x = � �Ω �q∇�u : M d�x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Consequently, collecting the above equalities we derive (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9) from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3 Energy stability Finally, we are ready to show the stability of the FSI problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Indeed, any solution to its weak formulation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8) (or equivalently (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9)) satisfies the following energy stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3 (Stability of the continuous problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let (p, u, ξ, η) be a solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8) (or (�p, �u, ξ, η) be a solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then the following energy estimates hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ∂t � Ωη 1 2ϱf|u|2 dx + ∂t � Σ 1 2 � ϱs|ξ|2 + γ1|∂x1η|2 + γ2|∂2 x1η|2� dx1 + 2µ � Ωη |(∇u)S|2 dx + � Σ γ3|∂x1ξ|2 dx1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' First, we derive by setting (q, ϕ, ψ) = (p, u, ξ) in the weak formulation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8) that ϱf � Ωη � ∂M t |u|2 2 + |u|2 2 div w � dx + 2µ � Ωη |(∇u)S|2 dx + ϱs � Σ ∂t |ξ|2 2 dx1 + � Σ � γ1∂x1η∂x1ξ + γ2∂2 x1η∂2 x1ξ + γ3|∂x1ξ|2� dx1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then we finish the proof after noticing ξ = ∂tη and employing the Reynolds transport theory (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5) with v = |u|2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 3 Numerical method In this section, we discretize the weak formulation introduced in the last section by a suitable finite element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 Time discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' We start with time discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let τ be the time increment and tk = kτ for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' , N(≡ T/τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then we denote by (pk h, uk h, ξk h, ηk h) the numerical approximation of the FSI problem at time tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Further, for any set of pointwise in time approximation {vk h}N k=0 we extend it to the whole time interval [0, T] in the following way vh(t) = vk h in [tk, tk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1) The discrete Eulerian domain Ωk ηh = Ωηk h at time tk is determined by the discrete ALE mapping: Ak ηh : �Ω �→ Ωk ηh, Ak ηh(�x) = (�x1, ηk h�x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Again Aηh is a piecewise constant in time function in the sense of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For a generic function vh (including test functions) defined on Ωηh we have �vh = vh ◦Aηh on �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Here we emphasize that �ηh = ηh and �ξh = ξh as their domain of definition Σ is time independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' To approximate the time derivatives ∂tv and ∂M t v we introduce Dtvk h = vk h − vk−1 h τ ≈ ∂tvk, DM t vk h = vk h − vk−1 h Xk−1 k τ ≈ ∂M t vk, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2) where Xj i = Aj ηh ◦ (Ai ηh)−1 denotes the mapping from Ωi ηh to Ωj ηh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 Discrete Reynolds transport theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Analogous to the continuous definitions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2) and the identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3) we have Fh = F(ηh) = ∇�xAηh = � 1 0 �x2∂x1ηh ηh � , det(Fh) = ηh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3) �wh = DtAηh = (0, Dtηh�x2), wh = �wh ◦ (Aηh)−1 = � 0, Dtηh ηh x2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4) and div wh = Dtηh ηh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5) Next, realizing the equality Dt�vk h = (DM t vk h) ◦ Ak ηh we observe the discrete analogue of the Reynolds transport theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5), see also [26, Lemma 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 (Discrete Reynolds transport).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let Dt and DM t be given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2), then for any k = 1, · · · , NT we have Dt � Ωkηh vk h dx = � Ωkηh � DM t vk h + div wk hvk−1 h Xk−1 k � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Choosing vh = 1Dh in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 for any Dh ⊂ Ωηh, we have the geometric conservation law, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=', Dt|Dh| = Dt � Ωηh 1 dx = � Ωηh div wh dx = � ∂Dh wh · n dS(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Further, as we keep in our numerical scheme that wh = uh on the boundary, where the velocity field is weakly divergence-free, we have Dt|Ωηh| = � ∂Ωηh wh · n dS(x) = � ∂Ωηh uh · n dS(x) = � Ωηh div uh dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Indeed, our method does fulfill the above equality, as our boundary condition is uk−1 h = wk h and we use a flat reference geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Therefore Dt|Ωk ηh| = � ∂Ωkηh wk h · n dS(x) = � Σ Dtηk h dx1 = � Σ ξk−1 h dx1 = � Ωk−1 ηh div uk−1 h = 0, where the last equality is due to the weakly divergence-free condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='13a) with the choice of test function q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3 Spatial discretization Let Th be a shape regular and quasi-uniform triangulation of the reference domain �Ω, where h stands for the maximum diameter of all elements of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let Σh be the surface mesh of Th on the top boundary �ΓS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' We denote by K ∈ Th a generic element in Th and by σ ∈ Σh a generic face element in Σh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Moreover, we introduce the following function spaces on �Ω �V f h = � �ϕ ∈ W 1,2(�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Rd) ��� �ϕ ∈ P1(K) ⊕ B1(K), ∀K ∈ Th, �ϕ|ΓD = 0 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='7) B1(K) = � φ ∈ P3(K) ��φ(ai) = 0, where ai, i = 1, 2, 3, are vertices of K ∈ Th � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8) �Qf h = � �q ∈ C0(�Ω) ����q ∈ P1(K), ∀K ∈ Th � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9) V s h = � ψ ∈ W 1,2 0 (Σ) ���ψ ∈ P1(σ), ∀σ ∈ Σh � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='10) 9 where Pn(K) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Pn(σ)) denote polynomials of degree not greater than n on K (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' on σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Further, we denote � Wηh = � ( �ϕ, ψ) ∈ �V f h × V s h ��� �ϕ(�x1, 1) = ψ(�x1)e2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Finally, we denote �V fsi h = �Qf h × � Wηh and V fsi h (t) = �V fsi h A−1 ηh (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let us point out that by using the linear finite element space V s h for the structure displacement η, we cannot directly discretize the bi-Laplacian term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Therefore, we decide to approximate the bi- Laplacian via duality, which still requires a discrete Laplace operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' To this end, we introduce the following discrete Laplace operator for ηh ∈ V s h by seeking ∂2 x1,hηh ∈ V s 0,h := V s h ∩ L2 0(Σ) such that � Σ ∂2 x1,hηh ψ dx1 + � Σ ∂x1ηh∂x1ψ dx1 = 0 for all ψ ∈ V s h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11) Here, we would like to point out that V s 0,h is a finite dimensional space and in view of the assumptions on the grid the stiffness matrix of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Thus, it admits a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4 The numerical method With the notations introduced above, we propose a monolithic finite element method for the dis- cretization of the weak formulation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Scheme-R(A monolithic finite element method on the reference domain �Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' , N we seek (�pk h, �uk h, ξk h, ηk+1 h ) ∈ �V fsi h × V s h with ξk h = Dtηk+1 h such that for all (�q, �ϕ, ψ) ∈ �V fsi h there hold � �Ω �q∇�uk h : Mk h d�x = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='12a) ϱf � �Ω Dt�uk h · �ϕηk h d�x + 1 2ϱf � �Ω Dtηk h�uk∗ h · �ϕ d�x + 1 2ϱf � �Ω � �ϕ · (∇�uk h) − �uk h · (∇ �ϕ) � (Fk h)−1 · �vk−1 h ηk h d�x + � �Ω �T(�uk h, �pk h) : � ∇ �ϕ(Fk h)−1� ηk h d�x + ϱs � Σ Dtξk hψ dx1 + as(ηk+1 h , ζk+1 h , ξk h, ψ) = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='12b) where �uk∗ h = 2�uk−1 h − �uk h, �vk−1 h = �uk−1 h − �wk h, Mk h = M(ηk h) is given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='10), as is given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='7), ζh = −∂2 x1,hηh is the (minus) discrete Laplace uniquely defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11), �T is given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='10), and the discrete initial data are given by u0 h = Pf hu0, ξ0 h = Rs hξ0, η0 h = Rs hη0, and η1 h = η0 h + τξ0 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Here Pf h : W 1,2(�Ω) �→ �V f h is a suitable projection operator and Rs h : W 1,2(�Ω) �→ V s h is a Riesz projection operator to be clarified in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Note that Scheme-R approximates the FSI problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4) based on the weak formula- tion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9) in the reference domain �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' It is linear and belongs to the monolithic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Practically, it is more convenient to work with the reference domain as it is time-independent and no need for re-meshing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Nevertheless, many researchers appreciate working with the current domain Ωηh (approx- imation of Ωη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' To this end, we present the following equivalent formulation of Scheme-R on the current domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Scheme-C(A monolithic finite element method on the current (push-forward) domain Ωηh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Given the initial data (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3) we set u0 h = Pf hu0, ξ0 h = Rs hξ0, η0 h = Rs hη0, and η1 h = η0 h + τξ0 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' , N we seek (pk h, uk h, ξk h, ηk+1 h ) ∈ V fsi h ×V s h with ξk h = Dtηk+1 h such that for all (q, ϕ, ψ) ∈ V fsi h there hold � Ωηk qdiv uk h dx = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='13a) 10 ϱf � Ωkηh � DM t uk h + div wk h uk∗ h 2 � ϕ dx + 1 2ϱf � Ωkηh � ϕ · (∇uk h) − uk h · (∇ϕ) � vk−1 h Xk−1 k dx + � Ωkηh T(uk h, pk h) : ∇ϕ dx + ϱs � Σ Dtξk hψ dx1 + as(ηk+1 h , ζk+1 h , ξk h, ψ) = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='13b) where vk−1 h = uk−1 h − wk h ◦ Xk k−1, uk∗ h = 2uk−1 h Xk−1 k − uk h, as is given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='7), and ζh = −∂2 x1,hηh is the (minus) discrete Laplace given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' We omit the proof on how to identify the equivalence of Scheme-R and Scheme-C as it is similar to the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In Scheme-C (or equivalently Scheme-R) we solve for each time step tk, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' , N}, the fluid variables (uk h, pk h) in an explicit domain Ωk ηh and solve the structure variable ηk+1 h , which determines the fluid domain Ωk+1 ηh of the next time step tk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' This differs from many monolithic schemes defined in an implicit domain Ωk ηh (or their equivalent form in the reference domain) when ηk h instead of ηk+1 h is unknown at time step tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Such a kind of solver “time splitting” helps us to define a linear scheme without destroying the stability of the numerical solutions, see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4 (On the extension to 3D/2D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Many parts of our analysis are also valid when con- sidering a three-dimensional fluid domain with a two-dimensional plate attached to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' However, the regularity of the (approximated) fluid domain is essentially weaker a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Observe that if the plate is two-dimensional, the discrete domain in space can not even be assumed to be uniformly Lipschitz continuous, as in two dimensions W 2,2 does not embed into Lipschitz functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Note that we approximate the boundary deformation η with a piecewise linear finite element space, resulting in a linear ALE mapping and a linear deformation of the fluid domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Therefore, the geometry of the fluid domain is automatically captured at every time step, as every element K ∈ Th is preserved as a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let us point out that the fourth order derivative in the structure (due to the bi-Laplacian term) is avoided by the introduction of a discrete Laplace operator, which maps a piecewise linear function space into the same space, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 4 Stability In this section, we show the stability of the Scheme-C (or equivalently Scheme-R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' We start with the following observation by recalling the discrete Laplace operator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' � Σ ∂x1ξk h∂x1(−∂2 x1,hηk+1 h ) dx1 = � Σ ∂2 x1,hξk h∂2 x1,hηk+1 h dx1 = � Σ Dt∂2 x1,hηk+1 h ∂2 x1,hηk+1 h dx1 = 1 2 � Σ � Dt|∂2 x1,hηk+1 h |2 + τ|Dt∂2 x1,hηk+1 h |2� dx1 = Dt � Σ 1 2|∂2 x1,hηk+1 h |2 dx1 + τ 2 � Σ |∂2 x1,hξk h|2 dx1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1) where we have used the algebraic equality (a − b)a = 1 2(a2 − b2) + 1 2(a − b)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2) 11 Then, recalling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='7) with the test function ψ = ξh and thanks to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1), we find as(ηk+1 h , −∂2 x1,hηk+1 h , ξk h, ξk h) = Dt �γ1 2 ���∂x1ηk+1 h ��� 2 L2(Σ) + γ2 2 ���∂2 x1,hηk+1 h ��� 2 L2(Σ) � + γ3 ���∂x1ξk h ��� 2 L2(Σ) + Dk s, where Dk s = γ1τ 2 � Σ |∂x1ξk h|2 dx1 + γ2τ 2 � Σ |∂2 x1,hξk h|2 dx1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3) Now we are ready to show the energy estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 (Energy estimates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let {(pk h, uk h, ξk h, ηk+1 h )}N k=1 be the solution of Scheme-C (or equivalently, let {(�pk h, �uk h, ξk h, ηk+1 h )}N k=1 be the solution of Scheme-R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then we have the following stability result for all m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' , N Em h + τ m � k=1 � 2µ ���(∇uk h)S��� 2 L2(Ωkηh) + γ3 ���∂x1ξk h ��� 2 L2(Σ) + Dk num � = E0 h (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4) where for any k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' , N the total energy Ek h and the numerical dissipation Dk num read Ek h = ϱf 2 ���uk h ��� 2 L2(Ωkηh) + ϱs 2 ���ξk h ��� 2 L2(Σ) + γ1 2 ���∂x1ηk+1 h ��� 2 L2(Σ) + γ2 2 ���∂2 x1,hηk+1 h ��� 2 L2(Σ) Dk num = ϱfτ 2 � Ωkηh ηk−1 h ηk h |DM t uk h|2 dx + ϱsτ 2 ���Dtηk+1 h ��� 2 L2(Σ) + γ1τ 2 ���∂x1ξk h ��� 2 L2(Σ) + γ2τ 2 ���∂2 x1,hξk h ��� 2 L2(Σ) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' We test the numerical method (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='13) by (q, ϕ, ψ) = (pk h, uk h, ξk h) to get ϱf � Ωkηh � DM t uk h · uk h + div wk h � uk−1 h Xk−1 k − uk h 2 � uk h � dx +2µ ���(∇uk h)S��� 2 L2(Ωkηh) + ϱs � Σ Dtξk hξk h dx1 + as(ηk+1 h , ζk+1 h , ξk h, ξk h) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5) Next, using the equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2) we get � Σ Dtξk hξk h dx1 = 1 2 � Σ Dt|ξk h|2 dx1 + τ 2 � Σ |Dtξk h|2 dx1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6) and � Ωkηh � DM t uk h · uk h + div wk h � uk−1 h Xk−1 k − uk h 2 � uk h � dx = � Ωkηh � DM t |uk h|2 2 + τ 2|DM t uk h|2 + div wk h(uk−1 h Xk−1 k uk h − |uk h|2 2 ) � dx = Dt � Ωkηh 1 2|uk h|2 dx + I0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='7) where we have also used the discrete Reynolds transport formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Here, the term I0 reads I0 = � Ωkηh � τ 2|DM t uk h|2 + div wk h � uk−1 h Xk−1 k uk h − |uk h|2 2 − |uk−1 h Xk−1 k |2 2 �� dx = � Ωkηh �τ 2 − τ 2 2 div wk h � |DM t uk h|2 dx = τ 2 � Ωkηh ηk−1 h ηk h |DM t uk h|2 dx, where we have used the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 12 Substituting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='7) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5) and owing to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3), we derive DtEk h + 2µ ���(∇uk h)S��� 2 L2(Ωkηh) + γ3 ���∂x1ξk h ��� 2 L2(Σ) + Dk num = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8) Finally, computing τ �m k=1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8) yields (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The above stability estimate can be rewritten in the reference domain as ϱf � �Ω 1 2ηm h |�um h |2 d�x + ϱs � Σ 1 2|ξm h |2 dx1 + γ1 2 ��∂x1ηm+1 h ��2 L2(Σ) + γ2 2 ��∂2 x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='hηm+1 h ��2 L2(Σ) + 2µτ m � k=1 � �Ω ηk h|(∇�uk h(Fk h)−1)S|2 d�x + γ3τ m � k=1 ���∂x1ξk h ��� 2 L2(Σ) + τ 2 2 m � k=1 ϱf � �Ω ηk−1 h |Dt�uk h|2 d�x + τ 2 2 m � k=1 ϱs � Σ |Dtξk h|2 dx1 + γ1τ 2 2 m � k=1 � Σ |∂x1ξk h|2 dx1 + γ2τ 2 2 m � k=1 � Σ |∂2 x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='hξk h|2 dx1 = ϱf � �Ω 1 2η0 h|u0 h|2 d�x + ϱs � Σ 1 2|ξ0 h|2 dx1 + γ1 2 ��∂x1η1 h ��2 L2(Σ) + γ2 2 ��∂2 x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='hη1 h ��2 L2(Σ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9) Note that ηh appears on the left-hand-side (LHS) of the energy estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4) (see also (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9)) and determines if all terms on the LHS of the energy balance are non-negative or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Therefore, it is important to preserve the positivity of ηh in order to get a priori estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Actually, there exists a T0 > 0 such that for all T ≤ T0 we have no contact between the upper surface and the bottom surface of Ωη, see [26, Lemma 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' More precisely, if ηh(0) > η, for every c there exists a T, such that ηh ≥ η − c ∀ t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='10) From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 and the above assumption, we have the following uniform estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let the initial data satisfy u0 ∈ W 1,2(Ωη(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' R2), η0 ∈ W 1,2(Σ), and ξ0 ∈ W 1,2(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let (ph, uh, ξh, ηh) = {(pk h, uk h, ξk h, ηk+1 h )}N k=1 be a solution to Scheme-C (or equivalently (�ph, �uh, ξh, ηh) be a solution to Scheme-R) with (τ, h) ∈ (0, 1)2 and let (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='10) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then we have the following uniform bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ∥ξh∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2(Σ)) + ∥∂x1ηh∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2(Σ)) + ��∂2 x1,hηh �� L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2(Σ)) <∼ 1, ��η−1 h �� L∞((0,T)×Σ) + ∥ηh∥L∞((0,T)×Σ) + ∥∂x1ηh∥L∞((0,T)×Σ) <∼ 1, ∥Fh∥L∞((0,T)×Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='R2×2) + ��F−1 h �� L∞((0,T)×Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='R2×2) <∼ 1, ∥�uh∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2(�Ω)) + ��(∇�uh(Fh)−1)S�� L2((0,T)×�Ω) <∼ 1, ∥∇�uh∥L2((0,T)×�Ω) <∼ 1, ∥�uh∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Lq1(�Ω)) <∼ 1, ∥ξh∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L∞(Σ)) <∼ ∥∇�uh∥L2((0,T)×�Ω) <∼ 1, ∥�wh∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2(�Ω)) + ∥�wh∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L∞(�Ω)) <∼ 1, ∥�vh∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Lq1(�Ω)) + ∥�vh∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2(�Ω)) <∼ 1, ∥�vh∥Lq2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Lq1(�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='R2)) <∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11) for any q1 ∈ [1, ∞) and q2 ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Noticing that η1 h = η0 h + τξ0 h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' τ < 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' and the algebraic inequality (a + b)2 ≤ 2(a2 + b2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' we know that γ1 2 ��∂x1η1 h ��2 L2(Σ) + γ2 2 ��∂2 x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='hη1 h ��2 L2(Σ) 13 ≤ γ1 ���∂x1η0 h ��2 L2(Σ) + τ 2 ��∂x1ξ0 h ��2 L2(Σ) � + γ2 ���∂2 x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='hη0 h ��2 L2(Σ) + τ 2 ��∂2 x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='hξ0 h ��2 L2(Σ) � < γ1 ���∂x1η0 h ��2 L2(Σ) + ��∂x1ξ0 h ��2 L2(Σ) � + γ2 ���∂2 x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='hη0 h ��2 L2(Σ) + ��∂2 x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='hξ0 h ��2 L2(Σ) � ≤ c(γ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' γ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ∥η0∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2(Σ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ∥ξ0∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2(Σ)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' where we have used the stability of the Riesz projection operator in the last step,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Therefore, the right-hand side of the energy estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4) is uniformly bounded by a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11)1 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11)4 after noticing ηh ≥ η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Further, by Korn’s inequality, Sobolev’s inequalities (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1), the assumption (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='10), and triangular inequality, we get all the rest estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 5 Interpolation operators A critical difficulty in convergence analysis is the appropriate choice of interpolation operators for the couple (u, ξ), that inherit not only the kinematic coupling condition at the fluid-structure interface but also the divergence-free property of the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Before digging into this issue, we recall some analytic estimates that we will use frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' First, the discrete Sobolev inequalities ∥�v∥Lq1(�Ω) <∼ ∥�v∥W 1,2(�Ω) for �v ∈ W 1,2(�Ω), q1 ∈ [1, ∞), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1a) ∥v∥L∞(Σ) <∼ ∥v∥W 1,2(Σ) for v ∈ W 1,2 0 (Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1b) Next, we recall the standard projection error, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Boffi et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' al [7] ���ΠQ h p − p ��� W k,s <∼ h ∥p∥W k+1,s , k = 1, 2, s ∈ [1, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2) where ΠQ h : L2(�Ω) �→ Qf h is any suitable projection operator satisfying the above (see for example [7, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2]), which we will use for the pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The projection operators for the velocities (both for the fluid and the solid) are much more complicated in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' It starts already with the necessity of a careful choice of the interpolation operator for solid deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 Interpolation operator for the solid deformation For the solid we will use the Riesz projection as the interpolation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let η ∈ W 1,2(Σ), our Riesz projection operator Rs h reads � Σ ∂x1(Rs hη − η)∂x1ψ dx1 = 0 ∀ ψ ∈ V s h with � Σ Rs hη dx1 = � Σ η dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3) Recalling the discrete Laplace (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11) we know that for any η ∈ W 2,2(Σ) it holds � Σ (∂2 x1,hRs hη − ∂2 x1η) ψ dx1 = 0 ∀ ψ ∈ V s h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4) Setting ψ as Rs hη and ∂2 x1,hRs hη respectively in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4), we obtain by H¨older’s inequality that ∥∂x1Rs hη∥L2(Σ) <∼ ∥∂x1η∥L2(Σ) and ��∂2 x1,hRs hη �� L2(Σ) <∼ ��∂2 x1η �� L2(Σ) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5) where by a <∼ b we mean a ≤ cb for a positive constant c that is independent of the computational parameters τ and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Further, if η ∈ W 3,2(Σ), we have ��∂2 x1,hRs hη − ∂2 x1η �� L2(Σ) <∼ h ∥η∥W 3,2(Σ) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6) see the detailed proof in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 for not only one-dimensional Σ but also a multi-dimensional domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' This concludes all necessary estimates that we need for the approximation of η as well as the structure velocity ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Building on these properties we have to extend this projection into the fluid- reference domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 Interpolation operator for the fluid velocity Given the divergence-free velocity field u with the boundary condition u|ΓS = ξe2, our aim here is to construct an interpolation operator Πf h : W 1,2(�Ω)) → �V f h , such that for q ∈ [2, ∞) h1− 2 q ����u − Πf h�u ��� Lγ(�Ω) + h ���∇�u − ∇Πf h�u ��� L2(�Ω) ≤ �Ch2 ∥∆�u∥L2(�Ω) , which has to satisfy also the following two restrictions: Kinematic condition Pf hu|ΓS = Rs hξe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Weakly divergence-free condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Here, one may naively consider the form � �Ω �q∇Pf h �u : M d�x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' However, it is necessary to have � �Ω �q∇Pf h �u : Mh d�x = 0 for the convenience of convergence analysis, see Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The next theorem takes care of the first bullet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For it we need the following lemma of analytical extensions, see [19, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Assume that Ωηh is a given subgraph, with ηh > δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let φ ∈ C∞ 0 ((Σ × [0, δ/2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' [0, ∞)) such that � Σ×[0,δ/2) φ dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then there is an extension operator Eηh : W k,p(Σ) → W k,p(Ωη), for k ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='} and 1 < p < ∞, such that the following hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Eηh(ξ)(x1, x2) = (0, ξ(x1)) for x2 ∈ [δ, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' div Eηh(ξ) = ∂2φ � Σ ξ dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ��∇kEηh(ξ) �� Lp(Ωη) ≤ c ∥ξ∥W k,p(Σ), where c depends on p, k and δ only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Note that div u = 0, implying that 0 = � ΓS(t) v · ν dx1 = � Σ ∂tη dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' It implies that div Eηh(∂tη) = ∂2φ � Σ ∂tη dx1 = 0 for any ηh ≥ δ as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='7) Our construction follows tightly [7, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 and Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4], where more details on the notation and arguments can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Following the argumentation there it seems more natural to work on the computed Eulerian grid, that is the grid pushed forward by Aηh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For that reason, we first introduce the auxiliary operator Pf h on Ωηh that eventually becomes the basis for the desired operator Πf h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let the grids Th and Σh respectively defined on the domains �Ω and Σ be shape regular and quasi-uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Moreover, let ΓS = {(x1, ηh(x1)) : x1 ∈ Σ} satisfy minΣ ηh ≥ δ and ∥∂x1ηh∥L∞ ≤ L for some positive constants δ and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then there exists an interpolation operator Pf h : Wηh → Wηh, that satisfies for (ξ, �ϕ) ∈ � Wηh, γ < ∞ and ϕ := �ϕ ◦ A−1 ηh ���ϕ − Pf hϕ ��� Lγ(Ωηh) + h ���∇(ϕ − Pf hϕ) ��� L2(Ωηh) <∼ h2 ∥ �ϕ∥W 2,2(�Ω) + h2 ∥ξ∥W 2,2(Σ) , ���Pf hϕ ��� Lγ(Ωηh) + ���∇Pf hϕ ��� L2(Ωηh) <∼ ∥ �ϕ∥ W 1,2(�Ω) + ∥ξ∥W 1,2(Σ) , where the bounds depend linearly on 1 δ, L, L δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Moreover, we find Pf hϕ(x1, ηh(x1)) = (0, Rs hξ(x1)) on Σ and � Ωηh qdiv Pf hϕ dx = � Ωηh qdiv ϕ dx ∀ q ∈ Qf h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The above construction on the variable domain Ωηh implies the following corollary for the reference domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 15 Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Under the assumption of the Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2, there exists �Pf h : � Wηh → � Wηh satisfying for (ξ, �ϕ) ∈ � Wηh and γ < ∞ that ��� �ϕ − �Pf h �ϕ ��� Lγ(�Ω) + h ���∇( �ϕ − �Pf h �ϕ) ��� L2(�Ω) <∼ h2 ∥ �ϕ∥W 2,2(�Ω) + h2 ∥ξ∥W 2,2(Σ) , ��� �Pf h �ϕ ��� Lγ(�Ω) + ���∇ �Pf h �ϕ ��� L2(�Ω) <∼ ∥ �ϕ∥W 1,2(�Ω) + ∥ξ∥W 1,2(Σ) , where the bounds depend linearly on 1 δ, L, L δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Moreover, we find �Pf hϕ(x1, 1) = (0, Rs hξ(x1)) on Σ and � �Ω �q∇ �Pf h �ϕ : M(ηh) d�x = � �Ω �q∇ �ϕ : M(ηh) d�x ∀ �q ∈ �Qf h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The proof is split into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Part I: construction of a Fortin operator on Ωηh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' We start with the operator �Π1 which is the piecewise affine interpolation operator on the reference grid constructed in [7, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2], that naturally preserves zero boundary values component wisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In particular, we may define Ps hξ as �Π1( �ϕ)(x1, 1) = � �Π1(�ϕ1)(x1, 1), �Π1(�ϕ2)(x1, 1) � =: (0, Ps hξ(x1)), which is by the construction of a function in V s h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Accordingly, we define Π1ϕ := �Π1( �ϕ) ◦ Aηh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In order to show the necessary bounds, we realize by [7, equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='20)] and by the uniform Lipschitz bounds of ηh that ∥∇(Π1ϕ − ϕ)∥L2(Ωηh) ≤ cγ,L ����Π1 �ϕ − �ϕ ��� W 1,2(�Ω) ≲ min � ∥ �ϕ∥W 1,2(�Ω) , h ∥ �ϕ∥W 2,2(�Ω) � and ∥Π1ϕ − ϕ∥L2(Ωηh) ≲ min � ∥ �ϕ∥L2(�Ω) , h ∥ �ϕ∥W 1,2(�Ω) , h2 ∥ �ϕ∥W 2,2(�Ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' This finishes the construction of Π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Next, we construct Π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' We start by recalling that �V f h is piecewise affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Hence composed with Aηh these objects are not any more piecewise affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' But as by our assumptions Aηh is bi-Lipschitz, all necessary bounds for Π1 are directly inherited from the bounds of �Π1 with an additional dependence on δ, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let us focus on a generic reference cell K ∈ Th with its bubble function bK ∈ P3(K) ∩ W 1,2 0 (K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' It is obvious that ∥bK∥Lp(K) ∼ h 2 p and ∥∇bK∥Lp(K) ∼ h 2 p −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Analog estimates with dependence on δ and L do also hold for bK ◦ Aηh as ηh is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Next, we show how to map the bubble function onto the current domain Ωηh according to the change of geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let Bηh = � � K aKbK ◦ Aηh : aK ∈ R2� be the set of the potential bubble functions pushed forward by Aηh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Our aim is to find a projector Π2 : W 1,2(Ωηh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' R2) → Bd ηh = � � K aKbK ◦ Aηh : aK ∈ R2� that satisfies � K � Aηh(K) (Π2ϕ − ϕ) · ∇q dx = 0, 16 for all q = �q ◦ Aηh, �q ∈ �V f h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let �q = c + a′x1 + ax2 on K for some constants a′, c, a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then (here we take ∇ as a column vector) ∇q(x1, x2) = � a′ − ax2 η2 h(x1)∂x1ηh(x1) a ηh(x1) � = � 1 − x2 η2 h(x1)∂x1ηh(x1) 0 1 ηh(x1) � �a′ a � =: Aηh(x1) �a′ a � and thus ϕ · ∇q = (ϕ1, ϕ2)Aηh �a′ a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' This allows us to define Π2(ϕ)|Aηh(K) = � K∈Th βKbK ◦ Aηh, where βK ∈ R2 is determined by the equation βT K � Aηh(K) bK ◦ AηhAηh dx = � Aηh(K) ϕTAηh dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' It is easy to check that Aηh is bounded from above and below by positive constants and c1h2 ∥ηh∥∞ ≤ � Aηh(K) bK ◦ Aηh(x1, x2) 1 ηh(x1) dx ≤ c2h2 δ Consequently the matrix MK = � Aηh(K) bK ◦ AηhAηh dx is invertible with |M−1 K | ≤ 1 | det( � Aηh(K) bK ◦ AηhAηh dx)| � Aηh(K) bK ◦ Aηh|Aηh| dx ≤ ch−2, where c depends linearly on L and L δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Further, we find |βK| ≤ |M−1 K | ∥Aηh∥∞ ∥ϕ∥L1(Aηh(K)) ≤ ch−2 ∥ϕ∥L1(Aηh(K)) ≤ ch− 2 p ∥ϕ∥Lp(Aηh(K)) where in the last step we have used Jensen’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Using the above estimate, we have ∥∇Π2ϕ∥Lp(Aηh(K)) ≤ |βK| ∥∇(bK ◦ Aηh)∥Lp(Aηh(K)) ≤ ch−1 ∥ϕ∥Lp(Aηh(K)) , and ∥Π2ϕ∥Lp(Aηh(K)) ≤ |βK| ∥(bK ◦ Aηh)∥Lp(Aηh(K)) ≤ c ∥ϕ∥Lp(Aηh(K)) , which allows us to follow the arguments at the end of [7, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4] to gain the expected estimates and bounds for the operator: Πf h(ϕ) := Π1(ϕ) + Π2(ϕ − Π1(ϕ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Part II: a Fortin operator with appropriate boundary values By construction Πf hϕ(x1, ηh(x1)) = Π1ϕ(x1, ηh(x1)) =: (0, Ps hξ(x1)), with Ps h being an interpola- tion operator for V s h with natural stability properties and error bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The problem is that, unlike Rs h, the operator Ps h does not have the required second-order estimates (in particular Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 does not hold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Nevertheless by the orthogonality of the error for Rs h and the estimates of first order for Ps hξ(x1), we find that ∥∂x1(Rs hξ − Ps hξ)∥L2(Σ) ≤ ∥∂x1(ξ − Ps hξ)∥L2(Σ) ≤ ch ��∂2 x1ξ �� L2(Σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8) The desired projector turns out to be the solution to a discrete Stokes problem: We derive it for ηh, ϕ and ξh fixed by minimizing � Ωηh |∇(ψh − ϕ)|2 dx 17 over the class of all ψh ∈ V f h , with ψh(x1, ηh(x1)) = (0, ξh(x1)) on Σ, which satisfy the discrete divergence-free property: � Ωηh div (ψh)·q dx for all q ∈ Qf h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The minimizer is then defined as Pf hϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The respective Euler-Lagrange equation becomes the discrete solution to an approximate Stokes problem � Ωηh ∇(Pf hϕ − ϕ) · ∇ψh dx = 0, for all ψh ∈ V f h with zero boundary values and which are discretely divergence-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The error of the projector is of two kinds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The first is the error stemming from the prescribed boundary values, and the second is the discretization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For the first, we take the linear divergence-free extension Eηh(Rs hξ − ξ) given by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Now we can take ψh = Πf h(ϕ + Eηh(Rs hξ − ξ)) as competitor in the minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Indeed, as (0, ξ(x1)) = ϕ(x1, ηh(x1)), we find that Πf h(ϕ + Eηh(Rs hξ − ξ))(x1, ηh(x1)) = Πf h(Eηh(ξ))(x1, ηh(x1)) = (0, Ps h(ξ)(x1) in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' This implies (as the projector is the minimizer) that ���∇(Pf hϕ − ϕ) ��� 2 L2(Ωηh) ≤ ���∇(Πf h(ϕ + Eηh(Rs hξ − ξ)) − ϕ) ��� 2 L2(Ωηh) ≤ c ���∇ϕ − ∇Πf hϕ ��� 2 L2(Ωηh) + c ���∇(Πf hEηh(Rs hξ − ξ)) ��� 2 L2(Ωηh) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The first term is estimated directly by the properties of the Fortin operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The second one is by the stability of the Fortin operator, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4 (On the importance of the proper choice of an interpolation operator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The deep rea- son why the interpolation has to be solved as a discrete PDE, is that the solid matter and the fluid matter have totally different properties, even so they are coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Our scheme follows the direct path that is also used in the existence theory, where already in the approximation the coupling and the different matters are simultaneously (monolithically) solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The fact that this uniform (and linear) approximation does indeed converge properly can only be revealed by imitating the coupling between two solutions of independent PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' This imitation is exactly performed by solving a discrete boundary value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The last step for the interpolation of u is the correction of the divergence due to the change of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For that, we use another analytic tool developed in [19, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' It is the so-called universal Bogovskij operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Universal it is, because it is independent of the particular (Lipschitz) geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' We cite the important estimate in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' There is an operator B : {f ∈ Lp(Ωηh) : � Ωηh f dx = 0} → W 1,p 0 (Ωηh) for any Ωηh for 1 < p < ∞ that is a given subgraph, with minΣ ηh > δ and ∥∂xηh∥L∞ ≤ L, such that the following hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' div B(f) = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ∥B(f)∥W 1,p(Ω) ≤ ∥f∥Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The above lemma and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3 lead to the final statement of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let Ωηh ⊂ R2 be a subgraph and let the assumptions of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then there exists Πf h : � Wη → � Wηh, satisfying for (ξ, u) ∈ Wη and γ < ∞ that ����u − Πf h�u ��� Lγ(�Ω) + h ���∇(�u − Πf h�u) ��� L2(�Ω) <∼ h2 ∥�u∥W 2,2(�Ω) + h2 ∥ξ∥W 2,2(Σ) + h ∥η − ηh∥W 1,2(Σ) , ���Πf h�u ��� Lγ(�Ω) + ���∇Πf h�u ��� L2(�Ω) <∼ ∥�u∥W 1,2(�Ω) + ∥ξ∥W 1,2(Σ) + ∥η − ηh∥W 2,2(Σ) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9) 18 where the bounds depend linearly on 1 δ, L, L δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Moreover, we find Πf hu(x1, 1) = (0, Rs hξ(x1)) on Σ and � �Ω �q∇Πf hu : M(ηh) dx = 0 ∀ �q ∈ �Qf h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The proof takes the function ϕ := u◦Aηh ◦A−1 η −B(div (u◦Aηh ◦A−1 η )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' By the Gauss theorem we note that � Ωηh div (u ◦ Aηh ◦ A−1 η ) dx = 0, hence B is well defined and so div ϕ = 0 on Ωηh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Further div (u ◦ Aηh ◦ A−1 η ) = � η ηh − 1 � ∂2u2 ◦ Aηh ◦ A−1 η + ∂1 � η ηh � x2∂2u1 ◦ Aηh ◦ A−1 η , which implies as W 1,∞(Σ) ⊂ W 2,2(Σ) that ��div u ◦ Aηh ◦ A−1 η �� L2(Ωηh) ≤ c ∥η − ηh∥W 2,2(Σ) ∥∇�u∥L2(�Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Hence by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5 and a change of variable we find ∥ �ϕ − �u∥W 1,2(�Ω) ≤ c ∥η − ηh∥W 2,2(Σ) ∥∇�u∥L2(�Ω) and �ϕ(x1, 1) = ξ(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then, we define Πf h�u = �Pf h �ϕ for which now the result follows from the previous estimates and Corol- lary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 6 Error estimates In this section, we study the error between the numerical solution (�ph, �uh, ξh, ηh) of Scheme-R and its target smooth solution (�p, �u, ξ, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Here we assume the existence of a smooth solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4) in the following class � � � � � � � � � � � η > η, η ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' W 3,2(Σ)) ∩ W 2,2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' W 2,2(Σ)), �u ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' W 1,2(�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' R2)) ∩ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' W 2,2(�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' R2)) ∂t�u ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' W 1,2(�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' R2)), �p ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' L2(�Ω)), ∇p ∈ L2((0, T) × �Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 The time projection Very relevant in this highly nonlinear coupled system is to choose a set of appropriate time-value tτ k, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' , NT , at which we will compare the continuous equation with its numerical approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For a given τ and k, we denote (�pk, �uk, ξk, ηk) := (�p, �u, ξ, η)(tτ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then, according to our smoothness assumption (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' we may choose the value tτ k ∈ [kτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (k + 1)τ) in such a way that τ � ∥�u(tτ k)∥2 W 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2(�Ω) + ∥∂t�u(tτ k)∥2 W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2(�Ω) + ��∂2 t �u(tτ k) ��2 L2(�Ω) + ∥η(tτ k)∥2 W 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3(Σ) + ∥ξ(tτ k)∥2 W 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4(Σ) + ∥∂tξ(tτ k)∥2 W 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2(Σ) � ≤ � (k+1)τ kτ � ∥�u(t)∥2 W 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2(�Ω) + ∥∂t�u(t)∥2 W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2(�Ω) + ��∂2 t �u(t) ��2 L2(�Ω) + ∥η(t)∥2 W 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3(Σ) + ∥ξ(t)∥2 W 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2(Σ) ∥∂tξ(t))∥2 W 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2(Σ) � dt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 19 which is possible to find by the continuity of the integral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' if the right-hand side is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In particular, we find that τ NT � k=1 � ����uk��� 2 W 2,2(�Ω) + ���∂t�uk��� 2 W 1,2(�Ω) + ���∂2 t �uk��� 2 L2(�Ω) + ���ηk��� 2 W 2,2(Σ) + ���∂tξk��� 2 W 2,2(Σ) � ≤ � T 0 � ∥�u∥2 W 2,2(�Ω) + ∥∂t�u∥2 W 1,2(�Ω) + ��∂2 t �u ��2 L2(�Ω) + ∥η∥2 W 2,2(Σ) + ∥∂tξ)∥2 W 2,2(Σ) � dt (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2) Actually, this right-hand side summarizes our regularity assumptions on the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' All the above regularity requirements do follow from these assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 (On the regularity assumptions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' When comparing the assumptions on the smooth so- lution with the theory for the heat/wave equation (or the 2D/Navier-Stokes equation), one realizes that we have the same regularity assumptions for the fluid as in the non-moving case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For the plate, which also deduces the domain essentially one more time-derivative has to be assumed, as nonlinear equations of a similar type can be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 Main result Before introducing the main result, let us denote the following error terms for each time step k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' , NT }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ek p = �pk h − �pk = (�pk h − ΠQ h �pk) + (ΠQ h �pk − �pk) =: δk p + Ik p , ek u = �uk h − �uk = (�uk h − Πf h�uk) + (Πf h�uk − �uk) =: δk u + Ik u, ek ξ = ξk h − ξk = (ξk h − Rs hξk) + (Rs hξk − ξk) =: δk ξ + Ik ξ , ek η = ηk h − ηk = (ηk h − Rs hηk) + (Rs hηk − ηk) =: δk η + Ik η , ek ζ = ζk h − ζk = (ζk h + ∂2 x1,hRs hηk) + (−∂2 x1,hRs hηk − ζk) =: δk ζ + Ik ζ , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3) where ζh = −∂2 x1,hηh and ζ = −∂2 x1η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Now we are ready to present the main result of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 (Convergence rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let {(�pk h, �uk h, ξk h, ηk+1 h )}NT k=1 be the solution of Scheme-R (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='12), and let (�u, �p, ξ, η)(t), t ∈ (0, T), be a strong solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4) belonging to the class (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then for any m ∈ {1, · · · , NT } it holds 1 2ϱf � �Ω |em u |2ηm h d�x + 1 2 � Σ � ϱs|em ξ |2 + γ1|∂x1em+1 η |2 + γ2|em+1 ζ |2� dx1 + 2µτ m � k=1 � �Ω ���∇ek u(Fk h)−1��� 2 d�x + γ3ϱsτ m � k=1 � Σ |∂x1δk ξ |2 dx1 <∼ τ 2 + h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In particular, we have the following convergence rates ∥eu∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2(�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='R2)) + ∥eξ∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2(Σ)) + ∥∂x1eη∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2(Σ)) + ∥eζ∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2(Σ)) + ∥∇eu∥L2((0,T)×�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='R2×2) + γ3 ∥∂x1eξ∥L2((0,T)×Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='R2) <∼ τ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' First, we subtract the weak formulation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9b) from the numerical scheme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='12b) and get � �Ω ϱf(ηk hDtek u + 1 2Dtηk hek∗ u ) · �ϕ d�x + 2µ � �Ω � ∇ek u(Fk h)−1�S : � ∇ �ϕ(Fk h)−1� ηk h d�x + ϱs � Σ Dtek ξψ dx1 + as(ek+1 η , ek+1 ζ , ek ξ, ψ) = − 7 � i=1 Rk i ( �ϕ, ψ), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5) 20 where Rk 1( �ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ψ) = ϱf � �Ω � ek η∂t�uk + ηk h(Dt�uk − ∂t�uk) � �ϕ d�x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Rk 2( �ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ψ) = 1 2ϱf � �Ω � (ek−1 ξ − τDtξk)�uk∗ − τ∂tηkDt�uk� �ϕ d�x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Rk 3( �ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ψ) = 1 2ϱf � �Ω � �ϕ · (∇ek u) − ek u · (∇ �ϕ) � (Fk h)−1�vk−1 h ηk h d�x + 1 2ϱf � �Ω � �ϕ · (∇�uk) − �uk · (∇ �ϕ) � � (Fk h)−1�vk−1 h ηk h − (Fk)−1�vkηk� d�x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Rk 4( �ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ψ) = � �Ω ek p∇ �ϕ : Mk h d�x + � �Ω �pk∇ �ϕ : � Mk h − Mk� d�x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Rk 5( �ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ψ) = 2µ � �Ω �� ∇�u(Fk h)−1�S : (∇ �ϕ(Fk h)−1)ηk h − � ∇�u(F)−1�S : (∇ �ϕ(Fk)−1)ηk� d�x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Rk 6( �ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ψ) = ϱs � Σ (Dtξk − ∂tξk)ψ dx1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Rk 7( �ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ψ) = −γ1 � Σ ∂2 x1(ηk+1 − ηk) ψ dx1 − γ2 � Σ ∂2 x1(ζk+1 − ζk) ψ dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6) The precise justification of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5) is given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' By setting ( �ϕ, ψ) = (δk u, δk ξ ) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5) and sum up from k = 1 to m we derive −τ m � k=1 7 � i=1 Rk i (δk u, δk ξ ) = τ m � k=1 � �Ω ϱf(ηk hDtek u + 1 2Dtηk hek∗ u ) · δk u d�x + 2µτ m � k=1 � �Ω � ∇ek u(Fk h)−1�S : � ∇δk u(Fk h)−1� ηk h d�x + τ m � k=1 ϱs � Σ Dtek ξδk ξ dx1 + τ m � k=1 as(ek+1 η , ek+1 ζ , ek ξ, δk ξ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='7) Further, applying the algebraic equalities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='10) to the above right-hand-side,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' we get (simi- larly as was performed for the stability estimate) − τ m � k=1 7 � i=1 Rk i (δk u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' δk ξ ) = τ m � k=1 � �Ω ϱf � ηk hDt(δk u + Ik u) + 1 2Dtηk h(δk∗ u + Ik∗ u ) � δk u d�x + 2µτ m � k=1 � �Ω � ∇(δk u + Ik u)(Fk h)−1�S : (∇δk u(Fk h)−1)ηk h d�x + τ m � k=1 ϱs � Σ Dt(δk ξ + Ik ξ ) δk ξ dx1 + τ m � k=1 as(ek+1 η ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ek+1 ζ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ek ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' δk ξ ) = δm E − δ0 E + τ m � k=1 Dk phys + τ m � k=1 Dk num + Gf + Gs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8) where δk E = � �Ω 1 2ϱfηk h|δk u|2 d�x + 1 2 � Σ � ϱs|δk ξ |2 + γ1|∂x1δk+1 η |2 + γ2|δk+1 ζ |2� dx1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' δk D =2µ � �Ω ηk h| � ∇δk u(Fk h)−1�S|2 d�x + γ3 � Σ |∂x1δk ξ |2 dx1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Dk num =τ 2ϱf � �Ω ηk−1 h |Dtδk u|2 d�x + τ 2 � Σ � ϱs|Dtδk ξ |2 + γ1|Dt∂x1δk+1 η |2 + γ2|Dtδk+1 ζ |2� dx1 ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Gf =τ m � k=1 � �Ω ϱf � ηk hDtIk u + 1 2Dtηk hIk∗ u � δk u d�x 21 + 2µτ m � k=1 � �Ω � ∇Ik u(Fk h)−1�S : (∇δk u(Fk h)−1)ηk h d�x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Gs =γ1τ m � k=1 � Σ ∂x1δk+1 η ∂x1(Dtηk+1 − ∂tηk) dx1 − γ2τ m � k=1 � Σ δk+1 ζ ∂2 x1(Dtηk+1 − ∂tηk) dx1 + τ m � k=1 � Σ ϱsDtIξδk ξ dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Next, we reformulate (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8) in the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' δm E − δ0 E + τ m � k=1 δk D + τ m � k=1 Dk num = −τ m � k=1 7 � i=1 Rk i − Gf − Gs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9) Then, by Young’s inequality, H¨older’s inequality, the interpolation error in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6, and the uniform bounds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11), we estimate the right-hand-side of the above equation as �����τ m � k=1 7 � i=1 Rk i + Gf + Gs ����� <∼ τ 2 + h2 + cτ m � k=1 δm E + 2αµτ m � k=1 � �Ω ���∇δk u(Fk h)−1��� 2 ηk h d�x, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='10) see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Further, substituting the above estimate into (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9) and noticing the initial error δ0 E = 0, we get (using also the lower bound of η, ηh) that δm E + (1 − α)2µτ m � k=1 � �Ω ���∇δk u(Fk h)−1��� 2 ηk h d�x + γ3ϱsτ m � k=1 � Σ |∂x1δξ|2 dx1 <∼ τ 2 + h2 + τ m � k=1 δk E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' By choosing any α ∈ (0, 1) and using Gr¨onwall’s inequality, we get δm E + τ m � k=1 δk D <∼ τ 2 + h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Recalling the interpolation errors (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6)) and the regularity of the strong solution (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1) we get 1 2ϱf � �Ω |Im u |2ηm h d�x + 1 2 � Σ � ϱs|Im ξ |2 + γ1|∂x1Im+1 η |2 + γ2|Im+1 ζ |2� dx1 + τ m � k=1 � 2µ � �Ω ���∇Ik u(Fk h)−1��� 2 ηk h d�x + γ3ϱs � Σ |∂x1Ik ξ |2 dx1 � <∼ h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Finally, due to the triangular inequality, we sum up the previous two estimates and get 1 2ϱf � �Ω |em u |2ηm h d�x + 1 2 � Σ � ϱs|eξ|2 + γ1|∂x1em+1 η |2 + γ2|em+1 ζ |2� dx1 + τ m � k=1 � 2µ � �Ω ���∇ek u(Fk h)−1��� 2 ηk h d�x + γ3ϱs � Σ |∂x1δk ξ |2 dx1 � <∼ τ 2 + h2, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11) which provides the proof for small T ≤ T0 such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='10) is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Next, we show that T can be arbitrarily large if η ≥ η on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' We start with a fixed T0 such that ηh ≥ η 2, this can be found by [26, Lemma 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then, by the above estimate (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11), we know that ∥eη(T0)∥L∞ ≤ c(τ + h), where the constant c depends on the lower bound η 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Recalling η(t) ≥ η we know that ηh(T0) ≥ η − c(τ + h), 22 which actually is much larger than η 2, if τ, h are small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Hence by [26, Lemma 5], there is a T1 > T0, such that ηh(T1) ≥ 2η 3 − c(τ + h) ≥ η 2 for τ and h small enough, where T1 depends only on the initial energy of the problem but is independent of τ and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Hence we can repeat the above argument with the same lower bound η 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' It implies for τ, h → 0 that this procedure can be repeated arbitrarily many times, thus (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='10) hold for any large T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 7 Numerical experiments In this section, we define a problem that we use to study the convergence rate of the linear semi-implicit Scheme-R (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='12) on a reference domain �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' This semi-implicit scheme is then compared with the nonlinear fully implicit scheme corresponding to the weak form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Both numerical implementations are described in detail in Appendices C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 Problem description In our experiments, the domain �Ω is a rectangle of dimensions 2×1 with periodic boundary conditions in the x1−direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' the solution on the left boundary coincides with the solution on the right boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' On the bottom we have no-slip boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' At t = 0, we prescribe zero initial conditions for all unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Moreover, we set ϱf = ϱs = 1, γ1 = γ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1, and γ3 = 0 since we wish to solve a problem with a non-dissipating elastic shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The flow is driven by the external force g periodic in x1 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The force is applied up to t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 such that a big amplitude of the structure deformation is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Next, the force is turned off and the system is left to relax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The force g reads g = �200t sin(2πx) t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2, 0 t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Snapshots of the simulation are given in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (a) (b) (c) (d) (e) (f) Figure 2: Snapshots of the simulation: (a) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 end of loading, (b) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='26 maximum of amplitude, (c) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='35, (d) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='41, (e) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='45, (f) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='53 another maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The color scale depicts pressure, arrows show the direction of the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 23 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 Convergence rates The simulation is computed for t ∈ [0, T], T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='0 for six different time steps τ = 5 × 10−3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5 × 10−3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='25 × 10−3, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='25 × 10−4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='125 × 10−4 and τmin = 1 × 10−4 on six different meshes with the mesh sizes h = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='83 × 10−1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='41 × 10−1, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='07 × 10−2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='54 × 10−2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='77 × 10−2 and hmin = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='84 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The solution with the finest mesh (corresponding to 410 880 degrees of freedom (dofs) in Step 1 of the implementation of Scheme-R, see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1) and the smallest time step is used as the reference solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The solutions for different mesh refinements and the smallest time step are compared to the reference solution, specifically, we record all summands of the right-hand-side of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2, these are: ∥eu∥L∞(L2), ∥eξ∥L∞(L2), ∥eη∥L∞(L2), ∥∇eη∥L∞(L2), ∥eζ∥L∞(L2) and ∥∇eu∥L2(L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The convergence with respect to the mesh size h is given in the Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The graphs depicting the convergence rate with respect to the mesh size h are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The convergence with respect to the time step τ is provided in Table 2 and Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' h ∥eu∥L∞(L2) ∥eξ∥L∞(L2) ∥eη∥L∞(L2) ∥∇eη∥L∞(L2) ∥eζ∥L∞(L2) ∥∇eu∥L2(L2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='83 × 10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='20 × 100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='84 × 100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='22 × 10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='41 × 100 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='22 × 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='23 × 101 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='41 × 10−1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='19 × 10−1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='80 × 10−1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='99 × 10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='79 × 10−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='42 × 100 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='51 × 100 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='07 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='05 × 10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='39 × 10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='52 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='34 × 10−1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='02 × 10−1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11 × 100 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='54 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='78 × 10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='31 × 10−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='65 × 10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='57 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='44 × 10−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='12 × 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='77 × 10−2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='91 × 10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='64 × 10−3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='32 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='94 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='89 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='04 × 100 Table 1: Convergence of errors with mesh refinement (using fixed time step τ = τmin);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' reference solution: hmin = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='84 × 10−3, τmin = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' τ ∥eu∥L∞(L2) ∥eξ∥L∞(L2) ∥eη∥L∞(L2) ∥∇eη∥L∞(L2) ∥eζ∥L∞(L2) ∥∇eu∥L2(L2) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='00 × 10−3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='55 × 10−1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='50 × 10−1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='23 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='66 × 10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='67 × 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='61 × 100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='50 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='36 × 10−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='87 × 10−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='21 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='39 × 10−1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='74 × 10−1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='52 × 10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='25 × 10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='87 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='43 × 10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='10 × 10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='91 × 10−2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='35 × 10−1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='28 × 10−1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='25 × 10−4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='25 × 10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='73 × 10−2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='17 × 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='25 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='05 × 10−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='02 × 10−1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='12 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='37 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='83 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='17 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='36 × 10−2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='60 × 10−2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='45 × 10−2 Table 2: Convergence of errors with time step refinement (using fixed mesh size h = hmin);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' reference solution: hmin = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='84 × 10−3, τmin = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 we proved that the convergence rate is linear both in h (space) and τ (time) for the sum of all errors mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' This is justified by the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In time the convergence is indeed linear for all summands (see Figure 4), in space we observe a quadratic convergence for ∥eu∥L∞(L2), ∥eξ∥L∞(L2), ∥eη∥L∞(L2), ∥eζ∥L∞(L2), but a linear convergence for ∥∇eη∥L∞(L2) and ∥∇eu∥L2(L2) (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3 Comparison between the semi-implicit Scheme-R and fully implicit scheme Since our proposed Scheme-R performs in accordance with the (optimal) predictions, we decided to test how well it behaves with respect to the fully implicit scheme, as many researchers believe that a monolithic scheme should be implemented fully implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In the fully implicit scheme we solve a fully implicit nonlinear problem based on the weak form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9), the details of the implementation are given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The main difference is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In the semi-implicit Scheme-R, as described in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1, every time step is solved in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' First, we solve a linear problem for velocity u, the second order derivative of the mesh displacement ζ, and the pressure p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' This is followed by a second step in which we update the mesh displacement η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In the fully implicit scheme we solve everything at once, which, however, requires to solve a more expensive nonlinear problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' It turns out that both schemes produce the same solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Since the main difference between the two schemes is in the time splitting, we compare the numeri- cal errors ∥∇eη∥L∞(L2) and ∥∇eη∥L∞(L2) for several different time steps τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Similarly as in the previous 24 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 10−1 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6 10−2 10−1 100 1 order 2 1 order 1 h Errors to reference solution ∥eu∥L∞(L2) ∥eξ∥L∞(L2) ∥eη∥L∞(L2) ∥∇eη∥L∞(L2) ∥eζ∥L∞(L2) ∥∇eu∥L2(L2) Figure 3: Mesh convergence comparison for ∥eu∥L∞(L2), ∥eξ∥L∞(L2), ∥eη∥L∞(L2), ∥∇eη∥L∞(L2), ∥eζ∥L∞(L2) and ∥∇eu∥L2(L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For a better comparison, the plots of the errors are shifted to start from the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 10−3 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 10−1 100 1 order 1 τ Errors to reference solution ∥eu∥L∞(L2) ∥eξ∥L∞(L2) ∥eη∥L∞(L2) ∥∇eη∥L∞(L2) ∥eζ∥L∞(L2) ∥∇eu∥L2(L2) Figure 4: Timestep convergence comparison ∥eu∥L∞(L2), ∥eξ∥L∞(L2), ∥eη∥L∞(L2), ∥∇eη∥L∞(L2), ∥eζ∥L∞(L2) and ∥∇eu∥L2(L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For a better comparison, the errors start at the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 25 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 10−3 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 10−1 100 τ Errors to reference solution ∥∇eu∥L2(L2) semi-implicit ∥∇eu∥L2(L2) fully implicit ∥∇eη∥L∞(L2) semi-implicit ∥∇eη∥L∞(L2) fully implicit Figure 5: Timestep convergence comparison semi-implicit vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' fully implicit for ∥∇eu∥L2(L2) and ∥∇eη∥L∞(L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' subsection, we compute the errors with respect to the same reference solution obtained by the semi- implicit Scheme-R with the finest mesh hmin = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='84 × 10−3 and smallest time step τ = 1 × 10−4 for the numerical solutions of these two schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The graph of convergence in time is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Note that the two discrete problems are computed on the same mesh h = hmin for both schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In case of the semi-implicit Scheme-R we solve a linear problem of size 410 880 dofs in Step 1 and a linear problem of size 153 920 dofs in Step 2 every time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In case of the fully implicit scheme we solve a nonlinear problem of size 564 800 dofs every time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' All problems are computed on a server equipped with Intel Xeon Gold 6240 CPU, and (although the code works in parallel) for the purpose of comparison we run them in serial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' We have recorded the CPU time of the computations for the largest and smallest time step τ, see Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For the largest time step τ = 5 × 10−3 the semi- implicit scheme is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='53 times faster than the fully implicit scheme that needs to solve three Newton iterations in average in every time step and solves a slightly larger problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For the smallest time step τ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='12 × 10−4 the semi-implicit scheme is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='88 times faster because the fully implicit scheme needs in average only two Newton iterations per time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Scheme τ Avg Newton its CPU time [min] Fully implicit 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='00 × 10−3 3 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5 Semi-implicit 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='00 × 10−3 – 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5 Fully implicit 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='12 × 10−4 2 1 310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='7 Semi-implicit 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='12 × 10−4 – 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='0 Table 3: Comparison of CPU times (in minutes) for semi-implicit and fully implicit schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 8 Conclusion We have introduced a novel semi-implicit and linear scheme for the approximation of the interaction between an incompressible fluid and elastic shell allowing for large deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' By this we mean that the domain of definition for the fluid is time changing and the changes of the domain can be arbitrarily large as long as no topological change appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' We have proved that the scheme is energy stable and it converges to the smooth solution linearly with respect to the mesh size h and time step τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' We have implemented the scheme in FEniCS and observed that the convergence rates are optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Possibly, the rates can be improved for some terms, where we observed quadratic growth which paves 26 the way for further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' We have compared our semi-implicit scheme with a fully implicit scheme that not only provides the same convergence rates as our scheme but does produce almost exactly the same solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Moreover, our scheme overperforms the fully nonlinear scheme several times in terms of consumed CPU time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The analysis presented here, in particular the development of the interpolation operators does form the basis for new theoretical numerical investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' It is shown that suitable interpolation operators for nonlinear equations coupled via their geometry can be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Hence they motivate a respective convergence analysis for the plethora of applications involving such couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' From their construction, it might also be possible to read where troubles of the convergence of schemes may be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For instance in case, when a topological change of geometry is approaching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Statements and Declarations Competing interests: On behalf of all authors, the corresponding author states that there is no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Alnæs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Blechta, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Hake, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Johansson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Kehlet, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Logg, C.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Tezduyar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Frontiers in Computational Fluid–Structure Interaction and Flow Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' series in Modeling and Simulation in Science, Engineering and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Birkh¨auser/Springer, xvi+480pp, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' [30] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Wang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Jimack, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Walkley and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Pironneau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' An energy stable one-field mono- lithic arbitrary Lagrangian–Eulerian formulation for fluid–structure interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Fluids and Structures 98: paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 103117, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 28 A Appendix: Interpolation operators In this part, we present some useful estimate/equality for the interpolation operators used in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' First, we show the approximation error of the discrete Laplace ∂2 x1,h given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11) of a Riesz projection operator Rs h defined by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let η ∈ W 3,2 ∩ W 1,2 0 (Σ), Σ ⊂ Rn, n = 2, 3, and V s h ⊂ W 1,2 0 (Σ) is a closed subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For any ψ ∈ V s h , let ∆hφ ∈ V s h satisfy − � Σ ∆hφψdz = � Σ ∇φ · ∇ψdz and Rs h satisfy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3) in n dimensions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=', � Σ (∇η − ∇Rs hη) · ∇ψ dz = 0 Moreover, we assume there exists a projection Ph : L2(Σ) → V s h satisfying ∥η − Phη∥2 ≤ ch ∥∇η∥2 ∀ η ∈ W 1,2(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then ∥∆η − ∆hRs hη∥2 ≤ ch ∥∇∆η∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' By the definition of the Riesz projection, and the discrete Laplace, we find ∥∆η − ∆hRs hη∥2 2 = � Σ (∆η − ∆hRs hη) (∆η − ∆hRs hη) dz = � Σ (∆η − ∆hRs hη) (Ph∆η − ∆hRs hη) dz + � Σ (∆η − ∆hRs hη) (∆η − Ph∆η) dz = − � Σ (∇η − ∇Rs hη) · (∇(Ph∆η − ∆hRs hη)) dz + � Σ (∆η − Ph∆η) (∆η − ∆hRs hη) dz ≤ ∥∆η − Ph∆η∥2 ∥∆η − ∆hRs hη∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' This implies the wanted estimate by the assumed property of Ph: ∥∆η − Ph∆η∥2 ≤ ch ∥∇∆η∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In our setting one possibility is to choose as Ph the L2-Projection into V s h defined by � (η − Phη) φh dz = 0 for all φ ∈ V s h , which is known to satisfy in our setting the needed estimate ∥η − Phη∥2 ≤ ch ∥∇η∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Note that as Phη ∈ V s h by definition, it is in particular a Lipschitz function and possesses a weak gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let �uh and �u be respectively the solution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='12a) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9a) with the test function �q ∈ �Qf h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then it holds 0 = � �Ω �q∇δu : Mh d�x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='12a) and the property of the fine constructed projection (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='10), we derive � �Ω �q∇δu : Mh d�x = � �Ω �q∇�uh : Mh d�x − � �Ω �q∇Πf h�u : Mh d�x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 29 B Appendix: Useful equalities and estimates B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 Proof of the error equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5) In this part, we show the details how to obtain the equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5) satisfied by the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' First, for any k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' , NT we subtract the weak formulation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9b) from the numerical scheme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='12b) and get 7 � i=1 T k i = 0, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1) where T k i reads (keeping in mind that Rk i , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' , 7, are given in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6)) T k 1 =ϱf � �Ω (ηk hDt�uk h − ηk∂t�uk) · �ϕ d�x =ϱf � �Ω � ηk hDt(�uk h − �uk) + ηk h(Dt�uk − ∂t�uk) + (ηk h − ηk)∂t�uk� �ϕ d�x =ϱf � �Ω ηk hDtek u · �ϕ d�x + Rk 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' T k 2 =1 2ϱf � �Ω (Dtηk h�uk∗ h − ∂tηk�uk) · �ϕ d�x =1 2ϱf � �Ω � Dtηk h(�uk∗ h − �uk∗) + (Dtηk h − ∂tηk)�uk∗ + ∂tηk(�uk∗ − �uk) � �ϕ d�x =1 2ϱf � �Ω Dtηk hek∗ u · �ϕ d�x + Rk 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' T k 3 =1 2ϱf � �Ω � �ϕ · (∇�uk h) − �uk h · (∇ �ϕ) � (Fk h)−1 · �vk−1 h ηk h d�x − 1 2ϱf � �Ω � �ϕ · (∇�uk) − �uk · (∇ �ϕ) � (Fk)−1 · �vkηk d�x =1 2ϱf � �Ω � �ϕ · (∇ek u) − ek u · (∇ �ϕ) � (Fk h)−1�vk−1 h ηk h d�x + 1 2ϱf � �Ω � �ϕ · (∇�uk) − �uk · (∇ �ϕ) � � (Fk h)−1�vk−1 h ηk h − (Fk)−1�vkηk� d�x =Rk 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' T k 4 = � �Ω � �pk h∇ �ϕ : Mk h − �pk∇ �ϕ : Mk� d�x = � �Ω ek p∇ �ϕ : Mk h d�x + � �Ω �pk∇ �ϕ : � Mk h − Mk� d�x = Rk 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' T k 5 =2µ � �Ω �� ∇�uk h(Fk h)−1�S : � ∇ �ϕ(Fk h)−1� ηk h − � ∇�uk(Fk)−1�S : � ∇ �ϕ(Fk)−1� ηk� d�x =2µ � �Ω � ∇ek u(Fk h)−1�S : � ∇ �ϕ(Fk h)−1� ηk h d�x + 2µ � �Ω �� ∇�uk(Fk h)−1�S : (∇ �ϕ(Fk h)−1)ηk h − � ∇�uk(Fk)−1�S : (∇ �ϕ(Fk)−1)ηk� d�x =2µ � �Ω � ∇ek u(Fk h)−1�S : � ∇ �ϕ(Fk h)−1� ηk h d�x + Rk 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' T k 6 =ϱs � Σ (Dtξk h − ∂tξk)ψ dx1 = ϱs � Σ Dtek ξψ dx1 + Rk 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' T k 7 =as(ηk+1 h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ζk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ξk h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ψ) − as(ηk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ζk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ξk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ψ) =as(ek+1 η ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ek+1 ζ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ek ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ψ) + γ1 � Σ ∂x1(ηk+1 − ηk)∂x1ψ dx1 + γ2 � Σ ∂x1(ζk+1 − ζk)∂x1ψ dx1 =as(ek+1 η ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ek+1 ζ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ek ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ψ) + Rk 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 30 Consequently, substituting the above expansions of the Ti-terms into (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1) and shifting the Ri-terms to the right-hand-side, we derive (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 Preliminary estimates In this part we show some preliminary estimates and equalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' First, we show the estimates related to the time discretization operator Dt given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let φ ∈ L2((0, T) × D) for D ∈ {Σ, �Ω}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then we have τ N � k=1 ���Dtφk��� 2 L2(D) <∼ ∥∂tφ∥2 L2((0,T)×D) , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2a) τ N � k=1 ���Dtφk − ∂tφk��� 2 L2(D) <∼ τ 2 ��∂2 t φ ��2 L2((0,T)×D) , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2b) τ N � k=1 ���Dtφk+1 − ∂tφk��� 2 L2(D) <∼ τ 2 ��∂2 t φ ��2 L2((0,T)×D) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2c) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' We only need to prove (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2c), the others follow analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 To begin, we recall the product rule (uv)′ = u′v + uv′ and apply it for u = (tk+1 − t)/τ and v = ∂tφ(t), which yields � tk+1 tk tk+1 − t τ ∂2 t φ(t) dt = �tk+1 − t τ ∂tφ(t) �tk+1 tk + 1 τ � tk+1 tk ∂tφ(t) dt = Dtφk+1 − ∂tφ(tk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Thanks to the above equality and H¨older’s inequality, we obtain τ N � k=1 ���Dtφk+1 − ∂tφk��� 2 L2(D) = τ N � k=1 � D ����� � tk+1 tk tk+1 − t τ ∂2 t φ(t) dt ����� 2 dx ≤ τ N � k=1 � D �� tk+1 tk ���� tk+1 − t τ ���� 2 dt � �� tk+1 tk ��∂2 t φ(t) ��2 dt � dx ≤ 1 3τ 2 � D N � k=1 � tk+1 tk ��∂2 t φ(t) ��2 dtdx = 1 3τ 2 ��∂2 t φ(t) ��2 L2((0,T)×D) , which proves (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let η ∈ W 2,2(Σ), ξ = ∂tη, ζ = −∂2 x1η, ηh ∈ V s h , ξk h = Dtηk+1 h , k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' , NT , ζh = −∂2 x1,hηh, and ψ ∈ V s h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let as be given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='7) and the notation of the errors be given by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then δk ξ = Dtδk+1 η + Rs h(Dtηk+1 − ∂tηk), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3) δζ = −∂2 x1,hδη, � Σ ψδζ dx1 = − � Σ ∂2 x1,hψ δη dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4) � Σ ∂x1Iη∂x1ψ dx1 = 0, � Σ ∂x1Iξ∂x1ψ dx1 = 0, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5) � Σ Iζψ dx1 = 0, � Σ ∂x1Iζ∂x1ψ dx1 = 0, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6) � Σ ∂x1δζ∂x1ψ dx1 = � Σ ∂2 x1,hδη∂2 x1,hψ dx1, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='7) 2For (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2a) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2b) and p = 2 = q a proof can be found in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3 of [10], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 31 � Σ ∂x1δk+1 η ∂x1δk ξ dx1 = � Σ � Dt |∂x1δk+1 η |2 2 + τ 2|Dt∂x1δk+1 η |2 � dx1 + � Σ ∂x1δk+1 η ∂x1(Dtηk+1 − ∂tηk) dx1, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8) � Σ ∂x1δk+1 ζ ∂x1δk ξ dx1 = � Σ � Dt |δk+1 ζ |2 2 + τ 2|Dtδk+1 ζ |2 � dx1 − � Σ δk+1 ζ ∂2 x1(Dtηk+1 − ∂tηk) dx1, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9) as(ek+1 η , ek+1 ζ , ek ξ, δk ξ ) = Dt � Σ �γ1 2 |∂x1δk+1 η |2 + γ2 2 |δk+1 ζ |2� dx1 + γ3 � Σ |δk ξ |2 dx1 + τ 2 � Σ � γ1|Dt∂x1δk+1 η |2 + γ2|Dtδk+1 ζ |2� dx1 + � Σ � γ1∂x1δk+1 η ∂x1(Dtηk+1 − ∂tηk) − γ2δk+1 ζ ∂2 x1(Dtηk+1 − ∂tηk) � dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Recalling equalities ξk = ∂tηk, ξk h = Dtηk+1 h and the errors defined in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3) we get (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3) δk ξ − Dtδk+1 η = (ξk h − Rs hξk) − (Dtηk+1 h − DtRs hηk+1) = DtRs hηk+1 − Rs hξk = Rs h(Dtηk+1 − ∂tηk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Moreover, it is easy to check δζ = ζh + ∂2 x1,hRs hη = ∂2 x1,hRs hη − ∂2 x1,hηh = −∂2 x1,hδη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Further, recalling the discrete Laplace operator, we complete the proof of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=', δζ = ζh + ∂2 x1,hRs hη = ∂2 x1,hRs hη − ∂2 x1,hηh = −∂2 x1,hδη, � Σ ψδζ dx1 = − � Σ ψ∂2 x1,hδη dx1 = � Σ ∂x1ψ∂x1δη dx1 = − � Σ ∂2 x1,hψ δη dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Next, recalling the Riesz projection (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3) we immediate get (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5) � Σ ∂x1Iη∂x1ψ dx1 = � Σ ∂x1(Rs hη − η)∂x1ψ dx1 = 0, � Σ ∂x1Iξ∂x1ψ dx1 = � Σ ∂x1(Rs hξ − ξ)∂x1ψ dx1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Analogously, we find � Σ Iζψ dx1 = � Σ (∂2 x1η − ∂2 x1,hRs hη)ψ dx1 = � Σ (∂x1Rs hη − ∂x1η)∂x1ψ dx1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then, setting ∂2 x1,hψ in the above equality, we get (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6) � Σ ∂x1Iζ∂x1ψ dx1 = − � Σ Iζ∂2 x1,hψ dx1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Further, recalling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4) we get (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='7) � Σ ∂x1δζ∂x1ψ dx1 = − � Σ δζ∂2 x1,hψ dx1 = − � Σ (ζh + ∂2 x1,hRs hη)∂2 x1,hψ dx1 = � Σ (∂2 x1,hηh − ∂2 x1,hRs hη)∂2 x1,hψ dx1 = � Σ ∂2 x1,hδη∂2 x1,hψ dx1 32 Using the above equalities (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='7), the algebraic inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2), and the Riesz projection (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3) we get (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8) � Σ ∂x1δk+1 η ∂x1δk ξ dx1 = � Σ ∂x1δk+1 η � ∂x1Dtδk+1 η + ∂x1Rs h(Dtηk+1 − ∂tηk) � dx1 = � Σ � Dt |∂x1δk+1 η |2 2 + τ 2|Dt∂x1δk+1 η |2 � dx1 + � Σ ∂x1δk+1 η ∂x1(Dtηk+1 − ∂tηk) dx1 Analogously, using the above equalities (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4), and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='7), the algebraic inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2), and the Riesz projection (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4), we get (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9) � Σ ∂x1δk+1 ζ ∂x1δk ξ dx1 = − � Σ δk+1 ζ ∂2 x1,hδk ξ dx1 = � Σ ∂2 x1,hδk+1 η ∂2 x1,hδk ξ dx1 = � Σ ∂2 x1,hδk+1 η ∂2 x1,hDtδk+1 η dx1 + � Σ ∂2 x1,hδk+1 η ∂2 x1,hRs h(Dtηk+1 − ∂tηk) dx1 = � Σ � Dt |∂2 x1,hδk+1 η |2 2 + τ 2|Dt∂2 x1,hδk+1 η |2 � dx1 + � Σ ∂2 x1,hδk+1 η ∂2 x1(Dtηk+1 − ∂tηk) dx1 = � Σ � Dt |δk+1 ζ |2 2 + τ 2|Dtδk+1 ζ |2 � dx1 − � Σ δk+1 ζ ∂2 x1(Dtηk+1 − ∂tηk) dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Using the equalities (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6) we know that as(Ik+1 η , Ik+1 ζ , Ik ξ , ψ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Consequently, collecting the above equality together with (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='8) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' we get as(ek+1 η ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ek+1 ζ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ek ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' δk ξ ) = as(δk+1 η ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' δk+1 ζ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' δk ξ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' δk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ξ ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='= γ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∂x1δk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∂x1δk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ξ dx1 + γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∂x1δk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ζ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∂x1δk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ξ dx1 + γ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∂x1δk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ξ ∂x1δk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ξ dx1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='= γ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='|∂x1δk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='+ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2|Dt∂x1δk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='dx1 + γ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∂x1δk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∂x1(Dtηk+1 − ∂tηk) dx1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='+ γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='|δk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ζ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='+ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2|Dtδk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ζ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='dx1 − γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='δk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ζ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='x1(Dtηk+1 − ∂tηk) dx1 + γ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='|δk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ξ |2 dx1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='= Dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�γ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 |∂x1δk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='|2 + γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 |δk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ζ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='|2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='dx1 + γ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='|δk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ξ |2 dx1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='+ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='γ1|Dt∂x1δk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='|2 + γ2|Dtδk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ζ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='|2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='dx1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Σ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='γ1∂x1δk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∂x1(Dtηk+1 − ∂tηk) − γ2δk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ζ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='x1(Dtηk+1 − ∂tηk) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='dx1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' which proves (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='10) and completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3 Secondary estimates Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let (p, u, ξ, η) be a target smooth solution of the FSI problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4) belonging to the class (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Let (ph, �uh, ξh, ηh) be a solution to the numerical scheme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then, ∥F∥L∞L∞ + ��F−1�� L∞L∞ + ∥M∥L∞L∞ + ��M−1�� L∞L∞ <∼ 1, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11) ∥eη∥LγL∞ ≤ ∥∂x1δη∥LγL2 + h ��∂2 x1η �� LγL2 , 1 ≤ γ ≤ ∞, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='12) ∥∂x1eη∥L2L∞ <∼ ∥δζ∥L2L2 + h ��∂3 x1η �� L2L2 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='13) 33 and ∥Mh − M∥L2L∞ <∼ ∥∂x1δη∥L2L2 + ∥δζ∥L2L2 + h ���∂2 x1η �� L2L2 + ��∂3 x1η �� L2L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='14) Further ∥Fh − F∥L2L∞ <∼ ∥∂x1δη∥L2L2 + ∥δζ∥L2L2 + h ���∂2 x1η �� L2L2 + ��∂3 x1η �� L2L2 � (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='15) and ��F−1 h − F−1�� L2L∞ <∼1 + ∥∂x1η∥L∞L∞ η2 ∥∂x1δη∥L2L2 + 1 η ∥δζ∥L2L2 + h �1 + ∥∂x1η∥L∞L∞ η2 ��∂2 x1η �� L2L2 + 1 η ��∂3 x1η �� L2L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='16) For p ∈ [1, ∞) we find ���ξk h ��� Lp ≤ ���∇uk h ��� L2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='17) and ∥Dtδη∥L2Lp <∼ ∥δξ∥L2Lp + τ ��∂2 t ∂xη �� L2L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='18) For p ∈ [1, ∞) we find ∥�uh − �u∥L2Lp <∼ ∥δu∥L2Lp + h ∥�u∥L2W 2,2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='19) ∥�wh − �w∥L2Lp <∼ ∥δξ∥L2Lp + τ ��∂2 t ∂xη �� L2L2 + h ∥η∥L2W 2,2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='20) and finally ϱfτ m � k=1 � � �Ω |�vk−1 h ηk h − Fk h(Fk)−1�vkηk|2 d�x � 2 p <∼ τ m � k=1 α � �Ω |∇δk u|2ηk h d�x + 1 α � �Ω |δk u|2ηk h d�x + ∥δξ∥2 L2Lp + c1 ∥δη∥2 L2Lp + c2 ∥∂x1δη∥2 L2Lp + c3τ 2 + c4h2, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='21) where c1 = ϱf/η ∥�v∥2 L∞L∞ � 1 + ∥M∥2 L∞L∞ � , c2 = ϱf/η ∥M∥2 L∞L∞ ∥�v∥2 L∞L∞ , c3 = ϱfη � ∥∂t�u∥2 L2Lp + ��∂2 t ∂xη ��2 L2L2 � , c4 = c1 ∥∂x1η∥L2Lp + c2 ��∂2 x1η �� L2L2 + ϱfη ∥∇�u∥2 L2Lp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='22) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Here we shall frequently recall the estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' First, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11) is obvious as ηh and η are bounded from above and below by positive constants, as well as ∂x1η and ∂x1ηh are bounded from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' By the triangular inequality and the Sobolev inequality we get (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='12) ∥ηh − η∥LγL∞ ≤ ∥δη∥LγL∞ + ∥Iη∥LγL∞ <∼ ∥∂x1δη∥LγL2 + h ∥∂x1η∥LγL∞ <∼ ∥∂x1δη∥LγL2 + h ��∂2 x1η �� LγL2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Analogously, we have (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='13) ∥∂x1ηh − ∂x1η∥L2L∞ ≤ ∥∂x1δη∥L2L∞ + ∥∂x1Iη∥L2L∞ <∼ ��∂2 x1,hδη �� L2L2 + h ��∂2 x1η �� L2L∞ = ∥δζ∥L2L2 + h ��∂2 x1η �� L2L∞ <∼ ∥δζ∥L2L2 + h ��∂3 x1η �� L2L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Recalling the definition of M and Mh, using triangular inequality and the estimates (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='12) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='13) we obtain (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='14) ∥Mh − M∥L2L∞ = ���� � eη −�x2∂x1eη 0 0 ����� L2L∞ <∼ ∥∂x1δη∥L2L2 + ∥δζ∥L2L2 + h ���∂2 x1η �� L2L2 + ��∂3 x1η �� L2L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 34 Analogously, we get (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='15) ∥Fh − F∥L2L∞ = ���� � 0 0 −�x2∂x1eη eη ����� L2L∞ <∼ ∥∂x1δη∥L2L2 + ∥δζ∥L2L2 + h ���∂2 x1η �� L2L2 + ��∂3 x1η �� L2L2 � , and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='16) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='��F−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='− F−1�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2L∞ = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='−�x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∂x1ηh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ηh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='+ �x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∂x1η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ηh − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2L∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ηh∂x1η − η∂x1ηh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ηηh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2L∞ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='η − ηh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ηηh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2L∞ = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='eη∂x1η + η∂x1eη ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ηηh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2L∞ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='eη ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ηηh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2L∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='<∼ 1 + ∥∂x1η∥L∞L∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='η2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∥eη∥L2L∞ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='η ∥∂x1eη∥L2L∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='<∼ 1 + ∥∂x1η∥L∞L∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='η2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∥∂x1δη∥L2L2 + h ∥∂x1η∥L2L∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∥δζ∥L2L2 + h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='��∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='x1η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2L∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='<∼ 1 + ∥∂x1η∥L∞L∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='η2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∥∂x1δη∥L2L2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='η ∥δζ∥L2L2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='+ h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�1 + ∥∂x1η∥L∞L∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='η2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∥∂x1η∥L2L∞ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='��∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='x1η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2L∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The estimate (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='17) is a consequence of the trace estimate and Sobolev embedding in 1-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='18) we first recall (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3) and the triangular inequality to get ���Dtδk+1 η ��� Lp ≤ ���δk ξ ��� Lp + ���Rs h(Dtηk+1 − ∂tηk) ��� Lp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Then the estimate follows from the continuity of Rs h and Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 as ���Rs h(Dtηk+1 − ∂tηk) ��� L2Lp <∼ ���∂x1(Dtηk+1 − ∂tηk) ��� L2L2 <∼ τ ��∂2 t ∂x1η �� L2L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The proof of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='19) is by Sobolev embedding and the interpolation estimate, ∥�uh − �u∥L2Lp ≤ ∥δu∥L2Lp + ∥Iu∥L2Lp <∼ ∥δu∥L2Lp + ∥∇Iu∥L2L2 <∼ ∥δu∥L2Lp + h ∥�u∥L2W 2,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Recalling the definition of �wh and �w, using again Sobolev embedding, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1, the estimate (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='18) and the interpolation inequality we get (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='20): ∥�wh − �w∥L2Lp <∼ ∥Dtηh − ∂tη∥L2Lp ≤ ∥Dt(ηh − η)∥L2Lp + ∥Dtη − ∂tη∥L2Lp <∼ ∥Dtδη∥L2Lp + ∥DtIη∥L2Lp + ∥Dt∂x1η − ∂t∂x1η∥L2L2 <∼ ∥Dtδη∥L2Lp + ∥∇DtIη∥L2L2 + τ ��∂2 t ∂x1η �� L2L2 <∼ ∥δξ∥L2Lp + h ∥η∥L2W 2,2 + τ ��∂2 t ∂x1η �� L2L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' By the triangular inequality, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='12), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='19), and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='20) we get (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='21) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ϱfτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='|�vk−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ηk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h − Fk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h(Fk)−1�vkηk|p d�x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='= ϱfτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='|(�vk−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='− �vk)ηk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h + �vk(ηk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h − ηk) + (I − Fk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h(Fk)−1)�vkηk|p d�x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='≤ ϱfτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='| − τDt�uk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h + ek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='u + �wk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h − �wk|pηk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h d�x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='+ ϱfτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='|�vk(ηk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h − ηk)|p/ηk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h d�x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='+ ϱfτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='|(I − Fk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h(Fk)−1)�vkηk|p/ηk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h d�x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='<∼ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ϱf|δk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='u|pηk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h d�x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='+ ∥δξ∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2Lp + c1 ∥δη∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2Lp + c2 ∥∂x1δη∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2Lp + c3τ 2 + c4h2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' where ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 4 are given above in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='22), interpolation implies for 2 p = θ, p < ∞ that � � �Ω ϱf|δk u|pηk h d�x � 2 p <∼ ���∇δk u ��� (1−θ)2 L2 ���δk u ��� 2θ L2 <∼ α ���∇δk u ��� 2 L2 + 1 α ���δk u ��� 2 L2 , which closes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4 Proof of estimates (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' First, by Young’s inequality, the interpolation error, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='6, and the uniform bounds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='11), we find |Gf| = ���τ m � k=1 � �Ω ϱf � ηk hDtIk u + 1 2Dtηk hIk∗ u � δk u d�x + 2µτ m � k=1 � �Ω � ∇Ik u(Fk h)−1�S : (∇δk u(Fk h)−1)ηk h d�x ���, <∼ m � k=1 τ � �Ω ϱfηk h|δk u|2 d�x + 2αµ m � k=1 τ � �Ω |∇δk u(Fk h)−1|2ηk h d�x + ch2 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='23) for any fixed α > 0, where c = 1 4ϱf � η ∥∂t∇�u∥2 L2L2 + 1 4η ∥ξh∥2 L2L∞ ∥∇�u∥2 L∞L2 � + 2µη 4α ∥Fh∥L∞L∞ ∥�u∥L2W 2,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Note that this constant is indeed bounded by the stability of the discrete solution, as ∥ξh∥2 L2L∞ can be bounded by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Second, by Young’s inequality, the time discretization error (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2c), we can control Gs in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' |Gs| = ���γ1τ m � k=1 � Σ ∂x1δk+1 η ∂x1(Dtηk+1 − ∂tηk) dx1 − γ2τ m � k=1 � Σ δk+1 ζ ∂2 x1(Dtηk+1 − ∂tηk) dx1 + τ m � k=1 � Σ ϱsDtIξδk ξ dx1 ��� <∼ m � k=1 τ � Σ � γ1|∂x1δk+1 η |2 + γ2|δk+1 ζ |2 + ϱs|δk ξ |2� dx1 + 1 4ϱs ∥∂tIξ∥2 L2L2 + 1 4 m � k=1 τ � Σ � γ1|Dt∂x1ηk+1 − ∂t∂x1ηk|2 + γ2|Dt∂2 x1ηk+1 − ∂t∂2 x1ηk|2� dx1 <∼ m � k=1 τ � Σ � γ1|∂x1δk+1 η |2 + γ2|δk+1 ζ |2 + ϱs|δk ξ |2� dx1 + c1τ 2 + c2h2, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='24) where c1 = 1 4γ1 ��∂2 t ∂x1η ��2 L2((0,T)×Σ) + 1 4γ2 ��∂2 t ∂2 x1η ��2 L2((0,T)×Σ) , c2 = 1 4ϱs ∥∂t∂x1ξ∥2 L2((0,T)×Σ) = 1 4ϱs ��∂2 t ∂x1η ��2 L2((0,T)×Σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Next, we analyze the Ri terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 36 Rk 1-term �����τ m � k=1 Rk 1 ����� = �����τ m � k=1 ϱf � �Ω � ek η∂t�uk + ηk h(Dt�uk − ∂t�uk) � δk u d�x ����� ≤ τ m � k=1 ϱf � �Ω ηk h|δk u|2 d�x + 1 2τ m � k=1 ϱf � �Ω |ek η|2/ηk h|∂t�uk|2 d�x + 1 2τ m � k=1 ϱf � �Ω ηk h|Dt�uk − ∂t�uk|2 d�x ≤ τ m � k=1 ϱf � �Ω ηk h|δk u|2 d�x + c1τ m � k=1 � Σ |∂x1δk η|2 dx1 + c2h2 + c3τ 2, where we have used (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='12) and the constants read c1 = ϱf η ∥∂t�u∥2 L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2(�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Rd)) , c2 = ϱf η ∥∂x1η∥2 L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L∞(Σ)) ∥∂t�u∥2 L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2(�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Rd)) , c3 = ϱfη 2 ��∂2 t �u ��2 L2((0,T)×�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Rd) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Rk 2-term �����τ m � k=1 Rk 2 ����� = ���� 1 2ϱf � �Ω � (ek−1 ξ − τDtξk)�uk∗ − τ∂tηkDt�uk� δk u d�x ���� ≤ τ m � k=1 ϱf � �Ω ηk h|δk u|2 d�x + τ 2 ϱf 2η ∥∂t�u∥2 L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2(�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Rd)) ∥∂tη∥2 L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L∞(Σ)) + ϱf η � τ 2 ∥∂tξ∥2 L2((0,T)×Σ) + τ m � k=1 � Σ |δk−1 ξ + Ik−1 ξ |2 dx1 � ∥�u∥L∞((0,T)×�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Rd) <∼ τ m � k=1 ϱf � �Ω ηk h|δk u|2 d�x + τ m � k=1 � Σ |δk ξ |2 dx1 + c1τ 2 + c2h2, where c1 = ϱf 2η ∥∂t�u∥2 L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2(�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Rd)) ∥∂tη∥2 L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L∞(Σ)) + ϱf η ��∂2 t η ��2 L2((0,T)×Σ) ∥�u∥L∞((0,T)×�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Rd) , c2 = ϱf η ∥∂t∂x1η∥L2((0,T)×Σ) ∥�u∥L∞((0,T)×�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Rd) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Rk 3-term For this term we recall Ladyzenskaja’s estimate in 2D ∥f∥2 L4 ≤ ∥∇f∥L2 ∥f∥L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='25) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='We use it to find by the previous arguments that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='����τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Rk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='���� ≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='����τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2ϱf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='δk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='u · (∇Ik ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='u) − Ik ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='u · (∇δk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='u) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='(Fk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h)−1�vk−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ηk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h d�x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='+ τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='m ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='���uk��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='<∼ c5h2τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='����vk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 + ατ m � k=1 ���∇δk u ��� 2 L2 + τ m � k=1 ����vk−1 h ηk h − Fk h(Fk)−1�vk��� L4 � ���δk u ��� L2 ���uk��� W 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 + ���∇δk u ��� L2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' with c5 = ��∇2u ��2 L2(L2) sup k ����vk��� 2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Now (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='21) and Young’s inequality implies that ����τ m � k=1 Rk 3 ���� <∼ ατ m � k=1 ���∇δk u ��� 2 L2 + 1 α � �Ω |δk u|2ηk h d�x + ∥δξ∥2 L2L4 + c1 ∥δη∥2 L2L4 + c2 ∥∂x1δη∥2 L2L4 + c3τ 2 + ˜c4h2 for c1, c2, c3, c4 given in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='22) with ˜c4 = c4 + c5h2τ α m � k=1 ����vk h ��� 2 W 1,2 + ∥u∥2 L2W 2,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Finally by using (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='17) we estimate ���δk ξ ��� 2 L4 <∼ ���δk ξ ��� L2 ���δk ξ ��� L∞ <∼ ���δk ξ ��� L2 ���∇δk u ��� L2 <∼ α ���∇δk u ��� 2 L2 + 1 α ���δk ξ ��� 2 L2 , With that and Sobolev embedding we conclude that ����τ m � k=1 Rk 3 ���� <∼ ατ m � k=1 ���∇δk u ��� 2 L2 + 1 α � �Ω |δk u|2ηk h d�x + 1 α ���δk ξ ��� 2 L2L2 + c1 ∥∂x1δη∥2 L2L2 + c2 ��∂2 x1δη ��2 L2L2 + c3τ 2 + ˜c4h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Rk 4-term By using the identity (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1), Young’s inequality, H¨older’s inequality, and the interpolation error (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2), we obtain �����τ m � k=1 Rk 4 ����� = �����τ m � k=1 � �Ω ek p∇δk u : Mk h d�x + � �Ω �pk∇δk u : � Mk h − Mk� d�x ����� = �����τ m � k=1 � �Ω � Ik p ∇δk u : Mk h + �pk∇δk u : � Mk h − Mk�� d�x ����� <∼ 2αµτ m � k=1 � �Ω |∇δk u(Fk h)−1|2ηk h d�x + η 4αµ ∥Ip∥2 L2((0,T)×�Ω) + + 1 4αµη ∥�p∥2 L∞L∞ ∥Fh∥2 L∞L∞ ∥Mh − M∥2 L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L∞(�Ω)) 38 <∼ 2αµτ m � k=1 � �Ω |∇δk u(Fk h)−1|2ηk h d�x + c1 � ∥∂x1δη∥2 L2((0,T)×Σ) + ��∂2 x1δη ��2 L2((0,T)×Σ) � + c2h2, where we have used (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='14) and the constants read c1 = 1 4αµη ∥�p∥2 L∞L2 ∥Fh∥2 L∞((0,T)×�Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Rd×d) , c2 = 1 4αµη ∥∇�p∥2 L2L2 + c1 � ∥∂x1η∥2 L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L∞(Σ)) + ��∂2 x1η ��2 L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L∞(Σ)) � Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Thanks to the nice interpolation operator which produces the divergence-free condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='10) with the covariance Mh instead of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Otherwise, we would lose the equality (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1) and has to estimate τ �m k=1 � �Ω δk p∇δk u : Mk h d�x, in which the pressure error δp is not available in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Rk 5-term Applying Young’s inequality, H¨older’s inequality, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='14), and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' we obtain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�����τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='Rk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='����� = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�����τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∇�uk(Fk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h)−1�S : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∇δk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='u(Fk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h)−1ηk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∇�uk(Fk)−1�S : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∇δk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='u(Fk)−1ηk�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='d�x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�����τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∇�uk� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='(Fk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h)−1 − (Fk)−1��S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=': ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∇δk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='u(Fk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h)−1ηk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='d�x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�����τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∇�uk(Fk)−1�S : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∇δk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='u(Fk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h)−1(ηk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h − Fk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h(Fk)−1ηk� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='d�x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='<∼ ατ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∇δk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='u(Fk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h)−1�S���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ηk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h d�x + η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2α ∥∇�u∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L∞L2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='��(Fh)−1 − (F)−1��2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2L∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='+ η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2α ∥∇�u∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L∞L2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='��F−1��2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L∞L∞ ∥Fh∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L∞L∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='��MT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h − MT��2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2L∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='<∼ ατ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='�Ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='∇δk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='u(Fk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h)−1�S���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='ηk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='h d�x + c1 ∥∂x1δη∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='L2((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='T)×Σ) + c2 ∥δζ∥2 L2((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='T)×Σ) + c3h2 where c1 = η 2α ∥∇�u∥2 L∞L2 ���F−1��2 L∞L∞ ∥Fh∥2 L∞L∞ + 1 + ∥∂x1η∥L∞L∞ η2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' c2 = η 2α ∥∇�u∥2 L∞L2 ���F−1��2 L∞L∞ ∥Fh∥2 L∞L∞ + 1 η � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' c3 =c1 ��∂2 x1η ��2 L2((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='T)×Σ) + c2 ��∂3 x1η ��2 L2((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='T)×Σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Rk 6-term By Young’s inequality and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2b) we obtain |τ m � k=1 Rk 6| = �����τ m � k=1 ϱs � Σ (Dtξk − ∂tξk)δk ξ dx1 ����� <∼ τ 2 4ϱs ��∂2 t ξ ��2 L2((0,T)×Σ) + τ m � k=1 � Σ ϱs|δξ|2 dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Rk 7-term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' �����τ m � k=1 Rk 7 ����� = �����τ m � k=1 γ1 � Σ ∂x1(ηk+1 − ηk)∂x1δk ξ dx1 + τ m � k=1 γ2 � Σ ∂x1(ζk+1 − ζk)∂x1δk ξ dx1 ����� <∼ τ m � k=1 τ � ���∂2 x1Dtηk��� + ���∂2 x1Dtζk��� L2 � ���δk ξ ��� L2 39 <∼ τ m � k=1 ���δk ξ ��� 2 L2 + τ 2( ��∂4 x1ξ ��2 L2L2 + ��∂2 x1ξ ��2 L2L2) Consequently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' collecting all the above estimates we get �����τ m � k=1 7 � i=1 Rk i + Gf + Gs ����� <∼ τ 2 + h2 + cτ m � k=1 δk E + 2αµτ m � k=1 � �Ω ���∇δk u(Fk h)−1��� 2 ηk h d�x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' which proves (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' C Numerical Implementation In this appendix we provide details of the numerical implementations of semi-implicit Scheme-R (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='12) and monolithic fully implicit (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9) both computed on the reference domain �Ω and both implemented using FEniCS finite element method [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Here, let us point out that, instead of implementing the height of the structure η, we take a shift η = η − 1 (independent of �x2) and then linearly extend it to the whole domain via η = η�x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Moreover, the structure velocity ξ on Γ is directly replaced by the second component of the fluid velocity ξ = u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Further, instead of ζ we shall use z as the second order derivative of the new η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Hereinafter, we shall frequently drop the superscript “�” for simplicity of the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='1 Implementation of semi-implicit Scheme-R We implemented Scheme-R (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='12), the monolithic method on the reference domain �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The domain �Ω is approximated by regular triangles K ∈ Th with the typical mesh size h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The problem comprises four global unknowns: velocity u, pressure p, mesh displacement in x2−direction η and its second order derivative z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The velocity-pressure pair is approximated with the inf-sup compatible MINI element [3], where the velocity is approximated by the piecewise linear continuous elements enlarged with the cubic bubbles, mesh displacement by the same elements as the velocity and the second order derivative of the mesh displacement is approximated by piecewise linear elements, for the definitions of the discrete function spaces see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' For the time stepping we use a backward Euler method with a fixed time step τ, we denote by uk, zk, pk and ηk the unknowns at the kth time step, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' at time t = kτ and u0, z0, p0, η0 are prescribed initial conditions (in our case equal to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Since in case of zero initial conditions, it holds η1 = η0, we may shift the time index k (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' k + 1) to k − 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' k) for the structure variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' This semi-implicit scheme is linear and the corresponding system of linear equations is solved with the direct solver MUMPS [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Components of the velocity u are denoted by (u1, u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Displacement ηk is computed explicitly using the y-component of the velocity uk 2 on the top bound- ary Γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ηk = ηk−1 + τ uk 2 on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The following quantities are used in the discretized weak form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The deformation gradient F is obtained from the displacement, J is its determinant and ˙J its time derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' All are evaluated at the (k−1)st time level Fk−1 = I + ∇(0, ηk−1)T, Jk−1 = det(Fk−1), ˙Jk−1 = DtJk−1 = Jk−1 − Jk−2 τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Finally, v is the relative velocity of the fluid and �T is the Cauchy stress tensor after the ALE trans- formation vk−1 = uk−1 − (0, ξk−1)T, �Tk = −pkI + 2µ � ∇uk(Fk−1)−1�S , where ξk−1 = Dtηk−1 is the mesh velocity that is computed after Step 2 when the displacement η is prolongated into the whole domain �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 40 The whole simulation consists of two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' In Step 1 we solve for velocity u, its Laplace z and pressure p, explicitly compute the value of η on the top boundary Γ and in Step 2 we linearly expand it to the whole domain �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Step 1 We solve for u, z and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' � �Ω Jk−1 tr � ∇uk � Fk−1�−1� q d�x = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' � �Ω � zb + � ∂x1ηk−1 + τ ∂x1uk 2 � ∂x1b � d�x = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ρf � �Ω Jk−1Dtuk · ϕϕϕ d�x + 1 2ρf � �Ω ˙Jk−1(2uk − uk−1) · ϕϕϕ d�x + 1 2ρf � �Ω Jk−1∇uk � Fk−1�−1 vk−1 · ϕϕϕ d�x − 1 2ρf � �Ω Jk−1∇ϕϕϕ � Fk−1�−1 vk−1 · uk d�x + � �Ω Jk−1 tr � �Tk∇ϕϕϕ(Fk−1)−1) � d�x + � Γ ρsDtuk 2 ϕ2 dS(x) + γ1 � Γ � ∂x1ηk−1 + τ ∂x1uk 2 � ∂x1ϕ2 dS(x) − γ2 � Γ ∂x1z ∂x1ϕ2 dS(x) + γ3 � Γ ∂x1u2 ∂x1ϕ2 dS(x) + � Γ fϕ2 dS(x) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' where ϕϕϕ = (ϕ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ϕ2) is the test function corresponding to the velocity u and ϕ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' ϕ2 its components,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' q is the test function for the pressure p and b the test function for z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Finally, f denotes the y-component of the force acting on the boundary Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Step 2 We linearly prolongate the displacement η to the whole domain �Ω by solving � �Ω ∂x2ηk ∂x2ψ d�x = 0, for all ψ ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Here, ηk = ηk−1 + τ uk 2, where uk 2 is obtained in Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='2 Implementation of fully implicit scheme In Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='3 we compare our Scheme-R to the fully implicit method based on the weak form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' As in the case of implementation of Scheme-R, the domain �Ω is approximated by regular triangles T ∈ Th and the problem comprises four global unknowns: velocity u, pressure p, mesh displacement in x2−direction η and its second derivative z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The velocity-pressure pair (u, p) is approximated with the MINI element, mesh displacement η is from the same space as velocity, and the second derivative of the displacement z is approximated by piecewise linear elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The time derivatives are approximated by the backward Euler time scheme, the nonlinearities are treated with the Newton solver, and the consequent set of linear equations by direct solver MUMPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Knowing the solution uk−1, ηk, zk, pk on the previous time level, we are solving the fully implicit nonlinear problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Thus, we solve for uk, ηk, zk and pk satisfying the continuity equation � �Ω Jk tr � ∇uk (Fk)−1� q d�x = 0, the coupled momentum equation ρf � �Ω JkDtuk · ϕϕϕ d�x + 1 2ρf � �Ω DtJk uk · ϕϕϕ d�x + 1 2ρf � �Ω Jk∇uk(Fk)−1vk · ϕϕϕ d�x − 1 2ρf � �Ω Jk∇ϕϕϕ(Fk)−1vk · uk d�x + � �Ω Jk tr � �Tk∇ϕϕϕ(Fk)−1) � d�x + � Γ ρsDtuk 2 ϕ2 dS(x) + γ1 � Γ ∂x1ηk ∂x1ϕ2 dS(x) 41 − γ2 � Γ ∂x1z ∂x1ϕ2 dS(x) + γ3 � Γ ∂x1u2 ∂x1ϕ2 dS(x) + � Γ fϕ2 dS(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Further, the discrete Laplace equation for zk and the harmonic extension of ηk � �Ω � zkb + ∂x1ηk ∂x1b � d�x = 0, � �Ω ∂x2ηk ∂x2ψ d�x = 0, for all test functions q, b,ϕϕϕ and ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' Here, we used the same notion as above Fk = I + ∇(0, ηk)T, Jk = det Fk, vk = uk − (0, Dtηk)T, �Tk = −pkI + 2µ � ∇uk(Fk)−1�S and the components of the velocity u are denoted by (u1, u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' The problem is periodic in x1 direction, with a homogeneous Dirichlet boundary conditions for u and η on the bottom boundary, and Dtηk = uk 2 on the top boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} +page_content=' 42' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfUQxX/content/2301.05014v1.pdf'} diff --git a/q9AyT4oBgHgl3EQfzvm4/content/tmp_files/2301.00707v1.pdf.txt b/q9AyT4oBgHgl3EQfzvm4/content/tmp_files/2301.00707v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e3279a428a67709a991a8e2f88c7510b2b429457 --- /dev/null +++ b/q9AyT4oBgHgl3EQfzvm4/content/tmp_files/2301.00707v1.pdf.txt @@ -0,0 +1,2753 @@ +1 +RIS-Assisted Receive Quadrature Spatial +Modulation with Low-Complexity +Greedy Detection +Mohamad H. Dinan, Member, IEEE, Marco Di Renzo, Fellow, IEEE, +and Mark F. Flanagan, Senior Member, IEEE +Abstract +In this paper, we propose a novel reconfigurable intelligent surface (RIS)-assisted wireless commu- +nication scheme which uses the concept of spatial modulation, namely RIS-assisted receive quadrature +spatial modulation (RIS-RQSM). In the proposed RIS-RQSM system, the information bits are conveyed +via both the indices of the two selected receive antennas and the conventional in-phase/quadrature (IQ) +modulation. We propose a novel methodology to adjust the phase shifts of the RIS elements in order to +maximize the signal-to-noise ratio (SNR) and at the same time to construct two separate PAM symbols +at the selected receive antennas, as the in-phase and quadrature components of the desired IQ symbol. +An energy-based greedy detector (GD) is implemented at the receiver to efficiently detect the received +signal with minimal channel state information (CSI) via the use of an appropriately designed one-tap +pre-equalizer. We also derive a closed-form upper bound on the average bit error probability (ABEP) of +the proposed RIS-RQSM system. Then, we formulate an optimization problem to minimize the ABEP in +order to improve the performance of the system, which allows the GD to act as a near-optimal receiver. +Extensive numerical results are provided to demonstrate the error rate performance of the system and +to compare with that of a prominent benchmark scheme. The results verify the remarkable superiority +of the proposed RIS-RQSM system over the benchmark scheme. +This work was funded by the Irish Research Council (IRC) under the Consolidator Laureate Award Programme (grant number +IRCLA/2017/209). The work of Marco Di Renzo was supported in part by the European Commission through the H2020 +ARIADNE project under grant agreement number 871464 and through the H2020 RISE-6G project under grant agreement +number 101017011. +Mohamad H. Dinan and Mark F. Flanagan are with the School of Electrical and Electronic Engineering, University College +Dublin, Belfield, Dublin 4, D04 V1W8 Ireland (email: mohamad.hejazidinan@ucdconnect.ie; mark.flanagan@ieee.org). +Marco Di Renzo is with Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes, 3 Rue Joliot- +Curie, 91192 Gif-sur-Yvette, France (email: marco.di-renzo@universite-paris-saclay.fr). +arXiv:2301.00707v1 [cs.IT] 2 Jan 2023 + +2 +Index Terms +6G, reconfigurable intelligent surface (RIS), spatial modulation (SM), quadrature spatial modulation +(QSM), greedy detector (GD). +I. INTRODUCTION +In the past few years, various wireless communication technologies have emerged with an +aim to support high demands for connectivity and an immense increase in mobile data traffic. +Among these, reconfigurable intelligent surfaces (RISs), also known as intelligent reflecting +surfaces (IRSs), represents a key innovation that has drawn significant attention from researchers +in both academia and industry [1] and is foreseen to be a potential candidate for 6th generation +(6G) networks [2], [3]. An RIS is a surface of electromagnetic meta-material consisting of a +large number of small, low-cost and energy-efficient reflecting elements that are able to control +the scattering and propagation in the channel by inducing a pre-designed phase shift to the +impinging wave. From this perspective, RIS technology represents a revolutionary paradigm +that can transform the uncontrollable disruptive propagation environment into a smart radio +environment [2], [4], thus enhancing the received signal quality [5], [6]. +On the other hand, spatial modulation (SM) [7], [8] and its variants such as generalized +spatial modulation (GSM) [9], receive spatial modulation (RSM) [10], [11], and quadrature +spatial modulation (QSM) [12], have been widely investigated in the last two decades as a +promising technology for beyond-5th-generation (B5G) networks. SM uses the indices of the +transmit/receive antennas to convey the information bits. It exploits the channel attributes to +simplify the transceiver structure in order to provide a more energy-efficient solution compared +with other conventional multiple-input multiple-output (MIMO) techniques [13]. +The implicit advantages of both RIS and SM technology have motivated researchers to combine +these two advanced technologies to obtain a reliable energy-efficient approach in order to achieve +so-called green or sustainable wireless communications. Specifically, in [14], two fundamental +RIS-based index modulation (IM) techniques were proposed, i.e., RIS-space-shift keying (RIS- +SSK) and RIS-spatial modulation (RIS-SM). In both scenarios, the RIS-access point (RIS-AP) +approach was implemented, in which the RIS forms part of the transmitter, and the index of the +receive antennas is used to convey the data bits. The numerical results confirm a significant +superiority of these RIS-aided schemes compared to conventional MIMO schemes. Various +principles of RIS-based SM (also known as metasurface-based modulation) were introduced in + +3 +[15]. The authors of [16] proposed an RIS-SSK system with multiple transmit antennas in which +the information bits map to the transmit antenna index and the single-antenna receiver receives +the signal reflected from the RIS. Various scenarios with ideal and non-ideal transceivers were +investigated and the error rate performance of each scenario was analyzed. The results indicate +that maximizing the signal-to-noise ratio (SNR) at the receiver is not a good approach for the +transmit RIS-SSK setting, and in fact shows a relatively poor performance. In light of this, in [17] +the authors proposed an optimization algorithm for the transmit RIS-SSK system to maximize the +minimum Euclidean distance among the received symbols. Using this approach, a performance +improvement is achieved at the expense of an increased computational complexity. Moreover, +in [18], adopting a similar approach, the authors proposed a joint optimization of the power +allocation matrix and the phase shifts of the RIS elements. An RIS-based SM system with both +the transmit and receive antenna index modulation was proposed in [19] to increase the spectral +efficiency. However, the results show that the error rate performance of the transmit SM bits is +extensively lower than that of the receive SM bits; this is due to a reduction in the resulting +channel-imprinted Euclidean distances. RIS-aided receive quadrature reflecting modulation (RIS- +RQRM) proposed in [20] is another interesting approach in which QSM is applied within the +receive antenna array. In this scenario, the RIS is divided into two halves, and each half targets +the real or imaginary part of the signal at the two selected receive antennas in order to double +the throughput; however, the SNR at the receiver is significantly reduced due the reduction in +the number of RIS elements per targeted antenna. Inspired by RIS-RQRM and [19], an IRS- +assisted transceiver QSM (IRS-TQSM) scheme was proposed in [21] which applies QSM at both +the transmitter and the receiver. In [22], [23], generalized SSK (GSSK) and GSM approaches +have been implemented in an RIS-assisted wireless system. In both scenarios, the RIS is divided +into multiple parts to target multiple antennas at the receiver; hence, the throughput can be +increased at the expense of a decrease in the SNR at the target antennas. The concept of SM +has also been applied within the RIS entity in [24], [25], [26] in order to transmit additional +data bits. This is an exciting approach to transmit the environmental data collected by the RIS; +however, experimental results show a very large degradation in the error rate performance of +the SM symbol, that is due to the similarity within the possible (noise-free) received signals. +In order to tackle the problem of the SNR decrease due to grouping of the RIS elements, in +[27] we proposed a new paradigm, namely RIS-assisted receive quadrature space-shift keying +(RIS-RQSSK) in order to simultaneously target two receive antennas. An optimization problem + +4 +was defined to maximize the SNR of the real part of the signal at one antenna and, at the +same time, of the imaginary part of the signal at the second antenna. The spectral efficiency of +this approach is increased without any degradation in the SNR. However, the throughput of the +RIS-RQSSK system is limited and can only be increased by increasing the number of receive +antennas which is not a viable option in practice. +Against this background, in this paper we introduce a new RIS-assisted quadrature scheme +in which, in addition to mapping the information bits independently to two indices of receive +antennas, additional bits are transmitted via conventional in-phase/quadrature (IQ) modulation. +The contributions of this paper are as follows: +• To improve the spectral efficiency of RIS-RQSSK while preserving its excellent perfor- +mance, we propose an RIS-assisted receive quadrature spatial modulation (RIS-RQSM) +system. In particular, all RIS elements target two independently selected receive antennas +to convey the information bits. In this scenario, we introduce a novel idea to optimize +the phase shifts of the RIS elements in order to not only maximize the SNR components +associated to the real and imaginary parts of the signal at the receive antennas, but also +to help in constructing the in-phase (I) and quadrature (Q) components of the symbol at +the two separate antennas. Specifically, the phase of the desired IQ symbol is created by +adjusting the phase shift of the RIS elements, while a positive symbol selected from a +specific pre-designed PAM constellation forms the amplitude of that IQ symbol. That is, in +the proposed RIS-RQSM system, in contrast to conventional IQ modulation, the transmitter +constructs the IQ symbol at the receiver with the aid of the RIS elements and a single radio +frequency (RF) chain. +• We propose an energy-based greedy detector (GD) at the receiver to detect the indices of +the selected antennas with low complexity. Then, the I and Q symbols can be detected +independently by using a one-dimensional maximum likelihood (ML) detector at each of +the detected antennas. We also propose and design a one-tap zero-forcing (ZF) pre-equalizer +which remarkably reduces the channel state information (CSI) requirement at the receiver. +This yields a significant reduction in the feedback payload. +• We analyze the average bit error probability (ABEP) of the proposed RIS-RQSM system +with the GD receiver and derive a closed-form upper bound which is tight, especially at high +SNR values. Then, we propose an optimization problem to design an IQ modulation scheme +in order to minimize the ABEP. We utilize some accurate approximations to reduce the + +5 +complexity of the optimization problem and derive an analytical solution. Indeed, optimizing +the IQ modulation enables the system to use the GD as an alternative to the ML detector. The +results show that the GD in the RIS-RQSM system with optimized constellation performs +considerably close to the ML detector, such that the performance gap is negligible. +• Finally, we compare the bit error rate (BER) performance results with those of the most +prominent benchmark scheme. The results show that the proposed RIS-RQSM system +substantially outperforms the benchmark scheme. This performance improvement improves +with an increasing number of receive antennas. +The rest of this paper is organized as follows. The RIS-RQSM system model is described in +Section II. In Section III, we summarize the transceiver design of the RIS-RQSSK system of [27], +which forms the baseline model for the proposed system. The transmitter and receiver structure +design for the proposed RIS-RQSM system is presented in detail in Section IV. The ABEP +performance of the proposed RIS-RQSM system is analyzed in Section V. In Section VI, we +formulate the optimization problem to minimize the system error rate performance and determine +its analytical solution. In Section VII, we provide numerical results and comparisons with the +benchmark scheme. Finally, Section VIII concludes this paper. +Notation: Boldface lower-case letters denote column vectors, and boldface upper-case letters +denote matrices. (·)R and (·)I denote the real and imaginary components of a scalar/vector, +respectively. (·)⋆ represents the optimum value of a scalar/vector variable. E {·} and V {·}, re- +spectively, denote the expectation and variance operator. N (µ, σ2) (resp., CN (µ, σ2)) represents +the normal (resp., complex normal) distribution with mean µ and variance σ2. For a real/complex +scalar s, |s| denotes the absolute value, while for a set S, |S| denotes its cardinality. sgn (·) +represents the sign function which determines the sign of a real variable, i.e., for x ̸= 0, it is +defined as sgn (x) = {+1 if x > 0, −1 if x < 0}. Finally, the set of complex matrices of size +m × n is denoted by Cm×n. +II. SYSTEM MODEL +In this section, we describe the system model for the proposed RIS-assisted receive quadrature +spatial modulation (RIS-RQSM) scheme. A schematic of the RIS-RQSM system is presented +in Fig. 1. We consider the RIS-AP model [5], [6], where the RIS forms part of the transmitter +and reflects the incident wave emitted from a single transmit antenna which is located in the +vicinity of the RIS such that the path loss and scattering of the link between the RIS and the + +6 +Fig. 1. A schematic representation of RIS-assisted receive quadrature spatial modulation (RIS-RQSM) system (in RIS-RQSSK +system, an RF source with constant energy is used). +transmit antenna is negligible. The RIS is comprised of N reflecting elements whose vector of +phase shifts θ ∈ CN×1 is controlled by the transmitter to convey information. Here we assume +lossless reflection from the RIS, i.e., |θi| = 1 for i = 1, 2, . . . , N. The receiver is equipped with +Nr antennas and is placed far from the transmitter. We assume that the receiver can only receive +the signal reflected from the RIS elements through the wireless fading channel H ∈ CNr×N, +whose elements are assumed to be independent and identically distributed (i.i.d.) according to +CN (0, 1). In this scenario, the input data stream is split into packets of log2 MN 2 +r bits. The first +log2 N 2 +r bits are used to independently select two receive antennas to convey the spatial symbol, +and the remaining log2 M bits determine the desired IQ symbol that is selected from an M-ary +QAM constellation. Unlike in conventional communication systems, in the RIS-RQSM system +the selected IQ symbol is not transmitted through a single-antenna transmitter, but is created +at the selected receive antennas via both adjusting the RIS phase shifts and emitting a specific +PAM symbol from the transmit antenna1, with a property that the I component appears on the +first selected antenna, while the Q component appears on the second selected antenna. Thus, the +RIS-RQSM scheme represents a significant generalization of the RIS-assisted receive quadrature +space-shift keying (RIS-RQSSK) system described in [27]. In RIS-RQSSK, an RF source is used +to transmit a constant signal toward the RIS; therefore, only a spatial symbol can be transmitted, +while the PAM signal in RIS-RQSM enables the transmitter to transfer additional data bits via +IQ modulation. In the next section, we will provide a brief overview of the RIS-RQSSK system. +Then, the proposed RIS-RQSM system will be described in Section IV. +1It is worth mentioning that in contrast to the conventional RIS-SM system, in the RIS-RQSM the RF source at the transmitter +only requires the hardware for the in-phase (I) signal component, which results in a lower hardware complexity. + +log2 MN? +Input Data +bits +RIS +Stream +Controller +H +RIS +RF Source/ +PAM Signal + Transmitter +Receiver7 +III. RIS-ASSISTED RECEIVE QUADRATURE SPACE-SHIFT KEYING [27] +In this section, we summarize the system model of the RIS-RQSSK scheme of [27] and +outline its phase shift optimization procedure. In the RIS-RQSSK system, the transmitter is +equipped with an RF source with constant energy Es. In this scenario, two receive antennas are +independently selected according to two packets of log2 Nr input data bits. Then, the transmitter +reflects the signal to the receiver through the RIS, aiming to simultaneously maximize the SNR +associated to the real part of the signal at the first selected receive antenna m, while also +maximizing the SNR associated to the imaginary part of the signal at the second selected receive +antenna n. For this system, the real and imaginary components of the baseband received signal +at the selected antennas m and n, respectively, are given by +yR +m = +� +Es +� +hR +mθR − hI +mθI� ++ nR +m, +(1) +yI +n = +� +Es +� +hR +n θI + hI +nθR� ++ nI +n, +(2) +where hl = [hl,1, hl,2, . . . , hl,N] is the l-th row of H, and nl ∈ C is the additive white Gaussian +noise at the l-th receive antenna that is distributed according to CN (0, N0). To maximize both +SNR components associated to the real and imaginary parts of the selected receive antennas m +and n, a max-min optimization problem was defined as +max +θR,θI min +���hR +mθR − hI +mθI�� , +��hR +n θI + hI +nθR��� +(3) +s.t. +� +θR +i +�2 + +� +θI +i +�2 = 1, for all i = 1, 2, . . . , N. +Taking the case where the noise-free signal components in (1) and (2) are positive, the optimal +values of +� +θR +i +� +and +� +θI +i +� +are given by +θR⋆ +i += +λAi + (1 − λ) Bi +� +(λAi + (1 − λ) Bi)2 + (λCi + (1 − λ) Di)2, +(4) +for all i = 1, 2, . . . , N, and +θI⋆ +i += +λCi + (1 − λ) Di +� +(λAi + (1 − λ) Bi)2 + (λCi + (1 − λ) Di)2, +(5) +for all i = 1, 2, . . . , N, where we define +Ai = hR +m,i, Bi = hI +n,i, Ci = −hI +m,i, and Di = hR +n,i, +(6) + +8 +to simplify the notation, and where, for N ≫ 1, the value of λ ∈ (0, 1) is the solution to +f(λ) ≜ +N +� +i=1 +(Ai − Bi) (λAi + (1 − λ) Bi) + (Ci − Di) (λCi + (1 − λ) Di) +� +(λAi + (1 − λ) Bi)2 + (λCi + (1 − λ) Di)2 += 0. +(7) +In addition, with the optimal phase shift values given in (4) and (5), the resulting SNR compo- +nents have the same value, i.e., we have +hR +mθR⋆ − hI +mθI⋆ = hR +n θI⋆ + hI +nθR⋆. +Finally, at the receiver, a simple but effective greedy detector (GD) is employed to detect the +selected receive antennas without the need for any knowledge of the CSI at the receiver. The +GD operates via +ˆm = arg +max +m∈{1,2,...,Nr} +�� +yR +m +�2� +, +(8) +ˆn = arg +max +n∈{1,2,...,Nr} +�� +yI +n +�2� +. +(9) +The performance results have demonstrated the superiority of the RIS-RQSSK system over +comparable benchmark schemes. This motivates us to extend this scheme to the context of QSM, +which is the subject of the next section. +IV. RIS-ASSISTED RECEIVE QUADRATURE SPATIAL MODULATION +In general, while the spectral efficiency of an SSK system can be increased by extending +it to the corresponding quadrature SSK system, it can be further improved by implementing a +conventional IQ modulation on top of the antenna index modulation. In the conventional receive +quadrature SM (RQSM), the transmit vector can be designed to place the real and imaginary +parts of the symbol separately at a specific position of the real and imaginary receive vector. On +the other hand, in the RIS-RQSM scheme, the transmitter is equipped with only one antenna +and therefore can only transmit one symbol in each symbol interval. In addition, since the real +and imaginary parts of the desired symbol needs to be separated at the receiver, the transmitter +can only perform amplitude modulation through the RF source to be detectable at the receiver +(as also suggested in [20] for the RIS-RQRM scheme), i.e., it is not feasible to transmit a +QAM symbol and receive the I and Q components separately at two different receive antennas. +To tackle this problem, in the proposed RIS-RQSM system we introduce a new paradigm in +order to construct an M-ary QAM symbol (in fact, two independent symbols from identical +√ +M-ary PAM constellations) at the receiver via the adjustment of both the amplitude of the RF + +9 +source and the phase shifts of the RIS elements. Therefore, in the RIS-RQSM system the rate is +R = log2 M + 2 log2 Nr bits per channel use (bpcu). In this scenario, the desired received signal +components are given by +yR +m = +� +hR +mθR − hI +mθI� +Gs + nR +m, +(10) +yI +n = +� +hR +n θI + hI +nθR� +Gs + nI +n, +(11) +where s is the transmit symbol selected from a specific positive real PAM constellation, denoted +by PRF. The amplitudes in PRF are the magnitudes of the complex symbols in an M-ary QAM +constellation M with average energy Es, i.e., s = |x| where x ∈ M is the desired IQ symbol, +and G > 0 is a one-tap zero-forcing (ZF) pre-equalizer to be defined later. +To produce the desired M-ary QAM signal at the receiver, we modify the problem in (3) to +accommodate both the index modulation and IQ modulation as +max +θR,θI min (YR, δYI) +(12a) +s.t. +YR = sgn +� +xR� � +hR +mθR − hI +mθI� +, +(12b) +YI = sgn +� +xI� � +hR +n θI + hI +nθR� +, +(12c) +� +θR +i +�2 + +� +θI +i +�2 = 1, for all i = 1, 2, . . . , N, +(12d) +where δ > 0 is the absolute value of the ratio of the real to the imaginary part of x, i.e., +δ = +��xR/xI��. It can be seen that this optimization problem is similar to the optimization problem +for the RIS-RQSSK scenario; hence, it can be solved by a similar approach to that used in [27] +(we omit the details for brevity). As a result, +� +θR⋆ +i +� +and +� +θI⋆ +i +� +are again given by (4) and (5), +respectively, and λ can also be evaluated by solving (7); however, it is required to re-define the +variables in (6) accordingly as +Ai = sgn +� +xR� +hR +m,i, Bi = δ sgn +� +xI� +hI +n,i, Ci = sgn +� +xR� � +−hI +m,i +� +, and Di = δ sgn +� +xI� +hR +n,i. +(13) +Note that the maximization problem forces YR and YI to be positive. As a result, the sign +functions in (12) determine the signs of the noise-free received signal components. To eluci- +date the functionality of the optimization problem above, we take symbol x = 1 − 3j as an +example; then, we have sgn +� +xR = 1 +� += +1 and sgn +� +xI = −3 +� += −1. Therefore, we obtain +YR = + +� +hR +mθR − hI +mθI� +> 0 and YI = − +� +hR +n θI + hI +nθR� +> 0, which indicates that the + +10 +real component of the constructed received symbol is positive and its imaginary component is +negative, similar to the selected symbol x. It is also worth pointing out that at the optimal point, +the values involved in the minimization are equal, i.e., with the values +� +θR⋆ +i +� +and +� +θI⋆ +i +� +we +have Y ⋆ +R = δY ⋆ +I , where Y ⋆ +R and Y ⋆ +I are the optimum values of YR and YI produced by (12). +Hence, we can conclude that the phase of the desired QAM symbol is correctly designed. Next, +in order to explain why the PAM constellation PRF must be utilized at the transmitter, we need +to ascertain how the RIS-aided channel acts for various values of δ. +Due to the presence of random variables in (7), λ also presents a random behavior. It is not +easy to determine the stochastic characteristics (e.g., mean and variance) of λ from (7); however, +experimental results provide strong evidence that the mean value of λ is E {λ} = ¯λ = +δ2 +1+δ2 and +that its variance tends to zero with an increasing number of RIS elements N. This observation +can be further used to approximate the average value of the optimum objective in (12), which +is provided in the following theorem. +Theorem 1. For large values of N, the means E {Y ⋆ +R} and E {Y ⋆ +I } can be closely approximated +by +E {Y ⋆ +R} ≈ +� +¯λN√π +2 +, E {Y ⋆ +I } ≈ +� +1 − ¯λN√π +2 +. +Proof: The proof is provided in Appendix A. +From Theorem 1, it can be observed that the mean value of the complex symbol created by +the received signal components at the selected antennas lies on a circle with radius β = N√π +2 +for any value of δ. Therefore, in addition to optimizing the phase angles of the RIS elements, +an appropriate positive PAM symbol s ∈ PRF is required to be modulated at the RF source in +order to adjust the magnitude of the received signal to accommodate the desired QAM symbol +in a predefined constellation. In other words, the phase of the QAM symbol is determined by +the RIS elements while its amplitude is determined by the PAM symbol. Therefore, the transmit +symbol s = |x| is required at the RF source. +The symbol s is then multiplied by G at the transmitter to ensure that the gain of the link +is constant at all times (i.e., for each symbol and for each channel realization). Therefore, we +design G via +G = E {Y ⋆ +R} +Y ⋆ +R += E +� +hR +mθR⋆ − hI +mθI⋆� +hR +mθR⋆ − hI +mθI⋆ +, +(14) + +11 +where θ⋆ is the optimum vector of phase shifts of the RIS elements associated to the desired +transmit symbol. Note that G has a value that is specific to each symbol x and channel realization +H. In fact, G can be realized as a one-tap ZF pre-equalizer. As a result, the receiver only needs +to know the effective gain of the RIS-assisted wireless channel, i.e, the gain of the equivalent +Gaussian channel which is obtained by the aid of the RIS elements, which is equal to β2; no +additional CSI is necessary for the GD detector, which significantly reduces the feedback payload +of the system. On the other hand, the CSI must be available at the transmitter in order to adjust +the phase shifts of the RIS elements and implement the one-tap pre-equalizer. +Theorem 2. Under the assumption of a large number of RIS elements, the mean values of G +and G2 both tend to unity, i.e., lim +N→∞E {G} = 1 and lim +N→∞E {G2} = 1. +Proof: Here we only prove that lim +N→∞E {G2} = 1. The convergence of the mean value of +G can be derived in a similar manner. The mean value of G2 is given by +E +� +G2� += E +�E2 {Y ⋆ +R} +(Y ⋆ +R)2 +� += µ2E +� +1 +(Y ⋆ +R)2 +� +, +where µ = E {Y ⋆ +R} = +√¯λ N√π +2 . According to the central limit theorem (CLT), Y ⋆ +R is distributed +as2 Y ⋆ +R ∼ N (µ, σ2), where σ2 ∝ N. Then, the average of G2 can be expressed as +E +� +G2� += µ2E +� +1 +(Y ⋆ +R)2 +� += +µ2 +√ +2πσ2 +� ∞ +−∞ +1 +y2e− (y−µ)2 +2σ2 dy += +1 +√ +2π +µ2 +σ2 +� ∞ +−∞ +1 +� +u + µ +σ +�2e− u2 +2 du, +where we used the change of variable u = y−µ +σ . Since µ +σ ∝ +√ +N → ∞ as N → ∞, we can write +lim +N→∞ E +� +G2� += lim +µ +σ →∞ +1 +√ +2π +µ2 +σ2 +� ∞ +−∞ +1 +� +u + µ +σ +�2e− u2 +2 du = +1 +√ +2π +� ∞ +−∞ +e− u2 +2 du = 1. +Note that in practice, the number of RIS elements is large enough so that the expressions in +Theorem 2 serve as accurate approximations for our design. Theorem 2 implies that the pre- +equalizer G does not change the average transmit power of the system, i.e., E +� +(Gs)2� += Es; +hence the SNR is simply given by Es/N0. +2This is proved in [27] for the RIS-RQSSK scenario, i.e., for δ = 1, however, the proof can be extended to the general case +where δ = +��xR/xI�� (for brevity, these details are omitted). Later (in Section V) we will show how the variance σ2 is related +to N. + +12 +Receiver Structure +Similar to the RIS-RQSSK scheme, the receiver can employ a GD to detect the selected +antenna indices via (8) and (9). After this, the receiver can demodulate the desired I and Q +symbols via +ˆxR = arg min +xR +�� +yR +m − βxR�2� +, +(15) +ˆxI = arg min +xI +�� +yI +n − βxI�2� +, +(16) +where β = N√π +2 +is the effective channel coefficient. +On the other hand, the maximum likelihood (ML) detector for the proposed RIS-RQSM system +operates via +( ˆm, ˆn, ˆx) = arg min +m,n,x +Nr +� +l=1 +(yl − hlθ⋆Gs)2 , +(17) +where we note that θ⋆ is a multi-variable function of (m, n, x), and s = |x|. While the GD is CSI- +free, the ML detector relies on having full CSI at the receiver. Furthermore, it can be seen that the +ML detector needs to compute θ⋆ for all combinations of the selected receive antennas and then +search over all possible combinations of the spatial symbols and IQ modulation symbols. These +facts make the ML detector significantly more complex than GD. Although the ML detector +provides an optimum receiver, we will show later in Sections VI and VII that optimizing the IQ +constellation, in addition to increasing the performance of the system, can also leverage the GD +efficiency such that it competes very strongly with the ML detector (i.e., the performance gap +is negligible). +V. PERFORMANCE ANALYSIS +In this section, we analyze the ABEP of the proposed RIS-RQSM system. This analysis +focuses on the GD receiver. Here we only perform the analysis for the detection of the antenna +m with active real part along with the real part of the corresponding modulated IQ symbol, xR; +due to the inherent symmetry in the expressions, it is easy to show that the ABEP expression +for the detection of the antenna n with active imaginary part along with the imaginary part of + +13 +the corresponding modulated IQ symbol xI is identical. An upper bound on the ABEP, which +is tight especially at high SNR, is given by +ABEP ≤ +1 − Pe (m) +√ +M log2 +�√ +MNr +� +� +xR +� +ˆxR̸=xR +PEP +� +xR → ˆxR|m = ˆm +� +e +� +xR → ˆxR� ++ 0.5Pe (m) , +(18) +where Pe (m) is the probability of erroneous detection of the selected receive antenna m, +PEP +� +xR → ˆxR|m = ˆm +� +is the pairwise error probability (PEP) associated with the real part of +the symbols x and ˆx conditioned on correct detection of the antenna index, and e +� +xR → ˆxR� +is the Hamming distance between the binary representations of the real parts of the symbols x +and ˆx. Here we assume that half of the bits are in error under the condition of erroneous index +detection (note that this assumption represents the worst-case scenario), so that Pe (m) can be +written as +Pe (m) = (Nr − 1) PEP (m → ˆm) , +(19) +where PEP (m → ˆm) is the average PEP associated with the antenna indices m and ˆm, and is +given by +PEP (m → ˆm) = +1 +√ +M +� +xR +PEP +� +m → ˆm|xR� += +1 +√ +M +� +xR∈MR +2 +√ +M +� +δ∈DxR +PEP +� +m → ˆm|xR, δ +� +, +(20) +where MR is the set consisting of all possible values of xR, the real component of symbols in +M, with |MR| = +√ +M, and Dξ = +���� xR +xI +��� |xR = ξ, xI ∈ MI +� +with |Dξ| = +√ +M +2 +(where MI is the +set consisting of all possible values of xI); for instance, for a conventional 16-QAM constellation +we have MR = MI = {−3, −1, 1, 3}, and for xR = {−1, 1} we have D−1 = D1 = {1, 1/3}, +while for xR = {−3, 3} we have D−3 = D3 = {1, 3}. Considering the use of GD at the receiver, +the PEP associated with the selected antenna m and the detected antenna ˆm ̸= m conditioned +on the selected symbol x (i.e., given xR and δ) is given by +PEP +� +m → ˆm|xR, δ +� += Pr +�� +yR +m +�2 < +� +yR +ˆm +�2 |xR, δ +� += Pr +��� +hR +mθR⋆ − hI +mθI⋆� +Gs + nR +m +�2 +< +�� +hR +ˆmθR⋆ − hI +ˆmθI⋆� +Gs + nR +ˆm +�2 |xR, δ +� +≈ Pr {|Z1| < |Z2|} , +(21) + +14 +where we define Z1 ≜ +� +hR +mθR⋆ − hI +mθI⋆� |xR| +√¯λ + nR +m and Z2 ≜ +� +hR +ˆmθR⋆ − hI +ˆmθI⋆� |xR| +√¯λ + nR +ˆm, +and we have used the approximations stated in Theorem 2, i.e., E {G} ≈ 1 and V {G} = +E {G2} − E {G}2 ≈ 0, and we know that s = |xR| +√¯λ , since ¯λ = +δ2 +1+δ2. To calculate the probability +above, the distributions of Z1, in the cases where m = n and m ̸= n, and Z2, in the cases +where ˆm = n and ˆm ̸= n, are required. In [27, Theorems 1-3], the distributions of the random +variables (RVs) Z1 and Z2 were derived for the case of RIS-RQSSK (in that case it was shown +that ¯λ = 1/2). The distributions of Z1 and Z2 for the more general case of RIS-RQSM can be +derived in a similar manner (we omit the details for brevity). +In the case where m = n, with reference to the CLT, Z1 is approximately distributed according +to N (µ1, σ2 +1), where µ1 = N√π +2 xR and σ2 +1 = N +� +xR�2 4−π +4 ++ N0 +2 . In the case where m ̸= n, the +mean µ1 is given by the same expression as in the case where m = n, and experimental results +provide strong evidence that the variance of Z1 is also exactly the same as in the case where +m = n. +On the other hand, Z2 is approximately distributed according to N (0, σ2 +2), where the variance +in each case of ˆm = n and ˆm ̸= n is given by +1) ˆm ̸= n: +σ2 +2 = ρ2 +1 ≜ N +� +xR�2 +2¯λ ++ N0 +2 , +(22) +2) ˆm = n: +σ2 +2 = ρ2 +2 ≜ N +� +xR�2 +2 ++ N0 +2 . +(23) +Therefore, to calculate the PEP, two different events need to be taken into consideration: i) +{E1 : m, ˆm ∈ {1, 2, . . . , Nr} , ˆm ̸= n}, and ii) {E2 : m ∈ {1, 2, . . . , Nr} , ˆm = n}. +It is worth pointing out that Z1 and Z2 represent the real part of the signal received at the +selected antenna m (having mean µ1 ∝ N ≫ 1) and at a non-selected antenna ˆm (having mean +zero), respectively. This is the reason that the GD is able to easily detect the index of the selected +receive antenna. +Next, we consider the instance where xR > 0 (it is clear that the PEP for xR < 0 is the +same). Considering the distribution of Z1, it can be seen that µ1 +σ1 ∝ +√ +N for relatively high SNR +values, so that µ1 +σ1 ≫ 1; as a result, we have Z1 > 0 with extremely high probability. Hence, the + +15 +PEP can be written as +PEP +� +m → ˆm|xR, δ +� +=PEP +� +m → ˆm|xR > 0, δ +� +≈Nr − 1 +Nr +Pr {Z1 < |Z2| |E1} + 1 +Nr +Pr {Z1 < |Z2| |E2} +=Nr − 1 +Nr +� ∞ +0 +Pr {Z1 = α, |Z2| > α|E1} dα ++ 1 +Nr +� ∞ +0 +Pr {Z1 = α, |Z2| > α|E2} dα. +The above two integrals can be evaluated in a unified manner via +Ii ≜ +� ∞ +0 +Pr {Z1 = α, |Z2| > α|Ei} dα = 2 +� ∞ +0 +pz1|Ei (α) Pr {Z2 > α|Ei} dα += +√ +2 +σ1 +√π +� ∞ +0 +e +− 1 +2 +� µ1−α +σ1 +�2 +Q +� α +ρi +� +dα, i = 1, 2. +(24) +Applying the exponential approximation of the Q-function as Q (x) ≈ +1 +12e− x2 +2 + 1 +4e− 2x2 +3 +from +[28], Ii is approximately given by +Ii ≈ +√ +2 +σ1 +√π +� ∞ +0 +e +− 1 +2 +� µ1−α +σ1 +�2 � 1 +12e +− 1 +2 +� +α +ρi +�2 ++ 1 +4e +− 2 +3 +� +α +ρi +�2� +dα, i = 1, 2. +After some manipulations we obtain +Ii ≈ +√ +2 +σ1 +√π +� +1 +12eu0,i +� ∞ +0 +e +− 1 +2 +� +α−m0,i +s0,i +�2 +dα + 1 +4eu1,i +� ∞ +0 +e +− 1 +2 +� +α−m1,i +s1,i +�2 +dα +� += 1 +σ1 +�1 +6eu0,is0,iQ +� +−m0,i +s0,i +� ++ 1 +2eu1,is1,iQ +� +−m1,i +s1,i +�� +, i = 1, 2, +(25) +where +u0,i = −1 +2 +µ2 +1 +σ2 +1 + ρ2 +i +, s0,i = +σ1ρi +� +σ2 +1 + ρ2 +i +, m0,i = +µ1ρ2 +i +σ2 +1 + ρ2 +i +, +u1,i = −2 +3 +µ2 +1 +4 +3σ2 +1 + ρ2 +i +, s1,i = +σ1ρi +� +4 +3σ2 +1 + ρ2 +i +, m1,i = +µ1ρ2 +i +4 +3σ2 +1 + ρ2 +i +, i = 1, 2. +It is easy to see that m0,i +s0,i and m1,i +s1,i , i = 1, 2, have relatively large values for large N, such that +the approximations Q +� +− m0,i +s0,i +� +≈ Q +� +− m1,i +s1,i +� +≈ 1 are very accurate; therefore, Ii can be written +as +Ii ≈ +ρi +6 +� +σ2 +1 + ρ2 +i +e +− 1 +2 +µ2 +1 +σ2 +1+ρ2 +i + +ρi +2 +� +4 +3σ2 +1 + ρ2 +i +e +− 2 +3 +µ2 +1 +4 +3 σ2 +1+ρ2 +i , i = 1, 2. +(26) +Hence, Pe (m) is given by +Pe (m) = 2 (Nr − 1) +M +� +xR +� +δ +�Nr − 1 +Nr +I1 + 1 +Nr +I2 +� +. +(27) + +16 +Finally, PEP +� +xR → ˆxR|m = ˆm +� +can be expressed as +PEP +� +xR → ˆxR|m = ˆm +� += Q +� +� +� +β2 (xR − ˆxR)2 +2N0 +� +� . +(28) +Substituting (27) and (28) into (18), an accurate closed-form approximation for the ABEP of +the RIS-RQSM system can be obtained. +VI. IQ MODULATION DESIGN +A significant advantage of the proposed RQSM system is that the receiver employs a simple +GD which can perform symbol detection with low complexity and with a minimal CSI require- +ment. However, as will be shown later, if a conventional QAM constellation is used, the system +shows a drop in error rate performance with higher modulation orders, since the symbols with +lowest energy in the QAM constellation dominate the performance of the GD. This phenomenon +has a greater impact in the case of RIS-RQSM than in the RIS-SM system of [14], as in the +former a higher average energy is received at the non-selected antennas, which results in reducing +the performance of the GD. This fact motivates us to design a new QAM constellation in order +to favor the GD3. Hence, in this section we optimize the constellation to minimize the BER +of the RIS-RQSM system with GD. In order to lower the complexity, we employ a number +of approximations in this section to simplify the ABEP upper bound which will then serve as +our objective function. However, the extensive numerical results included in Table I and in the +next section verify the accuracy of these approximations and show that the proposed approach +is practical and yields excellent results. +Thanks to the symmetry in the RIS-RQSM system, the real and imaginary dimensions of +the constellation can be designed separately following the same method, which simplifies the +optimization procedure. Hence, the optimization problem is defined as +min +MR ABEPub +(29) +s.t. +√ +M +� +i=1 +� +xR�2 ≤ +√ +MEs +2 +, +where ABEPub is the approximate upper bound on the ABEP expressed in (18). It is trivial +to observe that the signal constellation should be symmetric about the origin. Therefore, we +3Both the ML detector and the GD perform better with the proposed constellation, but the GD benefits more significantly. + +17 +Fig. 2. Normalized PAM constellation design for the RIS-RQSM system. +define the one-dimensional “normalized” +√ +M-PAM constellation for the real and imaginary +dimensions according to Fig. 2, such that the minimum-energy symbol has distance d0 +√Es +from the origin, while the distance between the i-th and (i+1)-th symbols is denoted by di +√Es, +i = 1, 2, . . . , +√ +M +2 − 1. Due to the symmetry about the origin, there exist +√ +M/2 parameters that +need to be optimized. For example, in 2-PAM, there is only one parameter d0; it is clear that +in this case d0 = 1/ +√ +2, so that this optimization framework is not necessary in that case. In a +4-PAM constellation there are two parameters d0 and d1 that should be optimized such that d0 +is increased and d1 is decreased with respect to the values for conventional PAM, i.e., the two +“inner” symbols are moved further away from the origin and the two “outer” symbols are moved +towards the origin; this adjustment of the constellation points provides a balance between the +spatial domain symbol error probability and the IQ modulation domain symbol error probability. +The expression for ABEPub in (18) is a relatively complex function of the parameters {di} due +to the summation over all symbols in calculating Pe (m) and in calculating PEP +� +xR → ˆxR|m = ˆm +� +associated with all of these distances. Hence, to simplify the solution for the optimization problem +in (29) we adopt some accurate approximations for evaluating the upper bound on the ABEP +that are valid at high SNR and with large N. +It is well-known that at high SNR values the IQ modulation domain bit error probability +(BEP) is dominated by the pairs of constellation points separated by the minimum Euclidean +distance, and it is also clear that the minimum-energy symbols control the BEP in the spatial +domain. Hence, considering Gray coding for the constellation, an approximate upper bound on +the ABEP is given by +ABEPub ≈ +4 +√ +M log2 +�√ +MNr +� +√ +M +2 −1 +� +i=1 +Q +� +� +� +β2Esd2 +i +2N0 +� +� + 0.5 ˜Pe (m) , +(30) + +18 +where ˜Pe (m) is the corresponding approximate value of Pe (m), given by +˜Pe (m) = 4 (Nr − 1) +M +� +δ∈Dd0 +√Es +PEP +� +m → ˆm|xR = d0 +� +Es, δ +� +, +(31) +where Dd0 +√Es = +� +1, +d0 +d0+d1, . . . , +d0 +d0+d1+···+d √ +M +2 +−1 +� +, and we use the fact that ˜Pe (m) ≪ 1, hence +1 − ˜Pe (m) ≈ 1 (note that the optimization function increases the distance between two inner +symbols, so that in (30), we did not consider the distance between the pair of inner symbols as +the minimum distance). Then, the optimization problem can be updated as +min +{di} ABEPub in (30) +(32) +s.t. +√ +M +2 −1 +� +i=0 +� +i +� +j=0 +dj +�2 +≤ +√ +M +4 . +Solving the above optimization problem is not a straightforward task and requires the use +of exhaustive search methods. However, standard lattice constellation structures, such as QAM +or PAM, suggest that equal distances between adjacent pairs of symbols admit a very simple +approach which provides a near-optimal solution in terms of the symbol error rate performance. +Hence, in the following, we assume that the distances between “positive” adjacent symbols are +equal (it is worth recalling that there is a symmetry about the origin, hence the distances between +negative adjacent symbols are also equal). +Special case where d1 = d2 = · · · = d √ +M +2 −1 +In this case, the problem consists of optimizing the two variables d0 and d1. Hence, the +optimization problem reduces to +min +{d0,d1} +4 +√ +M log2 +�√ +MNr +�M ′Q +� +� +� +β2Esd2 +1 +2N0 +� +� + 0.5 ˜Pe (m) , +(33) +s.t. +2d2 +0 + M ′ (2M ′ + 1) +3 +d2 +1 + 2M ′d0d1 ≤ 1, +where we define M ′ = +√ +M +2 −1. From the inequality constraint, d1 can be obtained as a function +of d0 (here we force equality in the constraint above to maximize the achievable SNR at the +receiver. It will be shown later in this section that equality indeed holds at the optimum point). +Then, by performing a grid search over variable d0, we can find the minimum value of the +ABEPub. However, taking the equal positive distance into account, it is more valuable to find +an analytical solution; this is the subject of the remainder of this section. + +19 +Analytical approach - asymptotic analysis: In order to find an efficient analytical solution for +the optimization problem, we analyze the distributions of Z1 and Z2 in more detail in order to +obtain a more tractable approximate expression for ABEPub. We see that the variance of Z2 +(i.e., the average received energy of the signal on a non-selected receive antenna) in the event E1 +increases with decreasing ¯λ = +δ2 +1+δ2, or equivalently, with decreasing δ = +��� xR +xI +���; in other words, +ρ2 +1 in (22) is maximized when δ is minimized. There are two consequences of this fact: first, the +BEP related to the spatial domain is dominated by those symbols bearing the minimum energy in +the real part while their corresponding imaginary parts have the maximum energy, i.e., the PEP +associated with δmin = +min|xR| +max|xI| = +d0 +d0+M′d1 dominates (31); secondly, comparing the two events +E1 and E2, the event E2 has a minor impact on the value of PEP +� +m → ˆm|xR, δ +� +, as the variance +of Z2 in the event E1 is significantly greater than that in the event E2 due to the appearance of +¯λ in the denominator. In summary, considering the above comments, the PEP associated to δmin +dominates and the event E2 can be eliminated from the PEP analysis, therefore ˜Pe (m) can be +approximated as +˜Pe (m) ≈ 4 (Nr − 1) +M +PEP +� +m → ˆm|xR = d0 +� +Es, δ = δmin +� +≈ 4 (Nr − 1)2 +MNr +I1 (µ1, ρ1, σ1) . +(34) +In addition, by substituting ¯λ = +δ2 +min +1+δ2 +min into (22) and performing some minor algebraic manipu- +lations, the variance of Z2 in the event E1 can be expressed as +ρ2 +1 = NEs +2 +� +d2 +0 + (d0 + M ′d1)2� ++ N0 +2 . +Note that ¯E ≜ d2 +0 + (d0 + M ′d1)2 is the sum of the energies associated with the symbols with +minimum and maximum distance from the origin. It is clear that 1 ≤ ¯E < ϵM, where equality +holds for M = 16, and ϵM is defined as the total energy of the inner and outer symbols in +the conventional +√ +M-PAM constellation (since the conventional constellation is the worst-case +scenario, ¯E can be upper bounded by ϵM), so that we obtain ϵM = 3(M−2 +√ +M+2) +2(M−1) +(note that the +average energy of the PAM constellation is 1/2). Therefore, we can write +NEs +2 ++ N0 +2 ≤ ρ2 +1 < NEs +2 +ϵM + N0 +2 . + +20 +In addition, it is easy to prove that (34) is monotonically increasing with respect to ρ1. Hence, +˜Pe (m) can be expressed as +˜Pe (m) ≈ ˜Pe (m) +��� +ρ2 +1= NEs +2 ++ N0 +2 +, M = 16, +˜Pe (m) ≲ ˜Pe (m) +��� +ρ2 +1= NEs +2 +ϵM+ N0 +2 +, M > 16. +Finally, from the formula σ2 +1 = N +� +xR�2 4−π +4 ++ N0 +2 applied to the minimum energy symbol +xR = d0 +√Es and considering the fact that d2 +0 ≪ 1, the variance of Z1 can be approximated as +σ2 +1 ≈ N0 +2 +4. +Therefore, after some manipulations we obtain ˜Pe (m) as +˜Pe (m) ≈2 (Nr − 1)2 +MNr +� +1 +3 +� +NEs + N0 +NEs + 2N0 +e− +πN2Esd2 +0 +4NEs+8N0 + +� +NEs + N0 +NEs + 7 +3N0 +e− +πN2Esd2 +0 +3NEs+7N0 +� +, M = 16, +˜Pe (m) ≲2 (Nr − 1)2 +MNr +� +1 +3 +� +NEsϵM + N0 +NEsϵM + 2N0 +e +− +πN2Esd2 +0 +4NEsϵM +8N0 + +� +NEsϵM + N0 +NEsϵM + 7 +3N0 +e +− +πN2Esd2 +0 +3NEsϵM +7N0 +� +, +M > 16. +Also applying the exponential approximation of the Q-function in (33), the optimization problem +becomes +min +{d0,d1} ABEPub ≈ a0e−b0d2 +0 + a1e−b1d2 +0 + M ′a2 +� 1 +12e−b2d2 +1 + 1 +4e− 4 +3 b2d2 +1 +� +(35) +s.t. +2d2 +0 + M ′ (2M ′ + 1) +3 +d2 +1 + 2M ′d0d1 ≤ 1, +where we define +a0 = (Nr − 1)2 +3MNr +� +NEsϵM + N0 +NEsϵM + 2N0 +, b0 = +πN 2Es +4NEsϵM + 8N0 +, +a1 = (Nr − 1)2 +MNr +� +NEsϵM + N0 +NEsϵM + 7 +3N0 +, b1 = +πN 2Es +3NEsϵM + 7N0 +, +a2 = +4 +√ +M log2 +�√ +MNr +�, b2 = πN 2Es +16N0 +. +The problem in (35) is not a convex optimization problem, as the objective function is not +convex in the domain of d0, d1 ∈ R+. However, it is easy to see that (35) satisfies the convexity +4Here we are assuming that N is sufficiently large so the SNR range +4−π +2 +NEsd2 +0 +N0 +≪ 1 is of interest, i.e., the BER is extremely +low outside of this SNR range. + +21 +condition ∇2ABEPub ≥ 0 when d0 ≥ +1 +√2bi, i = 0, 1, and d1 ≥ +1 +√2b2. For sufficiently high values +of N2Es +N0 +(note that N ≫ 1), it can be concluded that {b0, b1, b2} are sufficiently large such that +the optimized {d0, d1} lie in the convex region of the objective function. For such {b0, b1, b2}, +the problem is convex and can be solved using the following procedure. +The Karush-Kuhn-Tucker (KKT) [29] conditions associated to the above problem hold and +are given by +1. f1 (d⋆ +0, d⋆ +1) ≤ 0; +2. ν⋆ ≥ 0; +3. ν⋆f1 (d⋆ +0, d⋆ +1) = 0; +4. − 2a0b0d⋆ +0e−b0d⋆2 +0 − 2a1b1d⋆ +0e−b1d⋆2 +0 + ν⋆ (4d⋆ +0 + 2M ′d⋆ +1) = 0; +5. − 1 +6M ′a2b2d⋆ +1e−b2d⋆2 +1 − 2 +3M ′a2b2d⋆ +1e− 4 +3 b2d⋆2 +1 + ν⋆ +�2M ′ (2M ′ + 1) +3 +d⋆ +1 + 2M ′d⋆ +0 +� += 0; +where ν is the Lagrange multiplier associated with the inequality constraint. From condition 3, +we see that ν⋆ = 0 or f1 (d⋆ +0, d⋆ +1) = 0. However, if ν⋆ = 0, from conditions 4 and 5 we obtain +d⋆ +0 = d⋆ +1 = +∞, where clearly contradicts condition 1. Therefore, we have +2d⋆2 +0 + M ′ (2M ′ + 1) +3 +d⋆2 +1 + 2M ′d⋆ +0d⋆ +1 − 1 = 0, +which yields +d⋆ +0 = +−2M ′d⋆ +1 + +� +4M ′2d⋆2 +1 − 8 +� +M′(2M′+1) +3 +d⋆2 +1 − 1 +� +4 +. +(36) +Then, from conditions 4 and 5, we obtain +ν⋆ = a0b0d⋆2 +0 e−b0d⋆2 +0 + a1b1d⋆2 +0 e−b1d⋆2 +0 + 1 +12M ′a2b2d⋆2 +1 e−b2d⋆2 +1 + 1 +3M ′a2b2d⋆2 +1 e− 4 +3 b2d⋆2 +1 . +(37) +Substituting for ν⋆ from (37) and subsequently for d⋆ +0 from (36) into condition 5, the optimization +problem reduces to a single-variable equation in d⋆ +1. This equation does not admit a closed-form +analytical solution; however it is easy to solve numerically. +We conclude this section by providing a numerical example in Table I. In this table, we +compare the optimal {di} obtained by an exhaustive search to minimize the ABEP in (18) +with the corresponding values with equal positive distances obtained via the proposed analytical +approach, where N = 256, Nr = 4 and M = 64. It can be seen that positive distances {di}, +i > 0, obtained via exhaustive search are almost equal, and that these values become more +similar with increasing SNR. In addition, the ABEP values acquired by using the optimal values +from the proposed analytical approach are quite comparable to the equivalent ABEP obtained +by optimal values of the grid search, which serves as a proof that the assumptions we made to +offer a straightforward analytical solution to the optimization problem were indeed accurate. + +22 +TABLE I +COMPARISON BETWEEN OPTIMAL {di} VALUES OBTAINED VIA MINIMIZING (18) BY GRID SEARCH AND THE +CORRESPONDING VALUES OBTAINED BY THE ANALYTICAL APPROACH OF (35), WHERE N = 256, Nr = 4 AND M = 64. +Minimized ABEP based on (18) using grid search +Minimized ABEP by using analytical approach of (35) +SNR (dB) +d0 +d1 +d2 +d3 +ABEP +d0 +d1 +ABEP +-23 +0.2609 +0.250 +0.257 +0.272 +6.97 × 10−4 +0.2481 +0.2632 +7.62 × 10−4 +-21 +0.2695 +0.248 +0.253 +0.262 +6.46 × 10−5 +0.2661 +0.2543 +6.67 × 10−5 +-19 +0.2890 +0.240 +0.243 +0.248 +2.90 × 10−6 +0.2891 +0.2426 +2.96 × 10−6 +-17 +0.3169 +0.227 +0.228 +0.232 +5.71 × 10−8 +0.3179 +0.2278 +5.82 × 10−8 +VII. NUMERICAL RESULTS +In this section, we demonstrate the error rate performance of the proposed RIS-RQSM system +via numerical simulations. First, we investigate the performance of the proposed RIS-RQSM +system using conventional QAM constellations and provide comparisons with corresponding +systems using QAM constellations that are optimized based on the approach proposed in Sec- +tion VI. Next, we compare the results obtained by the optimized constellations with the error +rate performance of the most prominent recently proposed RIS-SM [14] system, which serves +as the benchmark scheme for the proposed approach. +Fig. 3 shows the BER performance of the proposed RIS-RQSM system with N = 256 for +the cases of Nr = 4 and Nr = 8. In this figure, we also compare the performance of the RIS- +RQSM system using conventional 16-QAM modulation with that of the system implementing +our optimized 16-QAM constellation. The curves demonstrate the effectiveness of the proposed +constellation design method; it can be observed that optimizing the design of the constellation +significantly enhances the performance of the system. The proposed constellation for RIS-RQSM +provides approximately 3.2 dB and 3.8 dB improvement over the conventional constellation in +systems with Nr = 4 and Nr = 8, respectively, at a BER of 10−5. We also compare the +performance of the GD with that of the ML detector. We see that there is a very large gap +between the performance of the GD and ML detector in the case of the conventional constellation, +while the performance of the GD in the system using the optimized constellation is considerably +close to that of the ML detector such that the performance gap is negligible. In order to observe +the effect of optimizing the constellation in a system with higher-order modulation, we present +the BER performance of the RIS-RQSM system with 64-QAM in Fig. 4. Here, we see that in + +23 +-34 +-32 +-30 +-28 +-26 +-24 +-22 +Es/N0 +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +BER +RIS-RQSM - Sim. - GD (Conventional 16-QAM) +RIS-RQSM - Ana. (Conventional 16-QAM) +RIS-RQSM - Sim. - ML (Conventional 16-QAM) +RIS-RQSM - Sim. - GD (Opt. 16-QAM) +RIS-RQSM - Ana. (Opt. 16-QAM) +RIS-RQSM - Asym. - Eq. (35) (Opt. 16-QAM) +RIS-RQSM - Sim. - ML (Opt. 16-QAM) +(a) +-34 +-32 +-30 +-28 +-26 +-24 +-22 +-20 +Es/N0 +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +BER +RIS-RQSM - Sim. - GD (Conventional 16-QAM) +RIS-RQSM - Ana. (Conventional 16-QAM) +RIS-RQSM - Sim. - ML (Conventional 16-QAM) +RIS-RQSM - Sim. - GD (Opt. 16-QAM) +RIS-RQSM - Ana. (Opt. 16-QAM) +RIS-RQSM - Asym. - Eq. (35) (Opt. 16-QAM) +RIS-RQSM - Sim. - ML (Opt. 16-QAM) +(b) +Fig. 3. Analytical and simulation BER results of the proposed RIS-RQSM system with and without optimized constellation. +Here M = 16, N = 256, and (a) Nr = 4 (R = 8 bpcu), (b) Nr = 8 (R = 10 bpcu). +systems with regular QAM constellations, an error floor occurs with the GD. This is due to the +fact that with critical symbols, i.e., minimum-energy symbols, ¯λ can attain a very small value; +hence, non-selected antennas can have a relatively high average received energy compared to +the selected antenna. However, we see that optimizing the constellation eliminates this error +floor and substantially improves the error rate performance. Similar to systems with 16-QAM +constellation, the performance of the GD is very close to that of ML detector with optimized +constellations. In fact, here the GD becomes feasible only with the optimized 64-QAM. In +Figs. 3 and 4, we also present the analytical ABEP performance of each system. For systems +with conventional QAM constellations, we evaluate and plot the analytical ABEP upper bounds +based on (18); we see that upper bound curves are quite tight and validate the accuracy of the +analytical results. For systems with optimized constellation we also plot the asymptotic result +in (35). These curves show that the utilized approximations in Section VI are completely valid +and accurate, especially at high SNR. +Next, in Fig. 5, we compare the BER performance of the proposed RIS-RQSM system with that +of the benchmark scheme, i.e., RIS-SM, in systems with N = 256 and Nr = 4. Fig. 5(a) shows + +24 +-30 +-28 +-26 +-24 +-22 +-20 +-18 +-16 +Es/N0 +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +BER +RIS-RQSM - Sim. - GD (Conventional 64-QAM) +RIS-RQSM - Ana. (Conventional 64-QAM) +RIS-RQSM - Sim. - ML (Conventional 64-QAM) +RIS-RQSM - Sim. - GD (Opt. 64-QAM) +RIS-RQSM - Ana. (Opt. 64-QAM) +RIS-RQSM - Asym. - Eq. (35) (Opt. 64-QAM) +RIS-RQSM - Sim. - ML (Opt. 64-QAM) +(a) +-30 +-28 +-26 +-24 +-22 +-20 +-18 +-16 +-14 +-12 +Es/N0 +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +BER +RIS-RQSM - Sim. - GD (Conventional 64-QAM) +RIS-RQSM - Ana. (Conventional 64-QAM) +RIS-RQSM - Sim. - ML (Conventional 64-QAM) +RIS-RQSM - Sim. - GD (Opt. 64-QAM) +RIS-RQSM - Ana. (Opt. 64-QAM) +RIS-RQSM - Asym. - Eq. (35) (Opt. 64-QAM) +RIS-RQSM - Sim. - ML (Opt. 64-QAM) +(b) +Fig. 4. Analytical and simulation BER results of the proposed RIS-RQSM system with and without optimized constellation. +Here M = 64, N = 256, and (a) Nr = 4 (R = 10 bpcu), (b) Nr = 8 (R = 12 bpcu). +the performance of the RIS-RQSM and RIS-SM systems where the bit rate is R = 8 bpcu. Hence, +the proposed RIS-RQSM system uses 16-QAM modulation, while the RIS-SM system uses +64-QAM modulation. The constellation used in the proposed RIS-RQSM system is optimized +to achieve the best performance. Fig. 5(b) presents the performance results in systems with +R = 10 bpcu, i.e., where RIS-RQSM and RIS-SM apply 64-QAM and 256-QAM, respectively. +The results show that the proposed RIS-RQSM system substantially outperforms the benchmark +scheme. This is mainly due to the fact that the RIS-SM system needs to employ a higher-order +modulation technique in order to compensate the additional bits transmitted by the quadrature +index modulation in the proposed RIS-RQSM system. Hence, the superiority over the benchmark +scheme increases by increasing number of receive antennas, as shown in Fig. 6. In this figure, +we provide comparisons between the BER performance of the RIS-RQSM and RIS-SM systems +where N = 256 and Nr = 8. As expected, the superiority over the RIS-SM system considerably +increases in a system with larger number of receive antennas, as a higher modulation order +is required for the RIS-SM system. The proposed RIS-RQSM system achieves approximately +4.3 dB and 7 dB performance improvement over the RIS-SM system for systems with Nr = 4 + +25 +-34 +-32 +-30 +-28 +-26 +-24 +-22 +-20 +Es/N0 +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +BER +RIS-RQSM - GD (Opt. 16-QAM) +RIS-SM - GD (64-QAM) +(a) +-30 +-28 +-26 +-24 +-22 +-20 +-18 +-16 +-14 +Es/N0 +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +BER +RIS-RQSM - GD (Opt. 64-QAM) +RIS-SM - GD (256-QAM) +(b) +Fig. 5. Comparison of the BER performance of the proposed RIS-RQSM system with that of RIS-SM system for N = 256, +Nr = 4, and (a) R = 8 bpcu, (b) R = 10 bpcu. +and Nr = 8, respectively, at a BER of 10−5. It is worth pointing out that the receiver in the +proposed RIS-RQSM system requires minimal CSI due to the use of the pre-equalizer G; this +CSI consists only of the average gain of the effective channel, which is simply a function of the +number of RIS elements, as shown in Section II. +VIII. CONCLUSION +The RIS-assisted receive quadrature spatial modulation (RIS-RQSM) system was proposed +in this paper as a general approach to RIS-assisted receive SM with excellent performance. +The proposed system increases the spectral efficiency by implementing both quadrature spatial +modulation and IQ modulation, while maintaining the signal quality at the receiver. In the +proposed RIS-RQSM system, the phase shifts of the RIS elements are designed to construct an +IQ symbol at the receiver; this enables the system to transmit two separate PAM symbols in the +presence of the RIS. We introduced a one-tap pre-equalizer to allow the proposed low-complexity +GD to detect the symbols with minimum CSI requirement. Analytical results and numerical +simulations both verify the excellent performance of the system and extensively demonstrate + +26 +-34 +-32 +-30 +-28 +-26 +-24 +-22 +-20 +-18 +Es/N0 +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +BER +RIS-RQSM - GD (Opt. 16-QAM) +RIS-SM - GD (128-QAM) +(a) +-30 +-28 +-26 +-24 +-22 +-20 +-18 +-16 +-14 +-12 +Es/N0 +10-6 +10-5 +10-4 +10-3 +10-2 +10-1 +100 +BER +RIS-RQSM - GD (Opt. 64-QAM) +RIS-SM - GD (512-QAM) +(b) +Fig. 6. Comparison of the BER performance of the proposed RIS-RQSM system with that of RIS-SM system for N = 256, +Nr = 8, and (a) R = 10 bpcu, (b) R = 12 bpcu. +its superiority over comparable benchmark schemes in the literature. The many advantages of +the RIS-RQSM system makes it a viable candidate for next-generation wireless communication +networks. +APPENDIX A +PROOF OF THEOREM 1 +Here we analyze the average of +Y ⋆ +R = sgn +� +xR� � +hR +mθR⋆ − hI +mθI⋆� += +N +� +i=1 +λA2 +i + λC2 +i + (1 − λ) AiBi + (1 − λ) CiDi +� +(λAi + (1 − λ) Bi)2 + (λCi + (1 − λ) Di)2. +As stated before, for large values of N, we have V {λ} ≈ 0; therefore, we replace λ by ¯λ in +calculating the average of Y ⋆ +R; this yields +E {Y ⋆ +R} ≊ E +� +� +� +N +� +i=1 +¯λA2 +i + ¯λC2 +i + +� +1 − ¯λ +� +AiBi + +� +1 − ¯λ +� +CiDi +��¯λAi + +� +1 − ¯λ +� +Bi +�2 + +�¯λCi + +� +1 − ¯λ +� +Di +�2 +� +� +� += NE +� +� +� +¯λA2 +i + ¯λC2 +i + +� +1 − ¯λ +� +AiBi + +� +1 − ¯λ +� +CiDi +��¯λAi + +� +1 − ¯λ +� +Bi +�2 + +�¯λCi + +� +1 − ¯λ +� +Di +�2 +� +� +� , + +27 +where we used the fact that each of the summands has an identical distribution. In the following, +we evaluate the average of the terms in the above summation individually and we omit the index +i to simplify the notation; hence we define +W1 ≜ ¯λ A2 +√ +Z +, W2 ≜ ¯λ C2 +√ +Z +, W3 ≜ +� +1 − ¯λ +� AB +√ +Z +, W4 ≜ +� +1 − ¯λ +� CD +√ +Z +, +where Z ≜ +�¯λA + +� +1 − ¯λ +� +B +�2 + +�¯λC + +� +1 − ¯λ +� +D +�2. +According to the law of total expectation, the expected value of W1 can be expressed as +E {W1} = EA +� +EW1|A {W1|A} +� += ¯λEA +� +A2EZ|A +� +Z− 1 +2|A +�� +, +(38) +where EZ|A +� +Z− 1 +2|A +� +is the inverse-fractional moment of Z where A is given, i.e., where A +is a constant. For a given A, using ¯λ = +δ2 +1+δ2 we have +�¯λA + +� +1 − ¯λ +� +B +� +∼ N +� +¯λA, +¯λ(1−¯λ) +2 +� +, +and +�¯λC + +� +1 − ¯λ +� +D +� +∼ N +� +0, +¯λ +2 +� +. Hence, the random variable (RV) (Z|A) is the sum of two +independent chi-square RVs each having one degree of freedom. The inverse-fractional moment +of (Z|A) can be computed by using the following equation [30] +EZ|A +� +Z−c|A +� += +1 +Γ (c) +� ∞ +0 +sc−1EZ|A +� +e−sZ|A +� +ds, +(39) +where EZ|A +� +e−sZ|A +� += Ls (fZ (Z|A)) is the Laplace transform (LT) of fZ (Z|A). We know that +the LT of the probability density function (PDF) of the sum of independent RVs is equal to the +product of the LTs of their individual PDFs, and that the LT of the PDF of an RV X = �n +i=1 X2 +i +with Xi ∼ N (µi, σ2) is given by +Ls (fX(X)) = +� +1 +1 + 2σ2s +� n +2 +exp +� +−µ2s +1 + 2σ2s +� +, +(40) +where µ2 = �n +i=1 µ2 +i . Hence, the LT of fZ (Z|A) is calculated as +Ls (fZ (Z|A)) = +� +1 +1 + ¯λs +� 1 +2 � +1 +1 + ¯λ +� +1 − ¯λ +� +s +� 1 +2 +exp +� +−¯λ2A2s +1 + ¯λ +� +1 − ¯λ +� +s +� +. +(41) +Then, (38) can be written as +E {W1} = +¯λ +Γ2 � 1 +2 +� +� ∞ +0 +s +1 +2 −1 +� +1 +1 + ¯λs +� 1 +2 � +1 +1 + ¯λ +� +1 − ¯λ +� +s +� 1 +2 +× +�� ∞ +−∞ +A2 exp +� +−A2 +1 + ¯λs +1 + ¯λ +� +1 − ¯λ +� +s +� +dA +� +ds, + +28 +where we used the fact that fA (A) = +1 +Γ( 1 +2) exp (−A2). Since +� ∞ +−∞ x2 exp +� +− x2 +2σ2 +� +dx = Γ +� 1 +2 +� +σ2√ +2σ2, +we have +� ∞ +−∞ +A2 exp +� +−A2 +1 + ¯λs +1 + ¯λ +� +1 − ¯λ +� +s +� +dA = Γ +� 1 +2 +� +2 +� +1 + ¯λ +� +1 − ¯λ +� +s +1 + ¯λs +� 3 +2 +. +It follows that +E {W1} = ¯λ +1 +2Γ +� 1 +2 +� +� ∞ +0 +s +1 +2 −11 + ¯λ +� +1 − ¯λ +� +s +� +1 + ¯λs +�2 +ds += ¯λ +� +1 − ¯λ +� +1 +2Γ +� 1 +2 +� +�� ∞ +0 +s +1 +2 −1 � +1 + ¯λs +�−1 ds + +¯λ +1 − ¯λ +� ∞ +0 +s +1 +2 −1 � +1 + ¯λs +�−2 ds +� +. +Recalling the definition of the type-2 beta function B (α, β) = +� ∞ +0 +tα−1 +(1+t)α+β dt = Γ(α)Γ(β) +Γ(α+β) , after +some minor manipulations we obtain +E {W1} = ¯λ +1 +2 � +1 − ¯λ +� +1 +2Γ +� 1 +2 +� +� +Γ +� 1 +2 +� +Γ +� 1 +2 +� +Γ (1) ++ +¯λ +1 − ¯λ +Γ +� 1 +2 +� +Γ +� 3 +2 +� +Γ (2) +� += +√π +4 +� +2¯λ +1 +2 − ¯λ +3 +2 +� +. +By symmetry it is clear that E {W2} = E {W1}. +Next we determine E {W3} = E +�� +1 − ¯λ +� AB +√ +Z +� +. Using the law of total expectation, we can +write +E {W3} = +� +1 − ¯λ +� +EA +� +AEB +� +BEZ|(A,B) +� +Z− 1 +2|(A, B) +��� +. +Given constant (A, B), we have +Ls (fZ (Z|(A, B))) = +� +1 +1 + ¯λs +� 1 +2 +exp +� +− +�¯λA + +� +1 − ¯λ +� +B +�2 s +� +. +(42) +Using (39), we have +E {W3} = +� +1 − ¯λ +� +EA +� +AEB +� +B +Γ +� 1 +2 +� +� ∞ +0 +s +1 +2 −1 +� +1 +1 + ¯λs +� 1 +2 +exp +� +− +�¯λA + +� +1 − ¯λ +� +B +�2 s +� +ds +�� +. +Then, using fA (A) = +1 +Γ( 1 +2) exp (−A2) and fB (B) = (1−¯λ) +1 +2 +¯λ +1 +2 Γ( 1 +2) exp +� +− 1−¯λ +¯λ B2� +, after some alge- +braic manipulations we obtain +E {W3} = +� +1 − ¯λ +� 3 +2 +¯λ +1 +2 +1 +Γ3 � 1 +2 +� +� ∞ +0 +s +1 +2 −1 +� +1 +1 + ¯λs +� 1 +2 +× +� +���� +� ∞ +−∞ +A exp +� +−A2 +1 + ¯λs +1 + ¯λ +� +1 − ¯λ +� +s +� +� +� +� +� +� +� ∞ +−∞ +B exp +� +� +� +� +�− +� +B + +¯λ2As +1+¯λ(1−¯λ)s +�2 +¯λ/(1−¯λ) +(1+¯λ(1−¯λ)s) +� +� +� +� +� dB +� +� +� +� +� dA +� +���� ds. (43) + +29 +The inner integral over B can be evaluated as +� ∞ +−∞ +B exp +� +� +� +� +�− +� +B + +¯λ2As +1+¯λ(1−¯λ)s +�2 +¯λ +1−¯λ +1+¯λ(1−¯λ)s +� +� +� +� +� dB = −Γ +�1 +2 +� +¯λ +5 +2 As +� +1 − ¯λ +� 1 +2 � +1 + ¯λ +� +1 − ¯λ +� +s +� 3 +2 . +Substituting this into (43), the average of W3 is given by +E {W3} = ¯λ2 � +1 − ¯λ +� +−1 +2Γ +� 1 +2 +� +� ∞ +0 +s +3 +2 −1 +1 +� +1 + ¯λs +�2ds += ¯λ +1 +2 � +1 − ¯λ +� +−1 +2Γ +� 1 +2 +�B +�3 +2, 1 +2 +� += − +√π +4 +¯λ +1 +2 � +1 − ¯λ +� +. +Also, by symmetry we have E{W4} = E{W3}. Finally, the average of Y ⋆ +R is given by +E {Y ⋆ +R} ≈ 2N (E {W1} + E {W3}) = ¯λ +1 +2 N√π +2 +. +Then, using ¯λ = +δ2 +1+δ2, E {Y ⋆ +I } is given by +E {Y ⋆ +I } = 1 +δE {Y ⋆ +R} ≈ (1 − ¯λ) +1 +2 N√π +2 +. +REFERENCES +[1] R. Liu, Q. Wu, M. Di Renzo, and Y. Yuan, “A path to smart radio environments: An industrial viewpoint on reconfigurable +intelligent surfaces,” IEEE Wireless Communications, vol. 29, pp. 202–208, Feb. 2022. +[2] M. 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Di Renzo, “Reconfigurable intelligent surface-aided quadrature reflection mod- +ulation for simultaneous passive beamforming and information transfer,” IEEE Transactions on Wireless Communications, +vol. 21, pp. 1469–1481, Mar. 2022. +[26] F. Shu, L. Yang, X. Jiang, W. Cai, W. Shi, M. Huang, J. Wang, and X. You, “Beamforming and transmit power design for +intelligent reconfigurable surface-aided secure spatial modulation,” IEEE Journal of Selected Topics in Signal Processing, +vol. 16, pp. 933–949, Aug. 2022. +[27] M. H. Dinan, N. S. Perovi´c, and M. F. Flanagan, “RIS-assisted receive quadrature space-shift keying: A new paradigm +and performance analysis,” IEEE Transactions on Communications, vol. 70, pp. 6874–6889, Oct. 2022. +[28] M. Chiani, M. Z. Win, and A. Zanella, “On the capacity of spatially correlated MIMO Rayleigh-fading channels,” IEEE +Transactions on Information Theory, vol. 49, pp. 2363–2371, Oct. 2003. +[29] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. +[30] A. M. Mathai and S. B. Provost, Quadratic Forms in Random Variables: Theory and Applications. New York, NY, USA: +Marcel Dekker, 1992. + diff --git a/q9AyT4oBgHgl3EQfzvm4/content/tmp_files/load_file.txt b/q9AyT4oBgHgl3EQfzvm4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6a87e20c21f5c7187030d58132ff6a216047b104 --- /dev/null +++ b/q9AyT4oBgHgl3EQfzvm4/content/tmp_files/load_file.txt @@ -0,0 +1,895 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf,len=894 +page_content='1 RIS-Assisted Receive Quadrature Spatial Modulation with Low-Complexity Greedy Detection Mohamad H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Dinan, Member, IEEE, Marco Di Renzo, Fellow, IEEE, and Mark F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Flanagan, Senior Member, IEEE Abstract In this paper, we propose a novel reconfigurable intelligent surface (RIS)-assisted wireless commu- nication scheme which uses the concept of spatial modulation, namely RIS-assisted receive quadrature spatial modulation (RIS-RQSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In the proposed RIS-RQSM system, the information bits are conveyed via both the indices of the two selected receive antennas and the conventional in-phase/quadrature (IQ) modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' We propose a novel methodology to adjust the phase shifts of the RIS elements in order to maximize the signal-to-noise ratio (SNR) and at the same time to construct two separate PAM symbols at the selected receive antennas, as the in-phase and quadrature components of the desired IQ symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' An energy-based greedy detector (GD) is implemented at the receiver to efficiently detect the received signal with minimal channel state information (CSI) via the use of an appropriately designed one-tap pre-equalizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' We also derive a closed-form upper bound on the average bit error probability (ABEP) of the proposed RIS-RQSM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Then, we formulate an optimization problem to minimize the ABEP in order to improve the performance of the system, which allows the GD to act as a near-optimal receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Extensive numerical results are provided to demonstrate the error rate performance of the system and to compare with that of a prominent benchmark scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The results verify the remarkable superiority of the proposed RIS-RQSM system over the benchmark scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' This work was funded by the Irish Research Council (IRC) under the Consolidator Laureate Award Programme (grant number IRCLA/2017/209).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The work of Marco Di Renzo was supported in part by the European Commission through the H2020 ARIADNE project under grant agreement number 871464 and through the H2020 RISE-6G project under grant agreement number 101017011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Mohamad H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Dinan and Mark F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Flanagan are with the School of Electrical and Electronic Engineering, University College Dublin, Belfield, Dublin 4, D04 V1W8 Ireland (email: mohamad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='hejazidinan@ucdconnect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='ie;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='flanagan@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Marco Di Renzo is with Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes, 3 Rue Joliot- Curie, 91192 Gif-sur-Yvette, France (email: marco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='di-renzo@universite-paris-saclay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='fr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='00707v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='IT] 2 Jan 2023 2 Index Terms 6G, reconfigurable intelligent surface (RIS), spatial modulation (SM), quadrature spatial modulation (QSM), greedy detector (GD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' INTRODUCTION In the past few years, various wireless communication technologies have emerged with an aim to support high demands for connectivity and an immense increase in mobile data traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Among these, reconfigurable intelligent surfaces (RISs), also known as intelligent reflecting surfaces (IRSs), represents a key innovation that has drawn significant attention from researchers in both academia and industry [1] and is foreseen to be a potential candidate for 6th generation (6G) networks [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' An RIS is a surface of electromagnetic meta-material consisting of a large number of small, low-cost and energy-efficient reflecting elements that are able to control the scattering and propagation in the channel by inducing a pre-designed phase shift to the impinging wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' From this perspective, RIS technology represents a revolutionary paradigm that can transform the uncontrollable disruptive propagation environment into a smart radio environment [2], [4], thus enhancing the received signal quality [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' On the other hand, spatial modulation (SM) [7], [8] and its variants such as generalized spatial modulation (GSM) [9], receive spatial modulation (RSM) [10], [11], and quadrature spatial modulation (QSM) [12], have been widely investigated in the last two decades as a promising technology for beyond-5th-generation (B5G) networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' SM uses the indices of the transmit/receive antennas to convey the information bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' It exploits the channel attributes to simplify the transceiver structure in order to provide a more energy-efficient solution compared with other conventional multiple-input multiple-output (MIMO) techniques [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The implicit advantages of both RIS and SM technology have motivated researchers to combine these two advanced technologies to obtain a reliable energy-efficient approach in order to achieve so-called green or sustainable wireless communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Specifically, in [14], two fundamental RIS-based index modulation (IM) techniques were proposed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', RIS-space-shift keying (RIS- SSK) and RIS-spatial modulation (RIS-SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In both scenarios, the RIS-access point (RIS-AP) approach was implemented, in which the RIS forms part of the transmitter, and the index of the receive antennas is used to convey the data bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The numerical results confirm a significant superiority of these RIS-aided schemes compared to conventional MIMO schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Various principles of RIS-based SM (also known as metasurface-based modulation) were introduced in 3 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The authors of [16] proposed an RIS-SSK system with multiple transmit antennas in which the information bits map to the transmit antenna index and the single-antenna receiver receives the signal reflected from the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Various scenarios with ideal and non-ideal transceivers were investigated and the error rate performance of each scenario was analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The results indicate that maximizing the signal-to-noise ratio (SNR) at the receiver is not a good approach for the transmit RIS-SSK setting, and in fact shows a relatively poor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In light of this, in [17] the authors proposed an optimization algorithm for the transmit RIS-SSK system to maximize the minimum Euclidean distance among the received symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Using this approach, a performance improvement is achieved at the expense of an increased computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Moreover, in [18], adopting a similar approach, the authors proposed a joint optimization of the power allocation matrix and the phase shifts of the RIS elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' An RIS-based SM system with both the transmit and receive antenna index modulation was proposed in [19] to increase the spectral efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' However, the results show that the error rate performance of the transmit SM bits is extensively lower than that of the receive SM bits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' this is due to a reduction in the resulting channel-imprinted Euclidean distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' RIS-aided receive quadrature reflecting modulation (RIS- RQRM) proposed in [20] is another interesting approach in which QSM is applied within the receive antenna array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In this scenario, the RIS is divided into two halves, and each half targets the real or imaginary part of the signal at the two selected receive antennas in order to double the throughput;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' however, the SNR at the receiver is significantly reduced due the reduction in the number of RIS elements per targeted antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Inspired by RIS-RQRM and [19], an IRS- assisted transceiver QSM (IRS-TQSM) scheme was proposed in [21] which applies QSM at both the transmitter and the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In [22], [23], generalized SSK (GSSK) and GSM approaches have been implemented in an RIS-assisted wireless system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In both scenarios, the RIS is divided into multiple parts to target multiple antennas at the receiver;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' hence, the throughput can be increased at the expense of a decrease in the SNR at the target antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The concept of SM has also been applied within the RIS entity in [24], [25], [26] in order to transmit additional data bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' This is an exciting approach to transmit the environmental data collected by the RIS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' however, experimental results show a very large degradation in the error rate performance of the SM symbol, that is due to the similarity within the possible (noise-free) received signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In order to tackle the problem of the SNR decrease due to grouping of the RIS elements, in [27] we proposed a new paradigm, namely RIS-assisted receive quadrature space-shift keying (RIS-RQSSK) in order to simultaneously target two receive antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' An optimization problem 4 was defined to maximize the SNR of the real part of the signal at one antenna and, at the same time, of the imaginary part of the signal at the second antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The spectral efficiency of this approach is increased without any degradation in the SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' However, the throughput of the RIS-RQSSK system is limited and can only be increased by increasing the number of receive antennas which is not a viable option in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Against this background, in this paper we introduce a new RIS-assisted quadrature scheme in which, in addition to mapping the information bits independently to two indices of receive antennas, additional bits are transmitted via conventional in-phase/quadrature (IQ) modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The contributions of this paper are as follows: To improve the spectral efficiency of RIS-RQSSK while preserving its excellent perfor- mance, we propose an RIS-assisted receive quadrature spatial modulation (RIS-RQSM) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In particular, all RIS elements target two independently selected receive antennas to convey the information bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In this scenario, we introduce a novel idea to optimize the phase shifts of the RIS elements in order to not only maximize the SNR components associated to the real and imaginary parts of the signal at the receive antennas, but also to help in constructing the in-phase (I) and quadrature (Q) components of the symbol at the two separate antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Specifically, the phase of the desired IQ symbol is created by adjusting the phase shift of the RIS elements, while a positive symbol selected from a specific pre-designed PAM constellation forms the amplitude of that IQ symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' That is, in the proposed RIS-RQSM system, in contrast to conventional IQ modulation, the transmitter constructs the IQ symbol at the receiver with the aid of the RIS elements and a single radio frequency (RF) chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' We propose an energy-based greedy detector (GD) at the receiver to detect the indices of the selected antennas with low complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Then, the I and Q symbols can be detected independently by using a one-dimensional maximum likelihood (ML) detector at each of the detected antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' We also propose and design a one-tap zero-forcing (ZF) pre-equalizer which remarkably reduces the channel state information (CSI) requirement at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' This yields a significant reduction in the feedback payload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' We analyze the average bit error probability (ABEP) of the proposed RIS-RQSM system with the GD receiver and derive a closed-form upper bound which is tight, especially at high SNR values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Then, we propose an optimization problem to design an IQ modulation scheme in order to minimize the ABEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' We utilize some accurate approximations to reduce the 5 complexity of the optimization problem and derive an analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Indeed, optimizing the IQ modulation enables the system to use the GD as an alternative to the ML detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The results show that the GD in the RIS-RQSM system with optimized constellation performs considerably close to the ML detector, such that the performance gap is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Finally, we compare the bit error rate (BER) performance results with those of the most prominent benchmark scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The results show that the proposed RIS-RQSM system substantially outperforms the benchmark scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' This performance improvement improves with an increasing number of receive antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The RIS-RQSM system model is described in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In Section III, we summarize the transceiver design of the RIS-RQSSK system of [27], which forms the baseline model for the proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The transmitter and receiver structure design for the proposed RIS-RQSM system is presented in detail in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The ABEP performance of the proposed RIS-RQSM system is analyzed in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In Section VI, we formulate the optimization problem to minimize the system error rate performance and determine its analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In Section VII, we provide numerical results and comparisons with the benchmark scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Finally, Section VIII concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Notation: Boldface lower-case letters denote column vectors, and boldface upper-case letters denote matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (·)R and (·)I denote the real and imaginary components of a scalar/vector, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (·)⋆ represents the optimum value of a scalar/vector variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' E {·} and V {·}, re- spectively, denote the expectation and variance operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' N (µ, σ2) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', CN (µ, σ2)) represents the normal (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', complex normal) distribution with mean µ and variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' For a real/complex scalar s, |s| denotes the absolute value, while for a set S, |S| denotes its cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' sgn (·) represents the sign function which determines the sign of a real variable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', for x ̸= 0, it is defined as sgn (x) = {+1 if x > 0, −1 if x < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Finally, the set of complex matrices of size m × n is denoted by Cm×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' SYSTEM MODEL In this section, we describe the system model for the proposed RIS-assisted receive quadrature spatial modulation (RIS-RQSM) scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' A schematic of the RIS-RQSM system is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' We consider the RIS-AP model [5], [6], where the RIS forms part of the transmitter and reflects the incident wave emitted from a single transmit antenna which is located in the vicinity of the RIS such that the path loss and scattering of the link between the RIS and the 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' A schematic representation of RIS-assisted receive quadrature spatial modulation (RIS-RQSM) system (in RIS-RQSSK system, an RF source with constant energy is used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' transmit antenna is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The RIS is comprised of N reflecting elements whose vector of phase shifts θ ∈ CN×1 is controlled by the transmitter to convey information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Here we assume lossless reflection from the RIS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', |θi| = 1 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The receiver is equipped with Nr antennas and is placed far from the transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' We assume that the receiver can only receive the signal reflected from the RIS elements through the wireless fading channel H ∈ CNr×N, whose elements are assumed to be independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=') according to CN (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In this scenario, the input data stream is split into packets of log2 MN 2 r bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The first log2 N 2 r bits are used to independently select two receive antennas to convey the spatial symbol, and the remaining log2 M bits determine the desired IQ symbol that is selected from an M-ary QAM constellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Unlike in conventional communication systems, in the RIS-RQSM system the selected IQ symbol is not transmitted through a single-antenna transmitter, but is created at the selected receive antennas via both adjusting the RIS phase shifts and emitting a specific PAM symbol from the transmit antenna1, with a property that the I component appears on the first selected antenna, while the Q component appears on the second selected antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Thus, the RIS-RQSM scheme represents a significant generalization of the RIS-assisted receive quadrature space-shift keying (RIS-RQSSK) system described in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In RIS-RQSSK, an RF source is used to transmit a constant signal toward the RIS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' therefore, only a spatial symbol can be transmitted, while the PAM signal in RIS-RQSM enables the transmitter to transfer additional data bits via IQ modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In the next section, we will provide a brief overview of the RIS-RQSSK system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Then, the proposed RIS-RQSM system will be described in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 1It is worth mentioning that in contrast to the conventional RIS-SM system, in the RIS-RQSM the RF source at the transmitter only requires the hardware for the in-phase (I) signal component, which results in a lower hardware complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' log2 MN?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Input Data bits RIS Stream Controller H RIS RF Source/ PAM Signal Transmitter Receiver7 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' RIS-ASSISTED RECEIVE QUADRATURE SPACE-SHIFT KEYING [27] In this section, we summarize the system model of the RIS-RQSSK scheme of [27] and outline its phase shift optimization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In the RIS-RQSSK system, the transmitter is equipped with an RF source with constant energy Es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In this scenario, two receive antennas are independently selected according to two packets of log2 Nr input data bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Then, the transmitter reflects the signal to the receiver through the RIS, aiming to simultaneously maximize the SNR associated to the real part of the signal at the first selected receive antenna m, while also maximizing the SNR associated to the imaginary part of the signal at the second selected receive antenna n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' For this system, the real and imaginary components of the baseband received signal at the selected antennas m and n, respectively, are given by yR m = � Es � hR mθR − hI mθI� + nR m, (1) yI n = � Es � hR n θI + hI nθR� + nI n, (2) where hl = [hl,1, hl,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' , hl,N] is the l-th row of H, and nl ∈ C is the additive white Gaussian noise at the l-th receive antenna that is distributed according to CN (0, N0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' To maximize both SNR components associated to the real and imaginary parts of the selected receive antennas m and n, a max-min optimization problem was defined as max θR,θI min ���hR mθR − hI mθI�� , ��hR n θI + hI nθR��� (3) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' � θR i �2 + � θI i �2 = 1, for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Taking the case where the noise-free signal components in (1) and (2) are positive, the optimal values of � θR i � and � θI i � are given by θR⋆ i = λAi + (1 − λ) Bi � (λAi + (1 − λ) Bi)2 + (λCi + (1 − λ) Di)2, (4) for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' , N, and θI⋆ i = λCi + (1 − λ) Di � (λAi + (1 − λ) Bi)2 + (λCi + (1 − λ) Di)2, (5) for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' , N, where we define Ai = hR m,i, Bi = hI n,i, Ci = −hI m,i, and Di = hR n,i, (6) 8 to simplify the notation, and where, for N ≫ 1, the value of λ ∈ (0, 1) is the solution to f(λ) ≜ N � i=1 (Ai − Bi) (λAi + (1 − λ) Bi) + (Ci − Di) (λCi + (1 − λ) Di) � (λAi + (1 − λ) Bi)2 + (λCi + (1 − λ) Di)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (7) In addition, with the optimal phase shift values given in (4) and (5), the resulting SNR compo- nents have the same value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', we have hR mθR⋆ − hI mθI⋆ = hR n θI⋆ + hI nθR⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Finally, at the receiver, a simple but effective greedy detector (GD) is employed to detect the selected receive antennas without the need for any knowledge of the CSI at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The GD operates via ˆm = arg max m∈{1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=',Nr} �� yR m �2� , (8) ˆn = arg max n∈{1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=',Nr} �� yI n �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (9) The performance results have demonstrated the superiority of the RIS-RQSSK system over comparable benchmark schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' This motivates us to extend this scheme to the context of QSM, which is the subject of the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' RIS-ASSISTED RECEIVE QUADRATURE SPATIAL MODULATION In general, while the spectral efficiency of an SSK system can be increased by extending it to the corresponding quadrature SSK system, it can be further improved by implementing a conventional IQ modulation on top of the antenna index modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In the conventional receive quadrature SM (RQSM), the transmit vector can be designed to place the real and imaginary parts of the symbol separately at a specific position of the real and imaginary receive vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' On the other hand, in the RIS-RQSM scheme, the transmitter is equipped with only one antenna and therefore can only transmit one symbol in each symbol interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In addition, since the real and imaginary parts of the desired symbol needs to be separated at the receiver, the transmitter can only perform amplitude modulation through the RF source to be detectable at the receiver (as also suggested in [20] for the RIS-RQRM scheme), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', it is not feasible to transmit a QAM symbol and receive the I and Q components separately at two different receive antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' To tackle this problem, in the proposed RIS-RQSM system we introduce a new paradigm in order to construct an M-ary QAM symbol (in fact, two independent symbols from identical √ M-ary PAM constellations) at the receiver via the adjustment of both the amplitude of the RF 9 source and the phase shifts of the RIS elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Therefore, in the RIS-RQSM system the rate is R = log2 M + 2 log2 Nr bits per channel use (bpcu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In this scenario, the desired received signal components are given by yR m = � hR mθR − hI mθI� Gs + nR m, (10) yI n = � hR n θI + hI nθR� Gs + nI n, (11) where s is the transmit symbol selected from a specific positive real PAM constellation, denoted by PRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The amplitudes in PRF are the magnitudes of the complex symbols in an M-ary QAM constellation M with average energy Es, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', s = |x| where x ∈ M is the desired IQ symbol, and G > 0 is a one-tap zero-forcing (ZF) pre-equalizer to be defined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' To produce the desired M-ary QAM signal at the receiver, we modify the problem in (3) to accommodate both the index modulation and IQ modulation as max θR,θI min (YR, δYI) (12a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' YR = sgn � xR� � hR mθR − hI mθI� , (12b) YI = sgn � xI� � hR n θI + hI nθR� , (12c) � θR i �2 + � θI i �2 = 1, for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' , N, (12d) where δ > 0 is the absolute value of the ratio of the real to the imaginary part of x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', δ = ��xR/xI��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' It can be seen that this optimization problem is similar to the optimization problem for the RIS-RQSSK scenario;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' hence, it can be solved by a similar approach to that used in [27] (we omit the details for brevity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' As a result, � θR⋆ i � and � θI⋆ i � are again given by (4) and (5), respectively, and λ can also be evaluated by solving (7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' however, it is required to re-define the variables in (6) accordingly as Ai = sgn � xR� hR m,i, Bi = δ sgn � xI� hI n,i, Ci = sgn � xR� � −hI m,i � , and Di = δ sgn � xI� hR n,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (13) Note that the maximization problem forces YR and YI to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' As a result, the sign functions in (12) determine the signs of the noise-free received signal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' To eluci- date the functionality of the optimization problem above, we take symbol x = 1 − 3j as an example;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' then, we have sgn � xR = 1 � = +1 and sgn � xI = −3 � = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Therefore, we obtain YR = + � hR mθR − hI mθI� > 0 and YI = − � hR n θI + hI nθR� > 0, which indicates that the 10 real component of the constructed received symbol is positive and its imaginary component is negative, similar to the selected symbol x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' It is also worth pointing out that at the optimal point, the values involved in the minimization are equal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', with the values � θR⋆ i � and � θI⋆ i � we have Y ⋆ R = δY ⋆ I , where Y ⋆ R and Y ⋆ I are the optimum values of YR and YI produced by (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Hence, we can conclude that the phase of the desired QAM symbol is correctly designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Next, in order to explain why the PAM constellation PRF must be utilized at the transmitter, we need to ascertain how the RIS-aided channel acts for various values of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Due to the presence of random variables in (7), λ also presents a random behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' It is not easy to determine the stochastic characteristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', mean and variance) of λ from (7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' however, experimental results provide strong evidence that the mean value of λ is E {λ} = ¯λ = δ2 1+δ2 and that its variance tends to zero with an increasing number of RIS elements N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' This observation can be further used to approximate the average value of the optimum objective in (12), which is provided in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' For large values of N, the means E {Y ⋆ R} and E {Y ⋆ I } can be closely approximated by E {Y ⋆ R} ≈ � ¯λN√π 2 , E {Y ⋆ I } ≈ � 1 − ¯λN√π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Proof: The proof is provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' From Theorem 1, it can be observed that the mean value of the complex symbol created by the received signal components at the selected antennas lies on a circle with radius β = N√π 2 for any value of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Therefore, in addition to optimizing the phase angles of the RIS elements, an appropriate positive PAM symbol s ∈ PRF is required to be modulated at the RF source in order to adjust the magnitude of the received signal to accommodate the desired QAM symbol in a predefined constellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In other words, the phase of the QAM symbol is determined by the RIS elements while its amplitude is determined by the PAM symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Therefore, the transmit symbol s = |x| is required at the RF source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The symbol s is then multiplied by G at the transmitter to ensure that the gain of the link is constant at all times (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', for each symbol and for each channel realization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Therefore, we design G via G = E {Y ⋆ R} Y ⋆ R = E � hR mθR⋆ − hI mθI⋆� hR mθR⋆ − hI mθI⋆ , (14) 11 where θ⋆ is the optimum vector of phase shifts of the RIS elements associated to the desired transmit symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Note that G has a value that is specific to each symbol x and channel realization H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In fact, G can be realized as a one-tap ZF pre-equalizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' As a result, the receiver only needs to know the effective gain of the RIS-assisted wireless channel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e, the gain of the equivalent Gaussian channel which is obtained by the aid of the RIS elements, which is equal to β2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' no additional CSI is necessary for the GD detector, which significantly reduces the feedback payload of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' On the other hand, the CSI must be available at the transmitter in order to adjust the phase shifts of the RIS elements and implement the one-tap pre-equalizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Under the assumption of a large number of RIS elements, the mean values of G and G2 both tend to unity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', lim N→∞E {G} = 1 and lim N→∞E {G2} = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Proof: Here we only prove that lim N→∞E {G2} = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The convergence of the mean value of G can be derived in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The mean value of G2 is given by E � G2� = E �E2 {Y ⋆ R} (Y ⋆ R)2 � = µ2E � 1 (Y ⋆ R)2 � , where µ = E {Y ⋆ R} = √¯λ N√π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' According to the central limit theorem (CLT), Y ⋆ R is distributed as2 Y ⋆ R ∼ N (µ, σ2), where σ2 ∝ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Then, the average of G2 can be expressed as E � G2� = µ2E � 1 (Y ⋆ R)2 � = µ2 √ 2πσ2 � ∞ −∞ 1 y2e− (y−µ)2 2σ2 dy = 1 √ 2π µ2 σ2 � ∞ −∞ 1 � u + µ σ �2e− u2 2 du, where we used the change of variable u = y−µ σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Since µ σ ∝ √ N → ∞ as N → ∞, we can write lim N→∞ E � G2� = lim µ σ →∞ 1 √ 2π µ2 σ2 � ∞ −∞ 1 � u + µ σ �2e− u2 2 du = 1 √ 2π � ∞ −∞ e− u2 2 du = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Note that in practice, the number of RIS elements is large enough so that the expressions in Theorem 2 serve as accurate approximations for our design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Theorem 2 implies that the pre- equalizer G does not change the average transmit power of the system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', E � (Gs)2� = Es;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' hence the SNR is simply given by Es/N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 2This is proved in [27] for the RIS-RQSSK scenario, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', for δ = 1, however, the proof can be extended to the general case where δ = ��xR/xI�� (for brevity, these details are omitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Later (in Section V) we will show how the variance σ2 is related to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 12 Receiver Structure Similar to the RIS-RQSSK scheme, the receiver can employ a GD to detect the selected antenna indices via (8) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' After this, the receiver can demodulate the desired I and Q symbols via ˆxR = arg min xR �� yR m − βxR�2� , (15) ˆxI = arg min xI �� yI n − βxI�2� , (16) where β = N√π 2 is the effective channel coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' On the other hand, the maximum likelihood (ML) detector for the proposed RIS-RQSM system operates via ( ˆm, ˆn, ˆx) = arg min m,n,x Nr � l=1 (yl − hlθ⋆Gs)2 , (17) where we note that θ⋆ is a multi-variable function of (m, n, x), and s = |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' While the GD is CSI- free, the ML detector relies on having full CSI at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Furthermore, it can be seen that the ML detector needs to compute θ⋆ for all combinations of the selected receive antennas and then search over all possible combinations of the spatial symbols and IQ modulation symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' These facts make the ML detector significantly more complex than GD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Although the ML detector provides an optimum receiver, we will show later in Sections VI and VII that optimizing the IQ constellation, in addition to increasing the performance of the system, can also leverage the GD efficiency such that it competes very strongly with the ML detector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', the performance gap is negligible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' PERFORMANCE ANALYSIS In this section, we analyze the ABEP of the proposed RIS-RQSM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' This analysis focuses on the GD receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Here we only perform the analysis for the detection of the antenna m with active real part along with the real part of the corresponding modulated IQ symbol, xR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' due to the inherent symmetry in the expressions, it is easy to show that the ABEP expression for the detection of the antenna n with active imaginary part along with the imaginary part of 13 the corresponding modulated IQ symbol xI is identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' An upper bound on the ABEP, which is tight especially at high SNR, is given by ABEP ≤ 1 − Pe (m) √ M log2 �√ MNr � � xR � ˆxR̸=xR PEP � xR → ˆxR|m = ˆm � e � xR → ˆxR� + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='5Pe (m) , (18) where Pe (m) is the probability of erroneous detection of the selected receive antenna m, PEP � xR → ˆxR|m = ˆm � is the pairwise error probability (PEP) associated with the real part of the symbols x and ˆx conditioned on correct detection of the antenna index, and e � xR → ˆxR� is the Hamming distance between the binary representations of the real parts of the symbols x and ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Here we assume that half of the bits are in error under the condition of erroneous index detection (note that this assumption represents the worst-case scenario),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' so that Pe (m) can be written as Pe (m) = (Nr − 1) PEP (m → ˆm) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (19) where PEP (m → ˆm) is the average PEP associated with the antenna indices m and ˆm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' and is given by PEP (m → ˆm) = 1 √ M � xR PEP � m → ˆm|xR� = 1 √ M � xR∈MR 2 √ M � δ∈DxR PEP � m → ˆm|xR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' δ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (20) where MR is the set consisting of all possible values of xR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' the real component of symbols in M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' with |MR| = √ M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' and Dξ = ���� xR xI ��� |xR = ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' xI ∈ MI � with |Dξ| = √ M 2 (where MI is the set consisting of all possible values of xI);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' for instance, for a conventional 16-QAM constellation we have MR = MI = {−3, −1, 1, 3}, and for xR = {−1, 1} we have D−1 = D1 = {1, 1/3}, while for xR = {−3, 3} we have D−3 = D3 = {1, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Considering the use of GD at the receiver, the PEP associated with the selected antenna m and the detected antenna ˆm ̸= m conditioned on the selected symbol x (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', given xR and δ) is given by PEP � m → ˆm|xR, δ � = Pr �� yR m �2 < � yR ˆm �2 |xR, δ � = Pr ��� hR mθR⋆ − hI mθI⋆� Gs + nR m �2 < �� hR ˆmθR⋆ − hI ˆmθI⋆� Gs + nR ˆm �2 |xR, δ � ≈ Pr {|Z1| < |Z2|} , (21) 14 where we define Z1 ≜ � hR mθR⋆ − hI mθI⋆� |xR| √¯λ + nR m and Z2 ≜ � hR ˆmθR⋆ − hI ˆmθI⋆� |xR| √¯λ + nR ˆm, and we have used the approximations stated in Theorem 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', E {G} ≈ 1 and V {G} = E {G2} − E {G}2 ≈ 0, and we know that s = |xR| √¯λ , since ¯λ = δ2 1+δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' To calculate the probability above, the distributions of Z1, in the cases where m = n and m ̸= n, and Z2, in the cases where ˆm = n and ˆm ̸= n, are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In [27, Theorems 1-3], the distributions of the random variables (RVs) Z1 and Z2 were derived for the case of RIS-RQSSK (in that case it was shown that ¯λ = 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The distributions of Z1 and Z2 for the more general case of RIS-RQSM can be derived in a similar manner (we omit the details for brevity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In the case where m = n, with reference to the CLT, Z1 is approximately distributed according to N (µ1, σ2 1), where µ1 = N√π 2 xR and σ2 1 = N � xR�2 4−π 4 + N0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In the case where m ̸= n, the mean µ1 is given by the same expression as in the case where m = n, and experimental results provide strong evidence that the variance of Z1 is also exactly the same as in the case where m = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' On the other hand, Z2 is approximately distributed according to N (0, σ2 2), where the variance in each case of ˆm = n and ˆm ̸= n is given by 1) ˆm ̸= n: σ2 2 = ρ2 1 ≜ N � xR�2 2¯λ + N0 2 , (22) 2) ˆm = n: σ2 2 = ρ2 2 ≜ N � xR�2 2 + N0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (23) Therefore, to calculate the PEP, two different events need to be taken into consideration: i) {E1 : m, ˆm ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' , Nr} , ˆm ̸= n}, and ii) {E2 : m ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' , Nr} , ˆm = n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' It is worth pointing out that Z1 and Z2 represent the real part of the signal received at the selected antenna m (having mean µ1 ∝ N ≫ 1) and at a non-selected antenna ˆm (having mean zero), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' This is the reason that the GD is able to easily detect the index of the selected receive antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Next, we consider the instance where xR > 0 (it is clear that the PEP for xR < 0 is the same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Considering the distribution of Z1, it can be seen that µ1 σ1 ∝ √ N for relatively high SNR values, so that µ1 σ1 ≫ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' as a result, we have Z1 > 0 with extremely high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Hence, the 15 PEP can be written as PEP � m → ˆm|xR, δ � =PEP � m → ˆm|xR > 0, δ � ≈Nr − 1 Nr Pr {Z1 < |Z2| |E1} + 1 Nr Pr {Z1 < |Z2| |E2} =Nr − 1 Nr � ∞ 0 Pr {Z1 = α, |Z2| > α|E1} dα + 1 Nr � ∞ 0 Pr {Z1 = α, |Z2| > α|E2} dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The above two integrals can be evaluated in a unified manner via Ii ≜ � ∞ 0 Pr {Z1 = α, |Z2| > α|Ei} dα = 2 � ∞ 0 pz1|Ei (α) Pr {Z2 > α|Ei} dα = √ 2 σ1 √π � ∞ 0 e − 1 2 � µ1−α σ1 �2 Q � α ρi � dα, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (24) Applying the exponential approximation of the Q-function as Q (x) ≈ 1 12e− x2 2 + 1 4e− 2x2 3 from [28], Ii is approximately given by Ii ≈ √ 2 σ1 √π � ∞ 0 e − 1 2 � µ1−α σ1 �2 � 1 12e − 1 2 � α ρi �2 + 1 4e − 2 3 � α ρi �2� dα, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' After some manipulations we obtain Ii ≈ √ 2 σ1 √π � 1 12eu0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='i � ∞ 0 e − 1 2 � α−m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='i s0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='i �2 dα + 1 4eu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='i � ∞ 0 e − 1 2 � α−m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='i s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='i �2 dα � = 1 σ1 �1 6eu0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='is0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='iQ � −m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='i s0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='i � + 1 2eu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='is1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='iQ � −m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='i s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='i �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (25) where u0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='i = −1 2 µ2 1 σ2 1 + ρ2 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' s0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='i = σ1ρi � σ2 1 + ρ2 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' m0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='i = µ1ρ2 i σ2 1 + ρ2 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='i = −2 3 µ2 1 4 3σ2 1 + ρ2 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='i = σ1ρi � 4 3σ2 1 + ρ2 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='i = µ1ρ2 i 4 3σ2 1 + ρ2 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' It is easy to see that m0,i s0,i and m1,i s1,i , i = 1, 2, have relatively large values for large N, such that the approximations Q � − m0,i s0,i � ≈ Q � − m1,i s1,i � ≈ 1 are very accurate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' therefore, Ii can be written as Ii ≈ ρi 6 � σ2 1 + ρ2 i e − 1 2 µ2 1 σ2 1+ρ2 i + ρi 2 � 4 3σ2 1 + ρ2 i e − 2 3 µ2 1 4 3 σ2 1+ρ2 i , i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (26) Hence, Pe (m) is given by Pe (m) = 2 (Nr − 1) M � xR � δ �Nr − 1 Nr I1 + 1 Nr I2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (27) 16 Finally, PEP � xR → ˆxR|m = ˆm � can be expressed as PEP � xR → ˆxR|m = ˆm � = Q � � � β2 (xR − ˆxR)2 2N0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (28) Substituting (27) and (28) into (18), an accurate closed-form approximation for the ABEP of the RIS-RQSM system can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' IQ MODULATION DESIGN A significant advantage of the proposed RQSM system is that the receiver employs a simple GD which can perform symbol detection with low complexity and with a minimal CSI require- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' However, as will be shown later, if a conventional QAM constellation is used, the system shows a drop in error rate performance with higher modulation orders, since the symbols with lowest energy in the QAM constellation dominate the performance of the GD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' This phenomenon has a greater impact in the case of RIS-RQSM than in the RIS-SM system of [14], as in the former a higher average energy is received at the non-selected antennas, which results in reducing the performance of the GD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' This fact motivates us to design a new QAM constellation in order to favor the GD3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Hence, in this section we optimize the constellation to minimize the BER of the RIS-RQSM system with GD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In order to lower the complexity, we employ a number of approximations in this section to simplify the ABEP upper bound which will then serve as our objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' However, the extensive numerical results included in Table I and in the next section verify the accuracy of these approximations and show that the proposed approach is practical and yields excellent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Thanks to the symmetry in the RIS-RQSM system, the real and imaginary dimensions of the constellation can be designed separately following the same method, which simplifies the optimization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Hence, the optimization problem is defined as min MR ABEPub (29) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' √ M � i=1 � xR�2 ≤ √ MEs 2 , where ABEPub is the approximate upper bound on the ABEP expressed in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' It is trivial to observe that the signal constellation should be symmetric about the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Therefore, we 3Both the ML detector and the GD perform better with the proposed constellation, but the GD benefits more significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 17 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Normalized PAM constellation design for the RIS-RQSM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' define the one-dimensional “normalized” √ M-PAM constellation for the real and imaginary dimensions according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 2, such that the minimum-energy symbol has distance d0 √Es from the origin, while the distance between the i-th and (i+1)-th symbols is denoted by di √Es, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' , √ M 2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Due to the symmetry about the origin, there exist √ M/2 parameters that need to be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' For example, in 2-PAM, there is only one parameter d0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' it is clear that in this case d0 = 1/ √ 2, so that this optimization framework is not necessary in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In a 4-PAM constellation there are two parameters d0 and d1 that should be optimized such that d0 is increased and d1 is decreased with respect to the values for conventional PAM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', the two “inner” symbols are moved further away from the origin and the two “outer” symbols are moved towards the origin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' this adjustment of the constellation points provides a balance between the spatial domain symbol error probability and the IQ modulation domain symbol error probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The expression for ABEPub in (18) is a relatively complex function of the parameters {di} due to the summation over all symbols in calculating Pe (m) and in calculating PEP � xR → ˆxR|m = ˆm � associated with all of these distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Hence, to simplify the solution for the optimization problem in (29) we adopt some accurate approximations for evaluating the upper bound on the ABEP that are valid at high SNR and with large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' It is well-known that at high SNR values the IQ modulation domain bit error probability (BEP) is dominated by the pairs of constellation points separated by the minimum Euclidean distance, and it is also clear that the minimum-energy symbols control the BEP in the spatial domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Hence, considering Gray coding for the constellation, an approximate upper bound on the ABEP is given by ABEPub ≈ 4 √ M log2 �√ MNr � √ M 2 −1 � i=1 Q � � � β2Esd2 i 2N0 � � + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='5 ˜Pe (m) , (30) 18 where ˜Pe (m) is the corresponding approximate value of Pe (m), given by ˜Pe (m) = 4 (Nr − 1) M � δ∈Dd0 √Es PEP � m → ˆm|xR = d0 � Es, δ � , (31) where Dd0 √Es = � 1, d0 d0+d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' , d0 d0+d1+···+d √ M 2 −1 � , and we use the fact that ˜Pe (m) ≪ 1, hence 1 − ˜Pe (m) ≈ 1 (note that the optimization function increases the distance between two inner symbols, so that in (30), we did not consider the distance between the pair of inner symbols as the minimum distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Then, the optimization problem can be updated as min {di} ABEPub in (30) (32) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' √ M 2 −1 � i=0 � i � j=0 dj �2 ≤ √ M 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Solving the above optimization problem is not a straightforward task and requires the use of exhaustive search methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' However, standard lattice constellation structures, such as QAM or PAM, suggest that equal distances between adjacent pairs of symbols admit a very simple approach which provides a near-optimal solution in terms of the symbol error rate performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Hence, in the following, we assume that the distances between “positive” adjacent symbols are equal (it is worth recalling that there is a symmetry about the origin, hence the distances between negative adjacent symbols are also equal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Special case where d1 = d2 = · · · = d √ M 2 −1 In this case, the problem consists of optimizing the two variables d0 and d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Hence, the optimization problem reduces to min {d0,d1} 4 √ M log2 �√ MNr �M ′Q � � � β2Esd2 1 2N0 � � + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='5 ˜Pe (m) , (33) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 2d2 0 + M ′ (2M ′ + 1) 3 d2 1 + 2M ′d0d1 ≤ 1, where we define M ′ = √ M 2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' From the inequality constraint, d1 can be obtained as a function of d0 (here we force equality in the constraint above to maximize the achievable SNR at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' It will be shown later in this section that equality indeed holds at the optimum point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Then, by performing a grid search over variable d0, we can find the minimum value of the ABEPub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' However, taking the equal positive distance into account, it is more valuable to find an analytical solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' this is the subject of the remainder of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 19 Analytical approach - asymptotic analysis: In order to find an efficient analytical solution for the optimization problem, we analyze the distributions of Z1 and Z2 in more detail in order to obtain a more tractable approximate expression for ABEPub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' We see that the variance of Z2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', the average received energy of the signal on a non-selected receive antenna) in the event E1 increases with decreasing ¯λ = δ2 1+δ2, or equivalently, with decreasing δ = ��� xR xI ���;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' in other words, ρ2 1 in (22) is maximized when δ is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' There are two consequences of this fact: first, the BEP related to the spatial domain is dominated by those symbols bearing the minimum energy in the real part while their corresponding imaginary parts have the maximum energy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', the PEP associated with δmin = min|xR| max|xI| = d0 d0+M′d1 dominates (31);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' secondly, comparing the two events E1 and E2, the event E2 has a minor impact on the value of PEP � m → ˆm|xR, δ � , as the variance of Z2 in the event E1 is significantly greater than that in the event E2 due to the appearance of ¯λ in the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In summary, considering the above comments, the PEP associated to δmin dominates and the event E2 can be eliminated from the PEP analysis, therefore ˜Pe (m) can be approximated as ˜Pe (m) ≈ 4 (Nr − 1) M PEP � m → ˆm|xR = d0 � Es, δ = δmin � ≈ 4 (Nr − 1)2 MNr I1 (µ1, ρ1, σ1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (34) In addition, by substituting ¯λ = δ2 min 1+δ2 min into (22) and performing some minor algebraic manipu- lations, the variance of Z2 in the event E1 can be expressed as ρ2 1 = NEs 2 � d2 0 + (d0 + M ′d1)2� + N0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Note that ¯E ≜ d2 0 + (d0 + M ′d1)2 is the sum of the energies associated with the symbols with minimum and maximum distance from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' It is clear that 1 ≤ ¯E < ϵM, where equality holds for M = 16, and ϵM is defined as the total energy of the inner and outer symbols in the conventional √ M-PAM constellation (since the conventional constellation is the worst-case scenario, ¯E can be upper bounded by ϵM), so that we obtain ϵM = 3(M−2 √ M+2) 2(M−1) (note that the average energy of the PAM constellation is 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Therefore, we can write NEs 2 + N0 2 ≤ ρ2 1 < NEs 2 ϵM + N0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 20 In addition, it is easy to prove that (34) is monotonically increasing with respect to ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Hence, ˜Pe (m) can be expressed as ˜Pe (m) ≈ ˜Pe (m) ��� ρ2 1= NEs 2 + N0 2 , M = 16, ˜Pe (m) ≲ ˜Pe (m) ��� ρ2 1= NEs 2 ϵM+ N0 2 , M > 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Finally, from the formula σ2 1 = N � xR�2 4−π 4 + N0 2 applied to the minimum energy symbol xR = d0 √Es and considering the fact that d2 0 ≪ 1, the variance of Z1 can be approximated as σ2 1 ≈ N0 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Therefore, after some manipulations we obtain ˜Pe (m) as ˜Pe (m) ≈2 (Nr − 1)2 MNr � 1 3 � NEs + N0 NEs + 2N0 e− πN2Esd2 0 4NEs+8N0 + � NEs + N0 NEs + 7 3N0 e− πN2Esd2 0 3NEs+7N0 � , M = 16, ˜Pe (m) ≲2 (Nr − 1)2 MNr � 1 3 � NEsϵM + N0 NEsϵM + 2N0 e − πN2Esd2 0 4NEsϵM +8N0 + � NEsϵM + N0 NEsϵM + 7 3N0 e − πN2Esd2 0 3NEsϵM +7N0 � , M > 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Also applying the exponential approximation of the Q-function in (33), the optimization problem becomes min {d0,d1} ABEPub ≈ a0e−b0d2 0 + a1e−b1d2 0 + M ′a2 � 1 12e−b2d2 1 + 1 4e− 4 3 b2d2 1 � (35) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 2d2 0 + M ′ (2M ′ + 1) 3 d2 1 + 2M ′d0d1 ≤ 1, where we define a0 = (Nr − 1)2 3MNr � NEsϵM + N0 NEsϵM + 2N0 , b0 = πN 2Es 4NEsϵM + 8N0 , a1 = (Nr − 1)2 MNr � NEsϵM + N0 NEsϵM + 7 3N0 , b1 = πN 2Es 3NEsϵM + 7N0 , a2 = 4 √ M log2 �√ MNr �, b2 = πN 2Es 16N0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The problem in (35) is not a convex optimization problem, as the objective function is not convex in the domain of d0, d1 ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' However, it is easy to see that (35) satisfies the convexity 4Here we are assuming that N is sufficiently large so the SNR range 4−π 2 NEsd2 0 N0 ≪ 1 is of interest, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', the BER is extremely low outside of this SNR range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 21 condition ∇2ABEPub ≥ 0 when d0 ≥ 1 √2bi, i = 0, 1, and d1 ≥ 1 √2b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' For sufficiently high values of N2Es N0 (note that N ≫ 1), it can be concluded that {b0, b1, b2} are sufficiently large such that the optimized {d0, d1} lie in the convex region of the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' For such {b0, b1, b2}, the problem is convex and can be solved using the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The Karush-Kuhn-Tucker (KKT) [29] conditions associated to the above problem hold and are given by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' f1 (d⋆ 0, d⋆ 1) ≤ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' ν⋆ ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' ν⋆f1 (d⋆ 0, d⋆ 1) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' − 2a0b0d⋆ 0e−b0d⋆2 0 − 2a1b1d⋆ 0e−b1d⋆2 0 + ν⋆ (4d⋆ 0 + 2M ′d⋆ 1) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' − 1 6M ′a2b2d⋆ 1e−b2d⋆2 1 − 2 3M ′a2b2d⋆ 1e− 4 3 b2d⋆2 1 + ν⋆ �2M ′ (2M ′ + 1) 3 d⋆ 1 + 2M ′d⋆ 0 � = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' where ν is the Lagrange multiplier associated with the inequality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' From condition 3, we see that ν⋆ = 0 or f1 (d⋆ 0, d⋆ 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' However, if ν⋆ = 0, from conditions 4 and 5 we obtain d⋆ 0 = d⋆ 1 = +∞, where clearly contradicts condition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Therefore, we have 2d⋆2 0 + M ′ (2M ′ + 1) 3 d⋆2 1 + 2M ′d⋆ 0d⋆ 1 − 1 = 0, which yields d⋆ 0 = −2M ′d⋆ 1 + � 4M ′2d⋆2 1 − 8 � M′(2M′+1) 3 d⋆2 1 − 1 � 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (36) Then, from conditions 4 and 5, we obtain ν⋆ = a0b0d⋆2 0 e−b0d⋆2 0 + a1b1d⋆2 0 e−b1d⋆2 0 + 1 12M ′a2b2d⋆2 1 e−b2d⋆2 1 + 1 3M ′a2b2d⋆2 1 e− 4 3 b2d⋆2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (37) Substituting for ν⋆ from (37) and subsequently for d⋆ 0 from (36) into condition 5, the optimization problem reduces to a single-variable equation in d⋆ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' This equation does not admit a closed-form analytical solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' however it is easy to solve numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' We conclude this section by providing a numerical example in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In this table, we compare the optimal {di} obtained by an exhaustive search to minimize the ABEP in (18) with the corresponding values with equal positive distances obtained via the proposed analytical approach, where N = 256, Nr = 4 and M = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' It can be seen that positive distances {di}, i > 0, obtained via exhaustive search are almost equal, and that these values become more similar with increasing SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In addition, the ABEP values acquired by using the optimal values from the proposed analytical approach are quite comparable to the equivalent ABEP obtained by optimal values of the grid search, which serves as a proof that the assumptions we made to offer a straightforward analytical solution to the optimization problem were indeed accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 22 TABLE I COMPARISON BETWEEN OPTIMAL {di} VALUES OBTAINED VIA MINIMIZING (18) BY GRID SEARCH AND THE CORRESPONDING VALUES OBTAINED BY THE ANALYTICAL APPROACH OF (35), WHERE N = 256, Nr = 4 AND M = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Minimized ABEP based on (18) using grid search Minimized ABEP by using analytical approach of (35) SNR (dB) d0 d1 d2 d3 ABEP d0 d1 ABEP 23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='2609 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='257 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='272 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='97 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='2481 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='2632 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='62 × 10−4 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='2695 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='248 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='253 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='262 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='46 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='2661 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='2543 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='67 × 10−5 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='2890 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='243 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='248 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='90 × 10−6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='2891 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='2426 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='96 × 10−6 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='3169 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='227 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='228 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='232 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='71 × 10−8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='3179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='2278 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='82 × 10−8 VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' NUMERICAL RESULTS In this section, we demonstrate the error rate performance of the proposed RIS-RQSM system via numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' First, we investigate the performance of the proposed RIS-RQSM system using conventional QAM constellations and provide comparisons with corresponding systems using QAM constellations that are optimized based on the approach proposed in Sec- tion VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Next, we compare the results obtained by the optimized constellations with the error rate performance of the most prominent recently proposed RIS-SM [14] system, which serves as the benchmark scheme for the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 3 shows the BER performance of the proposed RIS-RQSM system with N = 256 for the cases of Nr = 4 and Nr = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In this figure, we also compare the performance of the RIS- RQSM system using conventional 16-QAM modulation with that of the system implementing our optimized 16-QAM constellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The curves demonstrate the effectiveness of the proposed constellation design method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' it can be observed that optimizing the design of the constellation significantly enhances the performance of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The proposed constellation for RIS-RQSM provides approximately 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='2 dB and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='8 dB improvement over the conventional constellation in systems with Nr = 4 and Nr = 8, respectively, at a BER of 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' We also compare the performance of the GD with that of the ML detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' We see that there is a very large gap between the performance of the GD and ML detector in the case of the conventional constellation, while the performance of the GD in the system using the optimized constellation is considerably close to that of the ML detector such that the performance gap is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In order to observe the effect of optimizing the constellation in a system with higher-order modulation, we present the BER performance of the RIS-RQSM system with 64-QAM in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Here, we see that in 23 34 32 30 28 26 24 22 Es/N0 10-6 10-5 10-4 10-3 10-2 10-1 100 BER RIS-RQSM - Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' - GD (Conventional 16-QAM) RIS-RQSM - Ana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (Conventional 16-QAM) RIS-RQSM - Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' - ML (Conventional 16-QAM) RIS-RQSM - Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' - GD (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 16-QAM) RIS-RQSM - Ana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 16-QAM) RIS-RQSM - Asym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' - Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (35) (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 16-QAM) RIS-RQSM - Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' - ML (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 16-QAM) (a) 34 32 30 28 26 24 22 20 Es/N0 10-6 10-5 10-4 10-3 10-2 10-1 100 BER RIS-RQSM - Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' - GD (Conventional 16-QAM) RIS-RQSM - Ana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (Conventional 16-QAM) RIS-RQSM - Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' - ML (Conventional 16-QAM) RIS-RQSM - Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' - GD (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 16-QAM) RIS-RQSM - Ana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 16-QAM) RIS-RQSM - Asym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' - Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (35) (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 16-QAM) RIS-RQSM - Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' - ML (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 16-QAM) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Analytical and simulation BER results of the proposed RIS-RQSM system with and without optimized constellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Here M = 16, N = 256, and (a) Nr = 4 (R = 8 bpcu), (b) Nr = 8 (R = 10 bpcu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' systems with regular QAM constellations, an error floor occurs with the GD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' This is due to the fact that with critical symbols, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', minimum-energy symbols, ¯λ can attain a very small value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' hence, non-selected antennas can have a relatively high average received energy compared to the selected antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' However, we see that optimizing the constellation eliminates this error floor and substantially improves the error rate performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Similar to systems with 16-QAM constellation, the performance of the GD is very close to that of ML detector with optimized constellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In fact, here the GD becomes feasible only with the optimized 64-QAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 3 and 4, we also present the analytical ABEP performance of each system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' For systems with conventional QAM constellations, we evaluate and plot the analytical ABEP upper bounds based on (18);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' we see that upper bound curves are quite tight and validate the accuracy of the analytical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' For systems with optimized constellation we also plot the asymptotic result in (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' These curves show that the utilized approximations in Section VI are completely valid and accurate, especially at high SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Next, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 5, we compare the BER performance of the proposed RIS-RQSM system with that of the benchmark scheme, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', RIS-SM, in systems with N = 256 and Nr = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 5(a) shows 24 30 28 26 24 22 20 18 16 Es/N0 10-6 10-5 10-4 10-3 10-2 10-1 100 BER RIS-RQSM - Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' - GD (Conventional 64-QAM) RIS-RQSM - Ana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (Conventional 64-QAM) RIS-RQSM - Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' - ML (Conventional 64-QAM) RIS-RQSM - Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' - GD (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 64-QAM) RIS-RQSM - Ana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 64-QAM) RIS-RQSM - Asym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' - Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (35) (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 64-QAM) RIS-RQSM - Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' - ML (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 64-QAM) (a) 30 28 26 24 22 20 18 16 14 12 Es/N0 10-6 10-5 10-4 10-3 10-2 10-1 100 BER RIS-RQSM - Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' - GD (Conventional 64-QAM) RIS-RQSM - Ana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (Conventional 64-QAM) RIS-RQSM - Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' - ML (Conventional 64-QAM) RIS-RQSM - Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' - GD (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 64-QAM) RIS-RQSM - Ana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 64-QAM) RIS-RQSM - Asym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' - Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (35) (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 64-QAM) RIS-RQSM - Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' - ML (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 64-QAM) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Analytical and simulation BER results of the proposed RIS-RQSM system with and without optimized constellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Here M = 64, N = 256, and (a) Nr = 4 (R = 10 bpcu), (b) Nr = 8 (R = 12 bpcu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' the performance of the RIS-RQSM and RIS-SM systems where the bit rate is R = 8 bpcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Hence, the proposed RIS-RQSM system uses 16-QAM modulation, while the RIS-SM system uses 64-QAM modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The constellation used in the proposed RIS-RQSM system is optimized to achieve the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 5(b) presents the performance results in systems with R = 10 bpcu, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', where RIS-RQSM and RIS-SM apply 64-QAM and 256-QAM, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The results show that the proposed RIS-RQSM system substantially outperforms the benchmark scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' This is mainly due to the fact that the RIS-SM system needs to employ a higher-order modulation technique in order to compensate the additional bits transmitted by the quadrature index modulation in the proposed RIS-RQSM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Hence, the superiority over the benchmark scheme increases by increasing number of receive antennas, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In this figure, we provide comparisons between the BER performance of the RIS-RQSM and RIS-SM systems where N = 256 and Nr = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' As expected, the superiority over the RIS-SM system considerably increases in a system with larger number of receive antennas, as a higher modulation order is required for the RIS-SM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The proposed RIS-RQSM system achieves approximately 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='3 dB and 7 dB performance improvement over the RIS-SM system for systems with Nr = 4 25 34 32 30 28 26 24 22 20 Es/N0 10-6 10-5 10-4 10-3 10-2 10-1 100 BER RIS-RQSM - GD (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 16-QAM) RIS-SM - GD (64-QAM) (a) 30 28 26 24 22 20 18 16 14 Es/N0 10-6 10-5 10-4 10-3 10-2 10-1 100 BER RIS-RQSM - GD (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 64-QAM) RIS-SM - GD (256-QAM) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Comparison of the BER performance of the proposed RIS-RQSM system with that of RIS-SM system for N = 256, Nr = 4, and (a) R = 8 bpcu, (b) R = 10 bpcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' and Nr = 8, respectively, at a BER of 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' It is worth pointing out that the receiver in the proposed RIS-RQSM system requires minimal CSI due to the use of the pre-equalizer G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' this CSI consists only of the average gain of the effective channel, which is simply a function of the number of RIS elements, as shown in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' CONCLUSION The RIS-assisted receive quadrature spatial modulation (RIS-RQSM) system was proposed in this paper as a general approach to RIS-assisted receive SM with excellent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The proposed system increases the spectral efficiency by implementing both quadrature spatial modulation and IQ modulation, while maintaining the signal quality at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In the proposed RIS-RQSM system, the phase shifts of the RIS elements are designed to construct an IQ symbol at the receiver;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' this enables the system to transmit two separate PAM symbols in the presence of the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' We introduced a one-tap pre-equalizer to allow the proposed low-complexity GD to detect the symbols with minimum CSI requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Analytical results and numerical simulations both verify the excellent performance of the system and extensively demonstrate 26 34 32 30 28 26 24 22 20 18 Es/N0 10-6 10-5 10-4 10-3 10-2 10-1 100 BER RIS-RQSM - GD (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 16-QAM) RIS-SM - GD (128-QAM) (a) 30 28 26 24 22 20 18 16 14 12 Es/N0 10-6 10-5 10-4 10-3 10-2 10-1 100 BER RIS-RQSM - GD (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 64-QAM) RIS-SM - GD (512-QAM) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Comparison of the BER performance of the proposed RIS-RQSM system with that of RIS-SM system for N = 256, Nr = 8, and (a) R = 10 bpcu, (b) R = 12 bpcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' its superiority over comparable benchmark schemes in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The many advantages of the RIS-RQSM system makes it a viable candidate for next-generation wireless communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' APPENDIX A PROOF OF THEOREM 1 Here we analyze the average of Y ⋆ R = sgn � xR� � hR mθR⋆ − hI mθI⋆� = N � i=1 λA2 i + λC2 i + (1 − λ) AiBi + (1 − λ) CiDi � (λAi + (1 − λ) Bi)2 + (λCi + (1 − λ) Di)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' As stated before, for large values of N, we have V {λ} ≈ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' therefore, we replace λ by ¯λ in calculating the average of Y ⋆ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' this yields E {Y ⋆ R} ≊ E � � � N � i=1 ¯λA2 i + ¯λC2 i + � 1 − ¯λ � AiBi + � 1 − ¯λ � CiDi ��¯λAi + � 1 − ¯λ � Bi �2 + �¯λCi + � 1 − ¯λ � Di �2 � � � = NE � � � ¯λA2 i + ¯λC2 i + � 1 − ¯λ � AiBi + � 1 − ¯λ � CiDi ��¯λAi + � 1 − ¯λ � Bi �2 + �¯λCi + � 1 − ¯λ � Di �2 � � � , 27 where we used the fact that each of the summands has an identical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' In the following, we evaluate the average of the terms in the above summation individually and we omit the index i to simplify the notation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' hence we define W1 ≜ ¯λ A2 √ Z , W2 ≜ ¯λ C2 √ Z , W3 ≜ � 1 − ¯λ � AB √ Z , W4 ≜ � 1 − ¯λ � CD √ Z , where Z ≜ �¯λA + � 1 − ¯λ � B �2 + �¯λC + � 1 − ¯λ � D �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' According to the law of total expectation, the expected value of W1 can be expressed as E {W1} = EA � EW1|A {W1|A} � = ¯λEA � A2EZ|A � Z− 1 2|A �� , (38) where EZ|A � Z− 1 2|A � is the inverse-fractional moment of Z where A is given, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=', where A is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' For a given A, using ¯λ = δ2 1+δ2 we have �¯λA + � 1 − ¯λ � B � ∼ N � ¯λA, ¯λ(1−¯λ) 2 � , and �¯λC + � 1 − ¯λ � D � ∼ N � 0, ¯λ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Hence, the random variable (RV) (Z|A) is the sum of two independent chi-square RVs each having one degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' The inverse-fractional moment of (Z|A) can be computed by using the following equation [30] EZ|A � Z−c|A � = 1 Γ (c) � ∞ 0 sc−1EZ|A � e−sZ|A � ds, (39) where EZ|A � e−sZ|A � = Ls (fZ (Z|A)) is the Laplace transform (LT) of fZ (Z|A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' We know that the LT of the probability density function (PDF) of the sum of independent RVs is equal to the product of the LTs of their individual PDFs, and that the LT of the PDF of an RV X = �n i=1 X2 i with Xi ∼ N (µi, σ2) is given by Ls (fX(X)) = � 1 1 + 2σ2s � n 2 exp � −µ2s 1 + 2σ2s � , (40) where µ2 = �n i=1 µ2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Hence, the LT of fZ (Z|A) is calculated as Ls (fZ (Z|A)) = � 1 1 + ¯λs � 1 2 � 1 1 + ¯λ � 1 − ¯λ � s � 1 2 exp � −¯λ2A2s 1 + ¯λ � 1 − ¯λ � s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (41) Then, (38) can be written as E {W1} = ¯λ Γ2 � 1 2 � � ∞ 0 s 1 2 −1 � 1 1 + ¯λs � 1 2 � 1 1 + ¯λ � 1 − ¯λ � s � 1 2 × �� ∞ −∞ A2 exp � −A2 1 + ¯λs 1 + ¯λ � 1 − ¯λ � s � dA � ds, 28 where we used the fact that fA (A) = 1 Γ( 1 2) exp (−A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Since � ∞ −∞ x2 exp � − x2 2σ2 � dx = Γ � 1 2 � σ2√ 2σ2, we have � ∞ −∞ A2 exp � −A2 1 + ¯λs 1 + ¯λ � 1 − ¯λ � s � dA = Γ � 1 2 � 2 � 1 + ¯λ � 1 − ¯λ � s 1 + ¯λs � 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' It follows that E {W1} = ¯λ 1 2Γ � 1 2 � � ∞ 0 s 1 2 −11 + ¯λ � 1 − ¯λ � s � 1 + ¯λs �2 ds = ¯λ � 1 − ¯λ � 1 2Γ � 1 2 � �� ∞ 0 s 1 2 −1 � 1 + ¯λs �−1 ds + ¯λ 1 − ¯λ � ∞ 0 s 1 2 −1 � 1 + ¯λs �−2 ds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Recalling the definition of the type-2 beta function B (α, β) = � ∞ 0 tα−1 (1+t)α+β dt = Γ(α)Γ(β) Γ(α+β) , after some minor manipulations we obtain E {W1} = ¯λ 1 2 � 1 − ¯λ � 1 2Γ � 1 2 � � Γ � 1 2 � Γ � 1 2 � Γ (1) + ¯λ 1 − ¯λ Γ � 1 2 � Γ � 3 2 � Γ (2) � = √π 4 � 2¯λ 1 2 − ¯λ 3 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' By symmetry it is clear that E {W2} = E {W1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Next we determine E {W3} = E �� 1 − ¯λ � AB √ Z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Using the law of total expectation, we can write E {W3} = � 1 − ¯λ � EA � AEB � BEZ|(A,B) � Z− 1 2|(A, B) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Given constant (A, B), we have Ls (fZ (Z|(A, B))) = � 1 1 + ¯λs � 1 2 exp � − �¯λA + � 1 − ¯λ � B �2 s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (42) Using (39), we have E {W3} = � 1 − ¯λ � EA � AEB � B Γ � 1 2 � � ∞ 0 s 1 2 −1 � 1 1 + ¯λs � 1 2 exp � − �¯λA + � 1 − ¯λ � B �2 s � ds �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Then, using fA (A) = 1 Γ( 1 2) exp (−A2) and fB (B) = (1−¯λ) 1 2 ¯λ 1 2 Γ( 1 2) exp � − 1−¯λ ¯λ B2� , after some alge- braic manipulations we obtain E {W3} = � 1 − ¯λ � 3 2 ¯λ 1 2 1 Γ3 � 1 2 � � ∞ 0 s 1 2 −1 � 1 1 + ¯λs � 1 2 × � ���� � ∞ −∞ A exp � −A2 1 + ¯λs 1 + ¯λ � 1 − ¯λ � s � � � � � � � ∞ −∞ B exp � � � � �− � B + ¯λ2As 1+¯λ(1−¯λ)s �2 ¯λ/(1−¯λ) (1+¯λ(1−¯λ)s) � � � � � dB � � � � � dA � ���� ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' (43) 29 The inner integral over B can be evaluated as � ∞ −∞ B exp � � � � �− � B + ¯λ2As 1+¯λ(1−¯λ)s �2 ¯λ 1−¯λ 1+¯λ(1−¯λ)s � � � � � dB = −Γ �1 2 � ¯λ 5 2 As � 1 − ¯λ � 1 2 � 1 + ¯λ � 1 − ¯λ � s � 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Substituting this into (43), the average of W3 is given by E {W3} = ¯λ2 � 1 − ¯λ � −1 2Γ � 1 2 � � ∞ 0 s 3 2 −1 1 � 1 + ¯λs �2ds = ¯λ 1 2 � 1 − ¯λ � −1 2Γ � 1 2 �B �3 2, 1 2 � = − √π 4 ¯λ 1 2 � 1 − ¯λ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Also, by symmetry we have E{W4} = E{W3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} +page_content=' Finally, the average of Y ⋆ R is given by E {Y ⋆ R} ≈ 2N (E {W1} + E {W3}) = ¯λ 1 2 N√π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfzvm4/content/2301.00707v1.pdf'} 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[cond-mat.str-el] 6 Jan 2023 +Hydrodynamic approach to many-body systems: exact conservation laws +Boris N. Narozhnya,b +aInstitut f¨ur Theorie der Kondensierten Materie, Karlsruher Institut f¨ur Technologie, 76131 Karlsruhe, Germany +bNational Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 115409 Moscow, Russia +Abstract +In this paper I present a pedagogical derivation of continuity equations manifesting exact conservation +laws in an interacting electronic system based on the nonequilibrium Keldysh technique. The purpose of +this exercise is to lay the groundwork for extending the hydrodynamic approach to electronic transport to +strongly correlated systems where the quasiparticle approximation and Boltzmann kinetic theory fail. +Keywords: +Electronic hydrodynamics, graphene, viscosity, quantum conductivity, kinetic theory +Electronic hydrodynamics has evolved into a fast paced field with multiple experimental and theoretical +groups working to uncover observable signatures of hydrodynamic behavior of electronic systems [1–4] with +the primary focus on transport properties. In a “realistic” case of a weakly disordered conductor, hydrody- +namic equations encompass the conventional linear-response transport theory describing both the uniform +Ohmic current in macroscopic (“infinite”) systems and the nonuniform viscous flows of charge and energy +in constricted (“mesoscopic”) geometries [5–7]. +Similarly to the traditional transport theory, hydrodynamic equations can be derived from the kinetic +(Boltzmann) equation describing a system of weakly interacting quasiparticles [8]. +At the semiclassical +level, one often relies on the “scattering time approximation” (typically used to describe Drude-like transport +phenomena [9]) to simplify the collision integral. This approach can be further extended to include quantum +interference phenomena [10, 11] yielding the so-called “quantum corrections” to the leading semiclassical +behavior heralding the onset of low-temperature localization [12]. These additional features “correct” the +conductivity of the system, while the macroscopic description of the current flow remains Ohmic. +The +low-temperature Ohmic resistance is still determined by disorder, although electron-electron interaction +does affect the quantum corrections [11, 13]. +In contrast, the hydrodynamic behavior is dominated by +electron-electron interaction determining the viscosity coefficient [14]. +Systems dominated by electron-electron interactions, e.g., strongly correlated systems, “strange metals”, +etc., remain a formidable challenge for several decades. In simple terms, the difficulty lies in the failure of +the quasiparticle approach [15]. Moreover, even if quasiparticles could be defined the semiclassical kinetic +approach may fail in multi-component systems with non-Abelian degrees of freedom (e.g., spin or isospin) +due to the purely quantum nature of the latter [16]. It is then highly desirable to develop a macroscopic +theory of electronic transport in strongly interacting systems without reliance on the quasiparticle paradigm +and semiclassical approximation which is the ultimate motivation for this work. +In classical systems such a macroscopic theory is hydrodynamics. Indeed, the Navier-Stokes equation +[17–20] equally well describes water and air flows, while the Boltzmann kinetic theory allowing one to derive +this equation [21] is only justified for a dilute gas. The apparent universality of the hydrodynamic theory can +be attributed to two points: (i) the long time, long distance behavior is often assumed to be independent +of the details of short-distance scattering processes, and (ii) the conservation laws that are the basis of +hydrodynamics are equally applicable to all systems with the same symmetries. +Email address: boris.narozhny@kit.edu (Boris N. Narozhny) +Preprint submitted to Annals of Physics +January 9, 2023 + +Generalizing the hydrodynamic approach to systems beyond conventional fluids, one may consider it +in a broader sense meaning of a long-wavelength theory of small perturbations relative to an equilibrium +state [9]. This way both the conventional hydrodynamics and diffusion could be discussed on equal footing. +The difference between the two behaviors is momentum conservation which is assumed in hydrodynamics +and is broken in diffusive systems. +In solids, electronic momentum is never truly conserved (it can be +lost due to scattering off impurities, phonons, etc.). However, in ultra-pure materials it may be possible +to find an intermediate temperature range where electron-electron interaction is the dominant scattering +process [2–4] as reflected by the hierarchy of typical time scales τee ≪ τdis, τe−ph, ... (using self-evident +notations). Then it could be reasonable to neglect processes that do not conserve momentum, at least +as the “0-th” approximation describing the “ideal fluid” by means of macroscopic (differential) equations +essentially generalizing [2, 8] the Euler’s equation [20]. The non-conserving processes (electron-impurity or +electron-phonon coupling) can then be included perturbatively [2–4]. +The general problem of fermions with momentum-conserving interaction has been one of the most popular +in many body physics. Most generally, the system is described by a Hamiltonian comprising the one-particle +(“free”) and “interaction” parts +�H = �H0 + �Hint. +(1a) +The one-particle contribution can typically be separated into two contributions +�H0 = +� +ddr1 ˆψ†(r1) � +H(0) +1 +ˆψ(r1), +� +H(0) +1 += �K1 + U1, +(1b) +with �K1 representing the “kinetic energy” (possibly including multiple bands, spin-orbit interaction, etc.; +without loss of generality all additional quantum numbers are suppressed throughout this paper) and U1 +being the one-particle potential [the subscript “1” refers to the set of quantum numbers of the field ˆψ†(r1)]. +The interaction term is assumed to be translationally invariant (hence, momentum-conserving) +�Hint = 1 +2 +� +ddr1ddr2 ˆψ†(r1) ˆψ†(r2)V (r1−r2) ˆψ(r2) ˆψ(r1). +(1c) +The general problem represented by the Hamiltonian (1) cannot be solved exactly. However, the conservation +laws of particle number (charge), energy, and momentum are exact. +In this paper, I explore the emergence of exact conservation laws in the by now standard field-theoretic +approach to nonequilibrium systems, the Keldysh technique [10, 22]. This issue has been already extensively +discussed in literature on general many-body theory [23–25] and nuclear physics [26–28] establishing the +integral relations expressing the global symmetries of the system. The present paper explores a somewhat +different angle. I am interested in “deriving” the local continuity equations manifesting the conservation laws +that are the starting point of the hydrodynamic approach (e.g., conservation of the particle number, energy, +and momentum). The point is to express the macroscopic currents and densities in the most general form +(i.e., in terms of the exact quantities involved in the diagrammatic technique including Green’s functions, +self-energies, etc) allowing for a straightforward generalization to specific condensed matter system including +multiple bands and spin-orbit interaction. The requirement of the “exact” validity of the continuity equations +leads to general relations involving the self-energies and Green’s functions. These relations are satisfied by +the exact functions and serve as constraints on their approximate forms. At the same time, these relations +provide a blueprint for including additional, non-conserving terms to the Hamiltonian (e.g., electron-impurity +or electron-phonon scattering) leading to weak decay contributions to the resulting macroscopic equations +[2, 29, 30]. Finally, I compare the obtained expressions with those appearing as a result of the approximations +leading to the kinetic equation (semiclassical or quantum) as an intermediate step. The ultimate goal of this +work is to establish a hydrodynamic framework that does not rely on the quasiparticle paradigm (avoiding +the kinetic equation and specifically the concept of the semiclassical distribution function) and hence could +be useful for describing systems where quasiparticles are overdamped or altogether absent. +2 + +1. Equations of motion +In this paper I consider the conservation laws using the nonequilibrium Keldysh technique following +Ref. [25]. The notations for the Keldysh Green’s functions and general relations between them are summa- +rized in Appendix A, for a more detailed account of the Keldysh technique see Refs. [10, 22]. +All Green’s functions are defined in terms of the Heisenberg field operators and therefore it is important +to review the equations of motion governing their dynamics. Starting with the standard quantum-mechanical +definition of the time derivative, +i ∂ +∂t +ˆψ = +� +ˆψ, �H +� +. +(2a) +one finds [23] +i ∂ +∂t +ˆψ(r, t) − � +H(0) ˆψ(r, t) = +� +ddr′ ˆψ†(r′, t)V (r−r′) ˆψ(r′, t) ˆψ(r, t). +(2b) +Multiplying Eq. (2b) by i ˆψ† from the left and taking the thermodynamic average, one arrives at the Dyson’s +equation for the “12” component of the Keldysh Green’s function, see Eq. (A.5), but with the right-hand +side (RHS) expressed explicitly in terms of the interaction potential +i ∂ +∂t1 +G12 +1,2− �H(0) +1 G12 +1,2 = i +� +ddr3 +� +ˆψ†(r2, t2) ˆψ†(r3, t1)V (r1−r3) ˆψ(r3, t1) ˆψ(r1, t1) +� +. +(3a) +Here and throughout the paper I use the short-hand notation: G12 +1,2 = G12(r1, t1; r2, t2). +In contrast, the Dyson’s equation is expressed in terms of the self-energy, see Eq. (A.5) +i ∂ +∂t1 +G12 +1,2− � +H(0) +1 G12 +1,2 = +� +d3 +� +Σ11 +1,3G12 +3,2 − Σ12 +1,3G22 +3,2 +� +, +(3b) +where d3 = ddr3dt3. Alternatively [using Eqs. (A.10c) and the similar relation for the self-energy] +i ∂ +∂t1 +G12 +1,2− � +H(0) +1 G12 +1,2 = +� +d3 Ξ(1, 2; 3), +Ξ(1, 2; 3) = ΣR +1,3G12 +3,2+Σ12 +1,3GA +3,2. +(3c) +Comparing Eqs. (3a) and (3c), one arrives at the identity +i +� +ddr3 +� +ˆψ†(r2, t2) ˆψ†(r3, t1)V (r1−r3) ˆψ(r3, t1) ˆψ(r1, t1) +� += +� +d3 Ξ(1, 2; 3), +(3d) +relating the two-particle Green’s function in the RHS of Eq. (3a) to the single-particle quantities in the RHS +of Eq. (3c). +In what follows, I will also use the equation of motion for ˆψ† +i ∂ +∂t +ˆψ†(r, t) + ˆψ†(r, t) � +H(0),† = − +� +ddr′ ˆψ†(r, t)V (r−r′) ˆψ†(r′, t) ˆψ(r′, t). +(4) +Here � +H(0),† is the conjugate operator with any gradients acting on the coordinate dependence to the left. +Multiplying this equation by i ˆψ from the right one finds +i ∂ +∂t2 +G12 +1,2+ � +H(0),† +2 +G12 +1,2 = −i +� +ddr3 +� +ˆψ†(r2, t2) ˆψ†(r3, t2)V (r1−r3) ˆψ(r3, t2) ˆψ(r1, t1) +� +. +(5a) +On the other hand, conjugating the Dyson’s equation (3c) [see Eqs. (A.4), (A.9), (A.12), (A.13), and (A.14)] +and changing the variables 1 ↔ 2 one arrives at +i ∂ +∂t2 +G12 +1,2+ � +H(0),† +2 +G12 +1,2 = +� +d3 Ξ∗(2, 1; 3) = − +� +d3 +� +GR +1,3Σ12 +3,2 + G12 +1,3ΣA +3,2 +� +. +(5b) +Comparing Eqs. (5a) and (5b) yields the conjugate form of the identity (3d). +3 + +2. Continuity equation +Consider now the usual continuity equation +∂n +∂t + ∇·j = 0. +(6) +The continuity equation itself is well-known and does not need another derivation. This equation repre- +sents gauge invariance (i.e., the particle number conservation or charge conservation), the symmetry that +is typically assumed to be exact for all condensed matter systems (apart from the special case of supercon- +ductivity where this issue is more subtle, see Ref. [10]) and hence is independent of the particular form of +the Hamiltonian. The purpose of this section is to introduce notations for the particle number density, n, +and the current, j, and establish the constraint imposed on the self-energy by gauge invariance. +2.1. Continuity equation at the operator level +The particle number can be defined in the standard way using electronic field operators +ˆn(r, t) = ˆψ†(r, t) ˆψ(r, t), +n(r, t) = ⟨ˆn(r, t)⟩ , +(7a) +or the Green’s function [see Eq. (A.3a)] +n1 = −iG12 +1,1. +(7b) +The two definitions allow for two different derivations of the continuity equation starting either with the +equations of motion or the Dyson’s equations. +At the operator level, the continuity equation is just the equation of motion for the density operator that +can be obtained by combining the two equations of motion (2b) and (4) +∂ +∂t ˆn(r, t) = −i ˆψ†(r, t) �K ˆψ(r, t) + i +� +�K ˆψ(r, t) +�† ˆψ(r, t) = −∇·ˆj(r, t), +(8) +where the last step defines the current operator. The interaction potential does not appear in Eq. (8) due +to the standard commutation relations. +2.2. Continuity equation and the Keldysh Green’s functions +Combining Eq. (3c) with Eq. (5b) yields a Kadanoff-Baym equation [24] +� +i ∂ +∂t1 +− � +H(0) +1 ++ i ∂ +∂t2 ++ � +H(0),† +2 +� +G12 +1,2 = +� +d3 [ Ξ(1, 2; 3) + Ξ∗(2, 1; 3)] +(9) += +� +d3 +� +ΣR +1,3G12 +3,2 + Σ12 +1,3GA +3,2 − GR +1,3Σ12 +3,2 − G12 +1,3ΣA +3,2 +� +. +Comparing the time derivative terms in Eq. (9) to the definition of the particle density, see Eq. (7), one +notices the relation +� +i ∂ +∂t1 ++i ∂ +∂t2 +� +G12 +1,2 +���� +2→1 += −∂n1 +∂t1 +. +Now it becomes clear that in the limit 2 → 1 Eq. (9) can be written in the form of the conventional continuity +equation (6) where the divergence of the current is determined by the single-particle Hamiltonian +∇1·j1 = +� +�K1 − �K† +2 +� +G12 +1,2 +��� +2→1 , +(10) +while the self-energy satisfies the condition +� +d3 +� +ΣR +1,3G12 +3,1 + Σ12 +1,3GA +3,1 − GR +1,3Σ12 +3,1 − G12 +1,3ΣA +3,1 +� += 0. +(11) +The latter identity is satisfied by the exact self-energy and Green’s function and hence represents a constraint +on any approximate expressions. In fact, the identity (11) can be derived independently, following Refs. [24, +25], where the idea of “conserving approximations” was first suggested. +4 + +2.3. Conserving approximations +The need for a “conserving approximation” arises from the apparent arbitrariness of the diagrammatic +perturbation theory. Indeed, it may not be clear “a priori” that a given approximation for the self-energy +satisfies the exact conservation laws of the system (given that this is certainly not the case for at least some +individual diagrams; of course, any practitioner of the diagrammatic perturbation theory would make sure +that the calculation does not violate gauge invariance, although this might involve certain technical difficul- +ties – the point of a “conserving approximation” is that the conservation laws are satisfied automatically +without any need for special care). Baym suggested the self-consistent procedure where one starts with the +Luttinger-Ward functional +Φ[ ˇG] = +� +ln +� +�SC +�� +sk = +�� +�SC +� +− 1 +� +sk , +Φ∗ = Φ, +(12) +where the subscript “sk” indicates that only skeleton diagrams (i.e. diagrams without self-energy insertions +and with all Green’s functions replaced by full Green’s functions) are to be retained. Moreover, the logarithm +amounts to retaining only the connected diagrams. The important property of the functional is that the +exact self-energy can be obtained by the variation +Σij +1,2 = −(−1)i+j δΦ +δGji +2,1 +. +(13) +The “self-consistent” perturbation theory comprises an expansion of the functional Φ and a solution for ˇG +and ˇΣ using the Dyson’s equation (A.11) and Eq. (13). The latter step is self-consistent in the sense that +Eq. (A.11) determines the Green’s function in terms of the self-energy and Eq. (13) the other way around. +The key point of the self-consistent approach is that the resulting theory satisfies exact conservation laws +without any further approximation no matter how many diagrams are retained in the expansion of the +functional Φ [25]. The resulting approximations are known as “Φ-derivable”. While there can be many such +approximations (depending on the order to which Φ is expanded), all of them respect the conservation laws. +2.4. Φ-derivable approximations and gauge invariance +Applying a symmetry transformation to the exact Green’s function leads to a variation of the functional. +To the leading order, the variation δΦ is given by +δΦ = − +� +d1d2 Tr ˇτ3 ˇΣ1,2ˇτ3δ ˇG2,1, +(14) +which vanishes if the transformation corresponds to a true symmetry of the Hamiltonian. +Consider a gauge transformation (cf. the same argument of Ref. [25] but using the Matsubara Green’s +functions) which without loss of generality can be confined to the upper branch of the Keldysh contour +ˇG2,1 → eiˇχ2 ˇG2,1e−iˇχ1, +ˇχ = +�χ +0 +0 +0 +� +. +(15a) +Expanding to the leading order in χ and taking into account the matrix structure, one finds +δ ˇG2,1 = iχ2 +1+ˇτ3 +2 +ˇG2,1 − iχ1 ˇG2,1 +1+ˇτ3 +2 +(15b) +Substituting this expression into Eq. (14) and requiring that the functional is invariant under the gauge +transformation (since it is composed of closed particle lines) one finds (using the cyclic property of the trace +in each term separately) +� +d1χ1 +� +d2 Tr1+ˇτ3 +2 +� +ˇτ3 ˇΣ1,2ˇτ3 ˇG2,1 − ˇG1,2ˇτ3 ˇΣ2,1ˇτ3 +� += 0. +(16a) +5 + +Evaluating the trace and taking into account arbitrariness of χ1 one arrives at the identity +� +d2 +� +Σ11 +1,2G11 +2,1 − Σ12 +1,2G21 +2,1 − G11 +1,2Σ11 +2,1 + G12 +1,2Σ21 +2,1 +� += 0, +(16b) +which is a manifestation of gauge invariance. +Now, substituting Eqs. (A.10c) into Eq. (16b), one finds +Σ11 +1,2G11 +2,1 − Σ12 +1,2G21 +2,1 = ΣR +1,2G12 +2,1 + Σ12 +1,2GA +2,1 + ΣR +1,2GR +2,1. +(16c) +Comparing Eqs. (11) and (16b) I now conclude that in the limit 2 → 1 the integral in the right-hand side of +Eq. (9) takes the form +� +d3 +� +ΣR +1,3GR +3,1 − GR +1,3ΣR +3,1 +� += 0. +(16d) +This expression vanishes for the following reasons: (i) the self-energy has the same causality structure as +the Green’s function [22], therefore for any t1 ̸= t3 the product ΣR +1,3GR +3,1 vanishes; (ii) in the limit t3 → t1 +the retarded Green’s function has the form +GR(r1, r3; t1 = t3 + 0) = −iδ(r1 − r3), +(17) +so that even if the self-energy had a non-zero diagonal value ΣR(1, 1) it would be the same in both terms and +hence canceled in the difference. As a result, the identity (11) follows from Eq. (16b). The above argument +represents a proof of Eq. (11) and, by extension, confirms that the continuity equation (6) is consistent with +the Keldysh approach (exactly or within a Φ-derivable approximation). +2.5. Summary +To summarize this section, the continuity equation (6) manifesting particle number conservation follows +from the Heisenberg equations of motion due to the “density-density” interaction, see Eq. (1c). It is fully +preserved in the microscopic Keldysh approach (either while using the exact Green’s functions or within +a Φ-derivable – or any other conserving – approximation). At the same time, the continuity equation is +satisfied within the kinetic theory (that can be derived from the same microscopic theory using a series +of approximations, see Ref. [10] and Section 5) as well. Given that particle number conservation is the +exact symmetry, the continuity equation is valid independently of any (correctly applied) approximation. +Specifically, the arguments presented here do not rely on either the quasiparticle and semiclassical approx- +imations typically assumed to derive the kinetic equation or otherwise describe conventional metals and +semiconductors. +3. Momentum conservation +Consider now translational invariance. This is the crucial symmetry in conventional hydrodynamics, +where the Navier-Stokes equation [20] is a direct consequence of momentum conservation. +Since the Hamiltonian is explicitly translationally invariant, one should be able to express momentum +conservation by means of the continuity equation for the momentum density, g, similarly to Eq. (6) +∂gα +∂t + ∇βτβα = 0, +(18) +without any additional derivation (here ταβ is the momentum flux – or stress – tensor). However, there are +well documented difficulties along the way [23–25], primarily for long-ranged interactions. +Within the kinetic theory, one may derive the Navier-Stokes equation by multiplying the kinetic equation +by momentum and integrating over all single-particle states [2, 8, 21]. The equivalent procedure at the +microscopic level is to apply the momentum operator to the Dyson equations (3c) and (5b) followed by the +evaluating their sum in the limit 2 → 1. In the resulting equation [similar to Eq. (9)], the time derivative +terms combine into the time derivative of momentum density, while the rest should comprise the spatial +derivatives yielding the divergence of the momentum flux tensor and the “collision integral” terms vanishing +in the limit 2 → 1, essentially repeating the above calculation leading to the continuity equation (6). +6 + +3.1. Momentum density +The “momentum operator” mentioned above is the differential operator allowing one to define the mo- +mentum density. In quantum field theory, however, the definition of such operator is not unique [31, 32]. +The reason is that only the total momentum of the system is well-defined. While it can be expressed as +a volume integral over the momentum density, that integral remains unchanged if any contribution repre- +senting a surface term is added to the integrand. This freedom can be used to bring the stress tensor to a +symmetric form typically assumed in calculations of the viscosity tensor [33–36]. Taking into account the +possible additional terms (important for non rotationally invariant systems [36]), the most general form of +the momentum density can be written as [cf. Eqs. (7)] +g(r, t) = 1 +2 +� +ˆψ†(r, t)ˆp ˆψ(r, t) + +� +ˆp† ˆψ†(r, t) +� +ˆψ(r, t) +� +, +(19a) +where ˆp is the momentum operator appropriate for the system in question. Alternatively, the momentum +density can be expressed in terms of the Green’s function Eq. (A.3a) +g1 = − i +2 +� +ˆp1 + ˆp† +2 +� +G12 +1,2 +��� +2→1. +(19b) +In conventional systems with the parabolic spectrum the momentum operator has the usual form ˆp = −i∇ +and the resulting momentum density (19b) is proportional to the particle number current (10). This pro- +portionality does not hold in general (e.g., in the case of Dirac fermions in graphene [2, 4, 8]). +3.2. Global momentum conservation +Following the above procedure, I now apply the operator −(i/2)[ˆp1 + ˆp† +2] to the Dyson equations (3c) +and (5b), sum up the results, and take the limit 2 → 1. This yields +∂gi +1 +∂t1 ++ n1 +iˆpi +1U1−iˆp†,i +2 U2 +2 +����� +2→1 ++ ∇j +1τ ji +0 (1) = Ci +1, +(20a) +where +∇j +1τ ji +0 (1) = ˆpi +1+ ˆp†,i +2 +2 +� +�K1 − �K† +2 +� +G12 +1,2 +����� +2→1 +, +(20b) +and [see Eqs. (3) and (5b)] +C1 = − ˆp1+ˆp† +2 +2 +� +d3 +� +Ξ(1, 2; 3) + Ξ∗(2, 1; 3) +������ +2→1 +. +(20c) +Conservation of total momentum can be demonstrated by integrating Eq. (20a) over the system volume and +requiring that the volume integral of the RHS vanishes +∂ +∂t1 +� +ddr1 g1 + +� +ddr1 n1 +iˆp1U1−iˆp† +2U2 +2 +����� +2→1 += 0, +� +ddr1 C1 = 0. +(21) +The integral nature of the conservation law is consistent with the fact that it is the total momentum +of the system that is well defined and conserved. A local momentum flux might not be well defined if +interactions are long ranged. +7 + +3.3. Φ-derivable approximations and translational invariance +The last equality in Eq. (21) represents a constraint on the approximate self energies and Green’s func- +tions and can be proven similarly to Eq. (11). Consider a coordinate shift with the operator +�TR = eiR·ˆp. +(22) +Confining the shift to the upper branch of the contour in analogy with Eq. (15), the transformation of the +Green’s function can be expressed as +ˇG2,1 → 1+ˇτ3 +2 +�TR ˇG2,1 �T † +R +1+ˇτ3 +2 +. +(23a) +To the leading order in R, the variation of the Green’s function is given by +δ ˇG2,1 =i1+ˇτ3 +2 +R(t2)ˆp2 ˇG2,1 − iR(t1)ˆp† +1 ˇG2,1 +1+ˇτ3 +2 +. +(23b) +Substituting this expression into Eq. (14) and requiring that the functional is invariant under the shift of +coordinates (since only the system boundaries are shifted) one finds (using the cyclic property of the trace +in each term separately) +δΦ = i +� +d1R(t1) ˆp1+ˆp† +3 +2 +� +d2 +� +Σ11 +1,2G11 +2,3 − Σ12 +1,2G21 +2,3 − G11 +1,2Σ11 +2,3 + G12 +1,2Σ21 +2,3 +���� +3→1 = 0, +such that due to arbitrariness of R one arrives at +ˆp1+ˆp† +3 +2 +� +d2 +� +Σ11 +1,2G11 +2,3 − Σ12 +1,2G21 +2,3 − G11 +1,2Σ11 +2,3 + G12 +1,2Σ21 +2,3 +���� +3→1 = 0, +(24) +which is a manifestation of translational invariance. +The expression in the square brackets in Eq. (24) coincides with that in Eq. (16b), while the corresponding +combination of the self-energies and Green’s functions in Eq. (20c) is the same as in Eq. (11). Given that the +momentum operator does not affect time dependence and hence causality, one can use the same argument +as in Section 2 concluding that the terms containing products of two retarded (or two advanced) functions +vanish in the limit 2 → 1. Thus, the identity (24) proves the second identity in Eq. (21) and consequently, +the integral relation manifesting the momentum conservation. +3.4. Momentum conservation in the local approximation +A local (“differential”) version of the momentum conservation law can not be established without some +degree of approximation [23–25]. Within the kinetic approach, the local continuity equation for the momen- +tum density, see Eq. (18) is obtained by a straightforward integration of the kinetic equation multiplied by +momentum. This is possible because the distribution function, the central quantity the kinetic theory, is +already local. In contrast, “integrating” the Kadanoff-Baym equation (9) leads to Eq. (20a). This is not a +continuity equation since the quantity C in the RHS is not a divergence, see also Refs. [23, 25]. +The Kadanoff-Baym equation (9) can also be derived using the alternative form of the Dyson’s equations, +i.e., Eqs. (3a) and (5a). This leads to the expression for the quantity C in terms of the two-particle Green’s +function [one could also use the identity (3d) in Eq. (20c)] +C1 = − i +2 +� +ddr3 +� +ˆp1V (r1−r3)−ˆp† +2V (r2−r3) +���� +2→1 +� +ˆψ† +H(r1, t1) ˆψ† +H(r3, t1) ˆψH(r3, t1) ˆψH(r1, t1) +� +. +(25) +This form immediately proves that the volume integral of C vanishes, see Eq. (21): indeed, the integrand is +antisymmetric, which reflects the third Newton’s law [23]. +8 + +The quantity C is not a divergence since the interaction potential is nonlocal. However, for short-range +interactions (e.g., for sufficiently screened Coulomb potential in solids) it is possible to construct an effective +local interaction stress tensor by integrating C over a large enough volume [23] +� +V +ddr1C1 = − +� +V +ddr1 +� +ddr3 c+(r1, r3), +c+(r1, r3) = −c+(r3, r1), +(26a) +where c+(r1, r3) is the integrand in C, which can be expressed in terms of single-particle functions due to +the identity (3d) +c+(r1, r3) = ˆp1+ˆp† +2 +2 +� +dt3 +� +Ξ(1, 2; 3) + Ξ∗(2, 1; 3) +������ +2→1 +. +(26b) +Since c+(r1, r3) is antisymmetric, the integral over any identical region in r1 and r3 vanishes. Thus the +coordinate r3 in Eq. (26a) is effectively outside of the volume V , while r1 is inside V and the relative +coordinate, r13 = r1 − r3, is restricted by the interaction range. +Changing the integration variables in +Eq. (26a) to r13 and r3, the integral takes the form [where Vr1 indicates that the integration volume is V +in terms of the original variable r1 and R13 = (r1+r3)/2] +� +V +ddr1 +� +ddr3 c+(r1, r3) = +� +Vr1 +ddr13ddr3 c+(R13+r13/2, R13−r13/2). +Martin and Schwinger [23] introduced the hypothesis of “local uniformity” where expectation values of field +operators within a physically small region depend only on the relative coordinate. Then for a fixed r13 the +integration over r3 is restricted to a shell of thickness n·r13, where n is a unit vector normal to the surface +of V . Now the approximation of Ref. [23] can be asserted by setting R13 ≈ r3 ≈ const in that shell. The +integration measure over r3 can be replaced by −n·r13dS3 with the integral covering half the volume in r13 +−1 +2 +� +ddr13 +� +dS3(n·r13)c+(r3+r13/2, r3−r13/2) = −1 +2 +� +dS3ni +� +ddr13ri +13c+(r3+r13/2, r3−r13/2). +Now one can invoke the Euler’s theorem and approximate the quantity C by a divergence +Ci(r) ≈ −∇jτ ji +int, +τ ji +int = −1 +2 +� +ddr13rj +13ci ++(r+r13/2, r−r13/2). +(27) +The quantity τ ji +int represents the interaction contribution to the stress-tensor. +Formally, one can arrive at Eq. (27) by changing the integration variable in Eq. (20c) to r13, expressing +the integrand as c+(R13+r13/2, R13−r13/2), and expanding R13 = r1 − r13/2 in r13 such that +c+(R13+r13/2, R13−r13/2) ≈ c+(r1+r13/2, r1−r13/2) − 1 +2r13·∇1c+(r1+r13/2, r1−r13/2), +where the contribution of the first term vanishes due to the asymmetry. +3.5. Summary +To summarize this section, the continuity equation for the momentum density (18) can be derived from +the microscopic Keldysh approach (within a Φ-derivable approximation) by assuming “local uniformity” +within physically small volumes [23] (or the gradient approximation, see below). For short-ranged inter- +actions this assumption is clearly compatible with the hydrodynamic approach where one is interested in +long-wavelength properties of macroscopic observables (ideally, orders of magnitude longer than any micro- +scopic scale [20, 21]). On the other hand, for truly long-ranged interactions a local stress tensor cannot be +constructed leaving only the integral manifestation of momentum conservation. +9 + +4. Energy conservation +The Hamiltonian (1) does not explicitly depend on time and hence is invariant under time translations. +Hence, energy is conserved and one should be able to express this fact by means of a continuity equation +∂nE +∂t + ∇·jE = −j·∇U, +(28) +where nE is the energy density and jE is the energy current. Strictly speaking, Eq. (28) is only exact for +the case of local interactions, similarly to the case of momentum conservation. +Collisions between neutral molecules described by the traditional kinetic theory are typically assumed +to be local and hence it is not surprising that Eq. (28) can be straightforwardly obtained by integrating the +kinetic equation multiplied by energy [21]. At the same time, the kinetic theory description of plasma (i.e., a +system – or gas – of charged particles) is only approximate and can be justified at high enough temperatures +exceeding the average interaction energy or at high enough densities where the average interparticle distance +is much smaller than the typical screening radius [21]. The separation between the two length scales in the +problem allows one to distinguish between “collisions” – i.e. the short-distance scattering processes leading to +equilibration and hence contributing to the collision integral – and collective phenomena (involving distances +of the order of the screening length) that form the macroscopic fields responsible for the effective Lorentz +force appearing in the left-hand side (LHS) of the kinetic equation. +4.1. Global energy conservation +At the microscopic level, one can account for energy conservation by considering time translations. +Differentiating the Dyson equations (3c) and (5b) with respect to time and evaluating their difference, one +finds +� +i ∂ +∂t2 +� +i ∂ +∂t1 +− � +H(0) +1 +� +− i ∂ +∂t1 +� +i ∂ +∂t2 ++ � +H(0),† +2 +�� +G12 +1,2 = +(29) += i +� +d3 ∂ +∂t2 +� +ΣR +1,3G12 +3,2+Σ12 +1,3GA +3,2 +� ++ i +� +d3 ∂ +∂t1 +� +GR +1,3Σ12 +3,2+G12 +1,3ΣA +3,2 +� +. +Consider now the limit 2 → 1. In the LHS of Eq. (29) one immediately notices +� +− ∂ +∂t2 +∂ +∂t1 ++ ∂ +∂t1 +∂ +∂t2 +� +G12 +1,2 +���� +2→1 += 0, +and +� +−i ∂ +∂t2 +U1 − i ∂ +∂t1 +U2 +� +G12 +1,2 +���� +2→1 += U1 +∂n1 +∂t1 += −U1∇1·j1, +(30) +where the continuity equation (6) was used in the last step. The remaining term in the LHS contains exactly +the same operators as for a non-interacting system and hence can be brought to a form +� +−i ∂ +∂t2 +�K1 − i ∂ +∂t1 +�K† +2 +� +G12 +1,2 +���� +2→1 += ∇1·J 1 + ∂E(1) +1 +∂t1 +. +(31) +Here E(1) is the “one-particle” (“kinetic”) energy density and J is the corresponding current. +Specific +expressions for these quantities are determined by the quasiparticle spectrum and are easier to establish in +each particular case. Note, that the current J does contain an explicit interaction contribution, see below. +Using Eq. (31), I may re-write Eq. (29) in the limit 2 → 1 as +∂E(1) +1 +∂t1 ++ ∇1·J 1 − U1∇1·j1 = Υ1, +(32a) +10 + +where the RHS is denoted by +Υ1 = i +� ∂ +∂t2 +� +d3 Ξ(1, 2; 3)− ∂ +∂t1 +� +d3 Ξ∗(2, 1; 3) +����� +2→1 +. +(32b) +This quantity is related to the interaction potential, see Eq. (3d). Therefore, it is tempting to associate it with +the interaction energy density. This way the volume integral of Eq. (32a) yields the integral manifestation +of energy conservation +∂ +∂t +�� +ddr E(1) + Eint +� += − +� +ddr j·∇U. +(33) +4.2. Φ-derivable approximation and time translations +To prove the relation between the RHS of Eq. (29) and the interaction energy,consider a change of the +time variable on the “upper” branch of the Keldysh contour +t → θ(t) = t + ϕ(t). +(34) +Together with the change of variable, the Green’s function acquires an additional factor [25] +ˇG2,1 → ˇU(t2) ˇG2,1 ˇU(t1), +ˇU = +� +(∂θ/∂t)1/4 +0 +0 +1 +� +, +(35) +which is needed to cancel the Jacobian in Eq. (A.1b) so that the functional (12) remains invariant. Indeed, +every time integration in any diagram for the functional (12) involves four Green’s function and hence to +cancel the Jacobian, each of them has to be corrected by a factor of (∂θ/∂t)1/4 [25]. Expanding now in the +small variation, one finds similarly to Eq. (15) +δGij +2,1 = δi1 +�ϕ′(t2) +4 ++ϕ(t2) ∂ +∂t2 +� +Gij +2,1 + δj1 +�ϕ′(t1) +4 ++ϕ(t1) ∂ +∂t1 +� +Gij +2,1. +(36) +The expression (36) should now be substituted into the variation (14) of the functional Φ. This yields +δΦ = −1 +4 +� +d1 ˙ϕ(t1) +� +d2 +� +Σ11 +1,2G11 +2,1−Σ12 +1,2G21 +2,1+G11 +1,2Σ11 +2,1−G12 +1,2Σ21 +2,1 +� +(37) ++ +� +d1 ϕ(t1) +� +d2 +� +Σ11 +1,2 +∂ +∂t1 +G11 +2,1−Σ12 +1,2 +∂ +∂t1 +G21 +2,1 + Σ11 +2,1 +∂ +∂t1 +G11 +1,2−Σ21 +2,1 +∂ +∂t1 +G12 +1,2 +� +. +Since the time translation leaves the functional invariant, the above expression has to be set to zero, δΦ = 0. +The first term in Eq. (37) can be re-written with the help of Eqs. (3d), (16c), and (16d) as +−i +� +dt1 ˙ϕ(t1)Eint, +where +Eint = 1 +2 +� +ddr1 +� +ddr′� +ˆψ† +H(r1, t1) ˆψ† +H(r′, t1)V (r1−r′) ˆψH(r′, t1) ˆψH(r1, t1) +� +. +(38) +is the interaction energy of the system. Again, using Eqs. (16c) and (16d) one can identify the second term +in Eq. (37) with the quantity Υ1, see Eq. (32b). This way the variation of Φ takes the form +δΦ = −i +� +dt1 ˙ϕ(t1)Eint(t1) − i +� +d1 ϕ(t1)Υ1 = 0. +(39) +Integrating the first term by parts and using arbitrariness of ϕ(t1), one arrives at the identity +∂ +∂tEint + +� +ddrΥ = 0, +(40) +which is a manifestation of energy conservation and the justification for Eq. (33). +11 + +4.3. Energy conservation in the local approximation +I now construct the local expression of energy conservation using the same “local uniformity” approxi- +mation as in the case of momentum conservation. Starting with Eq. (32a), one should notice that its LHS +contains the time derivative of the one-particle energy density only. To arrive at the total energy density +one has to separate the “interaction energy” density from the RHS. To this end, let me re-write Υ with the +help of Eq. (3d) as +Υ1 = − +∂ +∂t2 +� +ddr3 +� +ˆψ†(r2, t2) ˆψ†(r3, t1)V (r1−r3) ˆψ(r3, t1) ˆψ(r1, t1) +����� +2→1 +(41) +− +∂ +∂t1 +� +ddr3 +� +ˆψ†(r2, t2) ˆψ†(r3, t2)V (r1−r3) ˆψ(r3, t2) ˆψ(r1, t1) +����� +2→1 +. +Singling out the interaction energy, see Eq. (38), this can be brought to the following form +Υ1 = −1 +2 +∂ +∂t1 +� +ddr3 +� +ˆψ†(r1, t1) ˆψ†(r3, t1)V (r1−r3) ˆψ(r3, t1) ˆψ(r1, t1) +� +(42) ++1 +2 +� +ddr3V (r1−r3) +� +ˆψ†(r1, t1) ∂ +∂t1 +� +ˆψ†(r3, t1) ˆψ(r3, t1) +� +ˆψ(r1, t1) +� +−1 +2 +� +ddr3V (r1−r3) +� +ˆψ†(r3, t1) ∂ +∂t1 +� +ˆψ†(r1, t1) ˆψ(r1, t1) +� +ˆψ(r3, t1) +� +. +Introducing the interaction energy density Eint in the first term and using the operator relation (8) in the +last two, I find +Υ1 = −∂Eint +∂t1 +− 1 +2 +� +ddr3V (r1−r3) +� +ˆψ†(r1, t1)∇3ˆj(r3, t1) ˆψ(r1, t1) +� ++1 +2 +� +ddr3V (r1−r3) +� +ˆψ†(r3, t1)∇1ˆj(r1, t1) ˆψ(r3, t1) +� +. +(43) +Substituting the above expression into Eq. (32a) one can now combine the time derivative terms into the +derivative of the total energy density, nE = E(1) + Eint. +To proceed further one need to specify the gradient term in Eq. (32a). Without loss of generality, I can +write the “kinetic” part of the Hamiltonian (1) as +�K = ∇·� +K, +(44) +where the vector � +K carries the dependence on all additional quantum numbers. The rational for Eq. (44) +is the following. Kinetic energy describes motion and hence the corresponding operator must not commute +with the coordinate. Therefore, �K must be a functional of the gradient operator, ∇, and moreover the +formal Taylor series in ∇ must start with the first power. Thus each term in the series is proportional to ∇ +leading to Eq. (44). Note that � +K may further depend on ∇ as in the case of the usual parabolic spectrum +where � +K = −∇/(2m), while for Dirac fermions in graphene � +K = −ivgσ, where vg is the Fermi velocity and +σ is the vector of the Pauli matrices [37]. +Using Eq. (44), one can arrive at the “explicit” expressions for E(1) and J by evaluating the LHS of +Eq. (31). These read +E(1) +1 += −i ∇1·� +K2G12 +1,2 +��� +2→1 , +J 1 = i +� ∂ +∂t1 +� +K2 + ∂ +∂t2 +� +K1 +� +G12 +1,2 +���� +2→1 +. +(45) +Taking into account the explicit form of G12 +1,2 in terms of the field operators, Eq. (A.3a) and using the +equations of motion (2b) and (4) to remove the time derivative from J ǫ, one finds following expression +J 1 = j(1) +ǫ (1) + U1j + +� +ddr3V (r1−r3) +� +ˆψ†(r3, t1)∇1ˆj(r1, t1) ˆψ(r3, t1) +� +, +(46) +12 + +where the one-particle contribution to the energy current is given by +j(1) +ǫ (1) = +� +(∇2·� +K2)� +K1 − (∇1·� +K1)� +K2 +� +G12 +1,2 +��� +2→1 . +(47) +Substituting all of the above results in Eq. (32a) yields +∂nE +∂t + ∇·j(1) +ǫ += −j·∇U + ˜Υ, +(48) +where +˜Υ1 = −1 +2 +� +ddr3V (r1−r3) +� +ˆψ†(r1, t1)∇3ˆj(r3, t1) ˆψ(r1, t1) +� ++1 +2 +� +ddr3V (r1−r3) +� +ˆψ†(r3, t1)∇1ˆj(r1, t1) ˆψ(r3, t1) +� +−∇1 +� +ddr3V (r1−r3) +� +ˆψ†(r3, t1)ˆj(r1, t1) ˆψ(r3, t1) +� +, +which can be re-written in a more symmetric form +˜Υ1 = −1 +2∇1 +� +ddr3V (r1−r3) +� +ˆψ†(r3, t1)ˆj(r1, t1) ˆψ(r3, t1) +� +−1 +2 +� +ddr3 [∇1V (r1−r3)] +� +ˆψ†(r1, t1)ˆj(r3, t1) ˆψ(r1, t1) +� +−1 +2 +� +ddr3 [∇1V (r1−r3)] +� +ˆψ†(r3, t1)ˆj(r1, t1) ˆψ(r3, t1) +� +. +(49) +The last two terms in Eq. (49) are not gradients, but can be brought to a gradient form in the “local +uniformity” approximation [23] used above in the case of momentum conservation. Repeating the steps +leading to Eq. (27), I arrive at the continuity equation for the energy density (28), where the energy current +is defined as +jE = j(1) +ǫ ++ 1 +2 +� +ddr3V (r1−r3) +� +ˆψ†(r3, t1)ˆj(r1, t1) ˆψ(r3, t1) +� +(50) +−1 +4 +� +ddr13r13 +� +∇i +13V (r1−r3) +� �� +ˆψ†(r1+r13/2, t1)ˆji(r1−r13/2, t1) ˆψ(r1+r13/2, t1) +� ++ +� +ˆψ†(r1−r13/2, t1)ˆji(r1+r13/2, t1) ˆψ(r1−r13/2, t1) +�� +. +4.4. Summary +In this section I have derived the continuity equation for the energy density (28) from the microscopic +Keldysh approach by assuming “local uniformity” within small physical volumes [23]. In that sense, the +above considerations closely follow the arguments presented in the previous section. The difference here is +that the resulting expression for the energy current is expressed in terms of a two-particle Green’s function. +While it is possible to express the energy current in terms of the derivatives of the self-energy and the single- +particle Green’s function using Eq. (3d), the resulting expression is somewhat cumbersome. At the same +time, the obtained expression (50) is not immediately related to the kinetic equation since the RHS of the +integrated Kadanoff-Baym equation, the quantity Υ, has to be split into the time derivative of the interaction +energy and a contribution towards the energy current. This point will be discussed in the subsequent section. +13 + +5. Kinetic equation +The local continuity equations Eqs. (6), (18), and (28) expressing conservation of the number of particles, +momentum, and energy, respectively can be straightforwardly obtained by integrating the Boltzmann kinetic +equation [21]. As direct consequence of the conservation laws, the collision integral vanishes upon integration. +In this section, I discuss the relation of this property to the identities Eqs. (16b), (24), and (40). +5.1. Kadanoff-Baym equation +Derivation of the kinetic equation is outlined in Ref. [10]. The calculation is very similar to at least some +of the above considerations, with a few notable points that should be clarified here. Firstly, Ref. [10] begins +with a Dyson’s equation for the Keldysh function GK instead of Eq. (3). Combining it with the conjugate +equation one finds the analogue of Eq. (9), which has the form +�� +i ∂ +∂t1 +− � +H0(1) +� ++ +� +i ∂ +∂t2 ++ � +H0(2) +�� +GK +1,2 = +� +d3 +� +ΣR +1,3GK +3,2 + ΣK +1,3GA +3,2 − GR +1,3ΣK +3,2 − GK +1,3ΣA +3,2 +� +. +(51) +Now, the Keldysh function is related to the function G12 by the identity +GK = 2G12 − iA, +(52) +where A is the spectral function defined in Eq. (A.15). The choice of the Keldysh functon as the “basis” +function for the nonequilibrium transport theory [as opposed to G12 which, after all, defines the particle +density, see Eq. (7)] is justified by the fact [10], that the spectral function does not depend on the state of +the system and hence its contribution to macroscopic quantities out of equilibrium is irrelevant. +Although Eq. (51) is distinct from Eq. (9), it has the same property [see Eq. (11)]: the RHS of Eq. (51) +vanishes in the limit 2 → 1. Indeed, combining Eqs. (9) and (51) according to Eq. (52), one finds the +equation for the spectral function +�� +i ∂ +∂t1 +− � +H(0) +1 +� ++ +� +i ∂ +∂t2 ++ � +H(0) +2 +�� +A1,2 = +� +d3 +� +ΣR +1,3A3,2 + Γ1,3GA +3,2 − GR +1,3Γ3,2 − A1,3ΣA +3,2 +� +, +(53a) +where Γ defined in Eq. (A.20) is the self-energy component analogous to the spectral function. Substituting +the definitions (A.15) and (A.20) into Eq. (53), one finds for the RHS +i +� +d3 +� +ΣR +1,3GR +3,2 − ΣA +1,3GA +3,2 − GR +1,3ΣR +3,2 + GA +1,3ΣA +3,2 +� +. +(53b) +This expression vanishes in the limit 2 → 1, see Eq. (16d), which proves that the RHS of Eq. (51) vanishes +in that limit as well. +Finally, one can re-write Eq. (51) introducing the quantities A and Γ in the RHS [10]. Introducing the +short-hand notations +�D12 = i ∂ +∂t1 ++ i ∂ +∂t2 +− � +H(0) +1 + � +H(0) +2 , +(54a) +(A ⊗ B)1,2 = +� +d3 A1,3B3,2, +(54b) +� +A ⊗, B +� +− = A ⊗ B − B ⊗ A, +� +A ⊗, B +� ++ = A ⊗ B + B ⊗ A, +(54c) +one arrives at the variant of the Kadanoff-Baym equation +�D12GK +1,2 = +� +ReΣ ⊗, GK� +− + +� +ΣK ⊗, ReGR� +− + i +2 +� +ΣK ⊗, A +� ++ − i +2 +� +Γ ⊗, GK� ++ . +(55) +This equation is equivalent to Eq. (51) and hence the RHS vanishes in the limit 2 → 1. The RHS of the +Kadanoff-Baym equation (9) can be brought to the same form as indicated above. +14 + +5.2. Kinetic equation +Let me briefly recall the standard steps of the derivation of the kinetic equation. This can be done in +two different ways [10]. +5.2.1. Quasiparticle and quasiclassical approximations +The first idea is to apply the gradient approximation to the Kadanoff-Baym equation (55). To do that +one first introduces the Wigner representation (using the relative and center of mass coordinates introduced +in section 3.4), see Appendix A. The Wigner representation is very physical, but unfortunately yields a +complicated expression for the convolution, the so-called Moyal product [38] +A ⊗ B = ei(∂A +ǫ ∂B +t −∇A +p·∇B +r −∂A +t ∂B +ǫ +∇A +r·∇B +p )AB. +(56) +The gradient approximation in the Kadanoff-Baym equation is achieved by keeping the first two terms in +the Taylor series for the exponential in Eq. (56), which yields +� +A ⊗, B +� +− = i [A, B]p , +� +A ⊗, B +� ++ = 2AB, +(57) +where +[A, B]p = (∂ǫA) (∂tB) − (∇pB)·(∇rB) − (∂tA) (∂ǫB) + (∇rA)·(∇pB) . +(58) +Applying the above approximation to Eq. (55) one finds +� +(ǫ − ξp − U − ReΣ) , GK� +p − +� +ΣK, ReGR +� +p = ΣKA − ΓGK. +(59) +Here all the derivatives are combined in the LHS, while the remaining RHS can be identified with the +collision integral. +The last step in the derivation is based on a further approximation. The “quasiparticle approximation” +relies on the Kadanoff-Baym solution [24] for the spectral function which can be approximated by a δ- +function, A = 2πδ(ǫ − ξp − U). Combined with the corresponding form of the Keldysh Green’s function, +GK = −2πihpδ(ǫ − ξp − U), where hp defines the conventional distribution function, fp = (1 − hp)/2, one +integrates Eq. (59) over ǫ and obtains the standard (Boltzmann) kinetic equation. The collision integral +(up to a numerical factor) takes the form I ∝ ΣK[hp] − (ΣR − ΣA)hp and is the function of p, r, and t. +Summing over all states now amounts to integrating over the momentum variable p. Given that it appears +as a Fourier transform inthe relative coordinate, such integration is equivalent to the limit 2 → 1 considered +above. +Alternatively, one can integrate over ξp (the “quasiclassical approximation”). The idea is that in the case +of, e.g., electro-phonon interaction the self-energy acquires energy dependence and hence the Kadanoff-Baym +solution for the spectral function can no lnger be reduced to the above δ-function. However, should the +momentum dependence of the self-energy remain weak (e.g., due to the Migdal theorem [39]) the spectral +function retains the form of a sharp peak in the variable ξp. This relies on the existence of the Fermi surface: +upon integration over ξp the Green’s functions are essentially restricted to the Fermi surface and depend +only on the orientation of momentum. While the quasiclassical approximation does not explicitly require +the mixed representation in the time variables, it is often invoked in order to reach the standard form of the +kinetic equation [10]. The procedure becomes rather similar to the previous case and yields the same form +of the collision integral, where both the self-energies and the distribution function now depend on ǫ instead +of the absolute value of p. Assuming the existence of quasiparticles, the two approximations can be related +by the formal introduction of the density of states. At the same time, the quasiclassical approximation does +not rely on the quasiparticle paradigm, which formally is expressed through the fact that the energy and +momentum variables are no longer related. A known limitation of the quasiclassical approximation is its +reliance on the particle-hole symmetry [10] which precludes one from describing, e.g., thermoelectric effects. +15 + +5.3. Beyond the quasiclassical approximation +The second approach to deriving the kinetic equation does not rely on the quasiparticle approximation. +Instead, one introduces the Ansatz [10] +GK = GR ⊗ h − h ⊗ GA. +(60) +Using this form in the Dyson’s equation (51) and taking into account the diagonal elements of Eq. (A.11), +one arrives at the equation +GR ⊗ B − B ⊗ GA = 0, +B = �D12h − +� +ReΣ ⊗, h +� +− + i +2 +� +Γ ⊗, h +� ++ + ΣK. +(61) +Solving this equation to the leading order of the gradient expansion amounts to setting B = 0 which +seemingly yields the same form of the collision integral, I ∝ ΣK[h] − (ΣR − ΣA)h, albeit obtained without +any recourse to the quasiparticle approximation. The difference is that here the “distribution function” +depends not only on p, r, and t as in the case of the quasiclassical approximation, but also on the energy +variable, i.e. h = h(p, ǫ; r, t). The quasiclassical (or quasiparticle) approximation allows one to integrate +over ǫ using the “δ-peak”-like form of the spectral function. The Ansatz (60) leads to the kinetic equation +without any additional assumptions on the form of A. +An alternative method of deriving the quantum kinetic equation was suggested in Ref. [27] on the basis +of the observation that all elements of the Keldysh Green’s function matrix could be expressed in terms of +two functions only, cf. Eqs. (A.4) and (A.12). Choosing the spectral function as one of the two and noticing +that it becomes real in the Wigner representation, one can introduce another real function in the Wigner +representation, h = h(p, ǫ; r, t), such that +G12(p, ǫ; r, t) = iA(p, ǫ; r, t)h(p, ǫ; r, t), +G21(p, ǫ; r, t) = −iA(p, ǫ; r, t) [1 − h(p, ǫ; r, t)] . +(62a) +This allows to use the functions A and h to express the Keldysh function GK +GK(p, ǫ; r, t) = −iA(p, ǫ; r, t) [1 − 2h(p, ǫ; r, t)] . +(62b) +Expressing the self-energies in a similar way +Σ12(p, ǫ; r, t) = iΓ(p, ǫ; r, t)γ(p, ǫ; r, t), +Σ21(p, ǫ; r, t) = −iΓ(p, ǫ; r, t) [1 − γ(p, ǫ; r, t)] , +(63a) +with +ΣK(p, ǫ; r, t) = −iΓ(p, ǫ; r, t) [1 − 2γ(p, ǫ; r, t)] , +(63b) +one can use the new notations to re-write Eq. (55) as +�DAh − +� +ReΣ ⊗, Ah +� +p − +� +Γγ ⊗, ReGR� +p = ΓA(γ − h), +(64) +where the RHS is essentially the same form of the collision integral expressed in the new notations. +The “quantum kinetic equation” (64) was obtained within the leading order of the gradient approximation +and hence provides a quantitative condition for its validity +|γ − h| ≪ 1. +(65) +Consequently, in the LHS of Eq. (64) one can replace γ by h. The resulting equation corresponds to the +Botermans and Malfliet [40] choice of the quantum kinetic equation (as opposed to the original Kadanoff- +Baym choice). Both variants are equivalent within the applicability range of the gradient approximation. +In comparison to the Ansatz (60), the variable choice (62) reintroduces the spectral function in the +definition of the distribution function h, while leaving the energy dependence of the latter. On the other +hand, the choice (62) is always possible [27], while the Ansatz (60) is guaranteed to be valid only within the +gradient approximation [10]. +16 + +5.4. Kadanoff-Baym equation and the continuity equation +Let me now compare the derivation of the continuity equation presented in section 2 to the well-known +approach of integrating the kinetic equation. Particle number conservation is manifested in the traditional +kinetic theory by the fact that collision integral vanishes after being summed up over all states [21]. The +quantum kinetic equation, regardless of the variant, cf. +Eqs. (59), (62b). +and (64), contains also the +renormalization terms in the LHS. Consequently, the derivation of the kinetic equation consists of making +sure that the integral of the collision term vanishes and at the same time that the renormalization terms do +not affect the particle density and current [10]. +In contrast, the argument presented in section 2 relies on the single identity, Eq. (11), where taking the +limit 2 → 1 is equivalent to integrating the collision integral over all energies and momenta, ǫ and p. The +combination of the self-energies and Green’s functions in Eq. (11) comprises both the collision integral and +renormalization terms (before the gradient approximation). However, vanishing of these terms together does +not in general guarantee that they should vanish individually although this does happen for most common +forms of the kinetic equation [10]. The fact that renormalization does not affect the particle density follows +from the operator definition, Eq. (7). Similarly, the current j is determined by the operator form of the +continuity equation, Eq. (8) and hence cannot be affected by interaction explicitly. This does not mean that +the density and current in an interacting system are the same as in non-interacting one: both definitions +involve the exact Green’s function G12, which can be very different from the free-particle one. +In that +sense, vanishing of the renormalization terms ensures consistency of definitions of macroscopic currents and +densities in the microscopic and kinetic theories. Of course, the total number of particles is the same as in +the free system since interaction does not “produce” or “destroy”any particles. +Finally, let me reiterate that the continuity equation (6) is exact as long as the interaction (and any +potential) is expressed in terms of particle density, as is the case with most typical models (electron-electron +Coulomb interaction, electron-phonon – or any other boson – coupling, electron-impurity scattering, etc). +The purpose of the identity (11) is to make sure that any approximation made for Green’s functions and +self-energies does not violate the conservation law. +5.5. Kadanoff-Baym equation and momentum conservation +The continuity equation for momentum density (18) is the central equation in the hydrodynamic theory +eventually yielding the Euler and Navier-Stokes equations. +In contrast to the continuity equation (6), +the equation (18) is not exact, but is valid within the gradient approximation. This is not a problem, +since hydrodynamics describes long-wavelength variations of macroscopic quantities. The same gradient +approximation is used to derive the kinetic equation. The equation (18) can then be obtained by multiplying +the kinetic equation by momentum and summing over all states without further approximations. As a +result of this procedure the RHS (i.e., the collision integral) of the kinetic equation vanishes which is the +manifestation of momentum conservation [21]. +Microscopically, momentum conservation is manifested through the identity (24). This directly leads +to vanishing of the quantity C integrated over all space and hence to the global (integral) relation (21). +The quantity C itself emerges from the RHS of the Kadanoff-Baym equation in the limit 2 → 1 which +is equivalent of integrating the RHS of the quantum kinetic equation. The gradient approximation used +to derive Eq. (18) is equivalent to the one needed to derive the quantum kinetic equation. In particular, +separating the integrand c+ into two parts corresponds to the distinction between the collision integral and +the renormalization terms in Eqs. (59) and (64). In this case it can be seen directly that the integrated +collision integral vanishes while the renormalization terms contribute to the momentum flux tensor, see +Eq. (27). The momentum density g is determined by the momentum operator, see Eq. (19), and hence +is unaffected by renormalizations similarly to the particle number density and current. +The derivation +presented in Sec. 3 is thus equivalent to the more standard route of going through the kinetic equation +(either the Boltzmann one or quantum), but has the advantage of being free of any additional approximation +beyond the gradient expansion. Taking into account additional interaction that do not conserve momentum +amounts to evaluating its contribution to the quantity C in the “0-th” approximation with respect to the +gradients, which is equivalent to evaluating the corresponding collision integral. +17 + +5.6. Kadanoff-Baym equation and energy conservation +Energy conservation is the most difficult part of the presented approach since the energy density at +the operator level is essentially a two-particle correlation function. +Global energy conservation can be +expressed in terms of the integral relation (33). As in the case of momentum conservation, the RHS of +the Kadanoff-Baym equation in the limit 2 → 1, i.e., the quantity Υ, determines the time derivative of the +interaction energy upon being integrated over all space. However, now both the energy density and current +are renormalized by interaction. Separating the time derivative of the interaction energy density from Υ +leaves the contribution to the energy current that has to be combined with the interaction contribution to the +“single-particle” current J . This should be contrasted with the standard kinetic theory derivation [2, 8, 21] +where a direct integration of the kinetic equation multiplied by energy yields the energy density and current +from the LHS, while the collision integral vanishes. At the same time, the “internal energy” appears though +thermodynamic identities [21]. This apparent complication in comparing the two approaches is reminiscent +of the common practice in conventional hydrodynamics where dissipative processes are taken into account +using the entropy flow equation rather than the continuity equation for the energy density [20, 21]. The +entropy flow equation will be discussed in a forthcoming publication [41]. +Recent literature on electronic hydrodynamics in graphene [2–4, 8] devotes little attention to the internal +energy. The role of electron-electron interaction is seen as being responsible for equilibration, although in +real materials equilibration is most likely to occur with the help of phonons. Taking into account electron- +phonon interaction leading to energy relaxation [30] would violate the identity (24). In the simplest case +(cf. the arguments of Ref. [30]), one would have to evaluate the phonon contribution to Υ establishing the +weak decay contribution to the continuity equation (28). +6. Discussion +In this paper I have presented a detailed derivation of the local continuity equations providing the basis of +the hydrodynamic theory of electronic transport. While the continuity equation manifesting gauge invariant +is exact, the corresponding equations for the momentum and energy density are obtained within the gradient +approximation. The presented derivation is more general than the kinetic theory approach since it relies +neither on additional approximations (such as the common quasiparticle or quasiclassical approximations) +nor on the concept of the distribution function. Although the latter can be introduced at the quantum level +[e.g., by Eqs. (60) or (62)], it is not always obvious how to generalize this quantity to more complicated +cases, e.g., involving spin-orbit interaction. Keeping the discussion in coordinate space allows for a direct +generalization for systems in confined geometries. +The idea that hydrodynamics is “more general” than the kinetic theory is not new and can be already +seen in the original hydrodynamic description of conventional fluids (none of which could be described by +a kinetic equation). Microscopic expressions for the momentum flux tensor, interaction energy density, and +energy current presented here open a direct pathway for evaluating these quantities using specific models of +the systems of interest. In particular, there is already a substantial literature on hydrodynamic approach to +“strange” or “bad” metals [42–46], where the excitation spectrum might not contain usual quasiparticles, +as could be seen in photoemission [47] and transport [48] experiments. Some of these materials exhibit +resistance that is linear in temperature over a wide range including both low and high temperatures [49], +the behavior that has been puzzling the community for decades. The analysis presented here could be seen +as a way of evaluating resistivity directly (similarly to the case of graphene [2–4]) without the need for a +Kubo formula and may prove helpful for describing less established systems such as “non-Fermi liquids”. +Acknowledgments +The author wishes to thank I.V. Aleiner, I.V. Gornyi, A.D. Mirlin, J. Schmalian, and A. Shnirman +for fruitful discussions. This work was supported by the German Research Foundation DFG project NA +1114/5-1 and the European Commission under the EU Horizon 2020 MSCA-RISE-2019 Program (Project +873028 HYDROTRONICS). +18 + +Appendix A. Non-equilibrium (or Keldysh) Green’s function formalism +Here I summarize the notations for the Keldysh Green’s functions and their standard relations to keep +the paper self-complete. For a detailed account of the Keldysh technique see Refs. [10, 22]. +Appendix A.1. Keldysh Green’s function +The central quantity of the formalism is the Green’s function that can be defined either in the Heisenberg +(subscript “H”) or “interaction (subscript “I”) representation on the Keldysh contour (subscript “C”) +G(1C, 2C) = −i +� +TC ˆψH(1C) ˆψ† +H(2C) +� += −i +� +TC �SC ˆψI(1C) ˆψ† +I(2C) +� +, +(A.1a) +where the latter expression retains only the “connected” diagrams. In Eq. (A.1), TC is the time-ordering +operator on the Keldysh contour and the “scattering matrix” is +�SC = TC exp + +−i +� +C +dtC �Hint(tC) + + . +(A.1b) +The Green’s function (A.1) can be more conveniently described in the matrix form +ˇG1,2 = +�G11 +1,2 +G12 +1,2 +G21 +1,2 +G22 +1,2 +� +, +(A.2) +where +G12 +1,2 = i +� +ˆψ† +H(2) ˆψH(1) +� +, +(A.3a) +G21 +1,2 = −i +� +ˆψH(1) ˆψ† +H(2) +� +, +(A.3b) +G11 +1,2 = θ(t1 − t2)G21 +1,2 + θ(t2 − t1)G12 +1,2, +(A.3c) +G22 +1,2 = θ(t1 − t2)G11 +1,2 + θ(t2 − t1)G21 +1,2. +(A.3d) +The four matrix elements are not independent and satisfy +ˇG1,2 = −ˇτ1 ˇG† +2,1ˇτ1, Tr ˇG1,2 = Tr ˇτ1 ˇG1,2, +(A.4a) +where ˇτi are the Pauli matrices in the “Keldysh space”. Explicitly, the later relation takes the form +G11 +1,2 + G22 +1,2 = G12 +1,2 + G21 +1,2. +(A.4b) +Appendix A.2. Self-energy +The Green’s function obeys the formally exact Dyson’s equation +� +i ∂ +∂t1 +− � +H(0) +1 +� +ˇG1,2 − +� +d3 ˇΣ1,3ˇτ3 ˇG3,2 = ˇτ3δ1,2, +(A.5) +where the self-energy is the matrix +ˇΣ1,2 = +�Σ11 +1,2 +Σ12 +1,2 +Σ21 +1,2 +Σ22 +1,2 +� +. +(A.6) +The above definition differs from that in Ref. [21], where the Pauli matrix in the integral in Eq. (A.5) +precedes the self-energy [21]. This amounts to the replacement +ˇΣ1,2 → ˇτ3 ˇΣ1,2ˇτ3, +19 + +or simply put, the extra minus sign for the off-diagonal elements. This can be made clearer by transforming +the integro-differential equation (A.5) to the integral form using the “free” Green’s function +ˇG(0) +1,2 = +� +i ∂ +∂t1 +− �H(0) +1 +�−1 +ˇτ3δ1,2. +(A.7) +Applying the operator +� +i∂t1 − � +H(0) +1 +�−1 +to Eq. (A.5) from the left, one finds +ˇG1,2 − +� +d3d4 ˇG(0) +1,4ˇτ3 ˇΣ4,3ˇτ3 ˇG3,2 = ˇG(0) +1,2, +(A.8) +in contrast to the corresponding equation in Ref. [21] where there are no Pauli matrices. +The rationale for the above notation is as follows. The self-energy has the same “symmetry” as the +Green’s function, see Eq. (A.4) +ˇΣ1,2 = −ˇτ1 ˇΣ† +2,1ˇτ1, Tr ˇΣ1,2 = Tr ˇτ1 ˇΣ1,2. +(A.9a) +The latter relation reads +Σ11 +1,2 + Σ22 +1,2 = Σ12 +1,2 + Σ21 +1,2, +(A.9b) +similarly to Eq. (A.4b) and without the extra minus sign in the right-hand side as in Ref. [21]. +Appendix A.3. Keldysh rotation +One may try to use the relations (A.4) to reduce the number of Green’s functions. This can be achieved +by a “rotation” [10, 50] +ˇG1,2 → 1 +2 +�1 +−1 +1 +1 +� +ˇτ3 ˇG1,2 +� 1 +1 +−1 +1 +� +. +(A.10a) +In the new basis, both the Green’s function and self-energy have the similar form (unlike the form suggested +in Ref. [21]) +ˇG = +�GR +GK +0 +GA +� +, +ˇΣ = +�ΣR +ΣK +0 +ΣA +� +. +(A.10b) +In terms of the original Green’s functions, the newly defined functions are given by +GR = G11 − G12 = G21 − G22 = θ(t1 − t2) +� +G21 +1,2 − G12 +1,2 +� +, +GA = G11 − G21 = G12 − G22 = −θ(t2 − t1) +� +G21 +1,2 − G12 +1,2 +� +, +(A.10c) +GK = G12 + G21 = G11 + G22. +In the rotated basis, the Dyson’s equation takes the form (same as in Ref. [21]) +� +i ∂ +∂t1 +− � +H(0) +1 +� +ˇG1,2 − +� +d3 ˇΣ1,3 ˇG3,2 = δ1,2. +(A.11) +The basis rotation does not completely eliminate the redundancy in the definitions of the Green’s function. +Indeed, the Green’s function in the rotated basis satisfies [cf. Eq. (A.4)] +ˇG1,2 = ˇτ2 ˇG† +2,1ˇτ2, +(A.12) +with the similar constraint on the self-energy [cf. Eq. (A.9)] +ˇΣ1,2 = ˇτ2 ˇΣ† +2,1ˇτ2. +(A.13) +20 + +As a result, only two functions are in either matrix are independent. +Suppose one chooses G12 +1,2 and G21 +1,2 as such independent functions. Then from Eq. (A.4) it follows that +they have the following property +� +G12 +1,2 +�∗ = −G12 +2,1, +� +G21 +1,2 +�∗ = −G21 +2,1. +(A.14) +Consider then their difference +A1,2 = i +� +G21 +1,2 − G12 +1,2 +� += i +� +GR +1,2 − GA +1,2 +� +. +(A.15a) +As follows from the symmetry of the Green’s functions, Eq. (A.14), the new function satisfies +A∗ +1,2 = A2,1. +(A.15b) +Appendix A.4. Wigner representation +The gradient approximation needed to derive quantum kinetic equations is most readily demonstrated in +the mixed or Wigner representation, i.e. the Fourier representation in the relative coordinate (the relative +and center of mass coordinates were introduced in section 3.4) +A(p, ǫ; R12, T12) = +� +ddr12dt12A1,2e−i(pr12−ǫt12), +(A.16) +where t12 = t1 − t2 and T12 = (t1 + t2)/2. +In the Wigner representation, the spectral function A is real [cf. Eq. 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JETP 41, 960 (1975)]. +23 + diff --git a/sdE0T4oBgHgl3EQfrgGT/content/tmp_files/load_file.txt b/sdE0T4oBgHgl3EQfrgGT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1778a7c62a408ba1752d4c5bba1cd0ee7b9d59fe --- /dev/null +++ b/sdE0T4oBgHgl3EQfrgGT/content/tmp_files/load_file.txt @@ -0,0 +1,1223 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf,len=1222 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='02567v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='str-el] 6 Jan 2023 Hydrodynamic approach to many-body systems: exact conservation laws Boris N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Narozhnya,b aInstitut f¨ur Theorie der Kondensierten Materie, Karlsruher Institut f¨ur Technologie, 76131 Karlsruhe, Germany bNational Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 115409 Moscow, Russia Abstract In this paper I present a pedagogical derivation of continuity equations manifesting exact conservation laws in an interacting electronic system based on the nonequilibrium Keldysh technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The purpose of this exercise is to lay the groundwork for extending the hydrodynamic approach to electronic transport to strongly correlated systems where the quasiparticle approximation and Boltzmann kinetic theory fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Keywords: Electronic hydrodynamics, graphene, viscosity, quantum conductivity, kinetic theory Electronic hydrodynamics has evolved into a fast paced field with multiple experimental and theoretical groups working to uncover observable signatures of hydrodynamic behavior of electronic systems [1–4] with the primary focus on transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In a “realistic” case of a weakly disordered conductor, hydrody- namic equations encompass the conventional linear-response transport theory describing both the uniform Ohmic current in macroscopic (“infinite”) systems and the nonuniform viscous flows of charge and energy in constricted (“mesoscopic”) geometries [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Similarly to the traditional transport theory, hydrodynamic equations can be derived from the kinetic (Boltzmann) equation describing a system of weakly interacting quasiparticles [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' At the semiclassical level, one often relies on the “scattering time approximation” (typically used to describe Drude-like transport phenomena [9]) to simplify the collision integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This approach can be further extended to include quantum interference phenomena [10, 11] yielding the so-called “quantum corrections” to the leading semiclassical behavior heralding the onset of low-temperature localization [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' These additional features “correct” the conductivity of the system, while the macroscopic description of the current flow remains Ohmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The low-temperature Ohmic resistance is still determined by disorder, although electron-electron interaction does affect the quantum corrections [11, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In contrast, the hydrodynamic behavior is dominated by electron-electron interaction determining the viscosity coefficient [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Systems dominated by electron-electron interactions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=', strongly correlated systems, “strange metals”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=', remain a formidable challenge for several decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In simple terms, the difficulty lies in the failure of the quasiparticle approach [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Moreover, even if quasiparticles could be defined the semiclassical kinetic approach may fail in multi-component systems with non-Abelian degrees of freedom (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=', spin or isospin) due to the purely quantum nature of the latter [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' It is then highly desirable to develop a macroscopic theory of electronic transport in strongly interacting systems without reliance on the quasiparticle paradigm and semiclassical approximation which is the ultimate motivation for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In classical systems such a macroscopic theory is hydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Indeed, the Navier-Stokes equation [17–20] equally well describes water and air flows, while the Boltzmann kinetic theory allowing one to derive this equation [21] is only justified for a dilute gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The apparent universality of the hydrodynamic theory can be attributed to two points: (i) the long time, long distance behavior is often assumed to be independent of the details of short-distance scattering processes, and (ii) the conservation laws that are the basis of hydrodynamics are equally applicable to all systems with the same symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Email address: boris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='narozhny@kit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='edu (Boris N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Narozhny) Preprint submitted to Annals of Physics January 9, 2023 Generalizing the hydrodynamic approach to systems beyond conventional fluids, one may consider it in a broader sense meaning of a long-wavelength theory of small perturbations relative to an equilibrium state [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This way both the conventional hydrodynamics and diffusion could be discussed on equal footing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The difference between the two behaviors is momentum conservation which is assumed in hydrodynamics and is broken in diffusive systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In solids, electronic momentum is never truly conserved (it can be lost due to scattering off impurities, phonons, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' However, in ultra-pure materials it may be possible to find an intermediate temperature range where electron-electron interaction is the dominant scattering process [2–4] as reflected by the hierarchy of typical time scales τee ≪ τdis, τe−ph, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (using self-evident notations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Then it could be reasonable to neglect processes that do not conserve momentum, at least as the “0-th” approximation describing the “ideal fluid” by means of macroscopic (differential) equations essentially generalizing [2, 8] the Euler’s equation [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The non-conserving processes (electron-impurity or electron-phonon coupling) can then be included perturbatively [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The general problem of fermions with momentum-conserving interaction has been one of the most popular in many body physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Most generally, the system is described by a Hamiltonian comprising the one-particle (“free”) and “interaction” parts �H = �H0 + �Hint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (1a) The one-particle contribution can typically be separated into two contributions �H0 = � ddr1 ˆψ†(r1) � H(0) 1 ˆψ(r1), � H(0) 1 = �K1 + U1, (1b) with �K1 representing the “kinetic energy” (possibly including multiple bands, spin-orbit interaction, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' without loss of generality all additional quantum numbers are suppressed throughout this paper) and U1 being the one-particle potential [the subscript “1” refers to the set of quantum numbers of the field ˆψ†(r1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The interaction term is assumed to be translationally invariant (hence, momentum-conserving) �Hint = 1 2 � ddr1ddr2 ˆψ†(r1) ˆψ†(r2)V (r1−r2) ˆψ(r2) ˆψ(r1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (1c) The general problem represented by the Hamiltonian (1) cannot be solved exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' However, the conservation laws of particle number (charge), energy, and momentum are exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In this paper, I explore the emergence of exact conservation laws in the by now standard field-theoretic approach to nonequilibrium systems, the Keldysh technique [10, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This issue has been already extensively discussed in literature on general many-body theory [23–25] and nuclear physics [26–28] establishing the integral relations expressing the global symmetries of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The present paper explores a somewhat different angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' I am interested in “deriving” the local continuity equations manifesting the conservation laws that are the starting point of the hydrodynamic approach (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=', conservation of the particle number, energy, and momentum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The point is to express the macroscopic currents and densities in the most general form (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=', in terms of the exact quantities involved in the diagrammatic technique including Green’s functions, self-energies, etc) allowing for a straightforward generalization to specific condensed matter system including multiple bands and spin-orbit interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The requirement of the “exact” validity of the continuity equations leads to general relations involving the self-energies and Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' These relations are satisfied by the exact functions and serve as constraints on their approximate forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' At the same time, these relations provide a blueprint for including additional, non-conserving terms to the Hamiltonian (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=', electron-impurity or electron-phonon scattering) leading to weak decay contributions to the resulting macroscopic equations [2, 29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Finally, I compare the obtained expressions with those appearing as a result of the approximations leading to the kinetic equation (semiclassical or quantum) as an intermediate step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The ultimate goal of this work is to establish a hydrodynamic framework that does not rely on the quasiparticle paradigm (avoiding the kinetic equation and specifically the concept of the semiclassical distribution function) and hence could be useful for describing systems where quasiparticles are overdamped or altogether absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Equations of motion In this paper I consider the conservation laws using the nonequilibrium Keldysh technique following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The notations for the Keldysh Green’s functions and general relations between them are summa- rized in Appendix A, for a more detailed account of the Keldysh technique see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' [10, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' All Green’s functions are defined in terms of the Heisenberg field operators and therefore it is important to review the equations of motion governing their dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Starting with the standard quantum-mechanical definition of the time derivative, i ∂ ∂t ˆψ = � ˆψ, �H � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (2a) one finds [23] i ∂ ∂t ˆψ(r, t) − � H(0) ˆψ(r, t) = � ddr′ ˆψ†(r′, t)V (r−r′) ˆψ(r′, t) ˆψ(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (2b) Multiplying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (2b) by i ˆψ† from the left and taking the thermodynamic average, one arrives at the Dyson’s equation for the “12” component of the Keldysh Green’s function, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='5), but with the right-hand side (RHS) expressed explicitly in terms of the interaction potential i ∂ ∂t1 G12 1,2− �H(0) 1 G12 1,2 = i � ddr3 � ˆψ†(r2, t2) ˆψ†(r3, t1)V (r1−r3) ˆψ(r3, t1) ˆψ(r1, t1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (3a) Here and throughout the paper I use the short-hand notation: G12 1,2 = G12(r1, t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r2, t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In contrast, the Dyson’s equation is expressed in terms of the self-energy, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='5) i ∂ ∂t1 G12 1,2− � H(0) 1 G12 1,2 = � d3 � Σ11 1,3G12 3,2 − Σ12 1,3G22 3,2 � , (3b) where d3 = ddr3dt3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Alternatively [using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='10c) and the similar relation for the self-energy] i ∂ ∂t1 G12 1,2− � H(0) 1 G12 1,2 = � d3 Ξ(1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 3), Ξ(1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 3) = ΣR 1,3G12 3,2+Σ12 1,3GA 3,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (3c) Comparing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (3a) and (3c), one arrives at the identity i � ddr3 � ˆψ†(r2, t2) ˆψ†(r3, t1)V (r1−r3) ˆψ(r3, t1) ˆψ(r1, t1) � = � d3 Ξ(1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 3), (3d) relating the two-particle Green’s function in the RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (3a) to the single-particle quantities in the RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In what follows, I will also use the equation of motion for ˆψ† i ∂ ∂t ˆψ†(r, t) + ˆψ†(r, t) � H(0),† = − � ddr′ ˆψ†(r, t)V (r−r′) ˆψ†(r′, t) ˆψ(r′, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (4) Here � H(0),† is the conjugate operator with any gradients acting on the coordinate dependence to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Multiplying this equation by i ˆψ from the right one finds i ∂ ∂t2 G12 1,2+ � H(0),† 2 G12 1,2 = −i � ddr3 � ˆψ†(r2, t2) ˆψ†(r3, t2)V (r1−r3) ˆψ(r3, t2) ˆψ(r1, t1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (5a) On the other hand, conjugating the Dyson’s equation (3c) [see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='4), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='9), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='12), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='13), and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='14)] and changing the variables 1 ↔ 2 one arrives at i ∂ ∂t2 G12 1,2+ � H(0),† 2 G12 1,2 = � d3 Ξ∗(2, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 3) = − � d3 � GR 1,3Σ12 3,2 + G12 1,3ΣA 3,2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (5b) Comparing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (5a) and (5b) yields the conjugate form of the identity (3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Continuity equation Consider now the usual continuity equation ∂n ∂t + ∇·j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (6) The continuity equation itself is well-known and does not need another derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This equation repre- sents gauge invariance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=', the particle number conservation or charge conservation), the symmetry that is typically assumed to be exact for all condensed matter systems (apart from the special case of supercon- ductivity where this issue is more subtle, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' [10]) and hence is independent of the particular form of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The purpose of this section is to introduce notations for the particle number density, n, and the current, j, and establish the constraint imposed on the self-energy by gauge invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Continuity equation at the operator level The particle number can be defined in the standard way using electronic field operators ˆn(r, t) = ˆψ†(r, t) ˆψ(r, t), n(r, t) = ⟨ˆn(r, t)⟩ , (7a) or the Green’s function [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3a)] n1 = −iG12 1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (7b) The two definitions allow for two different derivations of the continuity equation starting either with the equations of motion or the Dyson’s equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' At the operator level, the continuity equation is just the equation of motion for the density operator that can be obtained by combining the two equations of motion (2b) and (4) ∂ ∂t ˆn(r, t) = −i ˆψ†(r, t) �K ˆψ(r, t) + i � �K ˆψ(r, t) �† ˆψ(r, t) = −∇·ˆj(r, t), (8) where the last step defines the current operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The interaction potential does not appear in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (8) due to the standard commutation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Continuity equation and the Keldysh Green’s functions Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (3c) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (5b) yields a Kadanoff-Baym equation [24] � i ∂ ∂t1 − � H(0) 1 + i ∂ ∂t2 + � H(0),† 2 � G12 1,2 = � d3 [ Ξ(1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 3) + Ξ∗(2, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 3)] (9) = � d3 � ΣR 1,3G12 3,2 + Σ12 1,3GA 3,2 − GR 1,3Σ12 3,2 − G12 1,3ΣA 3,2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Comparing the time derivative terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (9) to the definition of the particle density, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (7), one notices the relation � i ∂ ∂t1 +i ∂ ∂t2 � G12 1,2 ���� 2→1 = −∂n1 ∂t1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Now it becomes clear that in the limit 2 → 1 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (9) can be written in the form of the conventional continuity equation (6) where the divergence of the current is determined by the single-particle Hamiltonian ∇1·j1 = � �K1 − �K† 2 � G12 1,2 ��� 2→1 , (10) while the self-energy satisfies the condition � d3 � ΣR 1,3G12 3,1 + Σ12 1,3GA 3,1 − GR 1,3Σ12 3,1 − G12 1,3ΣA 3,1 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (11) The latter identity is satisfied by the exact self-energy and Green’s function and hence represents a constraint on any approximate expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In fact, the identity (11) can be derived independently, following Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' [24, 25], where the idea of “conserving approximations” was first suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Conserving approximations The need for a “conserving approximation” arises from the apparent arbitrariness of the diagrammatic perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Indeed, it may not be clear “a priori” that a given approximation for the self-energy satisfies the exact conservation laws of the system (given that this is certainly not the case for at least some individual diagrams;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' of course, any practitioner of the diagrammatic perturbation theory would make sure that the calculation does not violate gauge invariance, although this might involve certain technical difficul- ties – the point of a “conserving approximation” is that the conservation laws are satisfied automatically without any need for special care).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Baym suggested the self-consistent procedure where one starts with the Luttinger-Ward functional Φ[ ˇG] = � ln � �SC �� sk = �� �SC � − 1 � sk , Φ∗ = Φ, (12) where the subscript “sk” indicates that only skeleton diagrams (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' diagrams without self-energy insertions and with all Green’s functions replaced by full Green’s functions) are to be retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Moreover, the logarithm amounts to retaining only the connected diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The important property of the functional is that the exact self-energy can be obtained by the variation Σij 1,2 = −(−1)i+j δΦ δGji 2,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (13) The “self-consistent” perturbation theory comprises an expansion of the functional Φ and a solution for ˇG and ˇΣ using the Dyson’s equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='11) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The latter step is self-consistent in the sense that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='11) determines the Green’s function in terms of the self-energy and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (13) the other way around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The key point of the self-consistent approach is that the resulting theory satisfies exact conservation laws without any further approximation no matter how many diagrams are retained in the expansion of the functional Φ [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The resulting approximations are known as “Φ-derivable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' While there can be many such approximations (depending on the order to which Φ is expanded), all of them respect the conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Φ-derivable approximations and gauge invariance Applying a symmetry transformation to the exact Green’s function leads to a variation of the functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' To the leading order, the variation δΦ is given by δΦ = − � d1d2 Tr ˇτ3 ˇΣ1,2ˇτ3δ ˇG2,1, (14) which vanishes if the transformation corresponds to a true symmetry of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Consider a gauge transformation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' the same argument of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' [25] but using the Matsubara Green’s functions) which without loss of generality can be confined to the upper branch of the Keldysh contour ˇG2,1 → eiˇχ2 ˇG2,1e−iˇχ1, ˇχ = �χ 0 0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (15a) Expanding to the leading order in χ and taking into account the matrix structure, one finds δ ˇG2,1 = iχ2 1+ˇτ3 2 ˇG2,1 − iχ1 ˇG2,1 1+ˇτ3 2 (15b) Substituting this expression into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (14) and requiring that the functional is invariant under the gauge transformation (since it is composed of closed particle lines) one finds (using the cyclic property of the trace in each term separately) � d1χ1 � d2 Tr1+ˇτ3 2 � ˇτ3 ˇΣ1,2ˇτ3 ˇG2,1 − ˇG1,2ˇτ3 ˇΣ2,1ˇτ3 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (16a) 5 Evaluating the trace and taking into account arbitrariness of χ1 one arrives at the identity � d2 � Σ11 1,2G11 2,1 − Σ12 1,2G21 2,1 − G11 1,2Σ11 2,1 + G12 1,2Σ21 2,1 � = 0, (16b) which is a manifestation of gauge invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Now, substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='10c) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (16b), one finds Σ11 1,2G11 2,1 − Σ12 1,2G21 2,1 = ΣR 1,2G12 2,1 + Σ12 1,2GA 2,1 + ΣR 1,2GR 2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (16c) Comparing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (11) and (16b) I now conclude that in the limit 2 → 1 the integral in the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (9) takes the form � d3 � ΣR 1,3GR 3,1 − GR 1,3ΣR 3,1 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (16d) This expression vanishes for the following reasons: (i) the self-energy has the same causality structure as the Green’s function [22], therefore for any t1 ̸= t3 the product ΣR 1,3GR 3,1 vanishes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (ii) in the limit t3 → t1 the retarded Green’s function has the form GR(r1, r3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' t1 = t3 + 0) = −iδ(r1 − r3), (17) so that even if the self-energy had a non-zero diagonal value ΣR(1, 1) it would be the same in both terms and hence canceled in the difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' As a result, the identity (11) follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (16b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The above argument represents a proof of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (11) and, by extension, confirms that the continuity equation (6) is consistent with the Keldysh approach (exactly or within a Φ-derivable approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Summary To summarize this section, the continuity equation (6) manifesting particle number conservation follows from the Heisenberg equations of motion due to the “density-density” interaction, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' It is fully preserved in the microscopic Keldysh approach (either while using the exact Green’s functions or within a Φ-derivable – or any other conserving – approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' At the same time, the continuity equation is satisfied within the kinetic theory (that can be derived from the same microscopic theory using a series of approximations, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' [10] and Section 5) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Given that particle number conservation is the exact symmetry, the continuity equation is valid independently of any (correctly applied) approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Specifically, the arguments presented here do not rely on either the quasiparticle and semiclassical approx- imations typically assumed to derive the kinetic equation or otherwise describe conventional metals and semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Momentum conservation Consider now translational invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This is the crucial symmetry in conventional hydrodynamics, where the Navier-Stokes equation [20] is a direct consequence of momentum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Since the Hamiltonian is explicitly translationally invariant, one should be able to express momentum conservation by means of the continuity equation for the momentum density, g, similarly to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (6) ∂gα ∂t + ∇βτβα = 0, (18) without any additional derivation (here ταβ is the momentum flux – or stress – tensor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' However, there are well documented difficulties along the way [23–25], primarily for long-ranged interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Within the kinetic theory, one may derive the Navier-Stokes equation by multiplying the kinetic equation by momentum and integrating over all single-particle states [2, 8, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The equivalent procedure at the microscopic level is to apply the momentum operator to the Dyson equations (3c) and (5b) followed by the evaluating their sum in the limit 2 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In the resulting equation [similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (9)], the time derivative terms combine into the time derivative of momentum density, while the rest should comprise the spatial derivatives yielding the divergence of the momentum flux tensor and the “collision integral” terms vanishing in the limit 2 → 1, essentially repeating the above calculation leading to the continuity equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Momentum density The “momentum operator” mentioned above is the differential operator allowing one to define the mo- mentum density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In quantum field theory, however, the definition of such operator is not unique [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The reason is that only the total momentum of the system is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' While it can be expressed as a volume integral over the momentum density, that integral remains unchanged if any contribution repre- senting a surface term is added to the integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This freedom can be used to bring the stress tensor to a symmetric form typically assumed in calculations of the viscosity tensor [33–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Taking into account the possible additional terms (important for non rotationally invariant systems [36]), the most general form of the momentum density can be written as [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (7)] g(r, t) = 1 2 � ˆψ†(r, t)ˆp ˆψ(r, t) + � ˆp† ˆψ†(r, t) � ˆψ(r, t) � , (19a) where ˆp is the momentum operator appropriate for the system in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Alternatively, the momentum density can be expressed in terms of the Green’s function Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3a) g1 = − i 2 � ˆp1 + ˆp† 2 � G12 1,2 ��� 2→1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (19b) In conventional systems with the parabolic spectrum the momentum operator has the usual form ˆp = −i∇ and the resulting momentum density (19b) is proportional to the particle number current (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This pro- portionality does not hold in general (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=', in the case of Dirac fermions in graphene [2, 4, 8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Global momentum conservation Following the above procedure, I now apply the operator −(i/2)[ˆp1 + ˆp† 2] to the Dyson equations (3c) and (5b), sum up the results, and take the limit 2 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This yields ∂gi 1 ∂t1 + n1 iˆpi 1U1−iˆp†,i 2 U2 2 ����� 2→1 + ∇j 1τ ji 0 (1) = Ci 1, (20a) where ∇j 1τ ji 0 (1) = ˆpi 1+ ˆp†,i 2 2 � �K1 − �K† 2 � G12 1,2 ����� 2→1 , (20b) and [see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (3) and (5b)] C1 = − ˆp1+ˆp† 2 2 � d3 � Ξ(1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 3) + Ξ∗(2, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 3) ������ 2→1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (20c) Conservation of total momentum can be demonstrated by integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (20a) over the system volume and requiring that the volume integral of the RHS vanishes ∂ ∂t1 � ddr1 g1 + � ddr1 n1 iˆp1U1−iˆp† 2U2 2 ����� 2→1 = 0, � ddr1 C1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (21) The integral nature of the conservation law is consistent with the fact that it is the total momentum of the system that is well defined and conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' A local momentum flux might not be well defined if interactions are long ranged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Φ-derivable approximations and translational invariance The last equality in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (21) represents a constraint on the approximate self energies and Green’s func- tions and can be proven similarly to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Consider a coordinate shift with the operator �TR = eiR·ˆp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (22) Confining the shift to the upper branch of the contour in analogy with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (15), the transformation of the Green’s function can be expressed as ˇG2,1 → 1+ˇτ3 2 �TR ˇG2,1 �T † R 1+ˇτ3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (23a) To the leading order in R, the variation of the Green’s function is given by δ ˇG2,1 =i1+ˇτ3 2 R(t2)ˆp2 ˇG2,1 − iR(t1)ˆp† 1 ˇG2,1 1+ˇτ3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (23b) Substituting this expression into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (14) and requiring that the functional is invariant under the shift of coordinates (since only the system boundaries are shifted) one finds (using the cyclic property of the trace in each term separately) δΦ = i � d1R(t1) ˆp1+ˆp† 3 2 � d2 � Σ11 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='2G11 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3 − Σ12 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='2G21 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3 − G11 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='2Σ11 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3 + G12 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='2Σ21 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3 ���� 3→1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' such that due to arbitrariness of R one arrives at ˆp1+ˆp† 3 2 � d2 � Σ11 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='2G11 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3 − Σ12 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='2G21 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3 − G11 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='2Σ11 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3 + G12 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='2Σ21 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3 ���� 3→1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (24) which is a manifestation of translational invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The expression in the square brackets in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (24) coincides with that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (16b), while the corresponding combination of the self-energies and Green’s functions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (20c) is the same as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Given that the momentum operator does not affect time dependence and hence causality, one can use the same argument as in Section 2 concluding that the terms containing products of two retarded (or two advanced) functions vanish in the limit 2 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Thus, the identity (24) proves the second identity in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (21) and consequently, the integral relation manifesting the momentum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Momentum conservation in the local approximation A local (“differential”) version of the momentum conservation law can not be established without some degree of approximation [23–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Within the kinetic approach, the local continuity equation for the momen- tum density, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (18) is obtained by a straightforward integration of the kinetic equation multiplied by momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This is possible because the distribution function, the central quantity the kinetic theory, is already local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In contrast, “integrating” the Kadanoff-Baym equation (9) leads to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (20a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This is not a continuity equation since the quantity C in the RHS is not a divergence, see also Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' [23, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The Kadanoff-Baym equation (9) can also be derived using the alternative form of the Dyson’s equations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=', Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (3a) and (5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This leads to the expression for the quantity C in terms of the two-particle Green’s function [one could also use the identity (3d) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (20c)] C1 = − i 2 � ddr3 � ˆp1V (r1−r3)−ˆp† 2V (r2−r3) ���� 2→1 � ˆψ† H(r1, t1) ˆψ† H(r3, t1) ˆψH(r3, t1) ˆψH(r1, t1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (25) This form immediately proves that the volume integral of C vanishes, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (21): indeed, the integrand is antisymmetric, which reflects the third Newton’s law [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 8 The quantity C is not a divergence since the interaction potential is nonlocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' However, for short-range interactions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=', for sufficiently screened Coulomb potential in solids) it is possible to construct an effective local interaction stress tensor by integrating C over a large enough volume [23] � V ddr1C1 = − � V ddr1 � ddr3 c+(r1, r3), c+(r1, r3) = −c+(r3, r1), (26a) where c+(r1, r3) is the integrand in C, which can be expressed in terms of single-particle functions due to the identity (3d) c+(r1, r3) = ˆp1+ˆp† 2 2 � dt3 � Ξ(1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 3) + Ξ∗(2, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 3) ������ 2→1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (26b) Since c+(r1, r3) is antisymmetric, the integral over any identical region in r1 and r3 vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Thus the coordinate r3 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (26a) is effectively outside of the volume V , while r1 is inside V and the relative coordinate, r13 = r1 − r3, is restricted by the interaction range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Changing the integration variables in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (26a) to r13 and r3, the integral takes the form [where Vr1 indicates that the integration volume is V in terms of the original variable r1 and R13 = (r1+r3)/2] � V ddr1 � ddr3 c+(r1, r3) = � Vr1 ddr13ddr3 c+(R13+r13/2, R13−r13/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Martin and Schwinger [23] introduced the hypothesis of “local uniformity” where expectation values of field operators within a physically small region depend only on the relative coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Then for a fixed r13 the integration over r3 is restricted to a shell of thickness n·r13, where n is a unit vector normal to the surface of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Now the approximation of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' [23] can be asserted by setting R13 ≈ r3 ≈ const in that shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The integration measure over r3 can be replaced by −n·r13dS3 with the integral covering half the volume in r13 −1 2 � ddr13 � dS3(n·r13)c+(r3+r13/2, r3−r13/2) = −1 2 � dS3ni � ddr13ri 13c+(r3+r13/2, r3−r13/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Now one can invoke the Euler’s theorem and approximate the quantity C by a divergence Ci(r) ≈ −∇jτ ji int, τ ji int = −1 2 � ddr13rj 13ci +(r+r13/2, r−r13/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (27) The quantity τ ji int represents the interaction contribution to the stress-tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Formally, one can arrive at Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (27) by changing the integration variable in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (20c) to r13, expressing the integrand as c+(R13+r13/2, R13−r13/2), and expanding R13 = r1 − r13/2 in r13 such that c+(R13+r13/2, R13−r13/2) ≈ c+(r1+r13/2, r1−r13/2) − 1 2r13·∇1c+(r1+r13/2, r1−r13/2), where the contribution of the first term vanishes due to the asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Summary To summarize this section, the continuity equation for the momentum density (18) can be derived from the microscopic Keldysh approach (within a Φ-derivable approximation) by assuming “local uniformity” within physically small volumes [23] (or the gradient approximation, see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' For short-ranged inter- actions this assumption is clearly compatible with the hydrodynamic approach where one is interested in long-wavelength properties of macroscopic observables (ideally, orders of magnitude longer than any micro- scopic scale [20, 21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' On the other hand, for truly long-ranged interactions a local stress tensor cannot be constructed leaving only the integral manifestation of momentum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Energy conservation The Hamiltonian (1) does not explicitly depend on time and hence is invariant under time translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Hence, energy is conserved and one should be able to express this fact by means of a continuity equation ∂nE ∂t + ∇·jE = −j·∇U, (28) where nE is the energy density and jE is the energy current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Strictly speaking, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (28) is only exact for the case of local interactions, similarly to the case of momentum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Collisions between neutral molecules described by the traditional kinetic theory are typically assumed to be local and hence it is not surprising that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (28) can be straightforwardly obtained by integrating the kinetic equation multiplied by energy [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' At the same time, the kinetic theory description of plasma (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=', a system – or gas – of charged particles) is only approximate and can be justified at high enough temperatures exceeding the average interaction energy or at high enough densities where the average interparticle distance is much smaller than the typical screening radius [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The separation between the two length scales in the problem allows one to distinguish between “collisions” – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' the short-distance scattering processes leading to equilibration and hence contributing to the collision integral – and collective phenomena (involving distances of the order of the screening length) that form the macroscopic fields responsible for the effective Lorentz force appearing in the left-hand side (LHS) of the kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Global energy conservation At the microscopic level, one can account for energy conservation by considering time translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Differentiating the Dyson equations (3c) and (5b) with respect to time and evaluating their difference, one finds � i ∂ ∂t2 � i ∂ ∂t1 − � H(0) 1 � − i ∂ ∂t1 � i ∂ ∂t2 + � H(0),† 2 �� G12 1,2 = (29) = i � d3 ∂ ∂t2 � ΣR 1,3G12 3,2+Σ12 1,3GA 3,2 � + i � d3 ∂ ∂t1 � GR 1,3Σ12 3,2+G12 1,3ΣA 3,2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Consider now the limit 2 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In the LHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (29) one immediately notices � − ∂ ∂t2 ∂ ∂t1 + ∂ ∂t1 ∂ ∂t2 � G12 1,2 ���� 2→1 = 0, and � −i ∂ ∂t2 U1 − i ∂ ∂t1 U2 � G12 1,2 ���� 2→1 = U1 ∂n1 ∂t1 = −U1∇1·j1, (30) where the continuity equation (6) was used in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The remaining term in the LHS contains exactly the same operators as for a non-interacting system and hence can be brought to a form � −i ∂ ∂t2 �K1 − i ∂ ∂t1 �K† 2 � G12 1,2 ���� 2→1 = ∇1·J 1 + ∂E(1) 1 ∂t1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (31) Here E(1) is the “one-particle” (“kinetic”) energy density and J is the corresponding current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Specific expressions for these quantities are determined by the quasiparticle spectrum and are easier to establish in each particular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Note, that the current J does contain an explicit interaction contribution, see below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (31), I may re-write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (29) in the limit 2 → 1 as ∂E(1) 1 ∂t1 + ∇1·J 1 − U1∇1·j1 = Υ1, (32a) 10 where the RHS is denoted by Υ1 = i � ∂ ∂t2 � d3 Ξ(1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 3)− ∂ ∂t1 � d3 Ξ∗(2, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 3) ����� 2→1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (32b) This quantity is related to the interaction potential, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Therefore, it is tempting to associate it with the interaction energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This way the volume integral of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (32a) yields the integral manifestation of energy conservation ∂ ∂t �� ddr E(1) + Eint � = − � ddr j·∇U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (33) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Φ-derivable approximation and time translations To prove the relation between the RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (29) and the interaction energy,consider a change of the time variable on the “upper” branch of the Keldysh contour t → θ(t) = t + ϕ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (34) Together with the change of variable, the Green’s function acquires an additional factor [25] ˇG2,1 → ˇU(t2) ˇG2,1 ˇU(t1), ˇU = � (∂θ/∂t)1/4 0 0 1 � , (35) which is needed to cancel the Jacobian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='1b) so that the functional (12) remains invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Indeed, every time integration in any diagram for the functional (12) involves four Green’s function and hence to cancel the Jacobian, each of them has to be corrected by a factor of (∂θ/∂t)1/4 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Expanding now in the small variation, one finds similarly to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (15) δGij 2,1 = δi1 �ϕ′(t2) 4 +ϕ(t2) ∂ ∂t2 � Gij 2,1 + δj1 �ϕ′(t1) 4 +ϕ(t1) ∂ ∂t1 � Gij 2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (36) The expression (36) should now be substituted into the variation (14) of the functional Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This yields δΦ = −1 4 � d1 ˙ϕ(t1) � d2 � Σ11 1,2G11 2,1−Σ12 1,2G21 2,1+G11 1,2Σ11 2,1−G12 1,2Σ21 2,1 � (37) + � d1 ϕ(t1) � d2 � Σ11 1,2 ∂ ∂t1 G11 2,1−Σ12 1,2 ∂ ∂t1 G21 2,1 + Σ11 2,1 ∂ ∂t1 G11 1,2−Σ21 2,1 ∂ ∂t1 G12 1,2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Since the time translation leaves the functional invariant, the above expression has to be set to zero, δΦ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (37) can be re-written with the help of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (3d), (16c), and (16d) as −i � dt1 ˙ϕ(t1)Eint, where Eint = 1 2 � ddr1 � ddr′� ˆψ† H(r1, t1) ˆψ† H(r′, t1)V (r1−r′) ˆψH(r′, t1) ˆψH(r1, t1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (38) is the interaction energy of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Again, using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (16c) and (16d) one can identify the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (37) with the quantity Υ1, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (32b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This way the variation of Φ takes the form δΦ = −i � dt1 ˙ϕ(t1)Eint(t1) − i � d1 ϕ(t1)Υ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (39) Integrating the first term by parts and using arbitrariness of ϕ(t1), one arrives at the identity ∂ ∂tEint + � ddrΥ = 0, (40) which is a manifestation of energy conservation and the justification for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Energy conservation in the local approximation I now construct the local expression of energy conservation using the same “local uniformity” approxi- mation as in the case of momentum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Starting with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (32a), one should notice that its LHS contains the time derivative of the one-particle energy density only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' To arrive at the total energy density one has to separate the “interaction energy” density from the RHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' To this end, let me re-write Υ with the help of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (3d) as Υ1 = − ∂ ∂t2 � ddr3 � ˆψ†(r2, t2) ˆψ†(r3, t1)V (r1−r3) ˆψ(r3, t1) ˆψ(r1, t1) ����� 2→1 (41) − ∂ ∂t1 � ddr3 � ˆψ†(r2, t2) ˆψ†(r3, t2)V (r1−r3) ˆψ(r3, t2) ˆψ(r1, t1) ����� 2→1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Singling out the interaction energy, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (38), this can be brought to the following form Υ1 = −1 2 ∂ ∂t1 � ddr3 � ˆψ†(r1, t1) ˆψ†(r3, t1)V (r1−r3) ˆψ(r3, t1) ˆψ(r1, t1) � (42) +1 2 � ddr3V (r1−r3) � ˆψ†(r1, t1) ∂ ∂t1 � ˆψ†(r3, t1) ˆψ(r3, t1) � ˆψ(r1, t1) � −1 2 � ddr3V (r1−r3) � ˆψ†(r3, t1) ∂ ∂t1 � ˆψ†(r1, t1) ˆψ(r1, t1) � ˆψ(r3, t1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Introducing the interaction energy density Eint in the first term and using the operator relation (8) in the last two, I find Υ1 = −∂Eint ∂t1 − 1 2 � ddr3V (r1−r3) � ˆψ†(r1, t1)∇3ˆj(r3, t1) ˆψ(r1, t1) � +1 2 � ddr3V (r1−r3) � ˆψ†(r3, t1)∇1ˆj(r1, t1) ˆψ(r3, t1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (43) Substituting the above expression into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (32a) one can now combine the time derivative terms into the derivative of the total energy density, nE = E(1) + Eint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' To proceed further one need to specify the gradient term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (32a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Without loss of generality, I can write the “kinetic” part of the Hamiltonian (1) as �K = ∇·� K, (44) where the vector � K carries the dependence on all additional quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The rational for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (44) is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Kinetic energy describes motion and hence the corresponding operator must not commute with the coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Therefore, �K must be a functional of the gradient operator, ∇, and moreover the formal Taylor series in ∇ must start with the first power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Thus each term in the series is proportional to ∇ leading to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Note that � K may further depend on ∇ as in the case of the usual parabolic spectrum where � K = −∇/(2m), while for Dirac fermions in graphene � K = −ivgσ, where vg is the Fermi velocity and σ is the vector of the Pauli matrices [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (44), one can arrive at the “explicit” expressions for E(1) and J by evaluating the LHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' These read E(1) 1 = −i ∇1·� K2G12 1,2 ��� 2→1 , J 1 = i � ∂ ∂t1 � K2 + ∂ ∂t2 � K1 � G12 1,2 ���� 2→1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (45) Taking into account the explicit form of G12 1,2 in terms of the field operators, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3a) and using the equations of motion (2b) and (4) to remove the time derivative from J ǫ, one finds following expression J 1 = j(1) ǫ (1) + U1j + � ddr3V (r1−r3) � ˆψ†(r3, t1)∇1ˆj(r1, t1) ˆψ(r3, t1) � , (46) 12 where the one-particle contribution to the energy current is given by j(1) ǫ (1) = � (∇2·� K2)� K1 − (∇1·� K1)� K2 � G12 1,2 ��� 2→1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (47) Substituting all of the above results in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (32a) yields ∂nE ∂t + ∇·j(1) ǫ = −j·∇U + ˜Υ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (48) where ˜Υ1 = −1 2 � ddr3V (r1−r3) � ˆψ†(r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' t1)∇3ˆj(r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' t1) ˆψ(r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' t1) � +1 2 � ddr3V (r1−r3) � ˆψ†(r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' t1)∇1ˆj(r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' t1) ˆψ(r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' t1) � −∇1 � ddr3V (r1−r3) � ˆψ†(r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' t1)ˆj(r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' t1) ˆψ(r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' t1) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' which can be re-written in a more symmetric form ˜Υ1 = −1 2∇1 � ddr3V (r1−r3) � ˆψ†(r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' t1)ˆj(r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' t1) ˆψ(r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' t1) � −1 2 � ddr3 [∇1V (r1−r3)] � ˆψ†(r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' t1)ˆj(r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' t1) ˆψ(r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' t1) � −1 2 � ddr3 [∇1V (r1−r3)] � ˆψ†(r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' t1)ˆj(r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' t1) ˆψ(r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' t1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (49) The last two terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (49) are not gradients, but can be brought to a gradient form in the “local uniformity” approximation [23] used above in the case of momentum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Repeating the steps leading to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (27), I arrive at the continuity equation for the energy density (28), where the energy current is defined as jE = j(1) ǫ + 1 2 � ddr3V (r1−r3) � ˆψ†(r3, t1)ˆj(r1, t1) ˆψ(r3, t1) � (50) −1 4 � ddr13r13 � ∇i 13V (r1−r3) � �� ˆψ†(r1+r13/2, t1)ˆji(r1−r13/2, t1) ˆψ(r1+r13/2, t1) � + � ˆψ†(r1−r13/2, t1)ˆji(r1+r13/2, t1) ˆψ(r1−r13/2, t1) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Summary In this section I have derived the continuity equation for the energy density (28) from the microscopic Keldysh approach by assuming “local uniformity” within small physical volumes [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In that sense, the above considerations closely follow the arguments presented in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The difference here is that the resulting expression for the energy current is expressed in terms of a two-particle Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' While it is possible to express the energy current in terms of the derivatives of the self-energy and the single- particle Green’s function using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (3d), the resulting expression is somewhat cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' At the same time, the obtained expression (50) is not immediately related to the kinetic equation since the RHS of the integrated Kadanoff-Baym equation, the quantity Υ, has to be split into the time derivative of the interaction energy and a contribution towards the energy current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This point will be discussed in the subsequent section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Kinetic equation The local continuity equations Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (6), (18), and (28) expressing conservation of the number of particles, momentum, and energy, respectively can be straightforwardly obtained by integrating the Boltzmann kinetic equation [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' As direct consequence of the conservation laws, the collision integral vanishes upon integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In this section, I discuss the relation of this property to the identities Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (16b), (24), and (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Kadanoff-Baym equation Derivation of the kinetic equation is outlined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The calculation is very similar to at least some of the above considerations, with a few notable points that should be clarified here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Firstly, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' [10] begins with a Dyson’s equation for the Keldysh function GK instead of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Combining it with the conjugate equation one finds the analogue of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (9), which has the form �� i ∂ ∂t1 − � H0(1) � + � i ∂ ∂t2 + � H0(2) �� GK 1,2 = � d3 � ΣR 1,3GK 3,2 + ΣK 1,3GA 3,2 − GR 1,3ΣK 3,2 − GK 1,3ΣA 3,2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (51) Now, the Keldysh function is related to the function G12 by the identity GK = 2G12 − iA, (52) where A is the spectral function defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The choice of the Keldysh functon as the “basis” function for the nonequilibrium transport theory [as opposed to G12 which, after all, defines the particle density, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (7)] is justified by the fact [10], that the spectral function does not depend on the state of the system and hence its contribution to macroscopic quantities out of equilibrium is irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Although Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (51) is distinct from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (9), it has the same property [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (11)]: the RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (51) vanishes in the limit 2 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Indeed, combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (9) and (51) according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (52), one finds the equation for the spectral function �� i ∂ ∂t1 − � H(0) 1 � + � i ∂ ∂t2 + � H(0) 2 �� A1,2 = � d3 � ΣR 1,3A3,2 + Γ1,3GA 3,2 − GR 1,3Γ3,2 − A1,3ΣA 3,2 � , (53a) where Γ defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='20) is the self-energy component analogous to the spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Substituting the definitions (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='15) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='20) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (53), one finds for the RHS i � d3 � ΣR 1,3GR 3,2 − ΣA 1,3GA 3,2 − GR 1,3ΣR 3,2 + GA 1,3ΣA 3,2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (53b) This expression vanishes in the limit 2 → 1, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (16d), which proves that the RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (51) vanishes in that limit as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Finally, one can re-write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (51) introducing the quantities A and Γ in the RHS [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Introducing the short-hand notations �D12 = i ∂ ∂t1 + i ∂ ∂t2 − � H(0) 1 + � H(0) 2 , (54a) (A ⊗ B)1,2 = � d3 A1,3B3,2, (54b) � A ⊗, B � − = A ⊗ B − B ⊗ A, � A ⊗, B � + = A ⊗ B + B ⊗ A, (54c) one arrives at the variant of the Kadanoff-Baym equation �D12GK 1,2 = � ReΣ ⊗, GK� − + � ΣK ⊗, ReGR� − + i 2 � ΣK ⊗, A � + − i 2 � Γ ⊗, GK� + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (55) This equation is equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (51) and hence the RHS vanishes in the limit 2 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The RHS of the Kadanoff-Baym equation (9) can be brought to the same form as indicated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Kinetic equation Let me briefly recall the standard steps of the derivation of the kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This can be done in two different ways [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Quasiparticle and quasiclassical approximations The first idea is to apply the gradient approximation to the Kadanoff-Baym equation (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' To do that one first introduces the Wigner representation (using the relative and center of mass coordinates introduced in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='4), see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The Wigner representation is very physical, but unfortunately yields a complicated expression for the convolution, the so-called Moyal product [38] A ⊗ B = ei(∂A ǫ ∂B t −∇A p·∇B r −∂A t ∂B ǫ +∇A r·∇B p )AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (56) The gradient approximation in the Kadanoff-Baym equation is achieved by keeping the first two terms in the Taylor series for the exponential in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (56), which yields � A ⊗, B � − = i [A, B]p , � A ⊗, B � + = 2AB, (57) where [A, B]p = (∂ǫA) (∂tB) − (∇pB)·(∇rB) − (∂tA) (∂ǫB) + (∇rA)·(∇pB) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (58) Applying the above approximation to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (55) one finds � (ǫ − ξp − U − ReΣ) , GK� p − � ΣK, ReGR � p = ΣKA − ΓGK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (59) Here all the derivatives are combined in the LHS, while the remaining RHS can be identified with the collision integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The last step in the derivation is based on a further approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The “quasiparticle approximation” relies on the Kadanoff-Baym solution [24] for the spectral function which can be approximated by a δ- function, A = 2πδ(ǫ − ξp − U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Combined with the corresponding form of the Keldysh Green’s function, GK = −2πihpδ(ǫ − ξp − U), where hp defines the conventional distribution function, fp = (1 − hp)/2, one integrates Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (59) over ǫ and obtains the standard (Boltzmann) kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The collision integral (up to a numerical factor) takes the form I ∝ ΣK[hp] − (ΣR − ΣA)hp and is the function of p, r, and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Summing over all states now amounts to integrating over the momentum variable p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Given that it appears as a Fourier transform inthe relative coordinate, such integration is equivalent to the limit 2 → 1 considered above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Alternatively, one can integrate over ξp (the “quasiclassical approximation”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The idea is that in the case of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=', electro-phonon interaction the self-energy acquires energy dependence and hence the Kadanoff-Baym solution for the spectral function can no lnger be reduced to the above δ-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' However, should the momentum dependence of the self-energy remain weak (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=', due to the Migdal theorem [39]) the spectral function retains the form of a sharp peak in the variable ξp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This relies on the existence of the Fermi surface: upon integration over ξp the Green’s functions are essentially restricted to the Fermi surface and depend only on the orientation of momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' While the quasiclassical approximation does not explicitly require the mixed representation in the time variables, it is often invoked in order to reach the standard form of the kinetic equation [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The procedure becomes rather similar to the previous case and yields the same form of the collision integral, where both the self-energies and the distribution function now depend on ǫ instead of the absolute value of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Assuming the existence of quasiparticles, the two approximations can be related by the formal introduction of the density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' At the same time, the quasiclassical approximation does not rely on the quasiparticle paradigm, which formally is expressed through the fact that the energy and momentum variables are no longer related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' A known limitation of the quasiclassical approximation is its reliance on the particle-hole symmetry [10] which precludes one from describing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=', thermoelectric effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Beyond the quasiclassical approximation The second approach to deriving the kinetic equation does not rely on the quasiparticle approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Instead, one introduces the Ansatz [10] GK = GR ⊗ h − h ⊗ GA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (60) Using this form in the Dyson’s equation (51) and taking into account the diagonal elements of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='11), one arrives at the equation GR ⊗ B − B ⊗ GA = 0, B = �D12h − � ReΣ ⊗, h � − + i 2 � Γ ⊗, h � + + ΣK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (61) Solving this equation to the leading order of the gradient expansion amounts to setting B = 0 which seemingly yields the same form of the collision integral, I ∝ ΣK[h] − (ΣR − ΣA)h, albeit obtained without any recourse to the quasiparticle approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The difference is that here the “distribution function” depends not only on p, r, and t as in the case of the quasiclassical approximation, but also on the energy variable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' h = h(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The quasiclassical (or quasiparticle) approximation allows one to integrate over ǫ using the “δ-peak”-like form of the spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The Ansatz (60) leads to the kinetic equation without any additional assumptions on the form of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' An alternative method of deriving the quantum kinetic equation was suggested in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' [27] on the basis of the observation that all elements of the Keldysh Green’s function matrix could be expressed in terms of two functions only, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='4) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Choosing the spectral function as one of the two and noticing that it becomes real in the Wigner representation, one can introduce another real function in the Wigner representation, h = h(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t), such that G12(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t) = iA(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t)h(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t), G21(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t) = −iA(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t) [1 − h(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (62a) This allows to use the functions A and h to express the Keldysh function GK GK(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t) = −iA(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t) [1 − 2h(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (62b) Expressing the self-energies in a similar way Σ12(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t) = iΓ(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t)γ(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t), Σ21(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t) = −iΓ(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t) [1 − γ(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t)] , (63a) with ΣK(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t) = −iΓ(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t) [1 − 2γ(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t)] , (63b) one can use the new notations to re-write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (55) as �DAh − � ReΣ ⊗, Ah � p − � Γγ ⊗, ReGR� p = ΓA(γ − h), (64) where the RHS is essentially the same form of the collision integral expressed in the new notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The “quantum kinetic equation” (64) was obtained within the leading order of the gradient approximation and hence provides a quantitative condition for its validity |γ − h| ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (65) Consequently, in the LHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (64) one can replace γ by h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The resulting equation corresponds to the Botermans and Malfliet [40] choice of the quantum kinetic equation (as opposed to the original Kadanoff- Baym choice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Both variants are equivalent within the applicability range of the gradient approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In comparison to the Ansatz (60), the variable choice (62) reintroduces the spectral function in the definition of the distribution function h, while leaving the energy dependence of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' On the other hand, the choice (62) is always possible [27], while the Ansatz (60) is guaranteed to be valid only within the gradient approximation [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Kadanoff-Baym equation and the continuity equation Let me now compare the derivation of the continuity equation presented in section 2 to the well-known approach of integrating the kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Particle number conservation is manifested in the traditional kinetic theory by the fact that collision integral vanishes after being summed up over all states [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The quantum kinetic equation, regardless of the variant, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (59), (62b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' and (64), contains also the renormalization terms in the LHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Consequently, the derivation of the kinetic equation consists of making sure that the integral of the collision term vanishes and at the same time that the renormalization terms do not affect the particle density and current [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In contrast, the argument presented in section 2 relies on the single identity, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (11), where taking the limit 2 → 1 is equivalent to integrating the collision integral over all energies and momenta, ǫ and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The combination of the self-energies and Green’s functions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (11) comprises both the collision integral and renormalization terms (before the gradient approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' However, vanishing of these terms together does not in general guarantee that they should vanish individually although this does happen for most common forms of the kinetic equation [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The fact that renormalization does not affect the particle density follows from the operator definition, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Similarly, the current j is determined by the operator form of the continuity equation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (8) and hence cannot be affected by interaction explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This does not mean that the density and current in an interacting system are the same as in non-interacting one: both definitions involve the exact Green’s function G12, which can be very different from the free-particle one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In that sense, vanishing of the renormalization terms ensures consistency of definitions of macroscopic currents and densities in the microscopic and kinetic theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Of course, the total number of particles is the same as in the free system since interaction does not “produce” or “destroy”any particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Finally, let me reiterate that the continuity equation (6) is exact as long as the interaction (and any potential) is expressed in terms of particle density, as is the case with most typical models (electron-electron Coulomb interaction, electron-phonon – or any other boson – coupling, electron-impurity scattering, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The purpose of the identity (11) is to make sure that any approximation made for Green’s functions and self-energies does not violate the conservation law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Kadanoff-Baym equation and momentum conservation The continuity equation for momentum density (18) is the central equation in the hydrodynamic theory eventually yielding the Euler and Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In contrast to the continuity equation (6), the equation (18) is not exact, but is valid within the gradient approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This is not a problem, since hydrodynamics describes long-wavelength variations of macroscopic quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The same gradient approximation is used to derive the kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The equation (18) can then be obtained by multiplying the kinetic equation by momentum and summing over all states without further approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' As a result of this procedure the RHS (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=', the collision integral) of the kinetic equation vanishes which is the manifestation of momentum conservation [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Microscopically, momentum conservation is manifested through the identity (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This directly leads to vanishing of the quantity C integrated over all space and hence to the global (integral) relation (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The quantity C itself emerges from the RHS of the Kadanoff-Baym equation in the limit 2 → 1 which is equivalent of integrating the RHS of the quantum kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The gradient approximation used to derive Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (18) is equivalent to the one needed to derive the quantum kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In particular, separating the integrand c+ into two parts corresponds to the distinction between the collision integral and the renormalization terms in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (59) and (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In this case it can be seen directly that the integrated collision integral vanishes while the renormalization terms contribute to the momentum flux tensor, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The momentum density g is determined by the momentum operator, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (19), and hence is unaffected by renormalizations similarly to the particle number density and current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The derivation presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 3 is thus equivalent to the more standard route of going through the kinetic equation (either the Boltzmann one or quantum), but has the advantage of being free of any additional approximation beyond the gradient expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Taking into account additional interaction that do not conserve momentum amounts to evaluating its contribution to the quantity C in the “0-th” approximation with respect to the gradients, which is equivalent to evaluating the corresponding collision integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Kadanoff-Baym equation and energy conservation Energy conservation is the most difficult part of the presented approach since the energy density at the operator level is essentially a two-particle correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Global energy conservation can be expressed in terms of the integral relation (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' As in the case of momentum conservation, the RHS of the Kadanoff-Baym equation in the limit 2 → 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=', the quantity Υ, determines the time derivative of the interaction energy upon being integrated over all space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' However, now both the energy density and current are renormalized by interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Separating the time derivative of the interaction energy density from Υ leaves the contribution to the energy current that has to be combined with the interaction contribution to the “single-particle” current J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This should be contrasted with the standard kinetic theory derivation [2, 8, 21] where a direct integration of the kinetic equation multiplied by energy yields the energy density and current from the LHS, while the collision integral vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' At the same time, the “internal energy” appears though thermodynamic identities [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This apparent complication in comparing the two approaches is reminiscent of the common practice in conventional hydrodynamics where dissipative processes are taken into account using the entropy flow equation rather than the continuity equation for the energy density [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The entropy flow equation will be discussed in a forthcoming publication [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Recent literature on electronic hydrodynamics in graphene [2–4, 8] devotes little attention to the internal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The role of electron-electron interaction is seen as being responsible for equilibration, although in real materials equilibration is most likely to occur with the help of phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Taking into account electron- phonon interaction leading to energy relaxation [30] would violate the identity (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In the simplest case (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' the arguments of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' [30]), one would have to evaluate the phonon contribution to Υ establishing the weak decay contribution to the continuity equation (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Discussion In this paper I have presented a detailed derivation of the local continuity equations providing the basis of the hydrodynamic theory of electronic transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' While the continuity equation manifesting gauge invariant is exact, the corresponding equations for the momentum and energy density are obtained within the gradient approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The presented derivation is more general than the kinetic theory approach since it relies neither on additional approximations (such as the common quasiparticle or quasiclassical approximations) nor on the concept of the distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Although the latter can be introduced at the quantum level [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=', by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (60) or (62)], it is not always obvious how to generalize this quantity to more complicated cases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=', involving spin-orbit interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Keeping the discussion in coordinate space allows for a direct generalization for systems in confined geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The idea that hydrodynamics is “more general” than the kinetic theory is not new and can be already seen in the original hydrodynamic description of conventional fluids (none of which could be described by a kinetic equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Microscopic expressions for the momentum flux tensor, interaction energy density, and energy current presented here open a direct pathway for evaluating these quantities using specific models of the systems of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In particular, there is already a substantial literature on hydrodynamic approach to “strange” or “bad” metals [42–46], where the excitation spectrum might not contain usual quasiparticles, as could be seen in photoemission [47] and transport [48] experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Some of these materials exhibit resistance that is linear in temperature over a wide range including both low and high temperatures [49], the behavior that has been puzzling the community for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The analysis presented here could be seen as a way of evaluating resistivity directly (similarly to the case of graphene [2–4]) without the need for a Kubo formula and may prove helpful for describing less established systems such as “non-Fermi liquids”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Acknowledgments The author wishes to thank I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Aleiner, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Gornyi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Mirlin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Schmalian, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Shnirman for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This work was supported by the German Research Foundation DFG project NA 1114/5-1 and the European Commission under the EU Horizon 2020 MSCA-RISE-2019 Program (Project 873028 HYDROTRONICS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' 18 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Non-equilibrium (or Keldysh) Green’s function formalism Here I summarize the notations for the Keldysh Green’s functions and their standard relations to keep the paper self-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' For a detailed account of the Keldysh technique see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' [10, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Keldysh Green’s function The central quantity of the formalism is the Green’s function that can be defined either in the Heisenberg (subscript “H”) or “interaction (subscript “I”) representation on the Keldysh contour (subscript “C”) G(1C, 2C) = −i � TC ˆψH(1C) ˆψ† H(2C) � = −i � TC �SC ˆψI(1C) ˆψ† I(2C) � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='1a) where the latter expression retains only the “connected” diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='1), TC is the time-ordering operator on the Keldysh contour and the “scattering matrix” is �SC = TC exp \uf8ee \uf8f0−i � C dtC �Hint(tC) \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='1b) The Green’s function (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='1) can be more conveniently described in the matrix form ˇG1,2 = �G11 1,2 G12 1,2 G21 1,2 G22 1,2 � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='2) where G12 1,2 = i � ˆψ† H(2) ˆψH(1) � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3a) G21 1,2 = −i � ˆψH(1) ˆψ† H(2) � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3b) G11 1,2 = θ(t1 − t2)G21 1,2 + θ(t2 − t1)G12 1,2, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3c) G22 1,2 = θ(t1 − t2)G11 1,2 + θ(t2 − t1)G21 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3d) The four matrix elements are not independent and satisfy ˇG1,2 = −ˇτ1 ˇG† 2,1ˇτ1, Tr ˇG1,2 = Tr ˇτ1 ˇG1,2, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='4a) where ˇτi are the Pauli matrices in the “Keldysh space”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Explicitly, the later relation takes the form G11 1,2 + G22 1,2 = G12 1,2 + G21 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='4b) Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Self-energy The Green’s function obeys the formally exact Dyson’s equation � i ∂ ∂t1 − � H(0) 1 � ˇG1,2 − � d3 ˇΣ1,3ˇτ3 ˇG3,2 = ˇτ3δ1,2, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='5) where the self-energy is the matrix ˇΣ1,2 = �Σ11 1,2 Σ12 1,2 Σ21 1,2 Σ22 1,2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='6) The above definition differs from that in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' [21], where the Pauli matrix in the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='5) precedes the self-energy [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This amounts to the replacement ˇΣ1,2 → ˇτ3 ˇΣ1,2ˇτ3, 19 or simply put, the extra minus sign for the off-diagonal elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This can be made clearer by transforming the integro-differential equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='5) to the integral form using the “free” Green’s function ˇG(0) 1,2 = � i ∂ ∂t1 − �H(0) 1 �−1 ˇτ3δ1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='7) Applying the operator � i∂t1 − � H(0) 1 �−1 to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='5) from the left, one finds ˇG1,2 − � d3d4 ˇG(0) 1,4ˇτ3 ˇΣ4,3ˇτ3 ˇG3,2 = ˇG(0) 1,2, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='8) in contrast to the corresponding equation in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' [21] where there are no Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The rationale for the above notation is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' The self-energy has the same “symmetry” as the Green’s function, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='4) ˇΣ1,2 = −ˇτ1 ˇΣ† 2,1ˇτ1, Tr ˇΣ1,2 = Tr ˇτ1 ˇΣ1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='9a) The latter relation reads Σ11 1,2 + Σ22 1,2 = Σ12 1,2 + Σ21 1,2, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='9b) similarly to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='4b) and without the extra minus sign in the right-hand side as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Keldysh rotation One may try to use the relations (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='4) to reduce the number of Green’s functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' This can be achieved by a “rotation” [10, 50] ˇG1,2 → 1 2 �1 −1 1 1 � ˇτ3 ˇG1,2 � 1 1 −1 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='10a) In the new basis, both the Green’s function and self-energy have the similar form (unlike the form suggested in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' [21]) ˇG = �GR GK 0 GA � , ˇΣ = �ΣR ΣK 0 ΣA � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='10b) In terms of the original Green’s functions, the newly defined functions are given by GR = G11 − G12 = G21 − G22 = θ(t1 − t2) � G21 1,2 − G12 1,2 � , GA = G11 − G21 = G12 − G22 = −θ(t2 − t1) � G21 1,2 − G12 1,2 � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='10c) GK = G12 + G21 = G11 + G22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In the rotated basis, the Dyson’s equation takes the form (same as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' [21]) � i ∂ ∂t1 − � H(0) 1 � ˇG1,2 − � d3 ˇΣ1,3 ˇG3,2 = δ1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='11) The basis rotation does not completely eliminate the redundancy in the definitions of the Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Indeed, the Green’s function in the rotated basis satisfies [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='4)] ˇG1,2 = ˇτ2 ˇG† 2,1ˇτ2, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='12) with the similar constraint on the self-energy [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='9)] ˇΣ1,2 = ˇτ2 ˇΣ† 2,1ˇτ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='13) 20 As a result, only two functions are in either matrix are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Suppose one chooses G12 1,2 and G21 1,2 as such independent functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Then from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='4) it follows that they have the following property � G12 1,2 �∗ = −G12 2,1, � G21 1,2 �∗ = −G21 2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='14) Consider then their difference A1,2 = i � G21 1,2 − G12 1,2 � = i � GR 1,2 − GA 1,2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='15a) As follows from the symmetry of the Green’s functions, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='14), the new function satisfies A∗ 1,2 = A2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='15b) Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Wigner representation The gradient approximation needed to derive quantum kinetic equations is most readily demonstrated in the mixed or Wigner representation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' the Fourier representation in the relative coordinate (the relative and center of mass coordinates were introduced in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='4) A(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' R12, T12) = � ddr12dt12A1,2e−i(pr12−ǫt12), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='16) where t12 = t1 − t2 and T12 = (t1 + t2)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' In the Wigner representation, the spectral function A is real [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='15b)] and satisfies the “sum rule” A∗(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t) = A(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t), ∞ � −∞ dǫ 2πA(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='17) At the same time, it completely determines the retarded and advanced functions through the relation GR(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t)= � GA(p, ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t) �∗ = ∞ � −∞ dǫ′ 2π A(p, ǫ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' r, t) ǫ − ǫ′ + i0+ , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='18) where (ǫ + i0+)−1 is the Fourier transform of the θ-function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='10c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' As a result, one may express these function as GR(A) = Re GR ∓ i 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='19) Similar relations can be defined for the self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Defining the analogue of the spectral function Γ1,2 = i � ΣR 1,2 − ΣA 1,2 � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='20) one finds in the Wigner representation ΣR(A) = ReΣ ∓ i 2Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='21) References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Polini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Geim, Viscous electron fluids, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' Today 73 (6) (2020) 28–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='1063/PT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE0T4oBgHgl3EQfrgGT/content/2301.02567v1.pdf'} +page_content='4497.' metadata={'source': 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quarkonia in the color evaporation +model +A.A. Chernyshev∗ +Samara National Research University, Samara, 443086, Russia +V.A. Saleev† +Samara National Research University, Samara, 443086, Russia and +Joint Institute for Nuclear Research, Dubna, 141980 Russia +Abstract +In the article, we study pair production of heavy quarkonia (J/ψJ/ψ, ΥΥ, ΥJ/ψ) in the im- +proved color evaporation model via the parton Reggeization approach. The last one is based on +high-energy factorization of hard processes in multi-Regge kinematics, the Kimber-Martin-Ryskin- +Watt model for unintegrated parton distribution functions, and the Lipatov effective field theory +of Reggezied gluons and quarks. We compare contributions from the single and double parton +scattering mechanisms in the pair production of heavy quarkonia. The numerical calculations are +performed with the Monte-Carlo event generator KaTie. +∗Electronic address: aachernyshoff@gmail.com +†Electronic address: saleev@samsu.ru +1 +arXiv:2301.04618v1 [hep-ph] 11 Jan 2023 + +I. +INTRODUCTION +Nowadays, pair production of heavy quarkonia in proton-proton collisions at the LHC is +studding intensively in the processes [1–5] +p + p → J/ψ + J/ψ + X, +(1) +p + p → Υ + Υ + X, +(2) +p + p → Υ + J/ψ + X. +(3) +These processes are very interesting for the theoretical study by the following reasons. First, +the production of four heavy quarks is very good test for a perturbative quantum chromo- +dynamics (QCD) [6]. The second ones, such processes of associated production are strongly +depended on the used model of heavy quark hadronization into the heavy quarkonium, even +more than in case of inclusive heavy quarkonium production [7–10]. The last, but not the +least reason is a possibility to study directly double parton scattering (DPS) production +mechanism [11] which contribution to the pair production cross-section estimates as more +large in compare with the conventional single parton scattering (SPS) scenario. To describe +in the collinear parton model (CPM) differential cross-section of the pair heavy quarkonia +with non-zero total transverse momenta, it is needed to take into account next-to-leading +order subprocesses with at least five final particles, like this g+g → c+¯c+b+¯b+g. It is very +complicated computing task in the pQCD and CPM. Taking in mind the last sentence, we +should use the high-energy factorization (HEF) approach or kTfactorization approach [12– +14] which have used successfully to describe a lot of data at the energy of the LHC. In the +HEF, the initial partons have non-zero transverse momenta and pair production of heavy +quarkonia may be described already in the leading order approximation of pQCD in the +subprocesses of gluon-gluon fusion via SPS production mechanism +R + R → c + ¯c + c + ¯c, +(4) +R + R → b + ¯b + b + ¯b, +(5) +R + R → c + ¯c + b + ¯b. +(6) +2 + +or quark-antiquark annihilation +Qq + ¯Qq → c + ¯c + c + ¯c, +(7) +Qq + ¯Qq → b + ¯b + b + ¯b, +(8) +Qq + ¯Qq → c + ¯c + b + ¯b. +(9) +Here R is the Reggeized gluon and Qq is the Reggeized quark (antiquark) with four-momenta +qµ = xP µ + qµ +T, P µ is the relevant proton momentum, qµ +T = (0, qTµ, 0) and q2 = q2 +T = +−qT2, q = u, d, s. We use HEF as it is formulated in the parton Reggeization approach +(PRA) [15–17]. Such approach has been used recently [18] for the description of the pair +J/ψ production cross section in the improved color evaporation (ICEM) model [9, 10] with +MC event generator KaTie [19]. At the presented paper, we study associated production of +ΥΥ and ΥJ/ψ in the similar manner. The results for pair J/ψ production are also presented +for completeness. +II. +THEORETICAL BASICS +A. +Parton Reggeization approach +The PRA is based on the HEF factorization justified in the leading logarithmic approxi- +mation of the QCD at high energies [12–14]. Dependent on transverse momentum, uninte- +grated parton distribution functions (unPDF) of Reggeized quarks and gluons are calculated +in the model proposed earlier by Kimber, Martin, Ryskin and Watt (KMRW)[20, 21], but +with our sufficient modifications [17]. Reggeized parton amplitudes are constructed accord- +ing to the Feynman rules of the L.N. Lipatov effective field theory (EFT) of Reggeized +gluons and quarks [22, 23]. A review of the PRA can be found in Refs. [15–17]. Inclusion of +high-order corrections in the PRA was studied in the Refs. [24–27]. +In the PRA, the cross section of the inclusive process p+p → Q+X via SPS mechanism +is related to the cross section of the parton subprocess by the factorization formula +dσSPS = +� +i,¯j +1 +� +0 +dx1 +x1 +� d2qT1 +π +Φi(x1, t1, µ2) +1 +� +0 +dx2 +x2 +� d2qT2 +π +Φj(x2, t2, µ2) · dˆσPRA, +(10) +where t1,2 = −q2 +T1,2, the cross section of the subprocess with Reggeized partons ˆσPRA is +expressed in terms of squared Reggeized amplitudes |APRA|2. The PRA hard-scattering +3 + +amplitudes are gauge invariant because the initial-state off-shell partons are considered as +Reggeized partons of the gauge-invariant EFT for QCD processes in the MRK limit [22, 23]. +The unPDFs in the modified KMRW model are calculated by the formula [17] +Φi(x, t, µ) = αs(µ) +2π +Ti(t, µ2, x) +t +� +j=q,¯q,g +1 +� +x +dz Pij(z)Fj +�x +z , t +� +θ (∆(t, µ) − z) , +(11) +where Fi(x, µ2 +F) = xfj(x, µ2 +F). Here and below, factorization and renormalization scales are +equal, µF = µR = µ, and ∆(t, µ2) = +√ +t/( +� +µ2 + +√ +t) is the KMRW-cutoff function [20]. The +modified unPDF Φi(x, t, µ) should be satisfied exact normalization condition: +µ2 +� +0 +dtΦi(x, t, µ2) = Fi(x, µ2), +(12) +or +Φi(x, t, µ2) = d +dt +� +Ti(t, µ2, x)Fi(x, t) +� +, +(13) +where Ti(t, µ2, x) is the Sudakov form–factor, Ti(t = 0, µ2, x) = 0 and Ti(t = µ2, µ2, x) = 1. +The explicit form of the Sudakov form factor in the (13) was first obtained in [17]: +Ti(t, µ2, x) = exp +� +��− +µ2 +� +t +dt′ +t′ +αs(t′) +2π +� +τi(t′, µ2) + ∆τi(t′, µ2, x) +� +� +�� , +(14) +where +τi(t, µ2) = +� +j +1 +� +0 +dz zPji(z)θ(∆(t, µ2) − z), +∆τi(t, µ2, x) = +� +j +1 +� +0 +dz θ(z − ∆(t, µ2)) +� +zPji(z) − Fj +� x +z, t +� +Fi(x, t) Pij(z)θ(z − x) +� +. +In contrast to the KMRW model, the Sudakov form factor (14) depends on x, which is +necessary to preserve the exact normalization (12) for any x and µ. The gauge invariance of +amplitudes with Reggeized partons in the PRA guaranteed allows you to study any processes +described non-Abelian QCD structures. PRA has been successfully used for descriptions of +angular correlations in two-jet events [15], production of the charm [28, 29] and beauty +mesons [16, 30], charmonium in the NRQCD [31, 32]. +4 + +B. +Improved color evaporation model +The relevant description of the ICEM can be found in the Ref. [33]. In the PRA, the +pT-spectra of single J/ψ(Υ) is possible at the leading order approximation of pQCD in the +parton subprocesses +R + R → c(b) + ¯c(¯b) +(15) +and +Qq + ¯Qq → c(b) + ¯c(¯b), +(16) +where q = u, d, s. +In the ICEM, the cross-section for the production of prompt J/ψ(Υ)-mesons is related +to the cross-section for the production of c¯c(b¯b) pairs in the single parton scattering (SPS) +as follows: +σSPS(p + p → J/ψ(Υ) + X) = Fψ(Υ) × +� 2mD(2mB) +mψ(mΥ) +dσ(p + p → c(b) + ¯c(¯b) + X) +dM +dM, (17) +where M is the invariant mass of the c¯c(b¯b) pair with 4–momentum pµ +c¯c(b¯b) = pµ +c(c) + pµ +¯c(¯b)), +mψ,Υ is the mass of the J/ψ(Υ) meson and mD(mB) is the mass of the lightest D(B) +meson. To take into account the kinematical effect associated with the difference between +the masses of the intermediate state and the final heavy quarkonium, the 4–momentum +of c¯c(b¯b) pair and J/ψ(Υ) meson are related by pµ +ψ(Υ) = (mψ(Υ)/M) pµ +c¯c(b¯b). The universal +parameter Fψ(Υ) is considered as a probability of transformation of the c¯c(b¯b) pair with +invariant mass mψ(Υ) < M < 2mD(B) into the prompt J/ψ(Υ) meson. +The cross section for the production of a pair of prompt J/ψ mesons (ΥΥ or ΥJ/ψ) +via the SPS is related to the cross section for the production of two pairs c¯c quarks in the +following way +σSPS(p + p → J/ψ + J/ψ + X) = +(18) += Fψψ × +� 2mD +mψ +� 2mD +mψ +dσ(p + p → c1 + ¯c1 + c2 + ¯c2 + X) +dM1dM2 +dM1dM2, +where M1,2 are the invariant masses of c¯c pairs with 4–momenta pµ +c¯c1 = pµ +c1 + pµ +¯c1 and pµ +c¯c2 = +pµ +c2 + pµ +¯c2. Parameter Fψψ is the probability of transformation of two pairs c¯c with invariant +masses mψ < M1,2 < 2mD into two J/ψ mesons. +5 + +In the DPS approach [11], the cross section for the production of a J/ψ pair is written in +terms of the cross sections for the production of single a J/ψ in two independent subprocesses +σDPS(p+p → J/ψ+J/ψ+X) = σSPS(p + p → J/ψ + X1) × σSPS(p + p → J/ψ + X2) +Nfσeff +, (19) +where Nf = 2 for the J/ψJ/ψ(ΥΥ) pair production, Nf = 1 for the J/ψΥ pair production +and the parameter σeff, which controls the contribution of the DPS mechanism, is considered +as free parameter. Thus, at fitting cross sections for the pair J/ψ(Υ)–meson production, +we assume that the parameter Fψ,Υ is fixed, and the parameters Fψψ,ΥΥ and σeff are free +parameters. +III. +NUMERICAL CALCULATIONS AND RESULTS +Recently, a new approach to obtaining gauge invariant amplitudes with off-shell initial- +state partons in hard subprocesses at high energies has been proposed. +The method is +based on the use of spinor amplitude formalism and recurrence relations of the BCFW +type [34, 35]. As it was demonstrated recently, this formalism [34, 35], which is implemented +in event generator KaTie, for numerical amplitude generation is equivalent to amplitudes +built according to Feynman rules of the Lipatov EFT at the level of tree diagrams [15, 16, 36]. +Such a way, at the stage of numerical calculations, we use the Monte-Carlo event generator +KaTie [19] for calculating the proton-proton cross sections, as well as it was done previously +in Ref. [18]. The accuracy of numerical calculations for total proton-proton cross sections is +equal to 0.1%. +A. +Associated J/ψJ/ψ production +The complete results for J/ψJ/ψ pair production is collected in our recent paper [18]. +Here we present only main ones. We obtain a quite satisfactory description for the single +prompt J/ψ pT−spectra and cross sections in the ICEM using the PRA at the wide range +of the collision energy. The obtained values of the hadronization parameter Fψ depend +on energy, and such dependence can be approximated by the formula Fψ(√s) = 0.012 + +0.952(√s)−0.525, see the Fig. 1. The data for the pair J/ψ production cross sections at the +6 + +energy range 7 − 13 TeV can be fitted self-consistently with two free parameters Fψψ and +σeff. We have found the best fit with Fψψ ≃ 0.02 and σeff ≃ 11.0 mb, see the Fig. 2, were +x = +n +� +k=1 +|σexp +k +− σtheor +k +| +∆σexp +k +and the sum is taken over all cross sections of three experiments: CMS [1], ATLAS [2] and +LHCb [3]. It is interesting that we find Fψψ ≈ Fψ [18]. Complete set of the plots for +differential cross sections in J/ψJ/ψ pair production are presented in the Ref. [18]. +B. +Associated ΥΥ production +As in the case of single prompt J/ψ production, we obtain a quite satisfactory description +for the single prompt Υ(1S) total cross sections in the ICEM using the PRA at the wide +range of the collision energy [37–45]. The obtained values of the hadronization probability +FΥ depend on energy by the formula FΥ(√s) = 0.012 + 4.166(√s)−0.677, see the Fig. 3. +Pair ΥΥ production cross-section was measured by the CMS Collaboration at the energy +√s = 13 TeV [46]. Taking into account the contributions of the SPS and the DPS production +mechanisms, parameter FΥΥ is obtained by the fit of the rapidity difference spectrum and the +azimuthal angle difference spectrum, as it is shown in Figs. 4 and 5. Here, the parameter +FΥ is fixed by the fitting of single Υ production and the universal parameter is taken +σeff = 11.0 ± 0.2 mb, as it was obtained by the fitting of single and pair J/ψ production +cross-sections. We find that FΥΥ ≃ FΥ ≃ 0.05, the same as it is for pair J/ψ production. +Such a way, as it is shown in Figs. 4 and 5, SPS contribution in the pair Υ production +cross-section is about 10 % only and DPS contribution dominates. +C. +Associated ΥJ/ψ production +The ΥJ/ψ pair production cross section and the azimuthal angle difference spectrum +were measured by D0 Collaboration at the energy √s = 1.96 TeV [5]. When the parameters +σeff = 11 mb, FΥ = 0.039, and Fψ = 0.044, have been fixed, we predict the hadronization +probability FψΥ approximately equal to Fψ × FΥ = 0.002. At this values of the hadroniza- +tion probabilities, the SPS contribution in J/ψΥ pair production is negligibly small, see +Fig. 6. +7 + +IV. +CONCLUSIONS +We obtain a quite satisfactory description for the single prompt J/ψ and Υ production +pT−spectra and cross sections in the ICEM using the PRA at the wide range of the collision +energy. It is obtained that the values of the hadronization probability Fψ,Υ are depended +on energy. +Both mechanisms, SPS and DPS, for the pair J/ψ and Υ pair production have been +considered. The data for the pair J/ψ and Υ production cross sections at the energy range +7 − 13 TeV can be fitted self-consistently with free parameters Fψψ, FΥΥ and σeff. We have +found the best fit with σeff ≃ 11.0 ± 0.2 mb, which is in a good agreement with another +theoretical estimates. We find the dominant role of the DPS mechanism in ΥΥ and J/ψΥ +pair production. In case of pair J/ψ production, the DPS mechanism absolutely dominates +only at the forward region of J/ψ rapidities. +The important phenomenological finding is that Fψψ ≃ Fψ ̸= Fψ × Fψ and FΥΥ ≃ +FΥ ̸= FΥ × FΥ, but FΥψ ≃ FΥ × Fψ . This result demonstrate the important property of +quantum identity and Pauli principle for quarks. +Acknowledgments +We are grateful to A. Van Hameren for advice on the program KaTie. The work was +supported by the Ministry of Science and Higher Education of the Russian Federation, +project FSSS-2020-0014. +8 + +101 +102 +103 +104 +√s [GeV] +0.1 +0.2 +0.3 +F ψ +a + b√s−c +(a, b, c) = (0.012, 0.952, 0.525) +NA3 - 1983 +AFS - 1980 +AFS - 1980 +PHENIX - 2012 +CDF - 2005 +LHCb - 2021 +ALICE - 2012 +ATLAS - 2011 +CMS - 2012 +LHCb - 2015 +FIG. 1: The hadronization probability Fψ as a function of proton collision energy √s. The corridor +between the upper and lower lines demonstrates the uncertainty from the hard scale variation by +the factor ξ = 2 and the c-quark mass from 1.2 to 1.4 GeV. +9.0 +9.5 +10.0 +10.5 +11.0 +11.5 +12.0 +12.5 +σeff [mb] +0.018 +0.019 +0.02 +0.021 +0.022 +0.023 +0.024 +F ψψ +x = � +k +|σexp +k −σtheor +k +| +∆σexp +k +x = 2.0 +x = 1.5 +x = 1.0 +FIG. 2: +Regions of the probabilities Fψψ and σeff in the ICEM for pair J/ψ production, obtained +as a result of data fitting. Isolines correspond to x = 1.0, 1.5 and 2.0. +9 + +102 +103 +104 +√s [GeV] +0.1 +0.2 +0.3 +0.4 +F Υ +a + b√s−c +(a, b, c) = (0.012, 4.166, 0.677) +AFS - 1980 +CDF - 2002 +LHCb - 2014 +ATLAS - 2013 +CMS - 2011 +LHCb - 2015 +ALICE - 2016 +LHCb - 2015 +LHCb - 2018 +FIG. 3: +The hadronization probability FΥ as a function of proton collision energy √s. The +corridor between the upper and lower lines demonstrates the uncertainty from the hard scale +variation by the factor ξ = 2 and the b-quark mass from 4.5 to 4.75 GeV. +0 +1 +2 +3 +4 +|∆yΥΥ| +10−1 +100 +101 +102 +103 +dσ +d|∆yΥΥ| [pb] +F ΥΥ = 0.005 +F Υ = 0.007 +σeff = 11 mb +√s = 13 TeV, |yΥ| < 2.0 +LO PRA, ICEM +SUM +SPS +DPS +CMS - 2020 +FIG. 4: +Different spectra of pair Υ production as a function of rapidity difference |∆yΥΥ|. The +data are from CMS collaboration at the √s = 13 TeV [46]. +10 + +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +|∆φΥΥ| +10−1 +100 +101 +102 +103 +dσ +d|∆φΥΥ| [pb] +F ΥΥ = 0.005 +F Υ = 0.007 +σeff = 11 mb +√s = 13 TeV, |yΥ| < 2.0 +LO PRA, ICEM +SUM +SPS +DPS +CMS - 2020 +FIG. 5: +Different spectra of pair Υ production as a function of azimuthal angle difference |∆φΥΥ|. +The data are from CMS collaboration at the √s = 13 TeV [46]. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +|∆φψΥ| +10−3 +10−2 +10−1 +100 +101 +102 +103 +B(J/ψ → µ+µ−) × B(Υ → µ+µ−) × dσ/d|∆φψΥ| [fb] +F ψ = 0.044 +F Υ = 0.039 +σeff = 11 mb +¯p + p → J/ψ + Υ(1S) + X +√s = 1.96 TeV +LO PRA, ICEM +SUM +SPS +DPS +D∅ - 2016 +FIG. 6: +Different spectra of pair J/ψΥ production as a function of azimuthal angle difference +|∆φJ/ψΥ|. The data are from D0 collaboration at the √s = 1.96 TeV [5]. +11 + +[1] V. Khachatryan et al. (CMS), JHEP 09, 094 (2014). +[2] M. Aaboud et al. (ATLAS), Eur. Phys. J. C 77, 76 (2017). +[3] R. Aaij et al. 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B 808, 135578 (2020). +13 + diff --git a/z9E3T4oBgHgl3EQfmwoZ/content/tmp_files/load_file.txt b/z9E3T4oBgHgl3EQfmwoZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b90749b8caff54bf66cf5709ccb38c81d102f726 --- /dev/null +++ b/z9E3T4oBgHgl3EQfmwoZ/content/tmp_files/load_file.txt @@ -0,0 +1,497 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf,len=496 +page_content='Pair production of heavy quarkonia in the color evaporation model A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Chernyshev∗ Samara National Research University, Samara, 443086, Russia V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Saleev† Samara National Research University, Samara, 443086, Russia and Joint Institute for Nuclear Research, Dubna, 141980 Russia Abstract In the article, we study pair production of heavy quarkonia (J/ψJ/ψ, ΥΥ, ΥJ/ψ) in the im- proved color evaporation model via the parton Reggeization approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The last one is based on high-energy factorization of hard processes in multi-Regge kinematics, the Kimber-Martin-Ryskin- Watt model for unintegrated parton distribution functions, and the Lipatov effective field theory of Reggezied gluons and quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' We compare contributions from the single and double parton scattering mechanisms in the pair production of heavy quarkonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The numerical calculations are performed with the Monte-Carlo event generator KaTie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' ∗Electronic address: aachernyshoff@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='com †Electronic address: saleev@samsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='ru 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='04618v1 [hep-ph] 11 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' INTRODUCTION Nowadays, pair production of heavy quarkonia in proton-proton collisions at the LHC is studding intensively in the processes [1–5] p + p → J/ψ + J/ψ + X, (1) p + p → Υ + Υ + X, (2) p + p → Υ + J/ψ + X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' (3) These processes are very interesting for the theoretical study by the following reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' First, the production of four heavy quarks is very good test for a perturbative quantum chromo- dynamics (QCD) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The second ones, such processes of associated production are strongly depended on the used model of heavy quark hadronization into the heavy quarkonium, even more than in case of inclusive heavy quarkonium production [7–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The last, but not the least reason is a possibility to study directly double parton scattering (DPS) production mechanism [11] which contribution to the pair production cross-section estimates as more large in compare with the conventional single parton scattering (SPS) scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' To describe in the collinear parton model (CPM) differential cross-section of the pair heavy quarkonia with non-zero total transverse momenta, it is needed to take into account next-to-leading order subprocesses with at least five final particles, like this g+g → c+¯c+b+¯b+g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' It is very complicated computing task in the pQCD and CPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Taking in mind the last sentence, we should use the high-energy factorization (HEF) approach or kTfactorization approach [12– 14] which have used successfully to describe a lot of data at the energy of the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' In the HEF, the initial partons have non-zero transverse momenta and pair production of heavy quarkonia may be described already in the leading order approximation of pQCD in the subprocesses of gluon-gluon fusion via SPS production mechanism R + R → c + ¯c + c + ¯c, (4) R + R → b + ¯b + b + ¯b, (5) R + R → c + ¯c + b + ¯b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' (6) 2 or quark-antiquark annihilation Qq + ¯Qq → c + ¯c + c + ¯c, (7) Qq + ¯Qq → b + ¯b + b + ¯b, (8) Qq + ¯Qq → c + ¯c + b + ¯b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' (9) Here R is the Reggeized gluon and Qq is the Reggeized quark (antiquark) with four-momenta qµ = xP µ + qµ T, P µ is the relevant proton momentum, qµ T = (0, qTµ, 0) and q2 = q2 T = −qT2, q = u, d, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' We use HEF as it is formulated in the parton Reggeization approach (PRA) [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Such approach has been used recently [18] for the description of the pair J/ψ production cross section in the improved color evaporation (ICEM) model [9, 10] with MC event generator KaTie [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' At the presented paper, we study associated production of ΥΥ and ΥJ/ψ in the similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The results for pair J/ψ production are also presented for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' THEORETICAL BASICS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Parton Reggeization approach The PRA is based on the HEF factorization justified in the leading logarithmic approxi- mation of the QCD at high energies [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Dependent on transverse momentum, uninte- grated parton distribution functions (unPDF) of Reggeized quarks and gluons are calculated in the model proposed earlier by Kimber, Martin, Ryskin and Watt (KMRW)[20, 21], but with our sufficient modifications [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Reggeized parton amplitudes are constructed accord- ing to the Feynman rules of the L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Lipatov effective field theory (EFT) of Reggeized gluons and quarks [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' A review of the PRA can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Inclusion of high-order corrections in the PRA was studied in the Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' [24–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' In the PRA, the cross section of the inclusive process p+p → Q+X via SPS mechanism is related to the cross section of the parton subprocess by the factorization formula dσSPS = � i,¯j 1 � 0 dx1 x1 � d2qT1 π Φi(x1, t1, µ2) 1 � 0 dx2 x2 � d2qT2 π Φj(x2, t2, µ2) · dˆσPRA, (10) where t1,2 = −q2 T1,2, the cross section of the subprocess with Reggeized partons ˆσPRA is expressed in terms of squared Reggeized amplitudes |APRA|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The PRA hard-scattering 3 amplitudes are gauge invariant because the initial-state off-shell partons are considered as Reggeized partons of the gauge-invariant EFT for QCD processes in the MRK limit [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The unPDFs in the modified KMRW model are calculated by the formula [17] Φi(x, t, µ) = αs(µ) 2π Ti(t, µ2, x) t � j=q,¯q,g 1 � x dz Pij(z)Fj �x z , t � θ (∆(t, µ) − z) , (11) where Fi(x, µ2 F) = xfj(x, µ2 F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Here and below, factorization and renormalization scales are equal, µF = µR = µ, and ∆(t, µ2) = √ t/( � µ2 + √ t) is the KMRW-cutoff function [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The modified unPDF Φi(x, t, µ) should be satisfied exact normalization condition: µ2 � 0 dtΦi(x, t, µ2) = Fi(x, µ2), (12) or Φi(x, t, µ2) = d dt � Ti(t, µ2, x)Fi(x, t) � , (13) where Ti(t, µ2, x) is the Sudakov form–factor, Ti(t = 0, µ2, x) = 0 and Ti(t = µ2, µ2, x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The explicit form of the Sudakov form factor in the (13) was first obtained in [17]: Ti(t, µ2, x) = exp � ��− µ2 � t dt′ t′ αs(t′) 2π � τi(t′, µ2) + ∆τi(t′, µ2, x) � � �� , (14) where τi(t, µ2) = � j 1 � 0 dz zPji(z)θ(∆(t, µ2) − z), ∆τi(t, µ2, x) = � j 1 � 0 dz θ(z − ∆(t, µ2)) � zPji(z) − Fj � x z, t � Fi(x, t) Pij(z)θ(z − x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' In contrast to the KMRW model, the Sudakov form factor (14) depends on x, which is necessary to preserve the exact normalization (12) for any x and µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The gauge invariance of amplitudes with Reggeized partons in the PRA guaranteed allows you to study any processes described non-Abelian QCD structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' PRA has been successfully used for descriptions of angular correlations in two-jet events [15], production of the charm [28, 29] and beauty mesons [16, 30], charmonium in the NRQCD [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Improved color evaporation model The relevant description of the ICEM can be found in the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' In the PRA, the pT-spectra of single J/ψ(Υ) is possible at the leading order approximation of pQCD in the parton subprocesses R + R → c(b) + ¯c(¯b) (15) and Qq + ¯Qq → c(b) + ¯c(¯b), (16) where q = u, d, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' In the ICEM, the cross-section for the production of prompt J/ψ(Υ)-mesons is related to the cross-section for the production of c¯c(b¯b) pairs in the single parton scattering (SPS) as follows: σSPS(p + p → J/ψ(Υ) + X) = Fψ(Υ) × � 2mD(2mB) mψ(mΥ) dσ(p + p → c(b) + ¯c(¯b) + X) dM dM, (17) where M is the invariant mass of the c¯c(b¯b) pair with 4–momentum pµ c¯c(b¯b) = pµ c(c) + pµ ¯c(¯b)), mψ,Υ is the mass of the J/ψ(Υ) meson and mD(mB) is the mass of the lightest D(B) meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' To take into account the kinematical effect associated with the difference between the masses of the intermediate state and the final heavy quarkonium, the 4–momentum of c¯c(b¯b) pair and J/ψ(Υ) meson are related by pµ ψ(Υ) = (mψ(Υ)/M) pµ c¯c(b¯b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The universal parameter Fψ(Υ) is considered as a probability of transformation of the c¯c(b¯b) pair with invariant mass mψ(Υ) < M < 2mD(B) into the prompt J/ψ(Υ) meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The cross section for the production of a pair of prompt J/ψ mesons (ΥΥ or ΥJ/ψ) via the SPS is related to the cross section for the production of two pairs c¯c quarks in the following way σSPS(p + p → J/ψ + J/ψ + X) = (18) = Fψψ × � 2mD mψ � 2mD mψ dσ(p + p → c1 + ¯c1 + c2 + ¯c2 + X) dM1dM2 dM1dM2, where M1,2 are the invariant masses of c¯c pairs with 4–momenta pµ c¯c1 = pµ c1 + pµ ¯c1 and pµ c¯c2 = pµ c2 + pµ ¯c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Parameter Fψψ is the probability of transformation of two pairs c¯c with invariant masses mψ < M1,2 < 2mD into two J/ψ mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 5 In the DPS approach [11], the cross section for the production of a J/ψ pair is written in terms of the cross sections for the production of single a J/ψ in two independent subprocesses σDPS(p+p → J/ψ+J/ψ+X) = σSPS(p + p → J/ψ + X1) × σSPS(p + p → J/ψ + X2) Nfσeff , (19) where Nf = 2 for the J/ψJ/ψ(ΥΥ) pair production, Nf = 1 for the J/ψΥ pair production and the parameter σeff, which controls the contribution of the DPS mechanism, is considered as free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Thus, at fitting cross sections for the pair J/ψ(Υ)–meson production, we assume that the parameter Fψ,Υ is fixed, and the parameters Fψψ,ΥΥ and σeff are free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' NUMERICAL CALCULATIONS AND RESULTS Recently, a new approach to obtaining gauge invariant amplitudes with off-shell initial- state partons in hard subprocesses at high energies has been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The method is based on the use of spinor amplitude formalism and recurrence relations of the BCFW type [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' As it was demonstrated recently, this formalism [34, 35], which is implemented in event generator KaTie, for numerical amplitude generation is equivalent to amplitudes built according to Feynman rules of the Lipatov EFT at the level of tree diagrams [15, 16, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Such a way, at the stage of numerical calculations, we use the Monte-Carlo event generator KaTie [19] for calculating the proton-proton cross sections, as well as it was done previously in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The accuracy of numerical calculations for total proton-proton cross sections is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Associated J/ψJ/ψ production The complete results for J/ψJ/ψ pair production is collected in our recent paper [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Here we present only main ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' We obtain a quite satisfactory description for the single prompt J/ψ pT−spectra and cross sections in the ICEM using the PRA at the wide range of the collision energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The obtained values of the hadronization parameter Fψ depend on energy, and such dependence can be approximated by the formula Fψ(√s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='012 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='952(√s)−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='525, see the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The data for the pair J/ψ production cross sections at the 6 energy range 7 − 13 TeV can be fitted self-consistently with two free parameters Fψψ and σeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' We have found the best fit with Fψψ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='02 and σeff ≃ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0 mb, see the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 2, were x = n � k=1 |σexp k − σtheor k | ∆σexp k and the sum is taken over all cross sections of three experiments: CMS [1], ATLAS [2] and LHCb [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' It is interesting that we find Fψψ ≈ Fψ [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Complete set of the plots for differential cross sections in J/ψJ/ψ pair production are presented in the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Associated ΥΥ production As in the case of single prompt J/ψ production, we obtain a quite satisfactory description for the single prompt Υ(1S) total cross sections in the ICEM using the PRA at the wide range of the collision energy [37–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The obtained values of the hadronization probability FΥ depend on energy by the formula FΥ(√s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='012 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='166(√s)−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='677, see the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Pair ΥΥ production cross-section was measured by the CMS Collaboration at the energy √s = 13 TeV [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Taking into account the contributions of the SPS and the DPS production mechanisms, parameter FΥΥ is obtained by the fit of the rapidity difference spectrum and the azimuthal angle difference spectrum, as it is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Here, the parameter FΥ is fixed by the fitting of single Υ production and the universal parameter is taken σeff = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='2 mb, as it was obtained by the fitting of single and pair J/ψ production cross-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' We find that FΥΥ ≃ FΥ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='05, the same as it is for pair J/ψ production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Such a way, as it is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 4 and 5, SPS contribution in the pair Υ production cross-section is about 10 % only and DPS contribution dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Associated ΥJ/ψ production The ΥJ/ψ pair production cross section and the azimuthal angle difference spectrum were measured by D0 Collaboration at the energy √s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='96 TeV [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' When the parameters σeff = 11 mb, FΥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='039, and Fψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='044, have been fixed, we predict the hadronization probability FψΥ approximately equal to Fψ × FΥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' At this values of the hadroniza- tion probabilities, the SPS contribution in J/ψΥ pair production is negligibly small, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 7 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' CONCLUSIONS We obtain a quite satisfactory description for the single prompt J/ψ and Υ production pT−spectra and cross sections in the ICEM using the PRA at the wide range of the collision energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' It is obtained that the values of the hadronization probability Fψ,Υ are depended on energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Both mechanisms, SPS and DPS, for the pair J/ψ and Υ pair production have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The data for the pair J/ψ and Υ production cross sections at the energy range 7 − 13 TeV can be fitted self-consistently with free parameters Fψψ, FΥΥ and σeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' We have found the best fit with σeff ≃ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='2 mb, which is in a good agreement with another theoretical estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' We find the dominant role of the DPS mechanism in ΥΥ and J/ψΥ pair production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' In case of pair J/ψ production, the DPS mechanism absolutely dominates only at the forward region of J/ψ rapidities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The important phenomenological finding is that Fψψ ≃ Fψ ̸= Fψ × Fψ and FΥΥ ≃ FΥ ̸= FΥ × FΥ, but FΥψ ≃ FΥ × Fψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' This result demonstrate the important property of quantum identity and Pauli principle for quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Acknowledgments We are grateful to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Van Hameren for advice on the program KaTie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The work was supported by the Ministry of Science and Higher Education of the Russian Federation, project FSSS-2020-0014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 8 101 102 103 104 √s [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='3 F ψ a + b√s−c (a, b, c) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='012, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='952, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='525) NA3 - 1983 AFS - 1980 AFS - 1980 PHENIX - 2012 CDF - 2005 LHCb - 2021 ALICE - 2012 ATLAS - 2011 CMS - 2012 LHCb - 2015 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 1: The hadronization probability Fψ as a function of proton collision energy √s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The corridor between the upper and lower lines demonstrates the uncertainty from the hard scale variation by the factor ξ = 2 and the c-quark mass from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='5 σeff [mb] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='024 F ψψ x = � k |σexp k −σtheor k | ∆σexp k x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0 x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='5 x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 2: Regions of the probabilities Fψψ and σeff in the ICEM for pair J/ψ production, obtained as a result of data fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' Isolines correspond to x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 9 102 103 104 √s [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='4 F Υ a + b√s−c (a, b, c) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='012, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='166, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='677) AFS - 1980 CDF - 2002 LHCb - 2014 ATLAS - 2013 CMS - 2011 LHCb - 2015 ALICE - 2016 LHCb - 2015 LHCb - 2018 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 3: The hadronization probability FΥ as a function of proton collision energy √s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The corridor between the upper and lower lines demonstrates the uncertainty from the hard scale variation by the factor ξ = 2 and the b-quark mass from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='5 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='75 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 0 1 2 3 4 |∆yΥΥ| 10−1 100 101 102 103 dσ d|∆yΥΥ| [pb] F ΥΥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='005 F Υ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='007 σeff = 11 mb √s = 13 TeV, |yΥ| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0 LO PRA, ICEM SUM SPS DPS CMS - 2020 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 4: Different spectra of pair Υ production as a function of rapidity difference |∆yΥΥ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The data are from CMS collaboration at the √s = 13 TeV [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0 |∆φΥΥ| 10−1 100 101 102 103 dσ d|∆φΥΥ| [pb] F ΥΥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='005 F Υ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='007 σeff = 11 mb √s = 13 TeV, |yΥ| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0 LO PRA, ICEM SUM SPS DPS CMS - 2020 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 5: Different spectra of pair Υ production as a function of azimuthal angle difference |∆φΥΥ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The data are from CMS collaboration at the √s = 13 TeV [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='0 |∆φψΥ| 10−3 10−2 10−1 100 101 102 103 B(J/ψ → µ+µ−) × B(Υ → µ+µ−) × dσ/d|∆φψΥ| [fb] F ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='044 F Υ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='039 σeff = 11 mb ¯p + p → J/ψ + Υ(1S) + X √s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='96 TeV LO PRA, ICEM SUM SPS DPS D∅ - 2016 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 6: Different spectra of pair J/ψΥ production as a function of azimuthal angle difference |∆φJ/ψΥ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' The data are from D0 collaboration at the √s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content='96 TeV [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 11 [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E3T4oBgHgl3EQfmwoZ/content/2301.04618v1.pdf'} +page_content=' 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