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+AdaSfM: From Coarse Global to Fine Incremental Adaptive
+Structure from Motion
+Yu Chen1, Zihao Yu2, Shu Song2, Tianning Yu3, Jianming Li3, Gim Hee Lee1
+Abstract— Despite the impressive results achieved by many
+existing Structure from Motion (SfM) approaches, there is still
+a need to improve the robustness, accuracy, and efficiency
+on large-scale scenes with many outlier matches and sparse
+view graphs. In this paper, we propose AdaSfM: a coarse-
+to-fine adaptive SfM approach that is scalable to large-scale
+and challenging datasets. Our approach first does a coarse
+global SfM which improves the reliability of the view graph by
+leveraging measurements from low-cost sensors such as Inertial
+Measurement Units (IMUs) and wheel encoders. Subsequently,
+the view graph is divided into sub-scenes that are refined in
+parallel by a fine local incremental SfM regularised by the result
+from the coarse global SfM to improve the camera registration
+accuracy and alleviate scene drifts. Finally, our approach uses
+a threshold-adaptive strategy to align all local reconstructions
+to the coordinate frame of global SfM. Extensive experiments
+on large-scale benchmark datasets show that our approach
+achieves state-of-the-art accuracy and efficiency.
+I. INTRODUCTION
+Structure from Motion (SfM) is an important topic that
+has been studied intensively over the past two decades. It
+has wide applications in augmented reality and autonomous
+driving for visual localization [1], [2], [3], and in multi-view
+stereo [4], [5] and novel view synthesis [6] by providing
+camera poses and optional sparse scene structures.
+Despite the impressive results from many existing works,
+SfM remains challenging in two aspects. The first challenge
+is outlier feature matches caused by the diversity of scene
+features, e.g. texture-less, self-similar, non-Lambertian, etc.
+These diverse features impose challenges in sparse feature
+extraction and matching which result in outliers that are detri-
+mental to the subsequent reconstruction process. Incremental
+SfM [7], [8] is notoriously known to suffer from drift due
+to error accumulation, though is robust in handling outliers.
+Global SfM methods [9], [10], [11] are proposed to handle
+drift, but fail to solve the scale ambiguities [12] of camera
+positions and are not robust to outliers [13], [14].
+The second challenge is sparse view graphs from some
+large-scale datasets. Incremental SfM is known to be ineffi-
+cient on large-scale datasets. Several works [16], [17], [18],
+[19] have been proposed to handle millions of images. These
+are divide-and-conquer SfM methods that deal with very
+large-scale datasets by grouping images into partitions. Each
+partition is processed by a cluster of servers that concurrently
+1School of Computing, National University of Singapore, {chenyu,
+gimhee.lee}@comp.nus.edu.sg
+2Segway-Ninebot Robotics Co., Ltd, yuzihao@buaa.edu.cn,
+songshu0905@gmail.com
+3Navimow B.V. Co., Ltd, tianning.yu@rlm.segway.com,
+jianming.li@ninebot.com
+Fig. 1. When combining with global SfM, our AdaSfM is more robust than
+traditional incremental SfM (tested on the public 4Seasons dataset [15]).
+circumvents the memory limitation. However, these methods
+[16], [17], [18], [19] are often limited to internet datasets
+or aerial images where the view graphs are very densely
+connected. The dense connections in the view graph ensure
+that there are sufficient constraints between the graph par-
+titions. Nonetheless, divide-and-conquer methods often fail
+in datasets with weak associations between images for local
+reconstruction alignments or lack of visual constraints for
+stable camera registration. An example of such a dataset
+is autonomous self-driving cars where the interval between
+consecutive images can be large.
+In view of the challenges from the outlier feature matches
+and sparse view graphs on the existing SfM approaches,
+we propose AdaSfM: a coarse-to-fine adaptive SfM pipeline
+to enhance the robustness of SfM in dealing with large-
+scale challenging scenes. Specifically, we first solve the
+global SfM at a coarse scale, and then the result of the
+global SfM is used to enhance the scalability of the local
+incremental reconstruction. Both the scale ambiguities and
+outlier ratio in global SfM can be significantly reduced
+by incorporating measurements from the IMU and wheel
+encoder, which are often available in mobile devices or
+autonomous self-driving cars. We preintegrate [20] the IMU
+measurements to get the relative poses of consecutive frames
+Pt
+= {Pt0, Pt1, · · · }, and use the measurements from
+the wheel encoder to constrain scale drifts of the IMU
+preintegration [21]. We then replace the relative poses of the
+consecutive frames in the view graph formed by two-view
+geometry [22], [8] with Pt estimated by the IMU and wheel
+encoder. This augmented view graph is then used to estimate
+arXiv:2301.12135v1  [cs.CV]  28 Jan 2023
+
+Fig. 2.
+The pipeline of our proposed SfM method. Our method takes images and measurements from low-cost sensors as inputs. The view graph is
+built after feature matching and refined by the result of global SfM. The absolute poses from the global SfM are used as priors in the subsequent local
+SfM process. The final reconstruction result is merged into the global SfM reference frame.
+the global poses. Consequently, we obtain a coarse scene
+structure and camera poses, where the latter can be used
+to filter wrong feature matches. Since that, we partition the
+view graph with the existing graph cut method [23] and then
+extend the sub-graphs with a novel adaptive flood-fill method
+to enhance the constraints of separators [24]. We define
+separators as images that connect different sub-graphs. For
+each local SfM, the poses from the global SfM are used for
+camera registration and to constrain the global refinement of
+3D points and camera poses. Finally, we design an adaptive
+global alignment strategy to merge local reconstructions with
+the coordinate frame of the global SfM set as the reference
+frame. We illustrate the pipeline of our method in Fig. 2.
+We evaluate our method extensively on large-scale chal-
+lenging scenes. Experimental results show that our AdaSfM
+is adaptive to different scene structures. Furthermore, we
+achieve better robustness and comparable efficiency in com-
+parison to existing state-of-the-art SfM methods.
+II. RELATED WORK
+Incremental SfM. Agarwal et al. [7] apply preconditioned
+conjugate gradient [25] to accelerate large-scale BA [26].
+The drift problem is alleviated in [27] with a re-triangulation
+(RT) step before global BA. Sch¨onberger and Frahm [8]
+augment the view graph by estimating multiple geometric
+models in geometric verification and improve the image
+registration robustness with next best view selection. In
+addition to the RT before BA [27], RT is also performed
+after BA in [8]. To reduce the time complexity of repetitive
+image registration, Cui et al [28] select a batch of images
+for registration, and select a subset of good tracks for BA.
+Global SfM. The simplest configuration of a global SfM
+method only requires 1) estimating the global rotations by
+rotation averaging (RA), 2) obtaining the global positions
+by TA, and 3) triangulating 3D points and performing a
+final global BA. Govindu [29] represents rotations by lie-
+algebra, and global rotations and global positions are es-
+timated simultaneously. Chatterjee and Govindu [30], [31]
+improve the rotation estimation of
+[29] by a robust l1
+initialization followed by a refinement of the rotations with
+iteratively reweighted least-squares (IRLS) [32]. To solve the
+TA problem, Wilson et al [33] project relative translations
+onto the 1D space to identify outliers. Relative translations
+that are inconsistent with the translation directions that have
+the highest consensus are removed. A nonlinear least-squares
+problem is then solved to get the global positions. Goldstein
+et al. [34] relax the scale constraints of
+[33] to linear
+scale factors, and the convex linear programming problem is
+solved by ADMM [35]. ¨Ozyesil and Singer [12] utilize the
+parallel rigidity theory to select the images where positions
+can be estimated uniquely and solved as a constrained
+quadratic programming problem. By minimizing the sin θ
+between two relative translations, Zhuang et al. [36] improve
+the insensitivity to narrow baselines of TA. The robustness of
+TA is also improved in [36] by incorporating global rotations.
+Hybrid SfM. Cui et al. [37] obtain orientations by RA
+and then register camera centers incrementally with the
+perspective-2-point (P2P) algorithm. Bhomick et al. [16]
+propose to divide the scene graph, where the graph is built
+from the similarity scores between images. Feature matching
+and local SfM can then be executed in parallel and local
+reconstructions are merged [16]. Zhu et al. [18], [19] adopt
+a similar strategy to divide the scene and the graph is
+constructed after feature matching. The relative poses are
+collected after merging all local incremental reconstruction
+results. The outliers are filtered during local reconstruction,
+global rotations are fixed by RA, and camera centers are reg-
+istered with TA at the cluster level. Based on [18], Chen et al.
+[17] find the minimum spanning tree (MST) to solve the final
+merging step. The MST is constructed at the cluster level,
+and the most accurate similarity transformations between
+clusters are given by the MST. Locher et al. [38] filtered
+wrong epipolar geometries by RA before applying the divide-
+and-conquer method [18]. Jiang et al. [39] use a visual-
+inertial navigation system (VINS)
+[40] to first estimate
+the camera trajectories with loop detection and loop closure
+[41]. Images are then divided into sequences according to
+timestamps. However, [39] requires two carefully designed
+systems: one for VINS with loop detection and the other for
+SfM. Loop detection is also a challenge in real-world scenes.
+III. NOTATIONS
+We denote the absolute camera poses as P = {Pi =
+[Ri|ti]}, where Ri, ti are the rotation and translation of the
+i-th image, respectively. The absolute camera poses project
+3D points X = {Xk} from the world frame to the camera
+frame. The camera centers are denoted by {Ci}. The relative
+pose from image i to image j are denoted as Pij = [Rij|tij],
+
+IMU +
+Images
+Wheel Encoders
+Global Alignment
+Global SfM
+Local SfM
+Local SfM
+Global
+View Graph
+Graph
+Bundle Adjustment
+:
+Partition
+Matches
+Refinement
+Local SfM
+Retriangulationwhere Rij, tij are the relative rotations and translations,
+respectively. We define the view graph as G = {V, E}, where
+V denotes the collection of images and E denotes the two
+view geometries, i.e. the relative poses and inlier matches
+between the image pairs. For two rotations Ri, Rj, we use
+log(Ri, Rj) = log(RjR⊤
+i ) to denote the angular error and
+∥Ri − Rj∥F to denote the chordal distance. Additionally,
+the keypoints and the normalized keypoints after applying
+the intrinsic matrix K are denoted by u and ˆu, respectively.
+IV. COARSE GLOBAL TO FINE INCREMENTAL SFM
+In this section, we introduce our method in detail. In
+Sec. IV-A, we introduce our global SfM that can effectively
+cope with outliers in challenging scenes. A refinement step
+is also introduced to remove outlier matches after global
+SfM. In Sec. IV-B, we describe our parallel incremental SfM
+approach that utilizes the results from coarse global SfM to
+mitigate the problems from sparse view graphs.
+A. Coarse Global SfM
+We first obtain the absolute rotations Ri by solving the
+rotation averaging problem:
+arg min
+{ ˆ
+Ri}
+�
+i∈V,
+(i,j)∈E
+d( ˆRj ˆR⊤
+i , Rij),
+(1)
+where ˆRi denotes the absolute poses obtained by rotation
+averaging, and d(·) = ∥ · ∥F denotes the chordal distance.
+Eq. (1) can be solved robustly and efficiently by [42]. We
+then obtain the absolute camera positions by solving the
+translation averaging problem. However, existing translation
+averaging methods often fail to recover the camera positions
+under challenging scenes due to two main factors: 1) The
+high ratio of outliers in the relative translations. 2) The
+view graph is solvable only when the parallel rigid graph
+condition [12] is satisfied. To alleviate the first problem, we
+first remove the erroneous matching pairs by checking the
+discrepancy of relative rotations: log(R⊤
+ij ˆRj ˆR⊤
+i ) > ϵR, and
+then the relative translations [12] are refined in parallel by:
+arg min
+tij
+∥ˆu′⊤([tij]×( ˆRj ˆR⊤
+i ))ˆu∥,
+s.t.
+∥tij∥ = 1.
+(2)
+We do not extract the rigid parallel graph [12] to solve the
+scale ambiguities since it is time-consuming to solve poly-
+nomial equations. Furthermore, the state-of-the-art method
+to establish the solvability of a view graph is only limited
+to 90 nodes [43]. We improve the solvability of the view
+graph by augmenting the relative translations in Pt of the
+consecutive frames from the IMU and wheel encoder. We
+do not augment the relative rotations because they are more
+accurate from the image-based two-view geometry. Note that
+errors can accumulate increasingly in the augmented relative
+poses during the motion of the devices due to the bias of
+the accelerometers and gyroscopes in the IMU, and drifts in
+the wheel encoder caused by friction and wheel slippages.
+To circumvent this problem, we only use the relative poses
+where the time difference is below a threshold ϵT .
+Since we obtained the augmented view graph Gaug =
+{V, Eaug}, the rigidity of the original view graph is aug-
+mented and the scale ambiguities of some images can be
+eliminated. We can then further solve the translation averag-
+ing problem below:
+arg min
+ˆ
+Ci,i∈V;
+sij,(i,j)∈Eaug
+�
+(i,j)∈Eaug
+∥sij( ˆCi − ˆCj) − R⊤
+j tij∥,
+(3)
+s.t.
+sij ≥ 0,
+∀(i, j) ∈ Eaug;
+�
+i∈V
+ˆCi = 0.
+(3) can be solved efficiently and robustly under the l1-
+norm by collecting all the constraints. Note all the relative
+translations are normalized in Eaug. The right of Fig. 3 shows
+our global SfM result by solving (3).
+After translation averaging, we triangulate the 3D points
+and perform an iterative global bundle adjustment to refine
+camera poses. It is worth mentioning that, global SfM can
+generate more tracks than incremental SfM, as its camera
+poses are less accurate and thus it fails to merge some tracks
+that are physically the same. Besides, according to [28],
+tracks are redundant for optimisation. Therefore, we can
+reduce the computation and memory burden with fewer
+tracks. Though a well-designed algorithm may help with
+the selection of tracks, we simply create tracks with a
+stricter threshold: only when the angle between the two rays
+respectively go through the 3D point and the two camera
+centers are larger than 5 degrees, it is deemed as a valid
+track. Note that for numerical stability during optimization,
+the coordinates are normalized after each iteration.
+Fig. 3.
+Comparison of global SfM results. Results from [12] (left) and
+Eq. (3) (right). Red and black colors respectively denote vehicle trajectories
+and sparse point clouds.
+1) Matches Refinement: The correct camera poses recov-
+ered by our global SfM with the relative poses from the
+low-cost sensors to eliminate the wrong two-view geometry
+estimates can be further utilized to filter out wrong image
+feature matches. For a calibrated camera with known intrin-
+sics, we can recover the essential matrix between images i
+and j from ˆE = [ˆtij]× ˆRij with the absolute rotations ˆRi and
+translations ˆti computed from rotation and translation aver-
+aging. (ˆtij, ˆRij) are computed from ( ˆRi, ˆRj) and (ˆti,ˆtj).
+The true matches ˆu′ ↔ ˆu must satisfy the check on the total
+point-to-epipolar line distance [22] over the two views, i.e.
+d⊥(ˆu, Eˆu′) + d⊥(ˆu′, Eˆu) ≤ ϵM.
+(4)
+d⊥(x, l) gives the shortest distance between a point x and
+a line l. The epipolar lines on the two images are given by
+l = Eˆu′ and l′ = Eˆu. ϵM is the threshold for the check.
+The effectiveness of global SfM to filter wrong matches
+can be seen in Fig. 7. We build a pseudo ground truth by
+
+COLMAP [8] to evaluate the accuracy of the global SfM.
+The ratio test is performed after NN by default. Fig. 4 shows
+the inlier ratio distribution after NN+RANSAC and matches
+refinement with relative poses obtained from global SfM
+and incremental SfM, respectively. Table. I gives the relative
+pose estimation AUC of NN+RANSAC and global SfM with
+respect to incremental SfM. It can be seen that our coarse
+global SfM can obtain comparable accuracy to COLMAP [8]
+in the refinement of the matches.
+Fig. 4.
+Inlier ratio distribution of NN+RANSAC, global SfM and
+incremental SfM (ground truth) on the 711 (left) and B6 (right) datasets.
+AUC
+NN+RANSAC
+Global SfM
+NN+RANSAC
+Global SfM
+R
+t
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+92.06
+35.41
+@10.0◦
+81.71
+20.21
+94.99
+33.03
+86.81
+32.46
+96.01
+51.07
+@20.0◦
+90.29
+29.97
+97.48
+49.87
+92.90
+46.95
+98.00
+64.65
+TABLE I
+RELATIVE POSE ESTIMATION AUC OF NN+RANSAC AND GLOBAL
+SFM WITH RESPECT TO INCREMENTAL SFM ON THE B6 (COLUMN 2-5)
+AND 711 (COLUMN 6-9) DATASETS.
+B. Finer Parallel Incremental SfM
+Although we have obtained the absolute camera poses by
+global SfM, these coarse poses are not accurate enough for
+localization. To improve the accuracy, we propose to refine
+the camera poses and scene structure with the divide-and-
+conquer incremental SfM.
+1) Adaptive Graph Partition: Existing approaches [18],
+[17] used a cut-and-expand schema to create overlapping
+areas between partitions. However, these approaches have
+two main drawbacks: : 1) The overlapping areas are not
+enough for final merging when the view graph becomes too
+sparse. This can be seen from Fig. 5(a). Edges (3, 20), (7,
+9), (8, 9), (8, 20), (16, 19), (17, 18) are collected after the
+graph cut, and then the images on these edges are added
+as separators of the partitions. In Fig. 5(a), only images
+{3, 7, 8, 9, 16, 17, 18, 19, 20} can be used to create the over-
+lapping areas (Fig. 5(b)). However, these separator images
+are insufficient to compute the similarity transformations for
+merging all local reconstructions due to the sparsity of the
+view graph. 2) Graph cut tends to separate partitions along
+edges with weak associations. This means the separators are
+often weakly constrained during reconstruction and thus their
+poses might not be accurate enough during reconstruction.
+We propose a flood-fill graph partition algorithm to over-
+come the above-mentioned disadvantages. We refer to the
+added nodes in each cluster after an expansion operation as
+a layer. The separators are collected to form a layer after
+the graph cut on the complete view graph. Fig. 5(a) shows
+examples of the separators marked green. We have separators
+S1 = {{3, 7, 8}, {9, 16, 17}, {18, 19, 20}} in the first layer.
+We then collect all the adjacent images of every separator
+for each partition. We find one adjacent image that does
+not belong to partition k, and add it to the second layer
+of separators S2 in partition k. Adjacent images are sorted
+in descending order according to the weights of the edges,
+i.e. the number of inlier matches. Fig. 5(b) shows that the
+separators S2 = {{9, 20}, {8, 18}, {8, 16}} at the second
+layer after traversing all separators in S1. The expansion step
+is repeated until the number of overlapping images reaches
+the overlapping threshold τot (e.g. 30%).Fig. 5(c) shows the
+separators S3 at the third layer.
+2) Local Incremental SfM: We perform incremental SfM
+in parallel after graph partitioning. For local incremental
+SfM, we utilize the result of global SfM ˆPglobal to improve
+the robustness of the image registration step, and to further
+constrain the camera poses during global optimization.
+a) Image Registration: We follow [8] for the two-view
+initialization. We then select a batch of the next-best images
+to register, where any image that sees at least vp scene points
+are put into one batch and sorted in descending order. For
+each candidate image i, we first use the P3P [44] to compute
+the initial pose Pp3p
+i
+. However, images can be registered
+wrongly due to wrong matches or scene degeneration. We
+propose to also compute the image pose Pgb
+i
+= [Rgb
+i
+| tgb
+i ]
+using ˆPglobal. We first collect the set of registered images that
+are co-visible to image i, and then the rotation of image i
+can be computed by a single rotation averaging [45]:
+arg min
+Rgb
+i
+�
+k
+∥ log( ˆRkiRk, Rgb
+i )∥,
+where
+ˆRki = ˆRi ˆR⊤
+k , (5)
+where k is the index of images that are co-visible to image
+i. For image translation, we first compute the translation
+of image i by each co-visible image and simply adopt the
+median of each dimension in translations tgb
+i :
+tgb
+i = median{ˆtki + ˆRkitk},
+where
+ˆtki = ˆti − ˆRkiˆtk.
+(6)
+To select the best initial pose, we reproject all visible 3D
+points of image i to compute the reprojection errors and mark
+the 3D point with the reprojection error less than 8px as an
+inlier. Finally, we select the pose which has the most inliers.
+b) Bundle Adjustment: To alleviate the drift problem
+for local incremental SfM, we perform global optimization
+using the classical bundle adjustment with the absolute
+poses obtained from global SfM as the supervision for the
+incrementally registered poses, i.e.
+arg min
+R,C,X
+� �
+i
+�
+k
+∥Π(Ri, Ci, Xk) − uik∥
++
+(7)
+�
+(i,j)∈Eaug
+�
+∥ log(Rij, ˆRij∥ + d∠(tij,ˆtij)
+��
+,
+
+30k
+Ground Truth
+GlobalSfM
+NN ransac
+25k
+20k
+15k
+10k
+5k
+Ok
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+inlier ratioGround Truth
+60k
+GlobalSfM
+NN ransac
+50k
+40k
+30k
+20k
+10k
+ok
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+inlier ratio(a) Initial graph cut.
+(b) The 1st graph expansion.
+(c) The 2nd graph expansion.
+Fig. 5.
+Pipeline of adaptive flood-fill graph partition. In the view graph, nodes are denoted by blue circles, edges are denoted by blue solid lines.
+Separators are marked by green circles.
+Fig. 6.
+Vehicle trajectories of different threshold trials when merging sub-reconstructions. The last figure is obtained by our method which starts
+from an initial inlier threshold τinit. Others are the results of using a fixed threshold during the alignment to merge all local reconstructions.
+where Π(·) reprojects a 3D point back to the image plane,
+d∠(·) denotes the angle between two vectors. Note that we do
+not make the hard constraint to force the translation part of
+ˆP−1
+ij Pij to be a zero-vector. Instead, we use d∠(tij,ˆtij) =
+d∠(Ci−Cj, ˆCi− ˆCj) to constrain the translation direction of
+camera poses. This is because the absolute positions obtained
+from global SfM are not sufficiently accurate.
+3) Adaptive Global Alignment: The global alignment step
+is crucial for the divide-and-conquer SfM since a wrong
+similarity transformation can cause catastrophic failure of
+the reconstruction. The difficulties in estimating a reliable
+similarity transformation are due to 1) The existence of
+outliers in registered camera poses. Although the outliers can
+be identified by RANSAC [46], the threshold that indicates
+outliers is hard to determine. This is due to the loss of the
+absolute scale of the real world in SfM without additional
+information such as GPS. It indicates that the optimal outlier
+threshold varies for each cluster. 2) The estimated similarity
+transformation can overfit wrongly with insufficient sample
+points. Existing divide-and-conquer methods
+[16], [18],
+[19], [47], [17] suffer from the two issues because the
+similarity transformations can only be estimated from the
+overlapping areas between the pairwise local partitions.
+To tackle the first issue, we propose an adaptive strategy
+to determine the inlier threshold τinlier. Given an initial inlier
+threshold τinit, we first estimate the similarity transformation
+by RANSAC [46]. We then compute the inlier ratio rinlier and
+increase the inlier threshold if rinlier < rmin. Furthermore,
+we decrease the threshold if rinlier ≥ rmax to prevent the
+threshold from becoming too large. A large threshold allows
+more outliers to be falsely selected and thus harming the
+similarity transformation estimation. The second issue can be
+solved easily within our framework. We set the coordinate
+frame of the global SfM as the reference frame, and align
+each local SfM into the reference frame. Therefore, for each
+partition, we can have as many sample points as the number
+of common registered images between a global SfM and a
+local partition to compute the similarity transformation. We
+also show the effectiveness of the algorithm to merge local
+reconstructions in Fig. 6. When zooming in, we can observe
+that our adaptive strategy perfectly closed the loop while
+other fixed threshold trials failed.
+V. EXPERIMENTAL RESULTS
+In this section, we perform extensive experiments to
+demonstrate the accuracy, efficiency, and robustness of our
+proposed methods.
+A. Implementation Details
+We use HFNet [48] as the default feature extractor and
+use the NN search for matching. A maximum of 500 feature
+points are extracted from each image and matched to the top
+30 most similar images based on the global descriptors from
+HFNet. We assume cameras are pre-calibrated and use the
+ceres-solver [49] for bundle adjustment. We did not compare
+our method against [39], as VINs [40] fails to find the right
+loops in our datasets. All methods are run on the same
+computer with 40 CPU cores and 96 GB RAM.
+Evaluation Datasets: We evaluate our method on our self-
+collected outdoor datasets and the 4seasons [15] datasets.
+Our self-collected datasets are collected by low-speed au-
+tonomous mowers, of which the running environments have
+many plants and texture-less areas. The 4seasons dataset is
+a cross-season dataset that includes multi-sensor data such
+as IMU, GNSS, and stereo images. It also provides camera
+poses computed by VI-Stereo-DSO [50], [51] and ground-
+truth camera poses by fusing multi-sensor data into a SLAM
+system. See our attached video for a more qualitative and
+quantitative evaluation of the 4Seasons dataset.
+
+1
+8
+9
+16
+21
+18
+231
+2
+8
+10
+16
+16
+2J
+13
+18
+231
+2
+8
+8
+10
+16
+15
+18
+13
+23
+18Tinlier = 0.5
+Tinlier = 1.0
+Tinlier = 1.5
+Tinlier = 2.0
+Tinit = 1.0Fig. 7.
+Vehicle trajectories after match refinement on B6 dataset. In Fig.(a) and Fig.(b), the visual results are respectively reconstructed without (left)
+and with (right) match refinement in each sub-figure. Fig.(c) shows some of the wrong matching pairs that are filtered by our method.
+Dataset
+N
+COLMAP [8]
+GraphSfM [17]
+Ours(Global SfM)
+Ours(Global+Inc.)
+Nc
+Np
+¯L
+RMSE
+T
+Nc
+Np
+¯L
+RMSE
+T
+Nc
+Np
+¯L
+T
+Nc
+Np
+¯L
+RMSE
+T
+high free
+48,753
+48,733
+567,030
+21.59
+1.47
+597,171
+48,491
+540,711
+22.73
+1.38
+88,896 (×6.7 ↑)
+48,758
+521,080
+14.51
+5,177
+48,694
+540,942
+22.79
+1.66
+105,163 (×5.7 ↑)
+711
+29,619
+27,175
+303,352
+25.35
+1.64
+160,322
+29,618
+259,292
+33.37
+1.46
+33,514 (×4.8 ↑)
+29,629
+249,673
+18.86
+3,499
+29,619
+256,495
+33.79
+1.61
+38,682 (×4.1 ↑)
+yht
+7,472
+7,470
+90,437
+20.81
+1.16
+20,428
+6,709
+78,659
+20.58
+1.17
+7,526 (×2.7 ↑)
+7,472
+132,167
+13.67
+524
+7,472
+108,711
+17.35
+1.43
+9,778 (×2.1 ↑)
+A4
+5,184
+5,132
+33,694
+41.92
+1.69
+18,104
+4,285
+28,726
+49.79
+1.55
+12,670 (×1.4 ↑)
+5,184
+24,193
+26.59
+1,349
+5,184
+34,007
+48.30
+1.43
+6,924 (×2.6 ↑)
+Htbd
+14,651
+14,645
+231,870
+24.62
+1.30
+56,888
+14,645
+232,441
+24.25
+1.37
+17,187 (×3.3 ↑)
+14,646
+190,904
+23.47
+1,523
+14,646
+238,035
+23.76
+1.36
+16,852 (×3.4 ↑)
+jy1
+32,484
+32,463
+534,117
+20.57
+1.44
+346,161
+32,466
+536,331
+20.18
+1.52
+28,673 (×12.1 ↑)
+32,484
+463052
+16.12
+3,077
+32,466
+621,437
+17.77
+1.53
+33,555 (×10.3 ↑)
+TABLE II
+COMPARISON OF RUNTIME AND ACCURACY ON REAL-WORLD DATASETS. FOR RUNTIME T (SECONDS), THE FIRST, SECOND AND THIRD THE BEST
+RESULTS ARE HIGHLIGHTED IN COLOR. Nc, Np DENOTE THE NUMBER OF REGISTERED IMAGES AND 3D POINTS, RESPECTIVELY, ¯L DENOTES THE
+AVERAGE TRACK LENGTH , AND RMSE DENOTES THE ROOT MEAN SQUARE ERROR IN PIXEL.
+Running Parameters: Empirically, we use the time
+threshold ϵT = 500 ms to adopt the fused relative poses
+in Gaug, and ϵR = 5 degree to check to relative rotation
+discrepancy. The point-to-epipolar line distance is ϵM =
+4 px. Besides, we set the overlapping ratio τot = 0.3 in
+the graph partition, vp = 10 for an image to be a candidate
+to register, and rmin = 0.7, rmax = 0.9, τinit = 1.0, αinc =
+0.2, αdec = 0.1 in global alignment.
+B. How Matching Refinement Saves SfM?
+In addition to running our experiments on HFNet, we
+also do evaluations on different trials. We first show the
+reconstruction results conducted on a challenging scene in
+Fig. 7, which is difficult for visual methods to identify the
+wrong feature matches due to specular issues.
+We use two different combinations of methods for feature
+extraction and matching in each scene. In the first combi-
+nation, we use HFNet [48] for feature extraction and NN
+search for feature matching. In the second combination, we
+use Superpoint [52] for feature extraction and Superglue [53]
+for feature matching. Both settings use RANSAC
+[46] to
+remove matching outliers that do not satisfy the point-to-
+epipolar line constraint. In each sub-figure, the left and right
+images are the results without and with matching refinement,
+respectively. It can be seen that for HFNet + NN, while both
+methods fail to reconstruct the two datasets, the result after
+our result is visually better than without matches refinement.
+For Superpoint + Superglue, the state-of-the-art methods
+respectively on feature extraction and matching, also fails
+on the dataset without refining matches. In contrast, our
+method can correctly identify the wrong matching pairs and
+then leverage the refined matchings to greatly improve the
+reconstruction quality for both settings.
+C. Qualitative Evaluation on Real-World Datasets
+We evaluated our full pipeline on several outdoor datasets.
+We use the registered images number Nc, the recovered 3D
+points Np, the average track length ¯L, and the root mean
+square error (RMSE) to evaluate the qualitative accuracy. As
+shown in Table. II, our method shows the most number of
+registered images in almost all the datasets, while [17] shows
+the least number of registered images. In terms of efficiency,
+our method is moderately slower than GraphSfM [17] in
+most datasets since our method requires an additional global
+SfM reconstruction step. Interestingly, GraphSfM [17] is
+almost 1× slower than our method on the A4 dataset. We
+conjecture that it is due to the frequent failure of GraphSfM
+in selecting suitable images to register and therefore more
+trials are required to register as many images as possible.
+On the other hand, our method is robust enough to deal with
+the case since we get the initial poses of the images from
+P3P or global SfM. Our explanation is validated in Table. II
+where GraphSfM [17] recovers only 4,235 poses out of 5,184
+images, which is almost 20% less than our method. We can
+further notice that the average track length of global SfM is
+remarkably shorter than other methods, which means poses
+from global SfM are not accurate.
+VI. CONCLUSION
+In this paper, we proposed a robust SfM method that
+is adaptive to scenes in different scales and environments.
+Integrating data from low-cost sensors, our initial global
+SfM can benefit from the augmented view graph, where the
+solvability of the original view graph is enhanced. The global
+SfM result is used as a reliable pose prior to improve the
+robustness of the subsequent local incremental SfM and the
+final global alignment steps. Comprehensive experiments on
+different challenging scenes demonstrated the robustness and
+adaptivity of our method, whilst taking more computation
+burden with an additional global SfM step.
+Acknowledgement. This research/project is supported by
+the National Research Foundation, Singapore under its AI
+Singapore Programme (AISG Award No: AISG2-RP-2021-
+024), and the Tier 2 grant MOE-T2EP20120-0011 from the
+Singapore Ministry of Education.
+
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+(b) Superpoint + Sup
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+
+VII. APPENDIX
+A. Adaptive Flood-Fill Graph Partition Algorithm
+The pseudo-code of our adaptive flood-fill graph partition
+algorithm is given in Alg. 1.
+Algorithm 1 Adaptive Flood-Fill Graph Partition Algorithm
+Input: Initial view graph G = {V, E}, Overlapping thresh-
+old τot, Partition number K
+Output: Sub-graphs {Gk = {Vi, Ei} | i ∈ [0, K]}
+1: Overlapping ratio τor := 0, Separators Vs := ∅, {Gk} :=
+GraphCut(G).
+2: while τor < τot do
+3:
+Update separators Vs = {Vs
+0, · · · , Vs
+K}.
+4:
+Edges Edis := ∅.
+5:
+for k ∈ [0, K] do
+6:
+Edis
+k
+= E − Ek and Edis
+k
+contains Vs
+k.
+7:
+Edis+ = Edis
+k .
+8:
+Sort Edis by descending order.
+9:
+for Edge e ∈ Edis do
+10:
+Select a partition Gk contains one of the nodes
+in e and has the smallest size.
+11:
+Add e to Gk.
+12:
+Update τor.
+B. Adaptive Global Alignment Algorithm
+The pseudo-code of our adaptive global alignment algo-
+rithm is given in Alg. 2.
+Algorithm 2 Adaptive Global Alignment Algorithm
+Input: Local
+reconstructions
+M
+=
+{Mi},
+τinit, rmin, rmax, iterNummax, αinc, αdec
+Output: Final reconstruction
+1: for i < |M| do
+2:
+τinlier := τinit, rinlier := 0, iterNum := 0
+3:
+while rinlier < rmin & iterNum < iterNummax do
+4:
+iterNum := iterNum + 1;
+5:
+Compute sim3 by τinlier;
+6:
+Compute rinlier by sim3;
+7:
+if rinlier < rmin then
+8:
+τinlier := τinlier + αinc;
+9:
+else if rinlier ≥ rmax then
+10:
+τinlier := τinlier − αdec;
+C. Visualization Results on Self-Collected Dataset
+The qualitative visualization results are shown in Fig. 8.
+We can see that our reconstruction results are better than
+COLMAP [8] and GraphSfM [17], especially when we zoom
+in to see the image poses. Moreover, GraphSfM [17] fails
+to correctly merge the sub-reconstructions. The misalignment
+can be observed from the zoom-in areas of side-view images,
+which further validates the robustness of our method.
+D. Ablations of Augmented View Graph
+We present more ablation of the augmented view graph
+on the 4Seasons dataset in Fig. 9. More visualization results
+on this dataset can be seen in our attached video.
+E. Quantitative Results on 4Seasons dataset.
+We present the quantitative results on the 4Seasons dataset
+in Table. III. The 4Seasons dataset provides ground truth
+camera poses and trajectories from VI-Stereo-DSO [50],
+[51]. The sensor data contain IMU, GNSS, and stereo
+images. In our experiment, we do not use the GNSS data.
+Besides, as this dataset does not provide wheel encode data,
+we perturb the VI-Stereo-DSO trajectories by Gaussian noise
+in the x-y-z axes to synthesize wheel encoder data. We
+strongly recommend readers refer to [15] for more details
+about the challenged dataset. As is expected, Our method
+outperforms COLMAP by a large margin in terms of both
+accuracy and efficiency. In the Old Town scene, COLMAP
+failed to reconstruct on sequence recording 2020-10-08 11-
+53-41 and sequence recording 2021-02-25 12-34-08 (we use
+- to denote the failed cases). As the two sequences contain
+severe motion blur and tunnels in images, which makes them
+very challenging to reconstruct. However, our method is also
+robust to these scenes since it can robustly fuse different
+sensor data.
+
+(a) Qualititve comparison on the 711 dataset.
+(b) Qualititve comparison on the A4 dataset.
+(c) Qualititve comparison on the high free dataset.
+Fig. 8.
+Reconstruction comparisons on our self-collected dataset. From left to right are the input images, top-view reconstruction, and side-view
+reconstruction.
+
+COLMAP
+SJnoCOLMAI
+s.inoCOLMAP
+sJIn.Fig. 9.
+Ablations of our augmented view graph on the 4Seasons dataset.
+Scene
+Sequence
+COLMAP [8]
+Ours (Global SfM)
+Ours (final)
+Nc
+Np
+∆R
+∆t
+T
+Nc
+Np
+∆R
+∆t
+T
+Nc
+Np
+∆R
+∆t
+T
+Neighborhood
+recording 2020-10-07 14-53-52
+6,326
+137,135
+0.65
+1.78
+334.90
+6,036
+66,777
+2.52
+1.17
+14.68
+6,033
+109,483
+0.74
+0.52
+123.96
+recording 2020-12-22 11-54-24
+6,518
+127,892
+0.55
+3.68
+354.35
+6,144
+64,405
+1.10
+0.86
+15.83
+6,144
+102,857
+0.51
+0.62
+151.88
+recording 2020-03-26 13-32-55
+7,414
+148,848
+0.61
+1.24
+603.13
+5,982
+70,066
+0.92
+0.79
+17.10
+5,982
+111,807
+1.11
+0.98
+157.76
+recording 2020-10-07 14-47-51
+6,688
+152,307
+0.56
+1.67
+359.03
+6,248
+76,305
+2.20
+1.17
+15.70
+6,248
+121,657
+0.75
+0.74
+152.85
+recording 2021-02-25 13-25-15
+6,174
+138,807
+0.75
+1.05
+325.65
+5,238
+62,879
+1.00
+1.14
+15.12
+5,238
+106,609
+0.46
+0.81
+202.85
+recording 2021-05-10 18-02-12
+7,784
+149,528
+3.04
+9.57
+444.85
+5,834
+61,889
+1.49
+1.38
+12.76
+5,834
+101,102
+0.47
+0.59
+153.36
+recording 2021-05-10 18-32-32
+7,174
+141,864
+2.77
+19.15
+416.34
+6,046
+89,010
+1.14
+1.03
+23.81
+6,046
+142,430
+1.49
+1.34
+264.75
+Business Park
+recording 2021-01-07 13-12-23
+8,016
+109,399
+0.72
+0.75
+643.22
+9,010
+72,096
+1.76
+1.60
+56.16
+9,010
+100,057
+0.66
+0.51
+465.34
+recording 2020-10-08 09-30-57
+11,520
+127,013
+0.37
+1.57
+1284.44
+8,278
+66,087
+1.59
+1.51
+48.72
+8,278
+108,000
+0.63
+0.45
+366.81
+recording 2021-02-25 14-16-43
+7,414
+148,848
+0.61
+1.24
+603.13
+5,982
+70,066
+0.92
+0.79
+17.10
+5,982
+111,807
+1.11
+0.98
+157.76
+Old Town
+recording 2020-10-08 11-53-41
+19,332
+279,989
+-
+-
+2454
+12,910
+181,569
+2.23
+2.81
+45.72
+12,048
+279,127
+0.55
+0.56
+254.71
+recording 2021-01-07 10-49-45
+16.420
+307,383
+8.63
+360.51
+1496.6
+12,728
+194,340
+2.56
+3.14
+53.18
+12,728
+327,348
+1.55
+1.03
+238.82
+recording 2021-02-25 12-34-08
+18,950
+305,461
+-
+-
+2392.98
+12,387
+182,940
+2.02
+3.14
+40.97
+12,387
+302,833
+0.63
+0.74
+683.97
+Office Loop
+recording 2020-03-24 17-36-22
+10,188
+209,942
+1.17
+3.40
+822.38
+9,522
+126,680
+2.28
+2.38
+31.87
+9,377
+214,285
+0.97
+0.98
+166.54
+recording 2020-03-24 17-45-31
+8,582
+195,738
+0.92
+3.04
+865.48
+9,186
+122,713
+2.79
+2.20
+33.91
+8,940
+205,790
+0.84
+0.85
+209.06
+recording 2020-04-07 10-20-31
+10,350
+223.649
+4.22
+42.44
+795.68
+10,184
+138,446
+2.53
+1.78
+39.83
+10,184
+224,499
+1.47
+1.14
+253.24
+recording 2020-06-12 10-10-57
+9,990
+236,593
+18.97
+83.94
+705.93
+10,150
+164,062
+1.92
+1.61
+37.32
+10,150
+246,516
+0.76
+0.87
+206.48
+recording 2021-01-07 12-04-03
+9,164
+475,950
+0.71
+2.58
+1000.75
+10,300
+143,715
+3.32
+2.39
+48.68
+10,300
+223,676
+1.08
+0.67
+249.42
+recording 2021-02-25 13-51-57
+9,574
+214,695
+0.84
+2.84
+773.32
+9,426
+122,746
+3.80
+2.68
+28.96
+9,426
+204,289
+1.01
+0.91
+173.29
+TABLE III
+COMPARISON OF RUNTIME AND ACCURACY ON THE 4SEASONS DATASETS. T DENOTES THE RUNTIME (IN MINUTES), Nc, Np DENOTE THE NUMBER
+OF REGISTERED IMAGES AND 3D POINTS, RESPECTIVELY, ∆R, ∆t DENOTES THE MEAN ROTATION ERROR (IN DEGREES) AND TRANSLATION ERROR
+(IN METERS), RESPECTIVELY, AND WE HIGHLIGHT THE BEST RESULTS IN BOLD.
+
+Global SfM from raw view graph
+gran

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+MNRAS 000, 1–10 (2022)
+Preprint 10 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Identifying meteorite droppers among the population of bright ’sporadic’
+bolides imaged by the Spanish Fireball Network during the spring of 2022
+E. Peña-Asensio,1,2★ J. M. Trigo-Rodríguez,2,3 A. Rimola,1 M. Corretgé-Gilart,4 and D. Koschny5
+1Departament de Química, Universitat Autònoma de Barcelona 08193 Bellaterra, Catalonia, Spain
+2Institut de Ciències de l’Espai (ICE, CSIC), Campus UAB, C/ de Can Magrans s/n, 08193 Cerdanyola del Vallès, Catalonia, Spain
+3Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Catalonia, Spain
+4Universitat Politècnica de Catalunya (UPC), Carrer de Jordi Girona, 31, 08034 Barcelona, Spain
+5TU Munich, Boltzmannstrasse 15, 85748 Garching, Germany
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+We take advantage of the extraordinary weather conditions available between February and March 2022 over Spain to analyze
+the brightest fireballs recorded by the monitoring stations of the Spanish Meteor Network (SPMN). We study the atmospheric
+flight of 15 large meteoroids to determine if they are meteorite dropper events to prepare campaigns to search for freshly fallen
+extraterrestrial material. We investigate their origins in the Solar System and their dynamic association with parent bodies and
+meteoroid streams. Employing our Python pipeline 3D-FireTOC, we reconstruct the atmospheric trajectory utilizing ground-
+based multi-station observations and compute the heliocentric orbit. In addition, we applied an ablation model to estimate the
+initial and terminal mass of each event. Using a dissimilarity criterion and propagating backward in time, we check the connection
+of these meteoroids with known complexes and near-Earth objects. We also calculate if the orbits are compatible with recent
+meteoroid ejections. We find that ∼27% of these fireballs are dynamically associated with minor meteoroid streams and exhibit
+physical properties of cometary bodies, as well as one associated with a near-Earth asteroid. We identify two meteorite-producing
+events; however, the on-site search was unsuccessful. By considering that these fireballs are mostly produced by cm-sized rocks
+that might be the fragmentation product of much larger meteoroids, our findings emphasize the idea that the population of
+near-Earth objects is a source of near-term impact hazards, existing large Earth-colliding meteoroids in the known complexes.
+Key words: meteorites, meteors, meteoroids – comets: general – minor planets, asteroids: general
+1 INTRODUCTION
+The interplanetary medium is composed of countless millimeter- and
+centimeter-sized objects called meteoroids, some of which eventually
+cross the path of our planet (Brown et al. 2002; Murad & Williams
+2002; Trigo-Rodríguez 2022). These small bodies are fragments pro-
+duced by the catastrophic disruption or collisions of comets, aster-
+oids, or even impacts on planets (Chapman 2010; Tóth et al. 2011;
+Gritsevich et al. 2012; Trigo-Rodriguez et al. 2014). Due to tidal
+forces and sublimation by high temperatures of the Sun, cometary ag-
+gregates and rubble pile asteroids with efficient disruption processes
+suffer fragmentations in their passage through the perihelion, scat-
+tering meteoroids throughout their orbit that constitute the so-called
+meteoroid streams (also known as meteor showers) (Jenniskens 1994,
+1998, 2006; Vaubaillon et al. 2019). Some of these meteoroid streams
+have Earth-intersecting orbits, so they are generally repeated in an-
+nual cycles. After experiencing different physical phenomena such as
+orbital perturbations, impacts with other objects, Yarkovsky, YORP,
+or Poynting-Robertson effect, other meteoroids suffer time scale de-
+coherence and end up their space travel impacting on our planet as
+sporadic events, that is, apparently not associated with any known
+★ E-mail: eloy.pena@uab.cat, eloy.peas@gmail.com
+complex (Olsson-Steel 1986; Bottke et al. 2000; Pauls & Gladman
+2005; Brož 2006; Koschny et al. 2019).
+The impact of these objects at high velocity with the upper part of
+our atmosphere produces a luminous phase in the visible range due
+to the collision with the atoms of the air and the consequent melt-
+ing, evaporation, and progressive ionization of the meteoroid mate-
+rial (Ceplecha et al. 1998; Silber et al. 2018). This phenomenon is
+known as a meteor and is called a fireball or bolide if its magnitude is
+greater than that of Venus. From the observation and analysis of fire-
+balls with ground-based multi-stations, more than 10 major showers
+have been established (Quadrantids, April Lyrids, 𝜂-Aquarids, South-
+ern Δ-Aquariids, Perseids, Orionids, Taurids, Leonids, Geminids and
+Ursids), that is, meteoroid streams that present activity of more than
+10-15 meteors per hour (Bagnall 2021). However, there are hundreds
+of minor showers with lower activities as well as near-Earth aster-
+oids, many of them poorly studied, that can produce bright fireballs
+and, therefore, potentially meteorite dropper events, just as being
+a source of impact hazard to the Earth (Voloshchuk & Kashcheev
+1996; Halliday 1987; Madiedo & Trigo-Rodríguez 2008; Borovička
+et al. 2015; Trigo-Rodríguez et al. 2017; Peña-Asensio et al. 2022).
+The months between January and April are especially relevant
+from the meteor science point of view as meteorite fall rates display
+a peak during the beginning of spring in either hemisphere (Halliday
+& Griffin 1982). Unfortunately, the weather during winter and spring
+© 2022 The Authors
+arXiv:2301.03515v1  [astro-ph.EP]  9 Jan 2023
+
+2
+E. Peña-Asensio et al.
+Table 1. Location of the fireball observation points involved in this work.
+Station
+Name
+Long (◦)
+Lat (◦)
+Alt (m)
+A
+Alpicat
+0.5568
+41.6676
+252
+B
+Barx
+-0.3041
+39.0146
+336
+C
+Benicàssim
+0.0386
+40.0342
+15
+D
+Calar Alto
+-2.549
+37.2212
+2152
+E
+Cebreros
+-4.3693
+40.4541
+700
+F
+Corbera
+1.8906
+41.4092
+501
+G
+Estepa
+-4.8766
+37.2914
+537
+H
+GranTeCan
+-17.8919
+28.7567
+2267
+I
+La Murta
+-1.6756
+38.0967
+469
+J
+Monfragüe
+-6.0108
+39.7736
+411
+K
+Morata de Jalón
+-1.4821
+41.474
+415
+L
+Olocau
+-0.5363
+39.6744
+225
+M
+Playa Blanca
+-13.8241
+28.8747
+10
+N
+Puertollano
+-4.1129
+38.7032
+697
+O
+Sant Mateu
+0.1758
+40.465
+349
+is usually not helpful for fireball monitoring and clouds generally
+prevent detailed trajectory reconstruction and strewn-field estimates.
+In this sense, the months of February and March 2022 were especially
+clement in the Spanish territory so the Spanish Meteor Network
+(SPMN) has been able to record and analyze several spectacular
+fireballs, many of them associated with minor meteoroid streams
+rather than being sporadic.
+In section 2, we first outline the SPMN network’s current in-
+frastructure that has allowed recording these events with multiple
+stations. We also mention the methodology applied for fireball anal-
+ysis. In section 3, we describe the results of the atmospheric flight
+reconstruction, terminal mass prediction, and heliocentric orbit cal-
+culation. In section 4, we analyze the dynamic associations with
+parent bodies, near-Earth asteroids and comets, and minor and major
+meteoroid streams. In addition, we examined the compatibility of
+these events being recently ejected meteoroids. Finally, we discuss
+the results in section 5 and offer our conclusions in section 6.
+2 DATA COLLECTION AND METHODOLOGY
+Since its creation in 2005, thanks to the operability of the SPMN net-
+work, the whole sky of continental Spain is monitored full time, the
+last decade also including the Balearic and Canary Islands. Currently,
+a total of 34 stations with charged-coupled device (CCD) video and
+all-sky cameras are operational, some of them equipped with spec-
+trometers. In addition, three forward-scatter detectors monitor radio
+meteors (Trigo-Rodríguez et al. 2004). The stations involved in the
+events analyzed in this work are shown in Table 1, also incorporat-
+ing the recently installed AllSky7 camera at European Space Agency
+Cebreros’ station. This camera array allowed us to record 169 bright
+meteors up to an apparent magnitude of -6 between February and
+March of 2022, from which we selected the 15 largest multi-station
+bolides for analysis.
+New video processing and trajectory calculation techniques allow
+the automation of the analysis process of meteors, bolides, and ar-
+tificial fireballs produced by atmospheric re-entries of human-made
+objects. We developed the 3D-FireTOC Python code that automates
+this study allowing the reconstruction of atmospheric trajectories and
+the calculation of heliocentric orbits from multiple recordings by us-
+ing the intersection of planes method (Peña-Asensio et al. 2021b,a).
+Unlike traditional analytical methods, which solve the orbit by cor-
+recting for zenith attraction and diurnal aberration (Ceplecha 1987),
+we have now implemented the accurate IAS15 high-order N-body
+integrator with an adaptive time step included in the REBOUND
+package to compute the heliocentric orbit (Rein & Spiegel 2015).
+The integrator is based on the RADAU-15 developed in Everhart
+(1985) and has a high performance resolving close encounters. We
+account for the Earth’s and Moon’s oblateness by including the J2
+and J4 gravitational harmonic coefficients thanks to the REBOUNDx
+module (Tamayo et al. 2020).
+For most cases, we performed the astrometric calibration by solv-
+ing the polynomial modification of Borovička (1992) proposed by
+Bannister et al. (2013), which exhibits a better convergence while
+ensuring a very excellent level of uncertainty. To achieve the best fit,
+we use a simplicial homology global optimization algorithm to find
+the absolute minimum (Endres et al. 2018). For recordings with suf-
+ficient background stars, we apply the method proposed in Borovicka
+et al. (1995), which produces even lower errors down to 0.01◦ for
+azimuth and elevation. All calibrations are also cross-checked with
+the quadratic model described in Peña-Asensio et al. (2021b).
+With the mean uncertainties obtained in the astrometry for the
+camera calibration fit, we generate 1,000 clones to perform a Monte
+Carlo simulation following a Gaussian distribution applied to each
+detected point. We propagate every clone backward starting with its
+pre-atmospheric velocity from the beginning of the detected lumi-
+nous phase until they are outside the Earth’s influence, specifically, at
+10 times the Earth Hill sphere. We then integrate forward to the date
+of impact but without taking into account the gravitational attraction
+of the Earth-Moon system to obtain the osculating orbital elements
+at the time of the detection (referred to the J2000 equinox).
+We further perform a backward integration over 10,000 years eval-
+uating the evolution of an orbital dissimilarity criterion to test the
+dynamic association with parent body candidates. This is necessary
+as the most favorable candidate at the time of impact is not always
+the most reliable because it may be the result of a coincidence at
+that precise date. The meteoroid is integrated with its correspond-
+ing 1,000 clones generated from the uncertainties and the meteoroid
+streams are modeled by 18 equally spaced distributed particles over
+the true anomaly. Based on the orbital dissimilarity criterion, we
+assume that an association is robust enough if it remains below the
+cutoff for 5,000 years, minimizing the probability of being a random
+association (Porubčan et al. 2004).
+Different techniques have been developed and discussed to estab-
+lish the association between meteors and meteor showers or parent
+bodies, and they are still a source of debate today. One of the most
+established and widely used criteria is 𝐷𝐷 (Drummond 1981), which
+is a semi-quantitative approach to measure the dissimilarity of two
+orbits as a function of their orbital parameters in the five-dimensional
+phase.
+Based on the 𝐷𝑆𝐻 criterion (Southworth & Hawkins 1963), the
+𝐷𝐷 criterion was defined as:
+𝐷2
+𝐷 =
+� 𝑒𝐵 − 𝑒𝐴
+𝑒𝐵 + 𝑒𝐴
+�2
++
+� 𝑞𝐵 − 𝑞𝐴
+𝑞𝐵 + 𝑞𝐴
+�2
++
+� 𝐼𝐵𝐴
+𝜋
+�2
++
++
+� 𝑒𝐵 + 𝑒𝐴
+2
+�2 � 𝜃𝐵𝐴
+𝜋
+�2
+,
+(1)
+where 𝑒 is the eccentricity, 𝑞 is the perihelion distance, 𝐼𝐵𝐴 is
+the angle between the orbital planes, 𝜋𝐵𝐴 is the difference between
+longitudes of perihelia measured from the intersection of both orbits,
+and 𝜃𝐵𝐴 is the orbit angle between the lines of apsides.
+The thresholds of the dissimilarity functions, far from defining an
+exact barrier, offer an approximation with fair statistical significance,
+which, in addition, may vary depending on the inclination of the orbits
+MNRAS 000, 1–10 (2022)
+
+Meteorite dropper spring 2022
+3
+and the population size. Therefore, they are not a defining indicator,
+and it is also necessary to verify that the orbits are not only similar
+at a given time but also that this similarity lasts over time. In this
+sense, we use 0.18 as a cut-off for 𝐷𝐷 (Galligan 2001). Although
+this threshold value is high, we use it as a first filter, but not as the
+only association condition as we also check its evolution over time.
+In addition, we evaluate if the separation of the meteoroid from
+its possible parent body could have occurred in relatively short
+timescales. For this purpose, during the orbital integration, we mon-
+itor the minimum distance between the objects and the change in
+the velocity vector that would be needed to move from one orbit to
+the other one. In this way, we can observe if the velocity change is
+compatible with typical collisional ejection processes between small
+bodies.
+We also examined Tisserand’s parameter with respect to Jupiter
+𝑇𝑗, which is helpful to determine the evolution of small bodies since
+it remains broadly constant for long periods. It is used to classify
+planet-crossing objects, usually, as Jupiter-family comets (JFCs) if
+2 < 𝑇𝑗 < 3 and asteroidal when 𝑇𝑗 > 3.
+We evaluate the catastrophic disruption for each event by obtaining
+the ram pressure at peak brightness, that is, the bulk aerodynamic
+strength (𝑠 = 𝜌·𝑣2) accordingly to the U.S. standard atmosphere 1976
+(Bronshten 1981). This parameter is typically used to mechanically
+characterize the meteoroid and to classify the material regarding the
+bulk density. For events that do not present an explosion, we evaluate
+the peak of maximum brightness, thus obtaining only an estimate of
+the lower limit for the composition.
+Additionally, assuming an isothermal atmosphere and applying
+the dynamic third-order time-dependent system for characterizing
+meteor deceleration based on the velocity (𝑣) and the height (ℎ),
+we compute the ballistic coefficient (𝛼) and mass loss parameter (𝛽)
+(Gritsevich & Stulov 2006; Gritsevich 2008, 2009; Gritsevich et al.
+2012; Turchak & Gritsevich 2014):
+𝐹𝑖(ℎ𝑖, 𝑣𝑖, 𝛼, 𝛽) = 2𝛼𝑒−ℎ𝑖 − Δ𝑖����−𝛽,
+(2)
+with Δ𝑖 = 𝐸𝑖(𝛽) − 𝐸𝑖(𝛽𝑣2
+𝑖 ), 𝑖 = 1, 2, ..., 𝑛, where
+𝐸𝑖(𝑥) =
+∫ 𝑥
+−∞
+𝑒𝑡𝑑𝑡
+𝑡
+𝑑𝑥.
+These adimensional parameters are defined as
+𝛼 = 1
+2𝑐𝑑
+𝜌0ℎ0𝑆0
+𝑀0 sin 𝛾 ,
+(3)
+and
+𝛽 = (1 − 𝜇)
+𝑐ℎ𝑣2
+0
+2𝑐𝑑𝐻∗ ,
+(4)
+where 𝑐𝑑 is the drag coefficient, 𝜌0 is the atmospheric density
+at sea level, ℎ0 is the scale height for a homogeneous atmosphere
+and 𝛾 is the slope of the fireball to the local horizon, 𝑀0 is the
+meteoroid mass before impacting the top of the atmosphere, 𝜇 is
+the dimensionless shape change parameter, 𝑐ℎ is the heat transfer
+coefficient, 𝑣0 is the entry velocity, and 𝐻∗ is the sublimation heat. 𝜇
+is a constant value that relates the cross-sectional area 𝑆 with the mass
+as follows: 𝑆/𝑆0 = (𝑀/𝑀0)𝜇 (Lyytinen & Gritsevich 2016). Note
+that as it is an atmospheric flight dynamics model with an asymptotic
+solution, the minimization problem itself yields an initial velocity at
+infinity that corresponds to the pre-atmospheric velocity.
+These parameters allow properly describing the atmospheric flight
+and estimating the meteor fate based on the so-called 𝛼 − 𝛽 criterion
+(Sansom et al. 2019). The boundaries that delimit the fall likelihood
+(with a terminal mass threshold of 50 g) are determined by the two
+extreme values of the shape change coefficient: 𝜇 = 0 when the
+meteoroid is not spinning and 𝜇 = 2/3 when the meteoroid surface
+is equally ablated due to the rotation.
+From the aerodynamic strength values, we assign a mete-
+oroid bulk density based on Chyba et al. (1993): cometary if
+𝑠 < 105 𝑃𝑎; carbonaceous if 105 𝑃𝑎 < 𝑠 < 106 𝑃𝑎; rocky if
+106 𝑃𝑎 < 𝑠 < 107 𝑃𝑎; and rocky-iron if its aerodynamic strength
+is greater than 107 𝑃𝑎. This allows us to fit the object size 𝐷, the
+pre-atmospheric mass 𝑀0, and the terminal mass 𝑀𝑡 (the final mass
+at the end of the luminous atmospheric phase), being
+𝑀0 =
+�
+1
+2
+𝑐𝑑 𝐴0𝜌0ℎ0
+𝛼𝜌2/3
+𝑚 sin 𝛾
+�3
+,
+(5)
+where 𝐴0 is the pre-atmospheric shape coefficient.
+The terminal mass can be computed using the last observed veloc-
+ity in the following instant mass equation
+𝑀(𝑡) = 𝑀0𝑒
+−
+𝛽
+1−𝜇
+�
+1−
+�
+𝑣 (𝑡)
+𝑣0
+�2�
+,
+(6)
+where 𝑣(𝑡) is the instantaneous velocity.
+3 ATMOSPHERIC FLIGHT AND HELIOCENTRIC ORBIT
+Once the most suitable recordings of each event have been selected,
+and the lenses of each camera have been calibrated to correct distor-
+tions and found the transformation between pixel and position in the
+sky, we can apply the triangulation using the weighted method of the
+intersection of planes for multiple stations to obtain the real position
+of the meteoroid in each frame. Each station recorded the events in a
+single shot, except for the grazing meteoroid SPMN080322, which
+moved out of the field of view. Therefore, we had to combine the
+recordings from two cameras to obtain the complete luminous trail.
+Figure 1 shows a composite of overlapping images of some of the
+events recorded and analyzed in the following section.
+In some images, like the one of the SPMN060222 fireball captured
+in color from Corbera, an intense reddish tone due to the glowing ion-
+ized air can be seen, although further color calibrations are necessary
+for a precise determination of the tone. In the trace drawn during the
+atmospheric flights, it can be seen how several of them show multiple
+brightness peaks, as a result of the rapid rotation and differentiated
+ablation, while others only exhibited a large final flare due to the
+catastrophic disruption. The beginning and ending position, distance
+flight, and direction of the luminous phase for each event are shown
+in Table 2. The initial heights range from ∼ 120 to 83 km and terminal
+heights (before starting the dark flight) range from ∼ 80 to 13 km.
+As expected, the azimuth and slope have a random distribution, with
+the average slope being around 45◦. Note that the slope is measured
+with respect to the local horizon, 0◦ corresponding to a fully grazing
+meteor. In this regard, we see how the event SPMN010322A traveled
+through the atmosphere a notably greater distance than the rest (∼198
+km), its slope being close to 10◦. Event SPMN080322A, although
+also with a shallow slope, underwent a rapid disruption at 70 km
+altitude, which did not allow it to cover a long distance.
+MNRAS 000, 1–10 (2022)
+
+4
+E. Peña-Asensio et al.
+Figure 1. Selection of blended frames of some of the events analyzed in this work: a) SPMN090322C from Calar Alto by José M. Serna García, b) SPMN060222 from Corbera, c) SPMN080222B from Barx, d)
+SPMN220222 from Alpicat, e) SPMN180222 from Estepa, and f) SPMN110222 from Madrid.
+Table 2. Recorded fireballs with the beginning and ending position, flight distance traveled, and direction of the atmospheric flight.
+SPMN code
+Datetime (UTC)
+Stations
+Long0 (◦)
+Lat0 (◦)
+h0 (km)
+Long𝑡 (◦)
+Lat𝑡 (◦)
+h𝑡 (km)
+Distance (km)
+Azimuth (◦)
+Slope (◦)
+060222
+2022-02-06 23:03:20
+A,F
+4.324±0.011
+42.848±0.004
+91.3±0.4
+4.392±0.009
+42.8570±0.0031
+69.16±0.26
+22.9±0.5
+80±5
+75±4
+080222A
+2022-02-08 01:09:54
+A,B,K
+-2.5529±0.0029
+41.2470±0.0012
+101.594±0.028
+-2.5542±0.0029
+41.6429±0.0014
+41.06±0.08
+77.54±0.32
+359.4±0.5
+51.33±0.11
+080222B
+2022-02-08 23:31:00
+A,B
+1.1353±0.0010
+38.9446±0.0008
+89.16±0.07
+1.1055±0.0009
+39.1885±0.0007
+36.134±0.024
+60.68±0.22
+353.97±0.32
+60.940±0.032
+110222
+2022-02-11 02:26:30
+B,E,O
+-3.625±0.009
+39.702±0.007
+89.8±0.8
+-3.731±0.005
+39.455±0.006
+37.50±0.21
+65.5±0.5
+198.7±2.8
+52.9±1.2
+140222B
+2022-02-14 20:59:07
+G,I,N
+-3.5864±0.0014
+37.8739±0.0004
+94.646±0.025
+-3.2628±0.0009
+37.78175±0.00030
+49.547±0.018
+60.51±0.22
+109.69±0.05
+48.18±0.06
+180222
+2022-02-18 01:02:45
+I,J,O
+-6.1642±0.0023
+39.380±0.004
+88.79±0.06
+-6.0776±0.0034
+39.5080±0.0034
+12.87±0.15
+82.9±0.8
+26.8±1.3
+66.43±0.18
+220222
+2022-02-22 04:34:24
+A,K
+-0.5435±0.0010
+42.3780±0.0005
+83.77±0.08
+0.1736±0.0004
+42.2556±0.0005
+38.92±0.04
+80.57±0.13
+102.42±0.08
+33.814±0.015
+010322A
+2022-03-01 00:48:01
+A,C,L
+2.6121±0.0034
+41.3954±0.0020
+95.79±0.07
+1.4258±0.0018
+39.9335±0.0015
+50.499±0.024
+197.0±0.4
+211.99±0.07
+13.293±0.024
+010322B
+2022-03-01 01:43:57
+B,O
+-2.793±0.008
+39.9817±0.0019
+101.07±0.30
+-3.258±0.010
+39.5159±0.0019
+71.70±0.24
+74.2±0.7
+217.7±1.0
+23.3±0.5
+080322A
+2022-03-08 00:36:59
+A,F
+0.8633±0.0008
+40.6421±0.0005
+96.82±0.08
+1.7423±0.0006
+41.00590±0.00030
+80.13±0.05
+87.00±0.12
+60.904±0.032
+11.06±0.06
+080322B
+2022-03-08 19:26:22
+A,L
+1.8383±0.0005
+40.4211±0.0005
+83.58±0.05
+1.8210±0.0005
+40.4374±0.0005
+36.786±0.021
+57.70±0.18
+320.613±0.024
+54.09±0.10
+090322B
+2022-03-09 03:01:46
+A,B,C
+-1.5107±0.0010
+39.8088±0.0004
+120.65±0.07
+-2.0243±0.0011
+39.94857±0.00035
+77.071±0.035
+71.19±0.13
+289.42±0.04
+37.75±0.12
+090322C
+2022-03-09 04:25:38
+D,I
+-2.0192±0.0007
+36.9452±0.0009
+92.94±0.15
+-2.1597±0.0006
+36.4849±0.0017
+58.54±0.10
+64.80±0.05
+193.73±0.15
+32.07±0.23
+100322
+2022-03-10 01:38:19
+H,M
+-15.540±0.014
+30.0550±0.0034
+85.4±0.8
+-15.600±0.022
+29.689±0.005
+29.2±0.6
+82.5±0.6
+188±5
+42.94±0.17
+120322
+2022-03-12 22:15:53
+A,L
+1.1473±0.0004
+40.7151±0.0007
+94.21±0.09
+1.09818±0.00035
+40.7597±0.0006
+67.85±0.05
+27.40±0.16
+319.4±0.4
+74.30±0.25
+MNRAS 000, 1–10 (2022)
+
+a
+b
+ldaia(
+d)
+ESTEPA-SEVILLA-SPAIN-@AJ_ROBLES
+NORTEUTC2022-02-18 01:02:5Meteorite dropper spring 2022
+5
+  0 °
+  60 °
+  120 °
+  180 °
+  240 °
+  300 °
+-90 °
+90 °
+-60 °
+-30 °
+0 °
+30 °
+60 °
+20
+40
+60
+Geocentric velocity (km/s)
+Figure 2. Sinusoidal projection of the geocentric (diamond) and apparent
+(gray cross) radiants. Radiant pairs are connected with a light blue line.
+Geocentric radiants are color-coded according to their geocentric velocity.
+Using the height at which the brightest flare occurs, the air density,
+and the velocity at that point, we calculate the aerodynamic strength.
+According to the value of this dynamic pressure, we estimate the
+bulk density as explained in Section 2, which is used to calculate
+the pre-atmospheric diameter assuming a perfect sphere. To obtain
+the ballistic coefficient and the mass loss parameter, we assume an
+aerodynamic drag coefficient of 1.3 and a shape change coefficient
+of 2/3 (Gritsevich & Koschny 2011). The geocentric velocities range
+from ∼ 63 to 11 km/s, and most of the radiants are in the northern
+hemisphere, as depicted in Figure 2 in sinusoidal projection. All the
+computed parameters are shown in Table 3 and 4.
+Two meteoroids penetrate up to ∼ 30 and 13 km altitude starting
+the dark flight at a velocity of ∼ 8 and 20 km/s, respectively. As can
+be seen in Figure 3, from the application of the 𝛼 − 𝛽 criterion and
+assuming 50 g as the minimum terminal mass to produce a recov-
+erable fall, event SPMN100322 had some possibility of generating
+a meteorite with a mass of ∼140 g, and event SPMN180222 was
+likely to be a ∼430 g meteorite dropper. Unfortunately, a field search
+campaign was prepared but no fragments were recovered.
+The computed osculating orbital elements at the time of impact
+of the analyzed fireballs are compiled in Table 5. As an example of
+the Monte Carlo simulation, Figure 4 shows a heat map of the semi-
+major axis and inclination distribution for the 1,000 clones of event
+SPMN010322A at the time of impact (t=0 year without Earth-Moon
+gravitational focusing correction) and at the end of the backward
+orbital integration (t=-10,000 year).
+Four orbits present very high eccentricity values with large semi-
+major axes, five can be classified as Jupiter-family comets, while
+four are asteroid-like orbits. As expected, the orbits tend to be of
+low inclination, with the exception of SPMN090322B which has an
+inclination of 122◦. None of the meteoroids had close encounters
+with the Moon prior to the impact.
+4 DYNAMIC ASSOCIATION WITH METEOROID
+STREAMS AND PARENT BODIES
+The study of the associations of meteoroids that impact our planet
+with parent bodies or meteoroid streams is not a trivial task. There
+1
+2
+3
+4
+5
+6
+7
+8
+ln( sin )
+4
+2
+2
+4
+6
+ln( )
+Likely fall
+Possible fall
+Unlikely fall
+20
+30
+40
+50
+60
+70
+80
+Terminal height (km)
+Figure 3. Distribution of the 15 fireballs analyzed over the Spanish territory
+during February and March 2022 according to the 𝛼 − 𝛽 criterion. The color
+bar shows the terminal height, the gray solid curve the boundary for a 50
+g meteorite assuming no spin of the meteoroid, and the black solid curve
+the boundary for a 50 g meteorite assuming equal ablation over the entire
+meteoroid surface. We assume 𝜇 = 2/3 for all meteoroids.
+are numerous mechanisms that prevent the correct linking of meteors
+with their origins, from the intrinsically chaotic behavior of plane-
+tary systems to non-gravitational effects and sporadic collisions and
+interactions (Trigo-Rodríguez et al. 2005). Because of the high prob-
+ability that two orbits are randomly associated (Wiegert & Brown
+2004), we have not only analyzed the similarity of the orbits at the
+time of impact but also studied their robustness over time. From the
+time evolution of the parent body dissimilarity criterion, we found
+some dynamic associations. Figure 5 shows the evolution of the dis-
+similarity criterion during the orbital integration of the 15 events
+analyzed in this work, along with their most favorable parent body
+candidates or meteor shower. Table 6 shows each event with its most
+likely association, along with the years of time it lasts under the 𝐷𝐷
+threshold, the minimum encounter distance, the required ejection ve-
+locity at the time of minimum distance, and the minimum required
+ejection velocity.
+5 out of 15 events, that is, about 30% of the bright fireballs, are
+below the cut-off for at least 5,000 years. 4 events would be associated
+with minor showers (∼27%) and 1 fireball associated with a near-
+Earth asteroid (∼7%). In all the associated cases, the required ejection
+velocity needed to transform the parent orbit into the meteoroid orbit
+is in good agreement with the estimated range for collisions between
+objects, which can produce a kick of a few kilometers per second
+(Melosh 1984).
+5 DISCUSSION
+In relation to the various ablation behaviors observed, it is impor-
+tant to note that this could be the result of the differences between
+chondritic meteoroid and cometary aggregate bulk properties. The
+low density and high porosity of the latter are directly related to
+their aerodynamic strengths (Blum et al. 2006). Cometary streams
+typically produce centimeter-sized projectiles causing fireballs with
+disruptive flares, and multiple sudden brightness increases or a catas-
+trophic final flare. Due to the heterogeneity of the meteoroid compo-
+nents, the evaporation temperature of each one is reached at different
+MNRAS 000, 1–10 (2022)
+
+6
+E. Peña-Asensio et al.
+Table 3. Recorded fireballs with aerodynamic strength, ballistic coefficient, mass loss parameter, pre-atmospheric diameter, pre-atmospheric mass, and terminal
+mass.
+SPMN code
+s (kPa)
+𝛼
+𝛽
+D (cm)
+M0 (g)
+M𝑡 (g)
+060222
+18.9±0.4
+(8.6±0.7)·102
+10.6±1.0
+1.17±0.08
+0.83±0.16
+<1
+080222A
+724±7
+195.1±3.4
+1.023±0.031
+6.35±0.10
+134±7
+11.3±0.5
+080222B
+501.06±0.25
+79.72±0.27
+2.244±0.004
+13.90±0.04
+1405±13
+5.363±0.025
+110222
+361.6±3.5
+16.1±1.9
+11.3±1.7
+76±8
+(2.3±0.7)·105
+<1
+140222B
+78.16±0.15
+253.7±1.5
+2.917±0.017
+5.121±0.027
+70.3±1.1
+<1
+180222
+1107±21
+11.94±0.23
+4.70±0.07
+25.3±0.5
+(2.96±0.16)·104
+432±34
+220222
+283.6±1.8
+102.2±0.6
+1.690±0.022
+17.02±0.10
+(2.58±0.05)·103
+33.9±1.2
+010322A
+209.4±1.1
+387±6
+2.13±0.04
+10.87±0.16
+673±29
+2.93±0.17
+010322B
+17.17±0.33
+(1.38±0.28)·103
+18±4
+1.8±0.4
+3.1±1.8
+<1
+080322A
+2.935±0.019
+6898±28
+3.60±0.07
+0.732±0.006
+0.205±0.005
+<1
+080322B
+364.7±2.2
+82.9±0.8
+1.709±0.026
+14.42±0.16
+(1.57±0.05)·103
+25.3±0.8
+090322B
+25.45±0.10
+4831±29
+5.974±0.015
+0.3274±0.0013
+0.01838±0.00022
+<1
+090322C
+(4.185±0.017)·105
+255±4
+9.11±0.17
+7.16±0.08
+192±7
+106±20
+100322
+(1.30±0.14)·103
+40.3±3.4
+1.03±0.33
+10.1±0.8
+(1.9±0.5)·103
+(1.4±0.8)·102
+120322
+19.55±0.26
+82±22
+140±34
+12±4
+(1.0±1.1)·103
+<1
+Table 4. Recorded fireballs with right ascension and declination of the radiant, apparent, geocentric, and heliocentric velocities.
+SPMN code
+RA𝑎 (◦)
+Dec𝑎 (◦)
+RA𝑔 (◦)
+Dec𝑔 (◦)
+RAℎ (◦)
+Decℎ (◦)
+V𝑎,0 (km/s)
+V𝑎,𝑡 (km/s)
+V𝑔 (km/s)
+Vℎ (km/s)
+060222
+108±4
+38.4±2.2
+106±4
+37.4±2.5
+65.6±0.9
+5.8±1.0
+19.61±0.09
+11.229±0.033
+16.32±0.09
+41.5±0.5
+080222A
+153.14±0.33
+2.69±0.11
+152.86±0.34
+1.70±0.12
+106.92±0.06
+-7.911±0.031
+37.17±0.24
+16.329±0.029
+35.46±0.25
+39.91±0.33
+080222B
+135.60±0.15
+10.09±0.04
+135.18±0.16
+8.23±0.05
+82.10±0.05
+-4.635±0.004
+23.475±0.005
+9.7592±0.0022
+20.666±0.006
+37.79±0.04
+110222
+211.58±0.34
+71.8±2.0
+217.9±1.1
+74.3±2.3
+60.9±0.8
+27.0±1.7
+20.1±0.4
+15.89±0.20
+16.8±0.4
+35.24±0.31
+140222B
+41.35±0.08
+39.312±0.028
+30.63±0.09
+35.822±0.023
+51.850±0.018
+5.641±0.017
+15.068±0.013
+8.905±0.004
+10.536±0.020
+39.904±0.019
+180222
+146.3±0.5
+17.90±0.28
+145.2±0.5
+16.39±0.29
+91.44±0.10
+1.29±0.07
+23.95±0.08
+20.045±0.013
+21.32±0.09
+38.79±0.23
+220222
+149.508±0.030
+30.58±0.06
+140.248±0.028
+23.63±0.11
+80.31±0.07
+2.559±0.022
+15.60±0.04
+5.9502±0.0031
+11.41±0.06
+35.225±0.034
+010322A
+299.19±0.11
+50.410±0.029
+304.55±0.12
+47.034±0.012
+49.777±0.031
+34.32±0.04
+25.56±0.05
+9.799±0.015
+22.97±0.05
+36.477±0.009
+010322B
+288.24±0.33
+54.1±1.1
+295.0±0.4
+51.9±1.1
+56.2±0.5
+35±4
+23.83±0.06
+19.43±0.13
+21.01±0.07
+34.99±0.34
+080322A
+113.58±0.07
+-13.672±0.026
+104.83±0.12
+-21.95±0.07
+83.91±0.06
+-13.79±0.07
+17.157±0.030
+13.560±0.006
+13.40±0.04
+39.542±0.029
+080322B
+121.84±0.05
+10.43±0.09
+123.60±0.04
+6.96±0.07
+91.87±0.07
+-4.541±0.026
+18.72±0.06
+8.266±0.013
+14.90±0.08
+41.49±0.06
+090322B
+259.88±0.09
+10.98±0.09
+260.24±0.09
+10.72±0.09
+259.93±0.27
+57.545±0.015
+63.937±0.007
+40.138±0.033
+62.749±0.008
+41.32±0.04
+090322C
+340.5±1.2
+77.58±0.05
+6.2±0.9
+73.90±0.14
+73.418±0.026
+18.82±0.17
+18.159±0.018
+17.96±0.07
+14.397±0.024
+38.81±0.05
+100322
+200±12
+75.3±1.7
+205±17
+79.1±1.8
+84.42±0.31
+24.54±0.33
+20.2±0.7
+8.358±0.030
+17.0±0.9
+38.3±1.4
+120322
+156.87±0.09
+28.29±0.26
+157.00±0.10
+27.05±0.26
+104.92±0.23
+7±34
+21.39±0.15
+14.27±0.06
+18.27±0.18
+40.48±0.09
+Table 5. Recorded fireballs with semi-major axis, eccentricity, inclination, perihelion distance, argument of the perihelion, ascending node, and Tisserand
+parameter (referred to the J2000 equinox). Uncertainty for the ascending node is 0.0001◦.
+SPMN code
+a (au)
+e
+i (◦)
+q (au)
+𝜔 (◦)
+Ω (◦)
+T 𝑗
+060222
+11±5
+0.92±0.04
+6.1±1.4
+0.892±0.011
+216.823±0.011
+317.8516
+1.60±0.21
+080222A
+4.3±0.6
+0.938±0.007
+14.69±0.10
+0.2658±0.0030
+120.6728±0.0030
+138.9337
+1.77±0.14
+080222B
+2.395±0.018
+0.7285±0.0015
+5.47±0.05
+0.6504±0.0015
+78.9636±0.0015
+139.8736
+3.015±0.014
+110222
+1.60±0.06
+0.403±0.020
+27.2±1.0
+0.954±0.006
+208.156±0.006
+322.0380
+4.02±0.11
+140222B
+4.344±0.033
+0.7739±0.0017
+5.655±0.010
+0.98217±0.00005
+170.87497±0.00005
+325.8715
+2.320±0.008
+180222
+3.05±0.18
+0.783±0.011
+1.53±0.13
+0.662±0.005
+255.517±0.005
+329.0900
+2.60±0.09
+220222
+1.605±0.007
+0.4698±0.0026
+2.68±0.05
+0.8510±0.0005
+235.6854±0.0005
+333.2960
+4.089±0.013
+010322A
+1.9276±0.0027
+0.5469±0.0006
+36.06±0.10
+0.87346±0.00008
+131.68867±0.00008
+340.1080
+3.413±0.004
+010322B
+1.57±0.06
+0.411±0.020
+35.36±0.14
+0.922±0.007
+139.445±0.007
+340.1481
+4.00±0.12
+080322A
+3.96±0.04
+0.7532±0.0026
+13.885±0.012
+0.97727±0.00023
+15.37103±0.00023
+167.1195
+2.392±0.012
+080322B
+13.5±1.0
+0.931±0.005
+4.68±0.04
+0.9330±0.0004
+28.9213±0.0004
+167.8948
+1.570±0.025
+090322B
+11.2±0.5
+0.911±0.004
+122.44±0.13
+0.99250±0.00009
+178.06764±0.00009
+348.2148
+1.108±0.020
+090322C
+3.16±0.04
+0.688±0.004
+18.90±0.07
+0.98494±0.00007
+168.69848±0.00007
+348.2915
+2.665±0.019
+100322
+2.8±0.9
+0.65±0.12
+24.60±0.17
+0.98426±0.00012
+192.22661±0.00012
+349.1653
+2.8±0.6
+120322
+6.05±0.30
+0.862±0.008
+7.92±0.04
+0.8339±0.0023
+229.2663±0.0023
+352.0240
+1.93±0.04
+altitudes, giving rise to the so-called differential ablation (Gómez
+Martín et al. 2017). The aerodynamic overpressure experienced by
+meteoroids when they fragment allows for estimating their aerody-
+namic strength. This, in turn, allows for deducing the bulk properties
+of their meteoroid stream (Kresak 1982; Trigo-Rodríguez & Llorca
+2006). These types of large fireballs associated with cometary ves-
+tiges are the result of rapid disruption in micrometric grains and
+the sudden ablation of volatile mineral phases driven by the thermal
+wave in the meteoroid head (Trigo-Rodríguez et al. 2019).
+Even in such circumstances, it is remarkable that the sporadic con-
+MNRAS 000, 1–10 (2022)
+
+Meteorite dropper spring 2022
+7
+Table 6. Most likely parent body and meteoroid stream candidates for each event with the minimum 𝐷𝐷 value, the years that fulfill the 𝐷𝐷 criterion threshold,
+the minimum encounter distance, the required ejection velocity at the time of minimum distance, and the minimum required ejection velocity during the orbital
+integration.
+SPMN code
+Association
+D𝑚𝑖𝑛
+t𝐷 (y)
+S𝑚𝑖𝑛 (au)
+V𝑆,𝑚𝑖𝑛 (km/s)
+V𝑚𝑖𝑛 (km/s)
+060222
+𝜌 Geminids
+0.176
+180
+0.186
+4.7
+4.7
+080222A
+o Leonids
+0.174
+90
+0.231
+4.6
+0.9
+080222B
+Southern 𝛿 Leonids
+0.018
+8720
+0.129
+0.8
+0.4
+110222
+𝜔 Cassiopeiids
+0.101
+10000
+0.087
+9.6
+1.4
+140222B
+March Cassiopeiids
+0.121
+1610
+0.145
+10.2
+0.5
+180222
+Southern 𝛿 Leonids
+0.07
+240
+0.278
+13.2
+2.0
+220222
+Northern 𝛼 Leonids
+0.09
+10000
+0.05
+5.8
+1.4
+010322A
+2019 CV2
+0.099
+2640
+0.264
+6.0
+1.7
+010322B
+2017 FM91
+0.092
+9990
+0.104
+6.6
+2.3
+080322A
+2007 DZ40
+0.073
+800
+0.144
+3.1
+1.1
+080322B
+February Hydrids
+0.168
+600
+0.37
+15.9
+3.0
+090322B
+72 Ophiuchids
+0.136
+9990
+0.811
+14.3
+0.4
+090322C
+March Cassiopeiids
+0.084
+110
+0.34
+10.9
+0.8
+100322
+𝜓 Draconids
+0.106
+2080
+0.37
+5.1
+2.0
+120322
+𝜆 Leonids
+0.125
+1300
+0.083
+7.4
+2.4
+2.590
+2.595
+2.600
+2.605
+2.610
+2.615
+2.620
+Semi-major axis (au)
+39.8
+39.9
+40.0
+40.1
+40.2
+Inclination (°)
+1,000 clones heatmap
+ t=0 year (impact)
+1.86
+1.88
+1.90
+1.92
+1.94
+1.96
+1.98
+Semi-major axis (au)
+43.4
+43.6
+43.8
+44.0
+44.2
+44.4
+44.6
+Inclination (°)
+1,000 clones heatmap
+ t=-10,000 year
+Figure 4. Typical heatmap of the inclination and semi-major axis distribution
+of the 1,000 clones for the SPMN010322A in the Monte Carlo simulation.
+The top figure corresponds to the time of impact (t=0 year) without Earth-
+Moon gravitational focusing correction. The bottom figure corresponds to the
+end of the backward orbital integration (t=-10,000 years).
+tribution is not dominant at all. We found a very significant percent-
+age of bright fireballs dynamically associated with minor showers.
+Although during the orbital integration there are no very close en-
+counters despite the reasonable ejection velocities, we must point
+out that we have propagated 18 particles distributed in true anomaly
+throughout the orbit of the meteoroid streams, but at their nominal
+values for the rest of the orbital elements. Due to the orbital perturba-
+tions accumulated over time and their violent origin, either by tidal
+forces disruption or catastrophic collisions, the meteoroid streams
+spread toroidally along their orbit and gradually disperse. Some re-
+gions even undergo more pronounced decoherence than others due
+to the gravitational influence of the Earth-Moon system or nearby
+planets.
+The minimum ejection velocities calculated to produce the me-
+teoroid orbit from the parent body have a standard deviation range
+between 0.16 and 1.4 km/s (with an average standard deviation of
+0.4 km/s) for the studied events. Although the ejection velocities
+found are compatible with collisions of small objects in the inner
+Solar System, this does not necessarily mean that these meteoroids
+have separated from their meteoroid stream or parent body recently;
+we just note it as a feasible possibility due to the usual disruption
+behavior of crumbling asteroids and comets.
+Although remarkable, the high number of minor showers produc-
+ing fireballs should not come as a surprise as such a percentage of me-
+teors associated with meteoroid streams is not unusual. For example,
+percentages up to 80% between November and January were already
+reported belonging to meteor showers (Rao & Murthy 1974). On the
+other hand, among the 2,401 records studied by Lindblad (1971),
+apparently, 37% were associated with meteoroid streams. A similar
+percentage (41%) was found by Southworth & Hawkins (1963). Of
+the orbits analyzed by Jacchia & Whipple (1961), 65% were linked to
+a meteor shower. Regarding the Meteorite Observation and Recovery
+Project (MORP) database, 37% of the fireballs could be associated
+with meteoroid stream (Halliday et al. 1996). Terentjeva (1990) per-
+formed a grouping according to event candidates to produce mete-
+orites, finding that 68% of 554 fireballs studied could be part of a
+shower. And also in good agreement with the results of this work,
+Babadjanov (1963) reported that of the 185 meteors studied, 73%
+appeared to be of cometary origin. Recent studies also show large
+percentages of meteors associated with meteor showers, for example,
+MNRAS 000, 1–10 (2022)
+
+8
+E. Peña-Asensio et al.
+Figure 5. Evolution of the dissimilarity function 𝐷𝐷 of the 15 meteoroids with their most favorable candidates during the orbital backward integration over
+10,000. The 1,000 clones of each event are also shown.
+MNRAS 000, 1–10 (2022)
+
+0.6
+0.5 
+0.4
+D0.3
+0.2
+0.1
+SPMN060222 - p Geminids
+SPMN080222A-
+0.0 -
+0.6
+ SPMN110222 - w Cassiopeiids
+0.5 
+0.4
+β0.3
+0.2
+0.1
+SPMN140222B - March Cassiopeids
+SPMN180222 - Southern 6 Leonid
+0.0 
+0.6
+ SPMN220222 - 209 Northern α Leonids
+SPMN010322A - 2019 CV2
+SPMN010322B - 2017 FM91
+0.5 
+0.4 -
+B0.3
+0.2
+0.1
+0.0 
+0.6 7
+SPMN090322B - 72 Ophiuchids
+0.5
+0.4
+β0.3
+0.1 
+SPMN080322A - 2007 DZ40
+SPMN080322B - February Hydrids
+0.0
+0.6
+2222727
+0.5
+0.4 
+D0.3
+0.2 
+0.1
+SPMN090322C - March Cassiopeids
+ SPMN100322 -  Draconids
+SPMN120322 - 入 Leonids
+0.0
+-10000
+-8000
+-6000
+-4000
+-2000
+10000 -8000
+-6000
+-4000
+-2000
+-10000
+-8000
+-6000
+-4000
+-2000
+Years
+Years
+YearsMeteorite dropper spring 2022
+9
+45% in Colas et al. (2020) and 35% in Drolshagen et al. (2021). Re-
+garding superbolides detected from space, 23% could be associated
+with meteoroid streams or near-Earth objects (Peña-Asensio et al.
+2022).
+Therefore, as previously studied, it is reasonable to expect that a
+large percentage of the meteors belong to minor meteoroid streams,
+but also, as we show in this work, some meteor showers can be a
+significant source of large projectiles for the Earth and the Moon.
+6 CONCLUSION
+The extraordinary meteorological conditions in Spain during the
+spring of 2022 have made it possible to obtain high-quality data re-
+lated to the fireball activity produced, to a large extent, by minor mete-
+oroid streams. Ground-based multi-station recordings were possible
+thanks to the ever-increasing atmospheric volume monitored by the
+SPMN network throughout Spain. We reported 15 bright bolides in
+February and March, two of them being potential meteorite dropper
+events. By applying novel computer vision techniques and improved
+methods of trajectory reconstruction and heliocentric orbit calcula-
+tion implemented in our software 3D-FireTOC, we have been able
+to study in detail the atmospheric flight and dynamic association of
+large cometary and asteroidal projectiles impacting our planet. Based
+on the trajectory data, we computed the initial and terminal mass, the
+aerodynamic strength, and the bulk density by means of an ablation
+model. In consequence, we claim that:
+• Among the 169 bright meteors recorded during the spring of
+2022 in Spain, 2 of them were potentially meteorite dropper events.
+• We identify the minor showers o Leonids, Southern 𝛿 Leonids,
+𝜔 Cassiopeiids, Northern 𝛼 Leonids, and 72 Ophiuchids, and the
+asteroid 2017 FM91 as sources of large projectiles during February
+and March.
+• Nearby meteoroid streams can be efficient producers of large
+projectiles as they account for the ∼27% of the fireballs.
+• Near-Earth objects may be a greater source of impact risk than
+previously thought.
+• It is needed to extend the study and cataloguing of minor show-
+ers, since, although they are not very active in terms of the number
+of meteors, our work indicates that they also produce large bolides
+annually.
+• These findings support the idea that certain meteoroid streams
+associated with comets or asteroids may represent a short-term im-
+pact hazard.
+Finally, we think that understanding the origin and mechanisms by
+which large meteoroids reach the Earth is of great scientific interest
+due to the possibility of associating complexes and parent bodies with
+fireballs and, ultimately, meteorites found on Earth and the Moon.
+The relevance of associations also reverts in outreach, as we can
+quickly inform the public about the origin of the fireballs reported
+by eyewitnesses.
+ACKNOWLEDGEMENTS
+This project has received funding from the European Research
+Council (ERC) under the European Union’s Horizon 2020 re-
+search and innovation programme (grant agreement No. 865657)
+for the project “Quantum Chemistry on Interstellar Grains”
+(QUANTUMGRAIN). JMT-R and E.P-A. acknowledge finan-
+cial support from project PID2021-128062NB-I00 funded by
+MCIN/AEI/10.13039/501100011033. AR acknowledge financial
+support from the FEDER/Ministerio de Ciencia e Innovación – Agen-
+cia Estatal de Investigación (PID2021-126427NB-I00, PI: AR). AR is
+indebted to DIUE (project 2017SGR1323). Cebreros #AMS81 ESA
+Ground station belongs to the AllSky7 fireball monitoring project).
+We also thank all station operators whose continuous dedication
+have allowed to record these bolides from multiple stations: Jordi
+Donet Donet, Vicent Ibáñez, Jose M. Serna, Rainer Kresken, Pablo
+Ramirez Moreta, Carlos Alcaraz, Antonio J. Robles, Ramón López,
+Agustín Núñez, José A. de los Reyes, Sensi Pastor, Antonio Fernán-
+dez Sánchez, Antonio Lasala, Álex Gómez, Juan Gómez, Ramón
+López, Francisco José García Rodríguez and Cesar Guasch Besal-
+duch.
+DATA AVAILABILITY
+The data underlying this article will be shared on reasonable request
+to the corresponding author.
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+

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@@ -0,0 +1,1082 @@
+Ballistic surface channels in fully in situ defined Bi4Te3 Josephson
+junctions with aluminum contacts
+D. Rosenbach,1, 2, ∗ A. R. Jalil,3, 4 T. W. Schmitt,1, 4 B. Bennemann,3, 2
+G. Mussler,1, 2 P. Schüffelgen,1, 2 D. Grützmacher,1, 2 and Th. Schäpers1, 2
+1Peter Grünberg Institute (PGI-9), Forschungszentrum Jülich, 52425 Jülich, Germany
+2JARA-Fundamentals of Future Information Technology, Jülich-Aachen Research Alliance,
+Forschungszentrum Jülich and RWTH Aachen University, Germany
+3Peter Grünberg Institute (PGI-10),
+Forschungszentrum Jülich, 52425 Jülich, Germany
+4JARA-FIT Institute Green IT, RWTH Aachen University, 52062 Aachen, Germany
+(Dated: January 11, 2023)
+1
+arXiv:2301.03968v1  [cond-mat.mes-hall]  10 Jan 2023
+
+Abstract
+In this letter we report on the electrical transport properties of Bi4Te3 in a Josephson junction
+geometry using superconducting Al electrodes with a Ti interdiffusion barrier. Bi4Te3 is proposed
+to be a dual topological insulator, for which due to time-reversal and mirror symmetry both a strong
+topological insulator phase as well as a crystalline topological phase co-exist. The formation of a
+supercurrent through the Bi4Te3 layer is explained by a two-step process. First, due to the close
+proximity of the Al/Ti electrodes a superconducting gap is induced within the Bi4Te3 layer right
+below the electrodes. The size of this gap is determined by analysing multiple Andreev reflections
+(MARs) identified within the devices differential resistance at low voltage biases. Second, based
+on the Andreev reflection and reverse Andreev reflection processes a supercurrent establishes in
+the weak link region in between these two proximity coupled regions. Analyses of the temperature
+dependency of both the critical current as well as MARs indicate mostly ballistic supercurrent
+contributions in between the proximitized Bi4Te3 regions even though the material is characterized
+by a semi-metallic bulk phase.
+The presence of these ballistic modes gives indications on the
+topological nature of Bi4Te3.
+I.
+INTRODUCTION
+Hybrid structures of three-dimensional topological insulators and superconductors are
+considered promising building blocks for the realization of topological quantum circuits [1–
+3]. A crucial optimization parameter is a sufficiently large coupling of the superconductor
+to the topological insulator. In order to probe the proximity coupling strength a Josephson
+junction with a topological insulator weak link bridging two superconducting electrodes
+can be employed [4–6]. By measuring the current-voltage characteristics of these junctions,
+the interface transparency as well as the underlying mode of transport, i.e. diffusive or
+ballistic, can be investigated [7, 8]. The supercurrent in a Josephson junction is carried by
+electron-hole bound states. Based on the nature of these bound states their energy phase
+relation (EΦR) has a fixed periodicity. Irradiating the junction with a radio frequency signal
+allows to investigate the Shapiro step response. As the Josephson voltage V0 = hf/2e in
+∗ rosenbach@ph2.uni-koeln.de
+present address: Physics Institute II, University of Cologne, 50937 Köln, Germany.
+2
+
+between two Shapiro steps depends on the periodicity of the the bound states EΦR they
+give indications on the nature of the bound states [4]. In junctions with topological weak
+link both Andreev bound states (ABS; diffusive bulk and surface modes) carrying 2e charge
+per cycle and Majorana bound states (MBS; ballistic, perfectly transmitted surface modes)
+carrying only a single 1e charge per cycle coexist. Hence the periodicity of the bound states
+EΦR and respective the Josephson voltage in between two Shapiro steps differ by a factor
+of 2 comparing MBSs to ABSs [9]. MBSs are crucial to probe the existence of zero energy
+states within topological Josephson junctions and are indicated by missing odd Shapiro
+steps in experiments [5–7].
+Topological Josephson junctions can be separated into two regions. The first region is the
+topological matter underneath the superconducting electrodes. Here, the proximity effect
+opens an effective induced superconducting gap within both the surface and bulk of the
+topological matter. The second region is in between these two laterally separated proxim-
+itized regions called the weak link defined by the non-proximitized part of the topological
+matter. For the investigation of novel topological matter the question arises what relevant
+transport channels exist and what is their main mode of transport, i.e. ballistic topological
+surface states or diffusive bulk states.
+For the weak link in between the superconducting electrodes we chose Bi4Te3, which is a
+natural superlattice of alternating Bi2 bilayers [10] and Bi2Te3 quintuple layers. Initially,
+Bi4Te3 has been reported to be a zero band gap semimetal, comprising a Dirac cone at
+the Γ-point [11]. More recently, band structure calculations supplemented with scanning
+tunneling spectroscopy and angular photoemission spectroscopy measurements showed that
+Bi4Te3 is a semimetal with topological surface states [12–14]. In advanced GW-band struc-
+ture calculations a band gap of about 0.2 eV was identified around the Γ-point. Owing to
+time-reversal and mirror symmetries, Bi4Te3 is a strong topological insulator (STI) as well
+as a topological crystalline insulators (TCI). Furthermore, it is predicted that it contains
+higher order topological states [14, 15].
+We report on the transport properties of Josephson junctions based on the Bi4Te3 ma-
+terial system as weak link material and Al/Ti as the superconducting electrodes. For the
+fabrication of the samples, we employed an all in situ method [7, 16, 17], meaning that
+3
+
+the Bi4Te3 weak link layer is grown by selective-area molecular beam epitaxy, while for the
+definition of the superconducting Al/Ti electrodes we use an in situ shadow evaporation
+technique. This approach allows to achieve a clean interface between the Bi4Te3 layer and
+the superconductor without any contamination. In our study the proximity strength of the
+Al/Ti superconducting electrodes towards the underlying Bi4Te3 nanoribbon is examined in
+low temperature transport experiments including current-voltage characteristics and differ-
+ential resistance measurements. From multiple Andreev reflections (MARs) identified within
+the differential resistance we gain information about the strength of the proximity effect in
+Bi4Te3 and the size of the induced superconducting gap. Furthermore, from the excess cur-
+rent and from the temperature dependence of both the junctions critical current and the
+MARs we are able to specify the dominant transport regime of the Josephson supercurrent.
+II.
+EXPERIMENTAL SETUP
+Nanoribbon Josephson junctions have been defined following an all in situ approach[7,
+16, 17]. Therefore, two independent masking techniques are used. The masks are defined
+using four alternating layers of SiO2 and Si3N4 deposited on a highly resistive Si (111)
+substrate (R > 2000 Ω·cm) [17]. The first two layers are 5 nm of oxidized SiO2 and 15 nm
+of low pressure chemical vapor deposited (LPCVD) Si3N4.
+They comprise the selective
+area growth (SAG) mask. Narrow (w = 1000 nm down to 100 nm) nanotrenches are etched
+into the top Si3N4 layer using a combination of electron beam lithography and reactive ion
+etching. Afterwards, a 300-nm-thick SiO2 layer and a 100-nm-thick Si3N4 layer are deposited
+using LPCVD. These layers comprise the stencil mask used to deposit the superconducting
+electrodes in situ. Free-hanging Si3N4 bridge structures are defined, as previously reported
+[7], by patterning the Si3N4 and subsequently removing the SiO2 underneath in hydrofluoric
+acid (HF). The HF dip also locally removes the SiO2 of the first oxidized layer of the SAG
+mask only within the Si3N4 nanotrenches. During molecular beam epitaxy the Bi4Te3 will
+selectively grow within these nanotrenches on top of the Si(111) that is revealed during SiO2
+removal. The free-hanging Si3N4 bridge structures will be used after the deposition of Bi4Te3
+to define the superconducting electrodes, without breaking the vacuum [7].
+Bi4Te3 is a stoichiometric state of the (Bi2Te3)m(Bi2)n family with (m : n) = (3 : 3).
+4
+
+A unit cell comprises an alternating stacking sequence of Bi2Te3 quintuple layers and Bi
+bilayers. The planar epitaxy of Bi4Te3 stoichiometric alloy is achieved via molecular beam
+epitaxy (MBE) by precisely controlling the Bi:Te beam flux ratio to 1:2 while keeping TBi at
+490◦C and TTe at 280◦C [15]. In order to acquire Bi4Te3 nanostructures, the optimum growth
+parameters are subjected to the pre-patterned substrates with combinational surfaces. The
+substrate rotation ensures a homogeneous growth of the Bi4Te3 layer also underneath the
+free-hanging Si3N4 bridges. The thickness of the Bi4Te3 nanoribbon depends on the geometry
+and width of the nanotrenches [15]. This is as also adatoms impinging on the Si3N4 within
+the limit of the adatom diffusion length can contribute to the growth of Bi4Te3 within the
+trenches. For the junctions investigated here their respective thicknesses are given in Tab. I.
+The superconducting electrodes are deposited within a nitrogen cooled chamber below 0◦C
+by turning off the substrate heater. The free-hanging Si3N4 bridges are aligned perpendicular
+to the effusion cells of the evaporated metal, such that the shadow defines the weak link area.
+Si
+Si3N4
+SiO2
+Ti
+Al2O3
+x
+z
+y
+Al
+Bi4Te3
+Δ
+Δ*
+a)
+b)
+x
+y
+z
+500 nm
+Si3N4
+Si3N4
+Bi4Te3
+Al/Ti
+Al/Ti
+FIG. 1. In situ deposited Bi4Te3 nanoribbon Josephson junction. a) shows a false-colored
+SEM graph of the top view of an Al/Ti - Bi4Te3 - Ti/Al Josephson junction as prepared in situ.
+The Al/Ti superconducting electrodes are highlighted in cyan/brown, while the Bi4Te3 nanoribbon
+is shown in green and the Si3N4 hard mask in blue. The cross section along the nanoribbon main
+axis is schematically depicted in b). Here, the Ti interdiffusion layer (brown), the Al2O3 dielectric
+capping layer (light grey), the Si substrate (dark grey) as well as the Si3N4/SiO2 (blue/yellow)
+SAG mask layers are visible. The Al/Ti contacts are attributed a composite superconducting pair
+parameter ∆ and the pair parameter of the proximity coupled region in the Bi4Te3 layer (dark
+green) is denoted by ∆∗.
+5
+
+# w [nm] L [nm] t [nm] Ic [nA] RN [Ω] IcRN [µeV] ∆∗ [µeV] Iexc [nA] IexcRN [µeV]
+α
+τ
+γB
+1
+1000
+130
+8.6
+176
+120
+21.12
+82.5
+500
+60
+0.72 0.65 0.52
+2
+500
+130
+10
+35
+310
+10.85
+95
+159
+49.3
+0.5 0.57 0.36
+3
+100
+140
+16.5
+30
+744
+22.32
+115
+75
+55.8
+0.49 0.56 0.24
+TABLE I. Overview of interface parameters of Josephson junctions with Bi4Te3 nanoribbon weak
+link and Al/Ti (30 nm/3 nm) superconducting contacts. Given are the geometric dimensions, the
+junction width w, the junction length/electrode separation length L and the mean thickness t of
+the nanoribbon. The proximity induced gap below the superconducting electrodes ∆∗, the excess
+current Ic as well as the dimensionless parameters α, τ and γB that describe the interfacial quality
+of the junctions.
+After electrode deposition devices are covered by a 5 nm thin Al2O3 dielectric layer electron
+beam evaporated from a stoichiometric target. A false-colored scanning electron micrograph
+of an as-prepared Josephson junction with aluminum superconducting contacts is shown in
+Fig. 1 a). Aluminum has previously been reported to diffuse into (Bi0.06Sb0.94)2Te3 thin
+films [18], which increases the interfacial resistance of junctions within the superconducting
+regime of the Al electrodes. In order to prevent diffusion of the Al into the underlying Bi4Te3
+layer, a 3 nm thin Ti layer is deposited first as an interdiffusion barrier, as depicted in the
+schematics of the junction cross section shown in Fig. 1 b). The critical temperature of
+the superconducting Al/Ti composite electrodes is determined to be Tc = 0.95 K from four-
+terminal measurements of the differential resistance as a function of the temperature T down
+to 23 mK base temperature of a dilution refrigerator. The magnitude of the superconducting
+pair parameter has been determined to measure ∆ = 145 µeV, following Bardeen-Cooper-
+Schrieffer theory [19].
+The electrodes of the in situ defined nanoribbon Josephson junctions are wire bonded
+to a chip carrier in a quasi-four terminal contact configuration. The junctions are cooled
+to a base temperature of T = 23 mK using a dilution refrigerator. At base temperature
+the sample resistance is measured using standard lock-in techniques and the potential drop
+across the junction is determined using a voltmeter.
+6
+
+III.
+EXPERIMENTAL RESULTS
+A.
+Multiple Andreev reflections
+We have measured three junctions of different width w = 100, 500, and 1000 nm (see
+Tab. I). Both dV/dI(I) as well as V (I) as a function of a d.c. current bias applied for the
+500,nm wide junction (junction #2) are shown in Fig. 2 a). For an applied bias current
+below the critical current of Ic = 35 nA (see also Tab. I) a Josephson supercurrent estab-
+lishes. The differential resistance dV/dI is zero below I < Ic and reaches a finite value as
+soon as the current bias exceeds I > Ic. The critical current is not hysteretic, whether the
+current bias is swept from positive to negative current biases or vice versa. In Tab. I the
+critical current Ic, the normal state resistance RN values are listed. The corresponding IcRN
+product values are found to be in the range between 10.85 and 22.32 µV.
+As mentioned before, we anticipate that establishing a supercurrent through the Bi4Te3
+a)
+-900
+-600
+-300
+0
+300
+600
+900
+-200
+0
+200
+0
+200
+400
+600
+V (μV)
+I (nA)
+dV/dI (Ω)
+0
+200
+400
+600
+dV/dI (Ω)
+Iexc
+-200
+-100
+0
+100
+200
+V (µV)
+b)
+0.0
+0.2
+0.4
+0.6
+0.8
+80
+160
+(μV)
+MAR ord. -1 (1/n)
+n=1
+2
+3
+4
+57
+Vn
+FIG. 2. IV -characteristics and differential resistance dV/dI of Josephson junction #2.
+a) IV -characteristics and differential resistance as a function of the applied d.c.
+bias current
+(dV/dI(I)).
+A linear extrapolation from the IV-characteristics above 2∆∗ to V = 0 is shown
+(red dashed line) to extract the excess current Iexc.
+b) Differential resistance as a function of
+the measured d.c. potential drop across the Josephson junction (dV/dI(V )), showing signatures
+of multiple Andreev reflections (MARs). The inset shows the position (Vn) of the MARs plotted
+against the inverse of the MAR order number (1/n). The linear fit is forced through the origin.
+layer is a two step process. First, the proximity to the superconducting metallic Al/Ti elec-
+7
+
+trodes induces a superconducting pair potential into the Bi4Te3 layer (dark green regions in
+Fig. 1 b)), which decays over a length scale given by the superconducting coherence length ξN
+within the Bi4Te3 layer. For the superconducting coherence length we have to consider two
+different cases. In the ’dirty limit’, the elastic scattering in the dissipative state of the Bi4Te3
+layer takes place on length scales smaller than the superconducting coherence length. When
+the distance between two elastic scattering events exceeds the superconducting coherence
+length, the transition is in the ’clean limit’. Using low-temperature magnetotransport data
+on nano-Hall structures, we find that the Bi4Te3 layer is (semi)metallic, in agreement with
+recent reports [12–14], with a carrier density of n2D ≈ 4 × 1014 cm−2 (see Supplementary
+Sec. A) and an elastic mean free path length of only le ≈ 4 nm. Furthermore, the Hall bar
+data does not show any significant increase of the magnetoresistance, as expected from a
+Dirac semimetal [20]. For given reasons we therefore assume that the proximitized regions
+of the Bi4Te3 film underneath the superconducting Al/Ti electrodes are in the dirty limit,
+since the estimated superconducting coherence length of ξN =
+�
+¯hDBulk/2πkBTc = 45 nm,
+with DBulk the diffusion constant of the bulk and Tc the critical temperature of the Ti/Al
+superconducting electrodes.
+When proximitizing the regions of the Bi4Te3 underneath the Al/Ti superconducting
+electrodes a Josephson supercurrent establishes in a next step between the two proximitized
+layers based on electron-hole bound states.
+When the applied current bias exceeds the
+critical current I > Ic the junctions resistance is modulated by Andreev reflection processes
+at the superconductor to normal conductor interface. Only beyond a current bias of about
+|I| > 740 nA the junctions resistance is mostly constant. At this point the potential drop
+across the junction measures 2∆∗, i.e. the size of the proximity induced superconducting
+gap, as indicated in Fig. 1 b). In order to quantify the size of the induced superconducting
+gap ∆∗ in the proximitized Bi4Te3 more precisely (cf.
+Fig. 1 b)) we analyzed multiple
+Andreev reflections (MARs) visible within the differential resistance dV/dI of the junction.
+These MARs occur at bias voltages below the size of the induced superconducting gap at
+voltages of V = 2∆∗/en, where n is an integer [21]. In junction #2 we observe MARs of
+the order n = 1, 2, 3, 4, 5, 6, 9, 11, 13. Missing signatures of intermediate order MARs (e.g.
+n = 7, 8) has been observed before in BiSbTeSe2 nanoribbon Josephson junctions [22] but
+an explanation is missing until now.
+8
+
+The size of the induced superconducting gap is determined by plotting the position (in volt)
+of each MAR against the inverse of the MAR order number (1/n). The induced supercon-
+ducting gap measures ∆∗ = 95 µeV (for n = 1, as indicated by a blue dot within Fig. 2
+b) at T = 50 mK), which is smaller than the gap of the Al/Ti superconducting electrodes
+(∆ = 145 µeV).
+B.
+Temperature dependency of Ic and MARs
+Figure 2 b) shows the differential resistance of junction #2 at different temperatures.
+The signatures of MARs vanish above the critical temperature of the Al/Ti superconducting
+electrodes. The temperature dependency of MARs of order n = 1 (blue dots), n = 2 (orange
+dots) and n = 3 (green dots) is shown in Figs. 2 b) and c). The temperature dependency of
+the induced superconducting gap is given by [23]
+∆∗(T) =
+∆Al/Ti(T)
+1 + γB
+�
+∆2
+Al/Ti(T) − ∆∗2(T)/kBTc
+,
+(1)
+where γB is a measure of the interfacial barrier strength in between the Al/Ti superconduct-
+ing electrodes and the Bi4Te3 nanoribbon layer. Above formula is fitted to the ∆∗(T) data
+and a value for γB = 0.36 has been determined. The γB values of the other two junctions
+are listed in Tab. I. The value of γB indicates that there is an effective barrier present
+between the Al/Ti layer and Bi4Te3 despite the in situ fabrication. It has been identified
+that the Bi4Te3 tends to be terminated by a Bi bi-layer underneath the Al/Ti layer while it
+is terminated by a Bi2Te3 layer otherwise [15]. A possible reason for the barrier identified
+might be the mismatch of Fermi energies in between these different regions on the surface
+of the proximitized and the non-proximitized regions of Bi4Te3 resulting in a potential step
+at their interface.
+As a next step, the Josephson supercurrent between the the proximitized regions with
+the superconducting gap ∆∗ is analyzed in detail. The supercurrent depends on the kind
+of transport, i.e. ballistic or diffusive, and on the transparency between the proximitized
+Bi4Te3 layers and the Bi4Te3 weak link (cf. Fig. 1 b). The transparency of the interfaces of
+the lateral Josephson junction are analyzed in two ways. The first method uses the excess
+9
+
+a)
+T=50mK
+T=800mK
+-200
+-100
+0
+100
+200
+200
+300
+400
+500
+600
+dV/dI (Ω)
+V (µV)
+b)
+c)
+V (μV)
+-200
+-150
+-100
+-50
+0
+50
+100
+150
+200
+-30
+0
+30
+60
+90
+120
+150
+I (nA)
+T = 50 mK
+T = 500 mK
+ 
+Ic (nA)
+30
+20
+10
+100
+200
+300
+T (mK)
+0
+40
+80
+120
+160
+200
+0.2
+0.4
+0.6
+0.8
+V (µV)
+T (K)
+2Δ*
+Δ*
+2/3Δ*
+FIG. 3.
+Temperature dependency of a) the differential resistance as a function of the bias
+potential of Josephson junction #2, dV/dI(V, T), in between 50 mK and 800 mK. Each trace is
+offset by 25 Ω. The size of the proximity induced superconducting gap (2∆∗ in µV) is highlighted
+by blue dots, while MARs of order n = 2 and 3 are highlighted by orange and green dots respectively.
+b) shows the temperature dependent position of 2∆∗ and MARs of order n = 2 and n = 3. c) shows
+the temperature dependent IV -characteristics from 50 mK to 500 mK, where each trace is offset by
+5 µV. The red dashed line indicates the critical current of the junction Ic = 35 nA at T = 50 mK.
+The temperature dependent critical current, Ic(T), is shown in the inset with the fit indicated by
+the dashed line.
+10
+
+600
+550
+500
+450
+400
+350
+300
+250
+200
+-200
+-100
+100
+200current of Iexc = 159 nA, which is determined from the junctions IV -characteristics by linear
+extrapolation above the superconducting gap V ≥ 2∆∗ (highlighted as dashed red line in
+Fig. 2 a)) to V = 0. The excess current displays the additional current due to successful
+Andreev reflections in the dissipative state of the junction and is directly related to the
+transparency of the junction. An analytical expression following Niebler, Cuniberti, and
+Novotny [24] is used to determine the junctions interfacial barrier strength Z = 0.86 using
+the parameter α = eIexcRN/∆∗ (cf. Tab. I). The barrier strength is related to the trans-
+parency via τ = 1/(1 + Z2) = 0.57. Note, that τ expresses the transparency between the
+proximitized and the non-proximitized regions of the Bi4Te3 layer in contrast to γB which
+quantifies the barrier between the Al/Ti superconducting electrodes and the proximitized
+Bi4Te3 region. Similar to the observation from Kunakova et al.[25] we find that the values for
+the interface transparency parameters τ and γB are interdependent. This effect can be at-
+tributed to a metallization effect the electrodes have on the surface states of the Bi4Te3 layer.
+An additional method to determine the interface transparency is by quasi-classical analy-
+sis of the temperature dependent critical current [7, 8]. Using a voltage criterion the critical
+current is extracted from the IV -characteristics at different temperatures shown in Fig. 2
+d), with the inset showing Ic(T). We used a ballistic model fit (shown as black dashed line
+in the inset of Fig. 1 b)) based on the Gor’kov equations with arbitrary junction length L
+[26] and barrier transparency D [27]. For the fit a value for the critical temperature of the
+gap within the Al/Ti electrodes Tc = 0.95 K and a Fermi velocity of vF = 3.8 × 105 m/s of
+the Bi2Te3 surface layer are used [15]. The best fit results in an interface transparency of
+D = 0.6, which is in good agreement with the transparency τ determined using the excess
+current analysis described before. We also performed a quasi-classical fit using a diffusive
+model based on the Usadel equations [28]. However, within a physically reasonable range of
+values we did not get a decent fit. Our analysis indicates that the supercurrent is carried by
+ballistic modes with increased superconducting coherence length rather than bulk modes.
+The superconducting coherence length of these ballistic modes can be estimated within
+the clean junction limit to measure ξN = ¯hvF/2πkBTc = 1.15 µm. The observation of a
+dominating ballistic channel might be attributed to highly conductive surface states of the
+Bi4Te3 layer, which overrules the diffusive transport in the bulk.
+11
+
+C.
+Shapiro steps
+We also performed differential resistance measurements under the influence of an exter-
+nally applied radio-frequency (rf) signal using a λ/4 antenna. In Figs. 4 a)-f) the differential
+resistance is displayed as a function of the applied rf power and the junctions potential differ-
+ence is scaled by hf/2e. Within the range of frequencies applied 1.7 GHz ≤ frf ≤ 14.25 GHz
+we observe full integer Shapiro steps. At frequencies frf = 14.25 GHz and 8.25 GHz, how-
+ever, additional sub-integer Shapiro steps have been measured. Fractional Shapiro steps can
+be caused by phase-slip centers inside the junction [29], phase instabilities introduced by
+Abrikosov vortices [30], magnetic disorder [31] or due to a non-sinusoidal or skewed current
+phase relation (CΦR) [32, 33]. In junctions of high transparencies or very short ballistic
+junctions the CΦR is expected to be non-sinusodial [30]. The relative measure of the junc-
+tion length over the superconducting coherence length within the Bi4Te3 layer (d/ξN) has
+influence on the maximum Josephson supercurrent and the shape of the CΦR. Already for
+values of ξN/d > 3 the CΦR is skewed and the maximum current density lies above a value of
+φ > π [30]. Skewed, non-sinusoidal CΦRs can be decomposed into sinusoidal components of
+lower periodicity, which can explain the evolution of sub-integer Shapiro steps. For ballistic
+modes of increased superconducting coherence length (ξN = 1.15 µm) this limit would need
+to be considered as the junction length of junction #2 (L = 130 nm) is much smaller. For
+the wider junction (junction #1, w = 1000 nm) as for the narrower junction (junction #3,
+w = 100 nm, see supplementary Sec. B) the sub-integer Shapiro steps have been observed
+as well, confirming the presence of ballistic modes independent of the junction geometry.
+For the presence of ballistic modes one would expect the supercurrent to be partially carried
+by MBSs, resulting in odd integer Shapiro steps to vanish [9]. Based on the fraction of the
+supercurrent that is carried by MBSs compared to the supercurrent carried by ABSs the
+cross-over frequency for the observation of missing odd integer Shapiro steps [4] should lie
+below fMBSs < 5.25 GHz, which is the cross over frequency considering the whole supercur-
+rent is carried only by MBSs. As no missing Shapiro steps have been recorded it is assumed
+that less then one third of the supercurrent is carried by MBSs. Therefore, a possible reason
+that we did not observe missing odd integer Shapiro steps as an indication of MBSs might
+be that the irradiated frequency was too large [9].
+12
+
+a)
+c)
+-4
+-2
+0
+2
+4
+6
+0
+1
+V (hf/2e)
+RF power (dBm)
+dV/dI (Ω)
+-2
+-1
+3
+2
+-3
+8
+10
+320
+240
+160
+80
+0
+-20
+-10
+0
+2
+V (hf/2e)
+RF power (dBm)
+dV/dI (Ω)
+-4
+-2
+6
+4
+-6
+0
+10
+320
+240
+160
+80
+0
+e)
+-6
+-4
+0
+2
+V (hf/2e)
+RF power (dBm)
+dV/dI (Ω)
+-4
+-2
+4
+-2
+0
+320
+240
+160
+80
+0
+-8
+f = 14.25 GHz
+f = 8.25 GHz
+f = 1.7 GHz
+1/2
+1/3
+1/5
+b)
+dV/dI (Ω)
+f = 14.25 GHz
+0
+1
+2
+3
+100
+200
+300
+V (hf/2e)
+400
+500
+1/2
+1/3
+1/5
+d)
+dV/dI (Ω)
+f = 8.25 GHz
+0
+1
+2
+3
+100
+200
+300
+V (hf/2e)
+400
+500
+P = 3 dBm
+P = -4 dBm
+P = -17 dBm
+P = -10 dBm
+P = -8 dBm
+P = -2 dBm
+f)
+dV/dI (Ω)
+0
+1
+2
+3
+200
+300
+400
+V (hf/2e)
+100
+f = 1.7 GHz
+FIG. 4. Shapiro response of Josephson junction #2 at different radio-frequencies ap-
+plied. a), c) and e) show the differential resistance as a function of the radio-frequency excitation
+amplitude/radio-frequency power (P in dBm) and the potential bias (V in hf/2e) of the junction
+(dV/dI(P, V )). The differential resistance is displayed in between values of dV/dI = 0 (red) and
+dV/dI = 320 Ω. b), d) and f) show line traces of the differential resistance as a function of the
+bias potential in a given range of radio-frequency powers, with the lowest power displayed in blue
+and the highest in red. Next to Shapiro steps at full integer values of V = n · hf/2e, there are
+sub-integer steps visible in line-cuts at f = 14.25 GHz and 8.25 GHz. The sub-integer steps are
+highlighted with their given fractions of the first integer Shapiro step.
+13
+
+3
+2
+1
+0
+-1
+-2
+3
+-4
+-2
+0
+2
+4
+9
+8
+100.000
+80.00
+160.0
+240.0
+320.00.000
+80.00
+160.0
+240.0
+320.00.000
+80.00
+160.0
+240.0
+320.0IV.
+CONCLUSIONS
+By characterizing Bi4Te3-based Josephson junctions we obtained a detailed picture of
+the different contributions taking part in establishing a supercurrent through the junctions
+weak link. By analysing MARs we found that the intimate contact of the Al/Ti layer on top
+of the Bi4Te3 layer results in an induced superconductive gap ∆∗ in the topological matter
+due to the proximity effect. In order to establish robust proximitzed regions underneath
+the Al/Ti electrodes the presence of bulk carriers are probably beneficial, if not essential.
+The Bi4Te3 has been identified to carry a large amount of bulk charges. The proximitized
+regions of the Bi4Te3 are coupled by the unproximitized Bi4Te3 weak link giving rise to
+a Josephson supercurrent.
+We anticipate that the Josephson supercurrent mainly flows
+in the topological surface channel rather than in the bulk of the Bi4Te3 link, similar to
+results obtained in junctions with a different topological insulator layer [7]. Analysing the
+temperature dependency of the critical current we indeed identified the transport regime in
+these junctions to be mainly ballistic. However, by analysing the temperature dependency
+of the MARs an effective barrier in between these regions, probably due to a different surface
+termination of both regions, has been identified. From our Shapiro step measurements we
+came to the conclusion that the current-phase relationship is non-sinusoidal, i.e. supporting
+our claim of ballistic modes in our junctions. However, we did not find a 4π contribution in
+the Shapiro step measurements indicating the presence of Majorana zero modes. One reason
+might be that our lowest rf frequency of f ≤ 1.7 GHz was too high, so that we could not
+enter the regime where the 4π contributions are visible. For future material combinations
+in hybrid Josephson junctions including topological matter it is important to consider our
+findings.
+ACKNOWLEDGEMENTS
+This work was partly funded by the Deutsche Forschungsgemeinschaft (DFG, German
+Research Foundation) under Germany’s Excellence Strategy - Cluster of Excellence Matter
+and Light for Quantum Computing (ML4Q) EXC 2004/1 - 390534769.
+This work was
+financially supported by the German Federal Ministry of Education and Research (BMBF)
+via the Quantum Futur project "MajoranaChips" (Grant No. 13N15264) within the funding
+14
+
+program Photonic Research Germany.
+Competing interests
+The author(s) declare no competing interests.
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+17
+
+SUPPLEMENTARY INFORMATION
+A.
+Magnetotransport
+Besides Josephson junctions with a Bi4Te3 weak-link we have fabricated nano Hall bars.
+Therefore we have selectively deposited Bi4Te3 in nanotrenches that have been arranged in
+a Hall bar layout with one main nanoribbon and three ribbons symmetrically on each side.
+The fabricated device is shown as a false color scanning electron micrograph in Fig. 5 a).
+The nanoribbons have a width of w = 100 nm and the spacing in between two side nanorib-
+bons is L = 1000 nm.
+Nano Hall bar devices have been cooled down to T = 1.5 K in a variable temperature in-
+sert cryostate equipped with a superconducting magnet that can apply magnetic fields up
+to Bmax = 13 T. The sample holder insert is equipped with an electromechanical stepper
+motor. The relative orientation of the magnetic field to the nano Hall bars can be changed
+from an alignment of the magnetic field parallel to the main nanoribbon axis to a mag-
+netic field oriented perpendicular out-of-plane. From Hall measurements in an out-of-plane
+applied magnetic field the Hall slope AH = dRxy/dB = 1.58 Ω/T and subsequently the two-
+dimensional sheet carrier density n2D = (AHe)−1 = 3.9 × 1014 cm−2 have been determined.
+The Bi4Te3 has a strong metallic character with high charge carrier density and low mobil-
+ities µ = L · (WRxxn2De)−1 = 215 cm2(V · s)−1
+Fig. 5 b) shows the longitudinal resistance of the nano Hall bar for different relative angles of
+the magnetic field applied to the surface of the substrate. Next to the weak antilocalization
+feature, typical for 3D bulk as well as 2D surfaces with strong spin-orbit coupling, the mag-
+netoresistance does not change by more then 2.5% over the whole range of applied magnetic
+fields B ≤ |±13T|. In a perpendicular applied magnetic field (Θ = 90◦, red curve) the mag-
+netoresistance does show a spectrum of universal conductance fluctuations. In Fig. 5 c) the
+amplitude of individual oscillations frequencies from a fast fourier transformation performed
+on the data from b), shows a set of prominent frequencies, limited by the phase coherence
+length (lφ = (φ0 · fmax) ≈ 25 nm).
+The temperature dependent magnetoresistance data in a perpendicular applied magnetic
+field for temperatures in between 2 K ≤ T ≤ 30 K is shown in Fig. 5 d). For each trace the
+root mean square of the oscillation amplitude is computed rms(δGxx) and the values are
+18
+
+shown in the inset as a function of temperature. The value for rms(δGxx) is constant up to
+a temperature of 9 K. For higher temperatures the values follow a T −3/2 dependency.
+-10
+-5
+0
+5
+10
+745
+750
+755
+760
+Rxx (Ω)
+B (T)
+-10
+10
+30
+50
+90
+70
+θ (°)
+Bi4Te3
+B (�=90Deg)
+B (�=0Deg)
+I
+0
+20
+40
+60
+80
+θ (°)
+100
+0
+2
+4
+6
+8
+10
+12
+1/B (1/T)
+FFT Amp. (a.u.)
+-10
+-5
+0
+5
+10
+B (T)
+745
+750
+755
+760
+Rxx (Ω)
+0
+5
+10
+15
+20
+25
+30
+T (K)
+10
+1
+0.01
+0.02
+0.03
+0.04
+T (K)
+rms(δGxx)(e2/h)
+a)
+b)
+c)
+d)
+T-3/2
+1 μm
+FIG. 5. Magnetotransport data on Bi4Te3 Hall bars. a) Layout of the selectively grown
+Bi4Te3 nano Hall bar investigated. b) Longitudinal magnetoresistance as a function of magnetic
+field for various tilt angles (Rxx(B, θ) of the devices main channel w.r.t the magnetic field. The
+orientation of the sample is schematically depicted. c) Fast-Fourier-transformation amplitude of
+the magnetoresistance traces from b) showing high frequent universal conductance fluctuations at
+a large range of angles in between 15◦ ≤ θ ≤ 90◦ and low frequent oscillations from coherent states
+within the nanoribbon cross section for a magnetic field applied parallel to the main axis of the
+nanoribbon. d) Temperature dependency of the longitudinal magnetoresistance (Rxx(B, T)) for
+temperatures in between 1.5 K≤ T ≤ 30 K. The inset shows the temperature dependency of the
+root mean square of the conductance fluctuation amplitude rms(δGxx(T)
+B.
+Shapiro response measurements
+Next to the 500 nm wide Bi4Te3 Josephson junction characterized in the main text, we
+have additionally measured a wide junction (w = 1000 nm, Junction #1) and a narrow
+junction (w = 100 nm, Junction #3). All the junction parameters are given within the table
+19
+
+BST2318 Bi4Te3 100nm HB Rxx(Q)
+100.0
+93.9
+87.8
+760
+81.7
+75.6
+69.4
+63.3
+57.2
+755
+51.1
+。
+45.0
+0
+38.9
+32.8
+750
+26.7
+20.6
+14.4
+8.3
+745
+2.2
+3.9
+-10.0
+-10
+-5
+0
+5
+10
+B (T)BST2318 Bi4Te3 100nm FFT Ampl.(°), L=1900nm
+12
+10
+10000.00
+8
+1000.00
+frequency (1/B)
+FFT Amp. (a.u)
+6
+100.00
+4
+10.00
+2
+1.00
+0
+0.10
+0
+20
+40
+60
+80
+100
+Deg (°)Mag = 32.58 K X
+1 μm
+WD = 3.4 mm
+EHT = 5.00 kV
+Date :27 Nov 2019
+IBN
+signal A = InLensa)
+b)
+-4
+-2
+0
+2
+4
+6
+0
+1
+V (hf/2e)
+RF power (dBm)
+dV/dI (Ω)
+-2
+-1
+3
+2
+-3
+8
+10
+600
+450
+300
+150
+0
+c)
+-20
+-16
+0
+2
+V (hf/2e)
+RF power (dBm)
+dV/dI (Ω)
+-2
+-12
+-8
+d)
+-12
+-8
+0
+V (hf/2e)
+RF power (dBm)
+dV/dI (Ω)
+-2
+2
+-4
+0
+-12
+-8
+0
+2
+V (hf/2e)
+RF power (dBm)
+dV/dI (Ω)
+-4
+-2
+4
+-16
+f = 9.75 GHz
+f = 5.25 GHz
+f = 3.0 GHz
+f = 2.1 GHz
+600
+450
+300
+150
+0
+600
+450
+300
+150
+0
+600
+450
+300
+150
+0
+-4
+0
+-6
+6
+FIG. 6. Shapiro response of Josephson junction #3 at different radio-frequencies ap-
+plied. a), b), c) and d) show the differential resistance as a function of the radio-frequency ex-
+citation amplitude/radio-frequency power (P in dBm) and the potential bias (V in hf/2e) of the
+junction (dV/dI(P, V )) at f = 9.75 GHz, f = 5.25 GHz, f = 3.0 GHz and f = 2.1 GHz, respectively.
+The differential resistance is displayed in between values of dV/dI = 0 (red) and dV/dI = 600 Ω.
+Next to Shapiro steps at full integer values of V = n · hf/2e, there are sub-integer steps visible for
+an applied radio-frequency of f = 9.75 GHz.
+in the main manuscript. Next to these standard junction characeristics we here show Shapiro
+step measurements of the narrow junction #3, shown in Fig. 6. Within a similar range of
+radiofrequencies applied, as for junction #2 in the main text, we observe a similar behavior
+w.r.t. the Shapiro step evolution in the differential resistance as a function of the applied RF
+power applied to and the d.c. potential bias applied across the junction (dV/dI(P, V )). For
+the largest frequency applied of f = 9.75 GHz, not only full integer Shapiro steps, but also
+half-integer Shapiro steps can be observed. This demonstrates that the existence of high
+coherent ballistic channels do not seem to change with the width of the nanoribbon, as they
+20
+
+0.000
+80.00
+160.0
+240.0
+320.00.000
+80.00
+160.0
+240.0
+320.00.000
+80.00
+160.0
+240.0
+320.00.000
+80.00
+160.0
+240.0
+320.0would in topological insulator nanoribbons, where a quantization of transverse momentum
+states would alter the surface state dispersion.
+21
+

3tE2T4oBgHgl3EQfOAZw/content/tmp_files/2301.03743v1.pdf.txt

@@ -0,0 +1,636 @@
+arXiv:2301.03743v1  [astro-ph.SR]  10 Jan 2023
+1
+LS And: WZ Sge-type outburst first time since the 1971 discovery
+Taichi Kato1
+tkato@kusastro.kyoto-u.ac.jp
+1 Department of Astronomy, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan
+Abstract
+LS And was a transient discovered in 1971 in the M 31 region and it has been argued whether it could
+be an intergalactic nova or a dwarf nova. Using the Zwicky Transient Facility (ZTF) data, I found that the
+object underwent the second known outburst in 2022 April. The behavior was that of a WZ Sge-type dwarf
+nova with a long fading tail and the light curves of the 1971 and 2022 outbursts matched very well. The
+light curves suggest that LS And is a typical WZ Sge-type dwarf nova near (but before reaching) the period
+minimum of cataclysmic variables. The true observed peak of the 1971 outburst was likely 12.2 mag. The
+outburst parameters were similar to those of other WZ Sge-type dwarf novae. The fading tail lasts more than
+a year and the object is still currently on this tail. There was a hint of 0.5-mag temporary brightening on the
+fading tail and the object appears still active after the outburst.
+LS And was discovered by van den Bergh et al. (1973) in the region of M 31 (named “m” in their paper).
+van den Bergh et al. (1973) stated that the object was visible only on a blue and on a yellow plate taken in
+immediate succession on 1971 August 26.
+van den Bergh et al. (1973) suggested that the variable might be
+either a supernova or a flare star. Although van den Bergh et al. (1973) did not give the brightness of this object,
+it was estimated to be 12.5 from their figure by Romano (1977).
+Sharov (1973) examined plates taken in the Crimean Station of Sternberg Astronomical Institute and Latvian
+Radio Astrophysical Observatory. Sharov (1973) succeeded in obtaining one observation near the maximum and
+the light curve of the fading part. Sharov (1973) noted the presence of a star of 21–22 mag on Palomar Observatory
+Sky Survey (POSS). Based on the large amplitude exceeding 8 mag, rapid fading (0.2 mag d−1) in the early fading
+part and the very slow (less than 0.001 mag d−1) fading rate in the late fading part, Sharov (1973) stated that
+the star was unlikely a supernova or a flare star. The light curve, however, did not resemble those of typical
+novae or dwarf novae and Sharov (1973) suggested that it might be a very distant nova (i.e. intergalactic nova)
+if it was indeed a nova.
+Romano (1977) examined Asiago plates and presented a rough light curve of the outburst (probably unaware
+of the work by Sharov 1973). Romano (1977) indicated that the variable was at the limit of visibility (∼20.5 mag)
+on POSS and that color was almost white. Romano (1977) excluded a flare star based on the light curve and
+also a supernova based on the absence of a galaxy near the star. Romano (1977) concluded that this object is
+probably a dwarf nova of UV Per type.1
+Following Romano (1977), Meinunger (1977) studied Sonneberg plates (probably also unaware of the work
+by Sharov 1973) and constructed a light curve. Meinunger (1977) concluded that the star was clearly a fast
+nova and could not be a supernova due to the absence of a galaxy near the star. Meinunger (1977) excluded a
+long-period dwarf nova (like UV Per) based on the facts: (1) the amplitude was larger than 8 mag [Meinunger
+(1977) even suggested that the object on POSS was a unrelated one], (2) the decline after the maximum was too
+fast and (3) no further outbursts were observed. Meinunger (1977) suggested that this object was probably a
+very bright nova in the halo of M 31.
+Sharov and Karimova (1978) and his colleagues examined materials and found new records during the out-
+burst close to the maximum in the collection of Odessa Observatory. Precise astrometry of the outbursting object
+using the materials at Latvian Radio Astrophysical Observatory indicated the identity with the object on POSS.
+Based on the large (9 mag) amplitude, exceeding those of dwarf novae, Sharov and Karimova (1978) considered
+that the object should be regarded as a fast nova despite its small amplitude for a nova. Sharov and Karimova
+(1978) also remarked that the supposed nova did not follow the maximum magnitude relation with decline time
+for M 31 novae (Sharov 1989), and suggested that either the relation was broken or the object was an intergalactic
+nova 100–150 kpc from the Sun. This classification by Sharov and Karimova (1978) was adopted in Duerbeck
+(1987) and LS And was classified as a fast nova in General catalogue of variable stars (GCVS: Kholopov et al.
+1985).
+In GCVS version 4.2 for extragalatic variables, LS And was also given a name M31V0002 probably
+reflecting the possibility of an object in M 31.
+1UV Per was considered to be the prototype of dwarf novae with large-amplitude and rare outbursts at that time (cf. Petit 1960).
+WZ Sge was considered as a recurrent nova and the concept of WZ Sge-type dwarf novae was not present. See Kato (2015) for a
+modern review of WZ Sge-type dwarf novae.
+
+2
+Table 1: Observations of the 1971 outburst of LS And.
+JD∗
+mag†
+source‡
+JD∗
+mag†
+source‡
+JD∗
+mag†
+source‡
+179
+[19.0
+3
+223.497
+18.30
+2
+292
+[19.0
+3
+183.468
+[20.0
+2
+224.511
+18.56
+2
+294
+[19.0
+3
+183.508
+[13.6
+5
+225.541
+18.30
+2
+296
+[19.0
+3
+187.479
+12.7:
+5
+235
+18.5
+3
+298
+[19.0
+3
+187.508
+11.7:
+5
+236.248
+18.80
+2
+300
+19.0:
+3
+190
+12.5
+1
+237.261
+18.83
+2
+302
+19.0
+3
+191
+13.8*
+4
+238.405
+18.83
+2
+304
+19.0
+3
+191.504
+13.60
+2
+239.392
+18.83
+2
+305.304
+18.83
+2
+193
+14.0*
+4
+240
+18.7
+3
+308
+19.0
+3
+193.476
+14.1:
+5
+240.407
+18.83
+2
+320
+19.0:
+3
+193.507
+14.1:
+5
+242
+18.7
+3
+324
+19.0
+3
+195.492
+14.5::
+5
+245
+18.5
+3
+332
+[19.0
+3
+195.515
+14.5::
+5
+245.339
+19.0
+2
+335.238
+18.8:
+2
+208
+15.85
+4
+246.254
+18.83
+2
+353.24
+19:
+2
+209
+14.9
+3
+249
+18.5
+3
+570.392
+19.2:
+2
+209.359
+15.80
+2
+249.276
+[18.8
+2
+575.408
+19.2:
+2
+210
+16.25
+4
+252.434
+18.8:
+2
+655.286
+[18.3
+2
+210.499
+15.98
+2
+254.519
+[18.3
+2
+681.291
+19.0
+2
+212
+16.4
+3
+263
+18.5:
+3
+682.168
+[19.2
+2
+213.486
+17.54
+2
+266.367
+18.8
+2
+684.233
+19.4
+2
+214
+17.6*
+4
+268.427
+18.83
+2
+685.201
+[19.2
+2
+215
+17.8*
+4
+271
+19.0
+3
+688.219
+19.4
+2
+217
+18.0*
+4
+276
+[19.0
+3
+983
+20.0:
+5
+217.367
+18.30
+2
+276.284
+18.83
+2
+987
+20.0:
+5
+220.358
+18.33
+2
+277.396
+18.8:
+2
+2105
+20:
+5
+221.545
+18.38
+2
+278.308
+18.9
+2
+222.4
+18.43
+2
+280
+[19.0
+3
+∗ JD−2441000.
+† [ upper limits. : uncertain. * eye estimate from the published figure.
+‡ 1: van den Bergh et al. (1973), 2: Sharov (1973), 3: Romano (1977),
+4: Meinunger (1977), 5: Sharov and Karimova (1978).
+
+3
+41180
+41200
+41220
+41240
+12
+14
+16
+18
+20
+vdB73
+S73
+R77
+M77
+S78
+Figure 1:
+Light curve of the 1971 outburst of LS And using the data in table 1.
+The sources are
+vdB73 (van den Bergh et al. 1973), S73 (Sharov 1973), R77 (Romano 1977), M77 (Meinunger 1977) and S78
+(Sharov and Karimova 1978). The “v” symbols represent upper limits.
+Although most professional astronomers considered or treated LS And as a nova (Downes and Shara 1993;
+Szkody 1994; Collazzi et al. 2009; Evans et al. 2014; Özdönmez et al. 2018), and some suspected to be an X-
+ray nova (Rosenbush 1999) or a recurrent nova (Duerbeck 1988; Pagnotta and Schaefer 2014), I may have been
+the first to become confident that this should be a large-amplitude dwarf nova after knowing this object in
+the freshly published work by Duerbeck (1987). A part of the atmosphere in the late 1980s among amateur
+astronomers was already told in Kato (2022a). Visual monitoring of LS And for a new outburst already started
+in 1987 by VSOLJ members, and then by observers around the world. Although results have not been fruitful
+for decades [now exceeding 6000 observations without detecting an outburst in the American Association of
+Variable Stars (AAVSO)2; I myself had more than 200 non-detection visual observations when I was an amateur
+astronomer], I consistently considered LS And as a candidate WZ Sge star (Kato et al. 2001, 2002). I expected
+that the Gaia satellite would clarify the nature of LS And, but there was no parallax information in Gaia DR2
+(Gaia Collaboration et al. 2018). The blue color (Gaia B − R=+0.25) and a large proper motion were, however,
+sufficient to convince me of the dwarf nova-type nature. The parallax in Gaia EDR3 (Gaia Collaboration et al.
+2021) was not conclusive, probably due to the faintness of this object. The color in Gaia EDR3 was even bluer
+(B − R=−0.06).
+The “moment” arrived like lightening when I was examining light curves obtained by the Zwicky Transient
+Facility (ZTF: Masci et al. 2019)3. It was when I started examining light curves of recent ZTF data. As usual,
+I was looking at the table of dwarf novae listed in alphabetical order, and almost unconsciously typed LS And
+(as a matter of fact, I already did not pay special attention to this object regularly since I knew that it had
+been well monitored by amateur observers and considered that no missed outburst would be expected in the
+ZTF data). The reason why I specially selected LS And was unknown, but the light curve on the display was a
+familiar one of a WZ Sge star. I initially considered that I entered a name of a different well-known WZ Sge star
+(almost unconsciously as a routine work), but realized that it was “LS And”. Unthinkable! I initially could not
+2<http://www.aavso.org/data-download>.
+3The
+ZTF
+data
+can
+be
+obtained
+from
+IRSA
+<https://irsa.ipac.caltech.edu/Missions/ztf.html>
+using
+the
+inter-
+face
+<https://irsa.ipac.caltech.edu/docs/program_interface/ztf_api.html>
+or
+using
+a
+wrapper
+of
+the
+above
+IRSA
+API
+<https://github.com/MickaelRigault/ztfquery>.
+
+4
+59700
+59720
+59740
+59760
+12
+14
+16
+18
+20
+ZTF r
+ZTF g
+ATLAS o
+ATLAS c
+ASN g
+Figure 2:
+Light curve of the 2022 outburst of LS And using ZTF, ATLAS and ASAS-SN data. There were no
+upper limit observations before the initial detection.
+41180
+41200
+41220
+41240
+12
+14
+16
+18
+20
+vdB73
+S73
+R77
+M77
+S78
+2022
+Figure 3:
+Comparison of light curves of the 1971 and 2022 outburst of LS And. The symbols for the 1971
+observations are the same as in figure 1. The 2022 data (ZTF r magnitudes) were shifted by 18503 d.
+
+5
+59500
+59600
+59700
+59800
+59900
+12
+14
+16
+18
+20
+22
+ZTF r
+ZTF g
+ATLAS o
+ATLAS c
+ASN g
+Figure 4:
+Long-term light curve of the 2022 outburst of LS And. The symbols are the same as in figure 2.
+believe my eyes, but it was indeed LS And and I almost automatically issued vsnet-alert 272674, even without
+sufficient patience for waiting the result of a query to the All-Sky Automated Survey for Supernovae (ASAS-SN)
+Sky Patrol data (Shappee et al. 2014; Kochanek et al. 2017). My emotion at that time may have been similar to
+a situation when I encountered a rare bird which I could not believe (cf. Kato 2022a). Birders will agree.
+In the world of birders, it must have become the busiest moment after any discovery — one needs to locate
+the bird and take images or recordings sufficient for a proof of the existence of a rare bird. The case for the
+detection of the 2022 outburst of LS And was different. There was no special care for preserving the data shown
+on the display, and I went to the library (fortunately very close) to search the light curve of the 1971 outburst,
+which still stayed deep in my memory even after decades.
+So it’s time to return to science. In table 1, I summarized photometric data for the 1971 outburst. The
+magnitudes were all photographic (equivalent to B). Magnitudes with * were estimated by my eyes from the
+figure in Meinunger (1977), which are probably correct to ±1 d and ±0.1 mag. The magnitude for JD=190 was
+similarly estimated from a published figure by Romano (1977). Meinunger (1977) claimed that the object was
+estimated too bright by Romano (1977). The light curve drawn from these data is presented in figure 1. This
+is not much different from the one published in Sharov and Karimova (1978), but is worth presenting here since
+Sharov and Karimova (1978) is difficult to reach.
+The 2022 light curve is shown in figure 2. It is very clear that the 1971 and 2022 light curves are very
+similar: plateau-type fading lasting for ∼20 d followed by rapid decline and subsequent slow fading. They are
+typical WZ Sge-type outbursts without rebrightening (type D superoutburst in Kato 2015). It is also well-known
+that the same WZ Sge star tends to repeat the same type of rebrightening (Kato 2015) and LS And is also
+the case. Although the mechanism of rebrightening(s) is not yet well understood, empirical relationship shows
+that WZ Sge stars without rebrightening are mostly objects near the period minimum of cataclysmic variables,
+but before reaching it (figure 17 in Kato 2015). The orbital period of LS And is thus expected to be within
+0.053–0.060 d. The fading rate of the plateau phase (BJD 2459696–2459714.5) was 0.089(1) mag d−1, which
+corresponds to log td=1.05, a typical value for a WZ Sge star without rebrightening and not resembling a period
+bouncer (see figure 87 in Kato et al. 2014). A comparison between the 1971 and 2022 outbursts is shown in figure
+3 (from now on, I treat all photometric bands in visual wavelengths almost identical with V , which is a good
+approximation for a WZ Sge star in outburst). These outbursts were almost exactly the same and the interval of
+4<http://ooruri.kusastro.kyoto-u.ac.jp/mailarchive/vsnet-alert/27267>.
+
+6
+these two outburst was 18503 d (=50.66 yr). This comparison suggests that the 2022 outburst would not have
+started before JD 2459682 (2022 April 12). Definitely a sigh! (particularly for amateur observers) considering
+the almost no evening visibility of this object in mid-April.
+People may wonder if these outburst could be those of an SU UMa star rather than a WZ Sge star, and how
+I can be confident about the classification without observation of early superhumps (cf. Kato 2015). I show a
+long-term light curve of the 2022 outburst in figure 2. The object was brighter by 1.5 mag after the outburst. The
+post-outburst phenomenon is a long fading tail, which is characteristic to a WZ Sge-type outburst and not seen
+in an SU UMa star. The presence of the same phenomenon was also reported after the 1971 outburst (Sharov
+1973).5 Before the outburst plateau, there was a phase with more rapid fading (more evident in the 1971 light
+curve and only one day in the 2022 one). This feature is commonly seen in WZ Sge-type outbursts and is referred
+to as a viscous decay phase. Early superhumps are expected during this phase if the binary has a sufficient
+inclination (Kato 2015, 2022b).
+The peak magnitude probably requires re-examination. Although most literature gives 11.7 mag as the
+maximum for LS And, it is evident from table 1 that this magnitude was uncertain (“:” usually means that
+the object is close to the limit of photographic materials or the quality of the photograph is poor) and was the
+brighter one of two uncertain observations (11.7 and 12.7 mag) only 40 min apart. It looks more likely that the
+true brightest observation was close to their average (12.2 mag). The outburst amplitude based on this value is
+8.8 mag using the ZTF data before the 2022 outburst. The true peak would have been brighter, though, since
+there was a 4 d observational gap before the first observation of the outburst (but see the discussion below).
+As seen from the 2022 observations, the magnitude when ordinary superhumps should appear following the
+viscous decay phase was 14.3 mag. In ordinary WZ Sge stars, the absolute magnitude (MV ) when ordinary
+superhumps appear is +5.4 (for an average inclination of 1 radian) (Kato 2022b).
+Using this value as the
+standard candle, the distance modulus of LS And is estimated to be 8.9. The observed peak (12.2 mag) in
+1971 corresponds to MV =+3.3. The quiescent magnitude (21.0 mag, ZTF data) corresponds to MV =+12.1.
+The difference (6.7 mag) between quiescent magnitude and the magnitude when ordinary superhumps appear is
+typical for a (non-period bouncer) WZ Sge star (see fig. 23 in Kato 2015; Tampo et al. 2020). Other properties
+of LS And are expected to be similar to those of typical WZ Sge stars.
+The detection of the 2022 outburst of LS And brought a some kind of despair to observers who had been
+expecting to see a fresh outburst for decades.
+Could there be a possibility that LS And silently underwent
+outbursts more frequently only around solar conjunctions? This was indeed the case of the SU UMa star VY Aqr
+located close to the ecliptic.
+Despite the mean interval of superoutbursts of less than 2 yr, this object was
+not recorded in superoutburst between 1994 and 2006, and between 2008 and 2020. It was most likely that
+superoutbursts in this object occurred around solar conjunctions and were not recorded. Although similar things
+may have happened in LS And at least in the past, modern deep observations such as ZTF should have detected
+the object during a fading tail if there was a missed superoutburst. There was no indication of such a detection in
+the ZTF data since 2018, and the outburst interval should be longer than 5 yr. The fading tail lasted more than a
+year (Sharov 1973). Sharov and Karimova (1978) described that the object returned to practically the same level
+before the outburst after 5.5 yr, although this description may have assumed a nova-type light curve and could
+have overestimated the duration of the fading tail. Considering these values and considering that parameters of
+LS And are similar to those of typical WZ Sge stars, the next major outburst would be expected after a decade or
+even more [see figure 5 in Kato (2015) for the distribution of outburst intervals in WZ Sge stars]. By comparing
+the recorded peak MV =+3.3 (in 1971) with the statistics of known WZ Sge stars (figure 10 in Tampo et al. 2020),
+it appears that the true peak in 1971 was not missed after a considerable delay (i.e. the object was unlikely to
+have reached 11.0 mag even at the true peak). The next superoutburst would also be around 12.2 mag. There
+are, however, exceptional objects like V3101 Cyg (Tampo et al. 2020; Hameury and Lasota 2021) and there may
+be an unexpected phenomenon even after the outburst. In the post-outburst data of LS And, 0.5 mag brightening
+lasting for 10–20 d and starting around JD 2459852 was present (figure 3). This might suggest that LS And is
+still active in the post-superoutburst phase and would be worth observing before it finally returns to quiescence.
+Acknowledgements
+This work was supported by JSPS KAKENHI Grant Number 21K03616. The author is grateful to the ZTF,
+ATLAS and ASAS-SN teams for making their data available to the public. I am grateful to VSOLJ, AAVSO and
+5It might be interesting to leave a remark that the figure in Sharov (1973) dealt with this phenomenon rather than the shape of
+the outburst. Please have a look at his figure if you have a chance too see this reference.
+
+7
+VSNET observers for reporting observations and to Naoto Kojiguchi for helping downloading the ZTF data.
+Based on observations obtained with the Samuel Oschin 48-inch Telescope at the Palomar Observatory as
+part of the Zwicky Transient Facility project. ZTF is supported by the National Science Foundation under Grant
+No. AST-1440341 and a collaboration including Caltech, IPAC, the Weizmann Institute for Science, the Oskar
+Klein Center at Stockholm University, the University of Maryland, the University of Washington, Deutsches
+Elektronen-Synchrotron and Humboldt University, Los Alamos National Laboratories, the TANGO Consortium
+of Taiwan, the University of Wisconsin at Milwaukee, and Lawrence Berkeley National Laboratories. Operations
+are conducted by COO, IPAC, and UW.
+The ztfquery code was funded by the European Research Council (ERC) under the European Union’s Horizon
+2020 research and innovation programme (grant agreement n◦759194 – USNAC, PI: Rigault).
+This work has made use of data from the Asteroid Terrestrial-impact Last Alert System (ATLAS) project.
+The Asteroid Terrestrial-impact Last Alert System (ATLAS) project is primarily funded to search for near earth
+asteroids through NASA grants NN12AR55G, 80NSSC18K0284, and 80NSSC18K1575; byproducts of the NEO
+search include images and catalogs from the survey area. This work was partially funded by Kepler/K2 grant
+J1944/80NSSC19K0112 and HST GO-15889, and STFC grants ST/T000198/1 and ST/S006109/1. The ATLAS
+science products have been made possible through the contributions of the University of Hawaii Institute for
+Astronomy, the Queen’s University Belfast, the Space Telescope Science Institute, the South African Astronomical
+Observatory, and The Millennium Institute of Astrophysics (MAS), Chile.
+List of objects in this paper
+LS And, VY Aqr, V3101 Cyg, UV Per, WZ Sge, SU UMa, M 31, M31V0002
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+van den Bergh, S., Herbst, E., & Pritchet, C. (1973) A search for faint variable objects. AJ 78, 375
+

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+arXiv:2301.00052v1  [math.GR]  30 Dec 2022
+Examples of left-orderable and
+non-left-orderable HNN extensions
+Azer Akhmedov, Cody Martin
+ABSTRACT: We prove that an HNN extension of a torsion-free nilpotent
+group is left-orderable. We also construct examples of non-left-orderable
+HNN extensions of left-orderable groups.
+1. Non-left-orderable HNN extensions of left-orderable
+groups
+It is well-known that an HNN extension of a torsion-free group is still
+torsion-free ([3], [1]). On the hand, for many classes of groups, existence
+of a torsion element is the only obstruction to left-orderability; for
+example, this is the case for the classes of one-relator groups, nilpotent
+groups, etc. Hence it is natural to study how left-orderability behaves
+under an HNN extension.
+In [2] (see Example 6.2 there), an example is constructed to show
+that left-orderability is not preserved under the HNN extension. In
+this section, we present systematic ways of producing non-left-orderable
+HNN extensions of left-orderable groups. The example of [2] is built
+as an HNN extension of a direct product of a free nilpotent group of
+class two with the fundamental group of Klein bottle.
+We produce
+examples of HNN extensions of groups such as non-Abelian free groups
+and virtually Abelian groups.
+We rely on the following well-known
+criterion about left-orderability of groups [4]
+Proposition 1.1. A group G is left-orderable if and only if for all
+k ≥ 1 and for all g1, . . . , gk ∈ G\{1}, there exist ǫ1, . . . , ǫk ∈ {−1, 1}
+such that the semigroup of G generated by gǫ1
+1 , . . . , gǫk
+k does not contain
+the identity element.
+Let us emphasize that we use the obvious “only if part” of this propo-
+sition; the harder “if part” is not needed.
+Given a group G, and subgroups A, B ≤ G with an isomorphism
+φ : A → B, the HNN extension (G, A, B, t, φ) is defined as the quo-
+tient of the free product G ∗ ⟨t⟩ by the normal closure of the subset
+{tat−1φ(a)−1 | a ∈ A}. We also write this HNN extension as (G, A, B, t)
+when φ is given in the context.
+Theorem 1.2. A free group of rank bigger than one admits a non-left-
+orderable HNN extension.
+1
+
+2
+Proof. By Britton’s Lemma, it suffices to prove the theorem for the
+group F2. Let a, b be the generators of F2. We can find positive expo-
+nents pi, qi, ri, si, 1 ≤ i ≤ 8 such that the elements
+u1 = ap1bq1, u2 = ap2bq2, u3 = ap3bq3, u4 = ap4bq4,
+u5 = ap5b−q5, u6 = ap6b−q6, u7 = ap7b−q7, u8 = ap8b−q8
+generate a free group of rank 8, and so do the elements
+v1 = ar1bs1, v2 = ar2b−s2, v3 = a−r3bs3, v4 = a−r4b−s4,
+v5 = ar5bs5, v6 = ar6b−s6, v7 = a−r7bs7, v8 = a−r8b−s8.
+(It suffices to take the sequences (pi)1≤i≤8, (qi)1≤i≤8, (ri)1≤i≤8, (si)1≤i≤8
+to be strictly increasing.) Let A, B be these free groups generated by
+u1, . . . , u8 and v1, . . . , v8 respectively, and φ : A → B be the isomor-
+phism such that φ(ui) = vi, 1 ≤ i ≤ 8.
+Then, by Proposition 1.1, the HNN extension (G, A, B, t) where
+t(a) = φ(a) for all a ∈ A is not left-orderable.
+□
+Remark 1.3. Let us remind that in the case of rank = 1, the claim
+does not hold anymore since any HNN extension of Z is isomorphic
+⟨t, a | tamt−1 = an⟩ for some non-zero integers m, n. All these groups
+(which include Z2, π1(Klein bottle) = ⟨a, b | aba−1 = b−1⟩, and the
+solvable Baumslag-Solitar group BS(1, n) ∼= Z ⋉ Z[ 1
+n]), are all left-
+orderable as torsion-free one-relator groups.
+Using similar ideas, we build a non-left-orderable HNN extension
+of a left-orderable solvable group. We again rely on the criterion of
+Proposition 1.1.
+Let n ≥ 2 and Γn be a group given by the presentation
+⟨s, x | [sn, x] = 1, [x, sixs−i] = 1, 1 ≤ i ≤ n − 1⟩.
+Let xi = sixs−i, i ∈ Z. Notice that xi = xj iff i ≡ j( mod n). The
+elements xi, 0 ≤ i ≤ n−1 generate a normal subgroup Nn isomorphic to
+Zn and the quotient by this subgroup is isomorphic to Z. Any element
+g of Γn can be written uniquely as siw(x0, . . . , xn−1) where i ∈ Z and
+w(x0, . . . , xn−1) = xp0
+0 . . . xpn−1
+n−1 for some integer exponents p0, . . . , pn−1.
+siw(x0, . . . , xn−1) will be called the canonical form of g. We also write
+Σ(g) = i + p0 + · · · + pn.
+Let us observe that Γn is torsion-free. Indeed, if g is a torsion element
+with a canonical form siw(x0, . . . , xn−1) as above then for all k ≥ 1,
+gk = sikw0(x0, . . . , xn−1)wi(x0, . . . , xn−1) . . . w(k−1)i(x0, . . . , xn−1)
+
+3
+where wj(x0, . . . , xn−1) = w(xj, xj+1, . . . , xn−1+j) hence it follows im-
+mediately that either i = 0; then, since Nn ∼= Zn, we obtain that
+w = 1.
+It turns out Γn is left-orderable (which also implies that it is torsion-
+free). We introduce a left order < on Γn as follows: An element g with
+the canonical form siw(x0, . . . , xn−1) as above will be called positive
+if either Σ(w) > 0 or Σ(w) = 0 and i > 0. If Σ(w) = 0 and i = 0,
+then we are in the group Nn ∼= Zn and there the order can be defined
+lexicographically. Then we see that a product of two positive elements
+is always positive and the inverse of a positive element is not positive.
+Hence < is a left-order.
+To state our next proposition we need to introduce some (well-
+known) terminology.
+Definition 1.4. Let G be a group generated by a subset S ⊆ G\{1}
+such that for all x ∈ G, if x ∈ S, then x−1 /∈ S (in particular, 1 /∈ S).
+We say that a non-trivial reduced word W(x1, . . . , xk) = xn1
+1 . . . xnk
+k
+is
+positive in the alphabet S if x1, . . . , xk ∈ S and all exponents ni, 1 ≤
+i ≤ k are positive.
+Proposition 1.5. In the group Γn let S1 = {s, x}, S2 = {s−1, x}, S3 =
+{s, x−1}, S4 = {s−1, x−1}. For n ≥ 12, there exists elements f1, . . . , f4,
+g1, . . . , g4 ∈ Γn such that the following conditions hold:
+i) ⟨f1, f2, f3, f4⟩ ∼= ⟨g1, g2, g3, g4⟩ ∼= Z4,
+ii) The elements f1, f2, f3, f4 can be represented with positive words
+in the alphabet S1,
+iii) For all 1 ≤ i ≤ 4, the element gi can be represented with a
+positive word in the alphabet Si.
+Proof. We define f1 = sn−1xs, f2 = sn−2(xs)2, f3 = sn−4(xs)4, f4 =
+sn−8(xs)8. Then f1, f2, f3, f4 belong to Nn and generate a subgroup
+isomorphic to Z4. We also define g1 = sn−1xs, g2 = sn−2(x−1s)2, g3 =
+s4−n(xs−1)4, f4 = s8−n(x−1s−1)8. The elements g1, g2, g3, g4 also belong
+to Nn and generate a subgroup isomorphic to Z4.
+□
+In the above proposition, the n ≥ 12 is not necessarily the best possi-
+ble. Using Proposition 1.5, we can now prove the following proposition
+which establishes the existence of a non-left-orderable HNN extension
+of a left-orderable virtually Abelian group.
+
+4
+Proposition 1.6. For all n ≥ 12, Γn admits a non-left-orderable HNN
+extension.
+Proof. Let f1, . . . , f4, g1, . . . , g4 ∈ Γn be elements satisfying conditions
+1)-3) of Proposition 1.5. Let φ : ⟨f1, f2, f3, f4⟩ → ⟨g1, g2, g3, g4⟩ be an
+isomorphism such that φ(fi) = gi, 1 ≤ i ≤ 4.
+We consider an HNN extension
+G := (Γn, ⟨f1, f2, f3, f4⟩, ⟨g1, g2, g3, g4⟩, t)
+by letting txt−1 = φ(x) for all x ∈ ⟨f1, f2, f3, f4⟩
+For any left-order on G, notice that the elements tfit−1, 1 ≤ i ≤ 4
+are either all positive or all negative. On the other hand, among the
+elements gi, 1 ≤ i ≤ 4 at least one is positive and one is negative. This
+is a contradiction. Hence G is not left-orderable.
+□
+2. HNN extensions of nilpotent groups
+The aim of this section is to prove that unlike solvable groups,
+an HNN extension of a left-orderable nilpotent group is always left-
+orderable. Let us recall that a nilpotent group is left-orderable iff it is
+torsion-free; this claim too does not hold for solvable groups.
+Let us first observe that, since a direct limit of left-orderable groups
+is left-orderable, an HNN extension an HNN extension (G, A.B, t) is
+left-orderable if for all finitely generated subgroups A0 and G0 of G,
+where A0 ≤ A, G0 ⊇ ⟨A0, B0⟩ and B0 = tA0t−1, the HNN extension
+(G0, A0, B0, t) is left-orderable. We will use this observation repeatedly
+in this section.
+We already observed that by classification, an HNN extension of
+an infinite cyclic group is left-orderable. The same holds for an HNN
+extension of any torsion free Abelian group.
+Indeed, it suffices to consider finitely generated Abelian groups so
+let G be a finitely generated torsion-free Abelian group, A, B ≤ G, φ :
+A → B be an isomorphism, and (G, A, B, t) be the HNN extension with
+respect to the isomorphism φ. Let G ∼= Zd and r = rankA = rankB.
+We will assume that G = Zd.
+Then for some linearly independent
+vectors u1, . . . , ur we have A = {c1u1 + · · · + crur : ci ∈ Z, 1 ≤ i ≤ r}
+and similarly for some linearly independent vectors v1, . . . , vr we have
+B = {c1v1 + · · · + crvr
+:
+ci ∈ Z, 1 ≤ i ≤ r}. We let G = R, A =
+{c1u1 + · · · + crur : ci ∈ R, 1 ≤ i ≤ r}, B = {c1v1 + · · · + crvr : ci ∈
+R, 1 ≤ i ≤ r} and φ : A → B be the extension of φ : A → B defined as
+φ(c1u1 + · · · + crur) = c1φ(u1) + · · · + crφ(ur) for all c1, . . . , cr ∈ R.
+
+5
+A key observation here is that even though the isomorphism φ :
+A → B cannot necessarily be extended to G, but one can extend the
+isomorphism φ : A → B to some automorphism F : G → G. Then the
+HNN extension (G, A, B, t) with respect to the isomorphism φ : A → B
+has a quotient isomorphic to the semidirect product Z⋉F G by a normal
+subgroup N ≤ G. Since N and Z ⋉F G are left-orderable we obtain
+that (G, A, B, t) is left-orderable (as an extension of a left-orderable
+group by a left-orderable group). By Britton’s Lemma, (G, A, B, t) is
+a subgroup of (G, A, B, t) hence it is also left-orderable.
+We now would like to carry the same argument for any torsion-free
+nilpotent group. The main issue here is that given a finitely generated
+torsion-free nilpotent group Γ, one needs to construct a completion Γ
+which would resemble the operation Zd → Rd so we can try to use the
+argument in the Abelian case.
+Let R be a commutative ring with identity and n ≥ 1.
+We let
+Un(R) be the group of n × n upper-triangular matrices with 1’s on the
+diagonal. The cases R = R and R = Z will be the most interesting to
+us.
+It is well-known that any finitely generated torsion-free nilpotent
+group Γ embeds in Un(Z) for some n ≥ 1. The Mal’cev completion of
+Un(Z) is Un(R) (and the Mal’cev completion of Zn is Rn) 1, however,
+given an isomorphism φ : A → B of subgroups of Un(Z), although it
+induces an isomorphism φ : A → B but one cannot necessarily extend
+this isomorphism to the entire G. For example, for n = 3, the group
+U3(Z) is isomorphic to the Heisenberg group
+⟨x, y, z | z = [x, y], [x, z] = [y, z] = 1⟩
+and if we let A = ⟨x⟩, B = ⟨z⟩ and φ(x) = z, then this isomorphism
+cannot be extended to the isomorphism of U3(Z) (or U3(R)). Thus we
+need to define a completion of Γ other than the Mal’cev completion.
+Let Xn,i, 1 ≤ i ≤ n − 1 be the matrix of Un(Z) where all off-diagonal
+entries are zero except the (i + 1, i)-th entry is equal to 1. In order to
+define a more suitable completion of Un(Z) we will extend it first, and at
+the end we will obtain a completion which is ”infinite-dimensional”. Let
+U∞(Z) be a group generated by xk, k ∈ Z such that for all k ∈ Z, n ≥ 1
+the subgroup generated by xk+1, . . . , xk+n−1 is isomorphic to Un(Z)
+through the isomorphism f(xk+j) = Xn,j, 1 ≤ j ≤ n − 1. Notice that
+U∞(Z) is well-defined this way and it contains isomorphic copies of all
+1in the literature, the term Mal’cev completion is used for some other related
+operations as well.
+
+6
+Un(Z), n ≥ 2. This group can be viewed as the group of infinite sized
+integral unipotent matrices. But to achieve our goal we extend U∞(Z)
+further as follows.
+Let us first observe that in the group Un(Z) viewed as the group of
+upper triangular unipotent integral matrices, [xi, xj] = 1 if |i − j| ≥ 2
+and for all 1 ≤ i ≤ n − 2, [xi, xi+1] is a unipotent matrix with all the
+off-diagonal entries zero, except the (i + 2, i + 1)-entry equals 1. Thus
+the elements [xi, xi+1], 1 ≤ i ≤ n−2 generate a subgroup isomorphic to
+Un−1(Z) with an isomorphism xi → [xi, xi+1], 1 ≤ i ≤ n − 2. Similarly,
+in the group U∞(Z), the elements [xi, xi+1], i ∈ Z generate a subgroup
+isomorphic to U∞(Z), and the homomorphism f : U∞(Z) → U∞(Z)
+defined as f(xi) = [xi, xi+1], i ∈ Z (it is sufficient to define it on the
+generators) establishes this isomorphism.
+The group U∞(Z) is a direct limit of the groups Un(Z), n ≥ 1.
+More precisely, let Hn, n ≥ 1 be the subgroup of U∞(Z) generated
+by x−n, x−n+1, . . . , xn−1, xn. Then Hn is isomorphic to U2n+1(Z), and
+U∞(Z) is a direct limit of the sequence Hn, n ≥ 1.
+In our construction of the completion, we will use a direct limit of
+groups each isomorphic to U∞(Z).
+Let Γk, k ∈ Z be a group gen-
+erated by zk,n, n ∈ Z with an isomorphism gk : Γk → U∞(Z) such
+that gk(zk,n) = xn. We have · · · ≤ Γ−1 ≤ Γ0 ≤ Γ1 ≤ Γ2 ≤ . . . and
+[zk,n, zk,n+1] = zk−1,n for all k, n ∈ Z. This defines an isomorphic em-
+bedding gk,k+1 : Γk → Γk+1, k ∈ Z where gk,k+1(zn,k) = zn,k+1. These
+inclusions define a direct limit U of Γk, k ∈ Z. The maps gk,k+1 induce
+a shift isomorphism θ : U → U, so, in particular, θ(x) = gk,k+1(x) for
+all x ∈ Γk, k ∈ Z
+In defining the completion U, first, let us recall the following facts
+about lattices of simply connected nilpotent Lie groups [5].
+Proposition 2.1. Let G be simply connected nilpotent Lie group, Γ be
+a discrete subgroup of G. The following are equivalent:
+(i) Γ is a lattice of G;
+(ii) Γ is Zariski dense in G;
+(iii) Γ is not contained in any proper connected closed subgroup of
+G;
+(iv) Γ is co-compact in G.
+Definition 2.2. Let m ≥ 2. For any subset Ω ⊆ Um(Z), we define
+Span(Ω) = ⟨Ω⟩Z where the latter denotes the Zariski closure.
+For
+example, Span(Um(Z)) = Um(R). Then, for any subset Ω ⊆ U∞(Z) we
+let
+Span(Ω) = ∪
+n≥1 Span(Ω ∩ Hn).
+
+7
+Then, for any subset Ω ⊆ U we define Span(Ω) = ∪
+k≥1 Span(Ω ∩ Γk).
+Finally, we define U = Span(U).
+The Lie subgroups of Un(R) (hence of U) are simply connected (in-
+deed contractible, as the exponential map determines a homeomor-
+phism to Rd with d being the dimension of the group) thus its iso-
+morphism type can be determined at the level of Lie algebras. The
+Lie algebra of every Lie subgroup of U is a finite-dimensional nilpotent
+Lie algebra. On the other hand, by Engel’s Theorem, for every finite-
+dimensional nilpotent Lie algebra g with the underlying vector space
+V , there exists an associated flag F(g) in the form {0} = V0 ≤ V1 ≤
+· · · ≤ Vn = V where dimVi = i, 0 ≤ i ≤ n and for all x ∈ g, 1 ≤ i ≤ n,
+ad(x)(Vi) ⊆ Vi−1.
+Thus ð can be faithfully represented by strictly
+upper-triangular matrices with respect to some basis of V . If g, h are
+finite-dimensional nilpotent Lie algebras and φ : g → h a Lie algebra
+isomorphism, then H = f(F) will be an associated flag of h. On the
+other hand, if g is a finite-dimensional nilpotent Lie algebra with un-
+derlying vector space V and I is an ideal of g faithfully represented
+in gl(V0) with strictly upper triangular matrices with respect to a ba-
+sis of a proper subspace V0, then by inductive process as in the proof
+of Engels’ Theorem, it follows that we can extend the basis of V0 to
+a basis of V such that g is faithfully represented with strictly upper
+triangular matrices. By this observation, any Lie group isomorphism
+Φ : G → H between finite-dimensional nilpotent Lie subgroups of U
+can be extended to the group automorphism of U, since for any Lie
+subgroups G1, G2 of U, G1 belongs to a Lie subgroup G3 which con-
+tains θk(G2) as a normal subgroup for some integer k (thus the Lie
+algebra g3 of G3 contains the Lie algebra of θk(G2) as an ideal.)
+We can now state and prove the following
+Proposition 2.3. An HNN extension of a torsion-free nilpotent group
+is left-orderable.
+Proof. Let Γ be a torsion-free nilpotent group. It is well-known that Γ
+is left-orderable (in fact, bi-orderable). Indeed, it suffices to prove this
+only for finitely generated subgroups, and any such subgroup embeds
+into Um(Z) for some m ≥ 2. The latter admits an easy bi-order. Indeed,
+more generally, we define a matrix A = (ai,j)1≤i,j≤n ∈ Um(R) as positive
+if d is the smallest positive integer such that ai,j ̸= 0, for some i, j ≥ 1
+with i + j = d, moreover, for this d, if p is the smallest positive integer
+with p + q = d and ap,q ̸= 0, then ap,q > 0. One easily checks that this
+
+8
+is in fact a genuine left-order (and even a bi-order). Then U∞(R) is
+also bi-orderable as a direct limit of Um(R), m ≥ 1 and so is U.
+To show that an HNN extension of Γ is also left-orderable, it again
+suffices to consider HNN extensions of finitely generated subgroups. So
+let us assume that Γ is also finitely generated, A, B ≤ Γ and φ : A → B
+be an isomorphism. Γ embeds in U∞(Z) and the latter is a subgroup
+of G = U∞(R).
+The isomorphism φ : A → B cannot necessarily be extended to
+G, but one can extend the isomorphism φ : Span(A) → Span(B)
+to some F : U → U where φ is an extension of φ by Mostow Strong
+Rigidity Theorem for lattices in solvable Lie groups [5]. Then the HNN
+extension (U, Span(A), Span(B), t) with respect to the isomorphism
+φ : Span(A) → Span(B) has a quotient isomorphic to the semidirect
+product Z⋉F U by a normal subgroup N ≤ U. Since N and Z⋉F U are
+left-orderable we obtain that (U, Span(A), Span(B), t) is left-orderable
+(as an extension of a left-orderable group by a left-orderable group). By
+Britton’s Lemma, (Γ, A, B, t) is a subgroup of (U, Span(A), Span(B), t)
+hence it is also left-orderable.
+□
+We would like to end this section with a torsion-free non-left-orderable
+example which will contain a class two nilpotent group as an index two
+subgroup; indeed, it will contain a subgroup of Heisenberg group H of
+3 × 3 integral unipotent matrices. This might be one of the simplest
+(smallest) examples of a torsion-free non-left-orderable group in litera-
+ture. It also shows that torsion-freeness does not imply left-orderability
+in the class of polycyclic groups.
+Let
+Γ = ⟨t, u, v | [u, v] = t4, tut−1 = u−1, tvt−1 = v−1⟩.
+The group Γ is related to the Heisenberg group
+H = ⟨x, y, z | [x, y] = z, [z, x] = [z, y] = 1⟩.
+Any element of H can be written uniquely as xmynzk where m, n, k ∈
+Z.
+H is bi-orderable.
+The elements x2, y, z generate an index two
+subgroup H0 of H.
+The group Γ will have an index two subgroup isomorphic to H0. We
+let u = x2, v = y, t2 = z. Let also G be a group given by the following
+presentation
+G = ⟨t, x, y, z | [x, y] = z, [z, x] = [z, y] = 1, t2 = z, txt−1 = x−1, tyt−1 = y−1⟩.
+Then Γ is a subgroup of G generated by t, x2, y.
+
+9
+Proposition 2.4. Γ is torsion-free and non-left-orderable.
+Proof. Assume that < is a left-order on Γ. Without loss of generality
+we may assume that t > 1. Then 1 < t < z thus z is also a positive
+element. On the other hand, let us observe that for all n ∈ 2Z, we have
+(txn)2 = t2 thus the element txn is positive for every even integer n.
+Let m be positive if y > 1 and negative if y < 1. Then, for all m ∈ 2Z,
+txnym is positive as a product of two positive elements txn and ym.
+Then (txnym)2 > 1. However,
+(txnym)2 = t2x−ny−mxnym = t2zmn = zmn+1.
+We can choose n such that mn+1 < 0. This yields that (txnym)2 < 1.
+Contradiction.
+To see torsion-freeness let us observe that any element g ∈ G can be
+written as g = x2pyqzr or g = tx2pyqzr. The element x2pyqzr is not a
+torsion since H is torsion-free. As for the element tx2pyqzr, we have
+(tx2pyqzr)2 = x−2py−qx2pyqz2r+1 = z±2pqz2r+1 ̸= 1. Thus Γ is torsion-
+free.
+□
+Acknowledgement:
+We are very thankful to Zipei Nie for reading
+the draft of this paper and correcting errors.
+References
+[1] J. Button, Topics in infinite groups, Lecture Notes.
+[2] V.V. Bludov and A.M.W.Glass, Word problems, embeddings, and free prod-
+ucts of right-ordered groups with amalgamated subgroup, Proceedings of the
+London Mathemtical Society, vol. 99, issue 3, (2009), 585-608
+[3] R. Lyndon and P. Schupp. Combinatorial Group Theory, Volume 89 of Ergeb-
+nisse der Mathematik und ihrer Grenzgebiete, Springer-Verlaq, 1977.
+[4] A. Navas, Groups of Circle Diffeomorphisms. The University of Chicago Press,
+2011.
+[5] M.S.Raghunathan, Discrete Subgroups of Lie Groups, Springer Berlin Heidel-
+berg, Nov 16, 1972
+Azer Akhmedov, Department of Mathematics, North Dakota State
+University, Fargo, ND, 58102, USA
+Email address: azer.akhmedov@ndsu.edu
+Cody Martin, Department of Mathematics, North Dakota State
+University, Fargo, ND, 58102, USA
+Email address: cody.martin@ndsu.edu
+

79AyT4oBgHgl3EQfQvZ4/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf,len=316
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='00052v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='GR]  30 Dec 2022  Examples of left-orderable and  non-left-orderable HNN extensions  Azer Akhmedov, Cody Martin  ABSTRACT: We prove that an HNN extension of a torsion-free nilpotent  group is left-orderable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' We also construct examples of non-left-orderable  HNN extensions of left-orderable groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Non-left-orderable HNN extensions of left-orderable  groups  It is well-known that an HNN extension of a torsion-free group is still  torsion-free ([3], [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' On the hand, for many classes of groups, existence  of a torsion element is the only obstruction to left-orderability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' for  example, this is the case for the classes of one-relator groups, nilpotent  groups, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Hence it is natural to study how left-orderability behaves  under an HNN extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  In [2] (see Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='2 there), an example is constructed to show  that left-orderability is not preserved under the HNN extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' In  this section, we present systematic ways of producing non-left-orderable  HNN extensions of left-orderable groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' The example of [2] is built  as an HNN extension of a direct product of a free nilpotent group of  class two with the fundamental group of Klein bottle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  We produce  examples of HNN extensions of groups such as non-Abelian free groups  and virtually Abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  We rely on the following well-known  criterion about left-orderability of groups [4]  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' A group G is left-orderable if and only if for all  k ≥ 1 and for all g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , gk ∈ G\\{1}, there exist ǫ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , ǫk ∈ {−1, 1}  such that the semigroup of G generated by gǫ1  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , gǫk  k does not contain  the identity element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Let us emphasize that we use the obvious “only if part” of this propo-  sition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' the harder “if part” is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Given a group G, and subgroups A, B ≤ G with an isomorphism  φ : A → B, the HNN extension (G, A, B, t, φ) is defined as the quo-  tient of the free product G ∗ ⟨t⟩ by the normal closure of the subset  {tat−1φ(a)−1 | a ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' We also write this HNN extension as (G, A, B, t)  when φ is given in the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' A free group of rank bigger than one admits a non-left-  orderable HNN extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  1  2  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' By Britton’s Lemma, it suffices to prove the theorem for the  group F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Let a, b be the generators of F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' We can find positive expo-  nents pi, qi, ri, si, 1 ≤ i ≤ 8 such that the elements  u1 = ap1bq1, u2 = ap2bq2, u3 = ap3bq3, u4 = ap4bq4,  u5 = ap5b−q5, u6 = ap6b−q6, u7 = ap7b−q7, u8 = ap8b−q8  generate a free group of rank 8, and so do the elements  v1 = ar1bs1, v2 = ar2b−s2, v3 = a−r3bs3, v4 = a−r4b−s4,  v5 = ar5bs5, v6 = ar6b−s6, v7 = a−r7bs7, v8 = a−r8b−s8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  (It suffices to take the sequences (pi)1≤i≤8, (qi)1≤i≤8, (ri)1≤i≤8, (si)1≤i≤8  to be strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=') Let A, B be these free groups generated by  u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , u8 and v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , v8 respectively, and φ : A → B be the isomor-  phism such that φ(ui) = vi, 1 ≤ i ≤ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Then, by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='1, the HNN extension (G, A, B, t) where  t(a) = φ(a) for all a ∈ A is not left-orderable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  □  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Let us remind that in the case of rank = 1, the claim  does not hold anymore since any HNN extension of Z is isomorphic  ⟨t, a | tamt−1 = an⟩ for some non-zero integers m, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' All these groups  (which include Z2, π1(Klein bottle) = ⟨a, b | aba−1 = b−1⟩, and the  solvable Baumslag-Solitar group BS(1, n) ∼= Z ⋉ Z[ 1  n]), are all left-  orderable as torsion-free one-relator groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Using similar ideas, we build a non-left-orderable HNN extension  of a left-orderable solvable group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' We again rely on the criterion of  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Let n ≥ 2 and Γn be a group given by the presentation  ⟨s, x | [sn, x] = 1, [x, sixs−i] = 1, 1 ≤ i ≤ n − 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Let xi = sixs−i, i ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Notice that xi = xj iff i ≡ j( mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' The  elements xi, 0 ≤ i ≤ n−1 generate a normal subgroup Nn isomorphic to  Zn and the quotient by this subgroup is isomorphic to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Any element  g of Γn can be written uniquely as siw(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , xn−1) where i ∈ Z and  w(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , xn−1) = xp0  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' xpn−1  n−1 for some integer exponents p0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , pn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  siw(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , xn−1) will be called the canonical form of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' We also write  Σ(g) = i + p0 + · · · + pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Let us observe that Γn is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Indeed, if g is a torsion element  with a canonical form siw(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , xn−1) as above then for all k ≥ 1,  gk = sikw0(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , xn−1)wi(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , xn−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' w(k−1)i(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , xn−1)  3  where wj(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , xn−1) = w(xj, xj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , xn−1+j) hence it follows im-  mediately that either i = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' then, since Nn ∼= Zn, we obtain that  w = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  It turns out Γn is left-orderable (which also implies that it is torsion-  free).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' We introduce a left order < on Γn as follows: An element g with  the canonical form siw(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , xn−1) as above will be called positive  if either Σ(w) > 0 or Σ(w) = 0 and i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' If Σ(w) = 0 and i = 0,  then we are in the group Nn ∼= Zn and there the order can be defined  lexicographically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Then we see that a product of two positive elements  is always positive and the inverse of a positive element is not positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Hence < is a left-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  To state our next proposition we need to introduce some (well-  known) terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Let G be a group generated by a subset S ⊆ G\\{1}  such that for all x ∈ G, if x ∈ S, then x−1 /∈ S (in particular, 1 /∈ S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  We say that a non-trivial reduced word W(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , xk) = xn1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' xnk  k  is  positive in the alphabet S if x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , xk ∈ S and all exponents ni, 1 ≤  i ≤ k are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' In the group Γn let S1 = {s, x}, S2 = {s−1, x}, S3 =  {s, x−1}, S4 = {s−1, x−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' For n ≥ 12, there exists elements f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , f4,  g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , g4 ∈ Γn such that the following conditions hold:  i) ⟨f1, f2, f3, f4⟩ ∼= ⟨g1, g2, g3, g4⟩ ∼= Z4,  ii) The elements f1, f2, f3, f4 can be represented with positive words  in the alphabet S1,  iii) For all 1 ≤ i ≤ 4, the element gi can be represented with a  positive word in the alphabet Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' We define f1 = sn−1xs, f2 = sn−2(xs)2, f3 = sn−4(xs)4, f4 =  sn−8(xs)8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Then f1, f2, f3, f4 belong to Nn and generate a subgroup  isomorphic to Z4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' We also define g1 = sn−1xs, g2 = sn−2(x−1s)2, g3 =  s4−n(xs−1)4, f4 = s8−n(x−1s−1)8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' The elements g1, g2, g3, g4 also belong  to Nn and generate a subgroup isomorphic to Z4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  □  In the above proposition, the n ≥ 12 is not necessarily the best possi-  ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Using Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='5, we can now prove the following proposition  which establishes the existence of a non-left-orderable HNN extension  of a left-orderable virtually Abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  4  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' For all n ≥ 12, Γn admits a non-left-orderable HNN  extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Let f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , f4, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , g4 ∈ Γn be elements satisfying conditions  1)-3) of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Let φ : ⟨f1, f2, f3, f4⟩ → ⟨g1, g2, g3, g4⟩ be an  isomorphism such that φ(fi) = gi, 1 ≤ i ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  We consider an HNN extension  G := (Γn, ⟨f1, f2, f3, f4⟩, ⟨g1, g2, g3, g4⟩, t)  by letting txt−1 = φ(x) for all x ∈ ⟨f1, f2, f3, f4⟩  For any left-order on G, notice that the elements tfit−1, 1 ≤ i ≤ 4  are either all positive or all negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' On the other hand, among the  elements gi, 1 ≤ i ≤ 4 at least one is positive and one is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' This  is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Hence G is not left-orderable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' HNN extensions of nilpotent groups  The aim of this section is to prove that unlike solvable groups,  an HNN extension of a left-orderable nilpotent group is always left-  orderable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Let us recall that a nilpotent group is left-orderable iff it is  torsion-free;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' this claim too does not hold for solvable groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Let us first observe that, since a direct limit of left-orderable groups  is left-orderable, an HNN extension an HNN extension (G, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='B, t) is  left-orderable if for all finitely generated subgroups A0 and G0 of G,  where A0 ≤ A, G0 ⊇ ⟨A0, B0⟩ and B0 = tA0t−1, the HNN extension  (G0, A0, B0, t) is left-orderable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' We will use this observation repeatedly  in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  We already observed that by classification, an HNN extension of  an infinite cyclic group is left-orderable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' The same holds for an HNN  extension of any torsion free Abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Indeed, it suffices to consider finitely generated Abelian groups so  let G be a finitely generated torsion-free Abelian group, A, B ≤ G, φ :  A → B be an isomorphism, and (G, A, B, t) be the HNN extension with  respect to the isomorphism φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Let G ∼= Zd and r = rankA = rankB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  We will assume that G = Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Then for some linearly independent  vectors u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , ur we have A = {c1u1 + · · · + crur : ci ∈ Z, 1 ≤ i ≤ r}  and similarly for some linearly independent vectors v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , vr we have  B = {c1v1 + · · · + crvr  :  ci ∈ Z, 1 ≤ i ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' We let G = R, A =  {c1u1 + · · · + crur : ci ∈ R, 1 ≤ i ≤ r}, B = {c1v1 + · · · + crvr : ci ∈  R, 1 ≤ i ≤ r} and φ : A → B be the extension of φ : A → B defined as  φ(c1u1 + · · · + crur) = c1φ(u1) + · · · + crφ(ur) for all c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , cr ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  5  A key observation here is that even though the isomorphism φ :  A → B cannot necessarily be extended to G, but one can extend the  isomorphism φ : A → B to some automorphism F : G → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Then the  HNN extension (G, A, B, t) with respect to the isomorphism φ : A → B  has a quotient isomorphic to the semidirect product Z⋉F G by a normal  subgroup N ≤ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Since N and Z ⋉F G are left-orderable we obtain  that (G, A, B, t) is left-orderable (as an extension of a left-orderable  group by a left-orderable group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' By Britton’s Lemma, (G, A, B, t) is  a subgroup of (G, A, B, t) hence it is also left-orderable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  We now would like to carry the same argument for any torsion-free  nilpotent group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' The main issue here is that given a finitely generated  torsion-free nilpotent group Γ, one needs to construct a completion Γ  which would resemble the operation Zd → Rd so we can try to use the  argument in the Abelian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Let R be a commutative ring with identity and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  We let  Un(R) be the group of n × n upper-triangular matrices with 1’s on the  diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' The cases R = R and R = Z will be the most interesting to  us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  It is well-known that any finitely generated torsion-free nilpotent  group Γ embeds in Un(Z) for some n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' The Mal’cev completion of  Un(Z) is Un(R) (and the Mal’cev completion of Zn is Rn) 1, however,  given an isomorphism φ : A → B of subgroups of Un(Z), although it  induces an isomorphism φ : A → B but one cannot necessarily extend  this isomorphism to the entire G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' For example, for n = 3, the group  U3(Z) is isomorphic to the Heisenberg group  ⟨x, y, z | z = [x, y], [x, z] = [y, z] = 1⟩  and if we let A = ⟨x⟩, B = ⟨z⟩ and φ(x) = z, then this isomorphism  cannot be extended to the isomorphism of U3(Z) (or U3(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Thus we  need to define a completion of Γ other than the Mal’cev completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Let Xn,i, 1 ≤ i ≤ n − 1 be the matrix of Un(Z) where all off-diagonal  entries are zero except the (i + 1, i)-th entry is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' In order to  define a more suitable completion of Un(Z) we will extend it first, and at  the end we will obtain a completion which is ”infinite-dimensional”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Let  U∞(Z) be a group generated by xk, k ∈ Z such that for all k ∈ Z, n ≥ 1  the subgroup generated by xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , xk+n−1 is isomorphic to Un(Z)  through the isomorphism f(xk+j) = Xn,j, 1 ≤ j ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Notice that  U∞(Z) is well-defined this way and it contains isomorphic copies of all  1in the literature, the term Mal’cev completion is used for some other related  operations as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  6  Un(Z), n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' This group can be viewed as the group of infinite sized  integral unipotent matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' But to achieve our goal we extend U∞(Z)  further as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Let us first observe that in the group Un(Z) viewed as the group of  upper triangular unipotent integral matrices, [xi, xj] = 1 if |i − j| ≥ 2  and for all 1 ≤ i ≤ n − 2, [xi, xi+1] is a unipotent matrix with all the  off-diagonal entries zero, except the (i + 2, i + 1)-entry equals 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Thus  the elements [xi, xi+1], 1 ≤ i ≤ n−2 generate a subgroup isomorphic to  Un−1(Z) with an isomorphism xi → [xi, xi+1], 1 ≤ i ≤ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Similarly,  in the group U∞(Z), the elements [xi, xi+1], i ∈ Z generate a subgroup  isomorphic to U∞(Z), and the homomorphism f : U∞(Z) → U∞(Z)  defined as f(xi) = [xi, xi+1], i ∈ Z (it is sufficient to define it on the  generators) establishes this isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  The group U∞(Z) is a direct limit of the groups Un(Z), n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  More precisely, let Hn, n ≥ 1 be the subgroup of U∞(Z) generated  by x−n, x−n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' , xn−1, xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Then Hn is isomorphic to U2n+1(Z), and  U∞(Z) is a direct limit of the sequence Hn, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  In our construction of the completion, we will use a direct limit of  groups each isomorphic to U∞(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Let Γk, k ∈ Z be a group gen-  erated by zk,n, n ∈ Z with an isomorphism gk : Γk → U∞(Z) such  that gk(zk,n) = xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' We have · · · ≤ Γ−1 ≤ Γ0 ≤ Γ1 ≤ Γ2 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' and  [zk,n, zk,n+1] = zk−1,n for all k, n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' This defines an isomorphic em-  bedding gk,k+1 : Γk → Γk+1, k ∈ Z where gk,k+1(zn,k) = zn,k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' These  inclusions define a direct limit U of Γk, k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' The maps gk,k+1 induce  a shift isomorphism θ : U → U, so, in particular, θ(x) = gk,k+1(x) for  all x ∈ Γk, k ∈ Z  In defining the completion U, first, let us recall the following facts  about lattices of simply connected nilpotent Lie groups [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Let G be simply connected nilpotent Lie group, Γ be  a discrete subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' The following are equivalent:  (i) Γ is a lattice of G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  (ii) Γ is Zariski dense in G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  (iii) Γ is not contained in any proper connected closed subgroup of  G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  (iv) Γ is co-compact in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Let m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' For any subset Ω ⊆ Um(Z), we define  Span(Ω) = ⟨Ω⟩Z where the latter denotes the Zariski closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  For  example, Span(Um(Z)) = Um(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Then, for any subset Ω ⊆ U∞(Z) we  let  Span(Ω) = ∪  n≥1 Span(Ω ∩ Hn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  7  Then, for any subset Ω ⊆ U we define Span(Ω) = ∪  k≥1 Span(Ω ∩ Γk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Finally, we define U = Span(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  The Lie subgroups of Un(R) (hence of U) are simply connected (in-  deed contractible, as the exponential map determines a homeomor-  phism to Rd with d being the dimension of the group) thus its iso-  morphism type can be determined at the level of Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' The  Lie algebra of every Lie subgroup of U is a finite-dimensional nilpotent  Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' On the other hand, by Engel’s Theorem, for every finite-  dimensional nilpotent Lie algebra g with the underlying vector space  V , there exists an associated flag F(g) in the form {0} = V0 ≤ V1 ≤  · · ≤ Vn = V where dimVi = i, 0 ≤ i ≤ n and for all x ∈ g, 1 ≤ i ≤ n,  ad(x)(Vi) ⊆ Vi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Thus ð can be faithfully represented by strictly  upper-triangular matrices with respect to some basis of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' If g, h are  finite-dimensional nilpotent Lie algebras and φ : g → h a Lie algebra  isomorphism, then H = f(F) will be an associated flag of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' On the  other hand, if g is a finite-dimensional nilpotent Lie algebra with un-  derlying vector space V and I is an ideal of g faithfully represented  in gl(V0) with strictly upper triangular matrices with respect to a ba-  sis of a proper subspace V0, then by inductive process as in the proof  of Engels’ Theorem, it follows that we can extend the basis of V0 to  a basis of V such that g is faithfully represented with strictly upper  triangular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' By this observation, any Lie group isomorphism  Φ : G → H between finite-dimensional nilpotent Lie subgroups of U  can be extended to the group automorphism of U, since for any Lie  subgroups G1, G2 of U, G1 belongs to a Lie subgroup G3 which con-  tains θk(G2) as a normal subgroup for some integer k (thus the Lie  algebra g3 of G3 contains the Lie algebra of θk(G2) as an ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=')  We can now state and prove the following  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' An HNN extension of a torsion-free nilpotent group  is left-orderable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Let Γ be a torsion-free nilpotent group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' It is well-known that Γ  is left-orderable (in fact, bi-orderable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Indeed, it suffices to prove this  only for finitely generated subgroups, and any such subgroup embeds  into Um(Z) for some m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' The latter admits an easy bi-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Indeed,  more generally, we define a matrix A = (ai,j)1≤i,j≤n ∈ Um(R) as positive  if d is the smallest positive integer such that ai,j ̸= 0, for some i, j ≥ 1  with i + j = d, moreover, for this d, if p is the smallest positive integer  with p + q = d and ap,q ̸= 0, then ap,q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' One easily checks that this  8  is in fact a genuine left-order (and even a bi-order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Then U∞(R) is  also bi-orderable as a direct limit of Um(R), m ≥ 1 and so is U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  To show that an HNN extension of Γ is also left-orderable, it again  suffices to consider HNN extensions of finitely generated subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' So  let us assume that Γ is also finitely generated, A, B ≤ Γ and φ : A → B  be an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Γ embeds in U∞(Z) and the latter is a subgroup  of G = U∞(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  The isomorphism φ : A → B cannot necessarily be extended to  G, but one can extend the isomorphism φ : Span(A) → Span(B)  to some F : U → U where φ is an extension of φ by Mostow Strong  Rigidity Theorem for lattices in solvable Lie groups [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Then the HNN  extension (U, Span(A), Span(B), t) with respect to the isomorphism  φ : Span(A) → Span(B) has a quotient isomorphic to the semidirect  product Z⋉F U by a normal subgroup N ≤ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Since N and Z⋉F U are  left-orderable we obtain that (U, Span(A), Span(B), t) is left-orderable  (as an extension of a left-orderable group by a left-orderable group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' By  Britton’s Lemma, (Γ, A, B, t) is a subgroup of (U, Span(A), Span(B), t)  hence it is also left-orderable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  □  We would like to end this section with a torsion-free non-left-orderable  example which will contain a class two nilpotent group as an index two  subgroup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' indeed, it will contain a subgroup of Heisenberg group H of  3 × 3 integral unipotent matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' This might be one of the simplest  (smallest) examples of a torsion-free non-left-orderable group in litera-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' It also shows that torsion-freeness does not imply left-orderability  in the class of polycyclic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Let  Γ = ⟨t, u, v | [u, v] = t4, tut−1 = u−1, tvt−1 = v−1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  The group Γ is related to the Heisenberg group  H = ⟨x, y, z | [x, y] = z, [z, x] = [z, y] = 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Any element of H can be written uniquely as xmynzk where m, n, k ∈  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  H is bi-orderable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  The elements x2, y, z generate an index two  subgroup H0 of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  The group Γ will have an index two subgroup isomorphic to H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' We  let u = x2, v = y, t2 = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Let also G be a group given by the following  presentation  G = ⟨t, x, y, z | [x, y] = z, [z, x] = [z, y] = 1, t2 = z, txt−1 = x−1, tyt−1 = y−1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Then Γ is a subgroup of G generated by t, x2, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  9  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Γ is torsion-free and non-left-orderable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Assume that < is a left-order on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Without loss of generality  we may assume that t > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Then 1 < t < z thus z is also a positive  element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' On the other hand, let us observe that for all n ∈ 2Z, we have  (txn)2 = t2 thus the element txn is positive for every even integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Let m be positive if y > 1 and negative if y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Then, for all m ∈ 2Z,  txnym is positive as a product of two positive elements txn and ym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Then (txnym)2 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' However,  (txnym)2 = t2x−ny−mxnym = t2zmn = zmn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  We can choose n such that mn+1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' This yields that (txnym)2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  Contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  To see torsion-freeness let us observe that any element g ∈ G can be  written as g = x2pyqzr or g = tx2pyqzr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' The element x2pyqzr is not a  torsion since H is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' As for the element tx2pyqzr, we have  (tx2pyqzr)2 = x−2py−qx2pyqz2r+1 = z±2pqz2r+1 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Thus Γ is torsion-  free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  □  Acknowledgement:  We are very thankful to Zipei Nie for reading  the draft of this paper and correcting errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  References  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Button, Topics in infinite groups, Lecture Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  [2] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Bludov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
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+page_content=' 99, issue 3, (2009), 585-608  [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Lyndon and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Schupp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content=' Combinatorial Group Theory, Volume 89 of Ergeb-  nisse der Mathematik und ihrer Grenzgebiete, Springer-Verlaq, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
+page_content='  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
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+page_content='edu  Cody Martin, Department of Mathematics, North Dakota State  University, Fargo, ND, 58102, USA  Email address: cody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}
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+page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfQvZ4/content/2301.00052v1.pdf'}

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+Layout-aware Webpage Quality Assessment
+Anfeng Cheng∗, Yiding Liu∗, Weibin Li∗, Qian Dong, Shuaiqiang Wang, Zhengjie Huang, Shikun
+Feng, Zhicong Cheng and Dawei Yin§
+Baidu Inc., Beijing, China
+{chenganfeng01,liweibin02,wangshuaiqiang,huangzhengjie,fengshikun01,chengzhicong01}@baidu.com
+liuyiding.tanh@gmail.com,dq22@mails.tsinghua.edu.cn,yindawei@acm.org
+ABSTRACT
+Identifying high-quality webpages is fundamental for real-world
+search engines, which can fulfil users’ information need with the
+less cognitive burden. Early studies of webpage quality assessment
+usually design hand-crafted features that may only work on par-
+ticular categories of webpages (e.g., shopping websites, medical
+websites). They can hardly be applied to real-world search engines
+that serve trillions of webpages with various types and purposes.
+In this paper, we propose a novel layout-aware webpage quality
+assessment model currently deployed in our search engine. Intu-
+itively, layout is a universal and critical dimension for the quality
+assessment of different categories of webpages. Based on this, we
+directly employ the meta-data that describes a webpage, i.e., Doc-
+ument Object Model (DOM) tree, as the input of our model. The
+DOM tree data unifies the representation of webpages with different
+categories and purposes and indicates the layout of webpages. To
+assess webpage quality from complex DOM tree data, we propose
+a graph neural network (GNN) based method that extracts rich
+layout-aware information that implies webpage quality in an end-
+to-end manner. Moreover, we improve the GNN method with an
+attentive readout function, external web categories and a category-
+aware sampling method. We conduct rigorous offline and online
+experiments to show that our proposed solution is effective in real
+search engines, improving the overall usability and user experience.
+KEYWORDS
+Webpage Quality Models, Graph Neural Network, Information Re-
+trieval, Search
+ACM Reference Format:
+Anfeng Cheng∗, Yiding Liu∗, Weibin Li∗, Qian Dong, Shuaiqiang Wang,
+Zhengjie Huang, Shikun Feng, Zhicong Cheng and Dawei Yin§. 2023. Layout-
+aware Webpage Quality Assessment. In SIGKDD ’23: ACM Special Interest
+Group on Knowledge Discovery and Data Mining, August 06-10, 2023, Long
+Beach, CA. ACM, New York, NY, USA, 11 pages. https://doi.org/XXXXXXX.
+XXXXXXX
+∗ Co-first authors.
+§ Dawei Yin is the corresponding author.
+Permission to make digital or hard copies of all or part of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full citation
+on the first page. Copyrights for components of this work owned by others than ACM
+must be honored. Abstracting with credit is permitted. To copy otherwise, or republish,
+to post on servers or to redistribute to lists, requires prior specific permission and/or a
+fee. Request permissions from permissions@acm.org.
+SIGKDD ’23, August 06–10, 2023, Long Beach, CA
+© 2023 Association for Computing Machinery.
+ACM ISBN 978-1-4503-XXXX-X/18/06...$15.00
+https://doi.org/XXXXXXX.XXXXXXX
+1
+INTRODUCTION
+Search engines, such as Google and Baidu, plays an important
+role in fulfilling users’ information need. Over the past decades,
+relevance modeling is the main concern of search engines, dedicated
+to putting the most relevant web content on top of the ranked
+results [23, 28, 38, 42]. However, the very fact that not all relevant
+contents are useful to users has become an increasingly serious
+symptom, where relevant webpages with low quality would induce
+a significant cognitive burden on the user. In such cases, useful
+information is hard to be identified, and the users need to take extra
+effort to understand the information to be conveyed. To reduce the
+cognitive burden, measuring the quality of webpages has become a
+critical concern, which can better benefit users with well-delivered
+information and improve the overall usability of a search engine. For
+example, given webpages with comparable relevance, high-quality
+webpages should be ranked higher than its competitors.
+Nevertheless, webpage quality assessment is a very important
+yet challenging task in web search, due to the complexity and
+diversity of webpages in the era of web 2.0. A webpage with a clear
+structure, tidy organization and concentrated delivery of crucial
+information is always preferable to one that only stacks content
+without proper presentation, even though each may contain similar
+information. Therefore, an accurate assessment of webpage quality
+can facilitate a search engine to reduce the cognitive burden and
+more effectively provide useful information for users.
+Previous attempts at webpage quality assessment mainly aim
+to manually design discriminative features [5, 14, 16, 24], where
+classification algorithms [7, 14] are applied subsequently. How-
+ever, modern search engines usually face trillions of webpages
+with various categories, where simple hand-crafted features and
+classification algorithms (e.g. Bayesian Networks [7]) can hardly
+capture the in-depth information that reveals the webpage quality.
+Moreover, most of them can only work on a particular category of
+webpages, e.g., shopping websites [8], medical website [25], web
+portals [6], and Wikipedia articles [17]. They are hard to be effec-
+tively applied to real-world search engines that serve trillions of
+heterogeneous webpages.
+To address the aforementioned limitations, we conduct the first
+work that investigates layout-aware webpage quality assessment on
+real-world web data. The intuition is based on the findings that the
+quality of a webpage is largely determined by its content layout [6,
+24], which is of a great influence on how users perceive textual and
+multi-modal content [20, 31, 34, 35, 41]. Modeling in-depth layout
+information is promising for webpage quality assessment in real-
+world web search scenario. However, it is also very challenging,
+where two crucial research questions need to be answered:
+RQ1: How to capture the layout information of different categories
+of webpages in a unified manner?
+arXiv:2301.12152v1  [cs.IR]  28 Jan 2023
+
+SIGKDD ’23, August 06–10, 2023, Long Beach, CA
+Cheng and Liu, et al.
+<html>
+<head>
+<body>
+<meta> <link>
+<title>
+<div>
+<p>
+<img>
+<html> 
+<head> 
+<meta name="viewport" content="width=device-
+width,initial-scale=1"> 
+<link href=" style.css" rel=" stylesheet”> 
+<title>Critical Path</title> 
+</head> 
+<body> 
+<p>Hello <span>web performance</span> 
+students!</p> 
+<div><img src=" awesome-photo.jpg"></div> 
+</body> 
+</html>
+Figure 1: A toy example that shows the webpage layout represented
+by the DOM tree.
+RQ2: How to encode webpage layout information for webpage quality
+assessment on large-scale heterogeneous webpages?
+To answer RQ1, we propose to extract layout information of
+webpages from Document Object Model (DOM) data. Specifically,
+DOM is a cross-platform and language-independent interface that
+treats a webpage as a tree structure wherein each node is an object
+representing a content piece of the webpage.
+Figure 1 shows a toy example of the hierarchical structure of a
+DOM tree, which is converted from its HTML source code. Each
+node in the tree is an object that contains partial content of the
+webpage and is associated with different attributes that describe the
+content (e.g., type and size). Such data indicates rich hierarchical in-
+formation on the content and its layout. And different categories of
+webpages can be represented in a unified manner. It is indisputable
+that inspecting the structure of DOM tree can help to measure the
+quality.
+For RQ2, it is very challenging as webpages in real search en-
+gines are highly diverse, where the modeling of layout information
+should be expressive to reveal the underlying patterns of hetero-
+geneous DOM tree data. Recent advances in deep representation
+learning [3, 22] have achieved great success on many web applica-
+tions [9, 13, 28], and also sheds new light on our task at hand.
+Notably, Graph Neural Networks [12, 15] has shown great per-
+formance in modeling structured text (e.g., word interactions) [18,
+37, 41], yet they are unexplored for complex DOM tree structure.
+Different from structure of textual document, the webpage layout
+represented by DOM tree is more complicated, which usually has
+hierarchical structure and the nodes usually have rich attributes.
+Existing methods that designed for text structure usually lack spe-
+cialized consideration for the problem of quality assessment on
+DOM tree data, and thus might be unsatisfactory for real search
+engines. To this end, we propose the first GNN-based method to
+learn the underlying semantics of webpage layout in an end-to-end
+manner, based on which we further make several improvements to
+advance its performance on the task of webpage quality assessment.
+To verify the effectiveness of our layout-aware webpage quality
+assessment model, we perform offline experiments on the dataset
+collected by the real-world search engine. Additionally, we deployed
+our model in the online ranking system and achieve good improve-
+ments. Last but not least, the proposed solution is currently fully
+deployed in the online system of Baidu Search. To illustrate how
+layout-aware webpage quality assessment facilitates the overall
+usability of our search engine, we further present the details of the
+model deployment.
+Overall, our main contributions can be summarized as follows.
+• We develop the largest application of deep learning for the
+problem of webpage quality assessment, which significantly
+improves the overall usability of real-world search engines.
+• We leverage DOM tree data and propose a GNN-based so-
+lution to learn the quality information of heterogeneous
+webpages in an end-to-end fashion.
+• We present the deployment of webpage quality assessment
+model in the real production environment, which effectively
+serves trillions of webpages with various categories and
+purposes.
+• We conduct rigorous offline and online experiments before
+fully deploying the model online. The experimental results
+show that the proposed solution is effective to be applied in
+real-world search engines.
+2
+RELATED WORK
+2.1
+Graph Neural Network
+Recent years have witnessed the success of graph neural networks
+(GNNs) for relational data. For example, Graph Convolutional Net-
+work (GCN) [21] is introduced to aggregate the one-hop neigh-
+bours of each node in the graph, followed by a linear projection and
+non-linear activation. GraphSAGE [15] is proposed to generalize
+GCN’s aggregation operation from average to sum, max and a RNN
+unit. Graph Attention Network (GAT) [30] employs the attention
+mechanism into GNNs, which allows GAT to assign different im-
+portance to nodes within the same neighbourhood. Generally, a
+GNN can be regarded as using the input graph structure as the com-
+putation graph for message passing [12], during which the local
+neighbourhood information is aggregated to get a more contextual
+representation. For more details, please refer to [32].
+Moreover, there are many applications across various domains
+that apply GNNs and achieve considerable improvements, such
+as protein model quality assessment [2, 26], fuel ignition quality
+assessment [27], advertising detection [36] and text classification
+[11, 19, 37, 41]. For example, Sanyal et al. [26] explore an alterna-
+tive approach and train a graph convolutional network with nodes
+representing protein atoms and edges connecting spatially adja-
+cent atom pairs. GraphQA [2] is a graph-based method to estimate
+the quality of protein models, that possesses favorable properties
+such as representation learning. Schweidtmann et al. [27] develop
+GNN models for predicting three fuel ignition quality indicators
+of oxygenated and non-oxygenated hydrocarbons. Yang et al. [36]
+propose WTAGRAPH, a web tracking and advertising detection
+framework based on graph neural networks.
+Notably, a handful of researches [18, 37, 41] leverage GNN to
+perceive text structure (e.g., word interactions) for down-stream
+tasks, which are the closest research to our study. However, they
+mainly consider the relationships between segments of text (e.g.,
+words and paragraphs), and do not consider the overall structure
+and layout of webpage, i.e., how the multi-modal content is orga-
+nized and presented. In addition, the expressiveness of GNN is not
+explored for the task of webpage quality assessment. In this paper,
+we develop the first GNN-based method for webpage quality assess-
+ment, which is further deployed in the real production environment
+that facilitate the usability of our search engine.
+
+Layout-aware Webpage Quality Assessment
+SIGKDD ’23, August 06–10, 2023, Long Beach, CA
+2.2
+Layout
+Layout, i.e., how the contents are organized and presented, is a
+critical dimension for document generation [4], scene recognition
+[1, 10] and webpage quality assessment [16, 24]. Substantial efforts
+have been made to explore the importance of layout in many AI
+areas. For example, Besides, Biswas et al. [4] design an automated
+deep generative model using graph neural networks to generate
+synthetic data with highly variable and plausible document layouts.
+Avetisyan et al. [1] use a message-passing graph neural network to
+model the inter-relationships between objects and layout, guiding
+the generation of a global object alignment in a scene. Chen et
+al. [10] build a Layout Graph Network (LGN) where regions in
+PaSL are defined as nodes and two kinds of independent relations
+between regions are encoded as edges. Zhang et al. [40] propose a
+new kind of classification method for lithography layout patterns
+based on graph convolution network.
+Recently, with the development of pre-trained language models,
+LayoutLM-style methods have achieved success in textual or multi-
+model document understanding. For instance, LAMPreT [31] en-
+codes each block with a multimodal transformer in the lower-level,
+and aggregates the block-level representations and connections
+utilizing a specifically designed transformer at the higher-level. Lay-
+outLM [34] jointly model interactions between text and layout in-
+formation across scanned document images, which is beneficial for
+a great number of real-world document image understanding tasks
+such as information extraction from scanned documents. LayoutLM-
+v2 [35] further uses the new text-image alignment and text-image
+matching tasks and integrates a spatial-aware self-attention mech-
+anism into the Transformer architecture. LayoutLM-v3 [20] pre-
+trains multi-modal Transformers for Document AI with unified text
+and image masking.
+Our work differs from the aforementioned studies, as our main
+focus is to model the layout of webpages for quality assessment.
+In particular, we model the layout of webpage with a graph con-
+struction method that represents a DOM tree as a layout graph and
+employ an expressive graph neural network to capture the underly-
+ing semantics of the layout graphs for webpage quality assessment.
+3
+PRELIMINARIES
+In this section, we introduce the basic concepts and formalize the
+problem of webpage quality assessment. We summarize the com-
+monly used notations in Table 1.
+3.1
+Layout-aware Webpage Quality
+Intuitively, high-quality webpages are those that clearly provide
+useful information for users in common. Specifically, given a set
+of webpages with comparable relevance under the same query, we
+consider the layout (i.e. structure design, content presentation) as
+the key dimension of measuring webpage quality and improving
+user experience [6, 24]. Based on this, we can construct a set of rules
+and principles for annotating webpage quality and utilize human
+annotation as the objective of our proposed method.
+The considering aspects of rules and principles to score the
+layout of a webpage are shown in table 2, including interactive
+Experience, paragraph and layout design. We give the definition
+Table 1: Commonly-used notations.
+Notations
+Descriptions
+G𝑝 = {N, E}
+A layout graph of webpage 𝑝
+N
+The node set of graph G𝑝
+E
+The edge set of graph G𝑝
+F = {F𝑡 }𝑇
+𝑡=1
+The layout-related feature sets
+𝑇
+The number of node type
+F𝑡 = {f𝑖}|F𝑡 |
+𝑖=1
+The feature set of node type 𝑡
+f𝑖
+The layout-related feature
+E(·)
+The embedding of its input
+�ℎ(0)
+𝑛
+The initialized embedding of node 𝑛
+𝑡𝑛
+The node type of node 𝑛
+𝜷𝑝
+The category of webpage 𝑝
+�ℎ(0)
+𝑣
+The initialized embedding of virtual node 𝑛
+𝜎(·)
+An activation function
+𝛼𝑛𝑚
+The attention score between nodes 𝑚 and 𝑛
+𝑒𝑛𝑗
+The attention coefficient between nodes 𝑚 and 𝑗
+𝑠𝑝
+The predicted assessment score of webpage 𝑝
+𝑦𝑝
+The manually assessment score of webpage 𝑝
+and some examples for each aspect. The rules and principles for
+annotators are defined as the following:
+• 0 means poor layout. On the basis of ordinary pages, points
+will be deducted for various flaws.
+• 1 means ordinary layout. 1 point is common, and annotators
+are required to add or deduct on this basis.
+• 1.5 means better structure. a certain gain compared to ordi-
+nary layout.
+• 2 means gainful layout. The user experience of this layout is
+significantly better than most layouts.
+Based on these principle, bonus and deduction rules can be for-
+mulated as: webpages with reasonable & beautiful layout or rich
+information will have an extra bonus, on the contrary, unreason-
+able & chaotic layout or valueless information will be deducted.
+Finally, annotators are required to score the give webpage from 0
+to 2 points based on the above rules and principles.
+3.2
+Layout Graph
+To extract quality information from webpage layout, we construct a
+layout graph for each webpage based on its DOM tree. In particular,
+a layout graph is denoted as G𝑝 = {N, E} that contains a node
+set N and an edge set E. Each node has a specific type (e.g., text,
+image and video), and is associated with several layout-related
+features F𝑡 = {f𝑖}|F𝑡 |
+𝑖=1 . The features of different types of nodes are
+denoted as F = {F𝑡 }𝑇
+𝑡=1, where𝑇 is the total number of node types.
+Besides, each layout graph is also associated with the category of
+
+SIGKDD ’23, August 06–10, 2023, Long Beach, CA
+Cheng and Liu, et al.
+Table 2: The considering aspects of rules and principles to score the layout of a webpage.
+Aspects
+Definition
+Examples
+Interactive Experience
+Whether the webpage has interactive function
+Click to call, swipe to browse pictures
+Paragraph
+Ways to split the document into paragraphs
+Using different heading, special font color to layering
+Layout Design
+The overall design of the webpage’s layout
+many additional functions, various modules, font section size is appropriate
+its webpage, which is denoted as 𝜷𝑝. The detailed construction
+process of layout graph is depicted in Section 4.2.
+3.3
+Webpage Quality Assessment
+Given a layout graph G𝑝 = {N, E}, its features F = {F𝑡 }𝑇
+𝑡=1 cate-
+gory 𝜷 and F𝑡 = {f𝑖}|F𝑡 |
+𝑖=1 , the task of webpage quality assessment is
+to estimate a score 𝑠𝑝 for a given webpage 𝑝 w.r.t. its quality, i.e,
+𝑠𝑝 = 𝑓𝜃 (G𝑝, F𝑡, 𝜷𝑝),
+(1)
+where 𝑓 (·) represents the quality model, and 𝜃 denotes its param-
+eters. The scores should be consistent with users’ perception of
+webpage quality, and reflect the rules and principles as we described
+above.
+4
+METHOD
+In this section, we first present the overview of our model. Then,
+we describe the graph formulation process for a webpage, including
+the construction of the layout graph and feature pre-processing.
+After that, we present a GNN-based solution for webpage quality
+assessment.
+4.1
+Overview
+Our solution mainly contains two components: layout graph for-
+mulation (i.e., Section 4.2) and quality assessment model (i.e., Sec-
+tion 4.3). In layout graph formulation, we first leverage the layout
+information encoded in DOM tree to construct a layout graph
+G𝑝 = {N, E} for every webpage 𝑝. Then, two types of features are
+designed for the quality assessment, as depicted in Figure 2, includ-
+ing those associated with each node in the graph, as well as the
+category of the corresponding webpage (i.e., 𝜷𝑝).
+Next, we propose a quality assessment model that leverages
+Graph Attention Network (GAT) to perform expressive message
+passing between nodes in the layout graph. Both local and global
+structure information of the layout graph can be encoded in latent
+representations, which are exploited for the quality assessment task.
+Moreover, we improve the vanilla GAT model by 1) introducing an
+attentive readout function via the virtual node, 2) incorporating
+graph-level category information in the scoring function, and 3)
+alleviating the data imbalance problem that is common in real-world
+applications.
+4.2
+Layout Graph Formulation
+Graph construction. The content layout has been viewed as
+one of the most critical dimensions for measuring webpage qual-
+ity [24]. To formulate layout information for various categories of
+webpages, we first construct layout graph based on DOM tree. In
+Table 3: The summary of selected features for each node type in our
+constructed graph.
+Classification
+Feature Name
+Location
+height, width, xpos, ypos, position type
+Content
+number of word, font size, font style,
+line height, font weight, alignment
+Layout
+border, padding, margin, visibility,
+display style, outline style, outline width
+Others
+tag name, webpage category
+particular, we leverage HTML parser Beautiful Soup 1 to parse the
+source code of a webpage, identifying the hierarchical structure
+of the webpage. Then, Depth First Search (DFS) is used for exact-
+ing adjacency relationships from the DOM tree. Specifically, we
+recursively record the nodes and the corresponding edges between
+parent and child nodes in the DOM tree, as shown in Algorithm 1.
+The layout graph G𝑝 of webpage 𝑝 can be expressed by the exacted
+nodes N and their relations E.
+Virtual node. It is worth noting that, we also include a global
+virtual node that connects to all the other nodes in the graph (as
+shown in Figure 2). It can be viewed as a super-hub [39] of the layout
+graph, which could be useful to aggregate the global information,
+and serves as hyperlinks that connect any two nodes in the layout
+graph. As such, we can capture global information of the given
+graph via the virtual node.
+Feature pre-processing. To capture the layout information of
+the webpage, we design a series of features for each node type.
+Taking the text node as an example, font style, font size, alignment
+and position in webpage are all represented by learnable embedding.
+The detailed list of features is presented in Table 3.
+More specifically, for continuous features (e.g., height, line height
+and margin), a non-uniform interval division strategy is employed
+to divide the continuous interval into several buckets, which can
+ensure that there are enough training samples in a single bucket.
+The uniform division of the whole interval leads to the data sparse
+issue since the continuous features typically obey a long-tail dis-
+tribution. Discrete features (e.g., font style, display style and tag
+name), are falling on a divided interval are mapped into a corre-
+sponding bucket, and this bucket is assigned a learnable embedding
+to represent the characteristics of its interval.
+1We parse webpages with the python library: https://www.crummy.com/software/
+BeautifulSoup/bs4/doc/
+
+Layout-aware Webpage Quality Assessment
+SIGKDD ’23, August 06–10, 2023, Long Beach, CA
+Layout-Related Features
+...
+Virtual Node
+Layout Graph
+Node Features
+Graph Features
+Projection
+Page Score
+Figure 2: The illustration of message passing in our model. The red node represents the virtual node in the constructed layout graph, which
+is utilized for capture the graph-level information.
+Algorithm 1: Layout Graph Construction
+Input: HTML DOM tree R𝑝 of webpage 𝑝
+Output: layout graph G𝑝 = {N, E} of webpage 𝑝
+% recursive graph construction;
+GraphConstruction(root) begin
+N𝑟 = {𝑣𝑖𝑟𝑡𝑢𝑎𝑙_𝑛𝑜𝑑𝑒, root.𝑛𝑜𝑑𝑒};
+E𝑟 = {(𝑣𝑖𝑟𝑡𝑢𝑎𝑙_𝑛𝑜𝑑𝑒, root.𝑛𝑜𝑑𝑒)};
+for child ∈ root.𝑐ℎ𝑖𝑙𝑑𝑟𝑒𝑛 do
+N𝑐 = child.𝑛𝑜𝑑𝑒;
+N𝑟 = N𝑟 ∪ N𝑐;
+E𝑐 = {(𝑣𝑖𝑟𝑡𝑢𝑎𝑙_𝑛𝑜𝑑𝑒, child.𝑛𝑜𝑑𝑒),
+(child.𝑛𝑜𝑑𝑒, root.𝑛𝑜𝑑𝑒)};
+E𝑟 = E𝑟 ∪ E𝑐;
+N𝑐, E𝑐 = GraphConstruction(child);
+end
+return N𝑟, E𝑟
+end
+G𝑝 = {GraphConstruction(Rp)};
+In addition to the node-level features, graph-level feature em-
+bedding is introduced to the layout graph (i.e., webpage category)
+to provide the model with the ability to perceive different cate-
+gories of webpages, which is vital to the quality assessment. One
+reason is that the same webpage category has a similar structure.
+With the development of webpage makers (like Dreamweaver, and
+Google Web Designer), large amounts of webpages are generated
+from templates and almost in the same layout. Therefore, with this
+graph-level embedding, the predicted assessment score shall be
+more robust in the online search engine. Another reason lies in
+that different webpage categories have different criteria for quality
+assessment. For example, a succinct and well-organized document
+layout without distracting pictures is preferred on a search page,
+but for a portal, a document layout with pictures and text is con-
+sidered to be better. In summary, it is meaningful and important to
+take the graph-level feature embedding into account for the layout
+graph.
+4.3
+Quality Assessment Model
+Given the constructed layout graph associated with rich features,
+the key of webpage quality assessment is to expressively reveal
+salient patterns underlying the graph. In particular, we consider
+two types of relationships in the graph that could be discriminative
+for the task:
+• Local relationships. Intuitively, the relationships between
+adjacent nodes in the layout graph are important to reveal
+content quality. For example, a node with <image> tag is usu-
+ally the illustration of its adjacent (e.g., parent) node with
+<div> tag, which contains textual description. The interac-
+tion of the two nodes indicates the web content has both
+visual and textual presentation, forming a strong signal of
+high-quality content.
+• Global relationships. Another important insight is that
+the relationships between local content and global layout
+should also be considered. For example, a node with textual
+description might be critical in a news article but is less
+important in a video webpage, whose quality largely depends
+on the node that contains the video.
+Attentive message passing. To achieve this, we leverage graph
+neural networks that are promising to capture such complicated
+patterns. In particular, we utilize the Graph Attention Network
+(GAT) [30] to model the interactions between nodes in the layout
+graph, where the modeling of node relationships can be viewed as
+message passing [12] among nodes.
+In particular, the architecture of GAT is composed by stacking
+multiple graph attention layers, each of which can be defined as
+�ℎ(𝑘+1)
+𝑛
+= 𝜎 ��
+�
+∑︁
+𝑚∈N𝑛
+𝛼𝑛𝑚W(𝑘)
+1
+�ℎ(𝑘)
+𝑚 ��
+�
+,
+(2)
+where 𝜎(·) is an activation function and 𝛼𝑛𝑚 is the attention value
+between node 𝑛 and node 𝑚. Here, �ℎ(𝑘)
+𝑛
+represents the embedding
+of node 𝑛 in the 𝑘-th layer. The attention value 𝛼𝑛𝑚 is learned
+to selectively propagate information from neighbour node 𝑚 to
+node 𝑛, and a node can attentively interact more with its important
+neighbours than those trivial ones. Formally, the attention value
+
+SIGKDD ’23, August 06–10, 2023, Long Beach, CA
+Cheng and Liu, et al.
+can be defined as
+𝛼𝑛𝑚 = softmax𝑚 (𝑒𝑛𝑚) =
+exp (𝑒𝑛𝑚)
+�
+𝑗 ∈N𝑛 exp �𝑒𝑛𝑗
+� ,
+(3)
+where the logits 𝑒𝑛𝑗 is computed as
+𝑒𝑛𝑗 = 𝜎
+�
+W(𝑘)
+3
+[W(𝑘)
+2
+�ℎ(𝑘)
+𝑛
+∥W(𝑘)
+2
+�ℎ(𝑘)
+𝑗
+]
+�
+.
+(4)
+Here, we use ∥ to represent the concatenation operation, and W(𝑘)
+2
+and W(𝑘)
+3
+are the weight matrices of the linear transformations
+at the 𝑘-th layer. Note that the weight matrices are shared across
+different nodes in a single graph attention layer.
+After 𝐾 times of message passing, the layout-aware patterns
+could be captured by node interactions (as defined in Eq. (2)) within
+𝐾-hops. It is worth noting that the virtual node also plays an im-
+portant role during the message passing process. The virtual node
+offers a pathway for nodes’ interaction with considering the global
+interactions in the graph, which is critical for the quality assess-
+ment task. Overall, the GAT-based message passing framework is
+able to comprehensively model both local and global relationships
+for the final task.
+Readout function. To compute the final quality score, we de-
+fine the readout function as mean-pooling [15, 32] to summarize
+all node representations as the final graph representation, and sub-
+sequently adopt a linear layer as
+𝑠𝑝 = W mean_pooling( �𝐻 (𝐾)
+N
+) + 𝑏,
+(5)
+where �𝐻 (𝐾)
+N
+is the set of node representations in 𝐾-th layer of GAT.
+Alternatively, we can apply a more reasonable readout function,
+which is to use the representation of the virtual node as the final
+graph representation, and rewrite Eq. (5) as
+𝑠𝑝 = W�ℎ(𝐾)
+𝑣
++ 𝑏,
+(6)
+where �ℎ(𝐾)
+𝑣
+is the virtual node representation in 𝐾-th layer (i.e. the
+last layer) of the model. In such case, the aggregation on the virtual
+node can be viewed as an attentive readout function, which has
+the capability of distinguishing the impact of different nodes in the
+graph for the final task.
+Category-aware quality assessment. The quality score defined
+in Eq. (6) is based on rich information aggregated from nodes. How-
+ever, graph-level information is critical yet not incorporated. There-
+fore, we further improve Eq. (6) with the category information of
+webpage. In particular, we denote the category embedding of a
+given webpage 𝑝 as E(𝜷𝑝), and further rewrite Eq. (6) as
+𝑠𝑝 = W(�ℎ(𝐾)
+𝑣
++ E(𝜷𝑝)) + 𝑏.
+(7)
+Note that the category embedding E(𝜷𝑝) has the same dimension-
+ality as the graph embedding �ℎ(𝐾)
+𝑣
+, such that the embeddings could
+be summed for the final assessment.
+Category-aware data sampling. As the graph-level category
+embedding is introduced in Eq.(7) to perceive different categories
+of webpages, the bias in different categories may affect the predic-
+tion of models. In particular, some webpages are highly similar in
+layout, such as some popular question-answering websites, which
+are generated from templates. Such webpages typically have sim-
+ilar layout scores. Consequently, the predicted assessment score
+may be dominated by the category-aware embedding (i.e. graph
+level embedding). To alleviate this issue, a category-aware sampling
+strategy is employed. Up-sampling is utilized to balance the number
+of two classes, based on which the bias could be mitigated and our
+model could learn a distinguishable quality assessment score for a
+single category of webpages.
+Optimization objective. After up-sampling, the model could
+be optimized through Mean Squared Error (MSE) loss. It can be
+defined as
+𝐽 = 1
+𝑃
+𝑃
+∑︁
+𝑝=1
+�𝑦𝑝 − 𝑠𝑝
+�2 ,
+(8)
+where 𝑃 is the total number of training samples after up-sampling
+and 𝑦𝑝 is the annotated layout score of webpage 𝑝.
+5
+DEPLOYMENT
+In this section, we show how the layout-aware webpage quality
+assessment model be applied to our online ranking system. We
+first introduce the input data construction process of the quality
+assessment model and then present the general picture of the quality
+score working in the ranking system. The overview of deployment
+is shown in Figure 3.
+5.1
+Offline Input Data Construction
+In the left component of Figure 3, we present the process of input
+data construction for our model. Firstly, each webpage on the world
+wide web will be parsed through our HTML parser. All features
+of the HTML are stored in a database. Secondly, we construct the
+layout graph based on DOM tree and extract the features needed
+for quality assessment model using the algorithm defined in Al-
+gorithm 1. Note that this process runs offline, it can significantly
+reduce the computing time of the online search system.
+We also list the features which are used in our webpage quality
+assessment model, details are shown in Table 3. We classify the
+features into three main categories w.r.t., location, content, and
+layout according to the different roles they play in building webpage.
+Category location is the primarily feature that locates the position
+of elements in the webpage e.g., height, width and position type.
+Category content contains text-related features e.g., the number of
+words, font style, and line height. Category layout is a feature that
+controls the layout of elements, e.g., border, padding, and margin.
+In addition, we add tag name, natural categorical information, and
+webpage category, which is used to balance the distribution of train
+data under different webpage forms.
+5.2
+Online System Workflow
+The online system workflow is presented in the right component of
+Figure 3. Our ranking system contains a wide variety of webpage
+features, where quality is one of the most important factors. To
+apply our layout-aware webpage quality assessment model in our
+online retrieval system, the new quality scores need to be loaded
+into the retrieval feature list. The online ranking system only needs
+to load the new quality assessment score and apply it to obtain the
+new ranking results with respect to the new ranking webpage list,
+
+Layout-aware Webpage Quality Assessment
+SIGKDD ’23, August 06–10, 2023, Long Beach, CA
+HTML Parser
+HTML
+Database
+Webpages
+Layout Graph
+Construction
+Input Data
+Database
+Webpage
+DOM Tree with Features
+Input Data of Model
+Virtual
+Node
+𝑵𝟏
+𝑵𝟐
+...
+height
+width
+paddling
+margin
+font border ...
+Features
+image
+AirTag
+head
+html
+body
+title
+iMac
+p
+div
+height、width、margin、
+border、padding、font
+size、font style、xpos、
+content length、ypos、
+overflow、visibility ......
+Layout-aware Quality
+Assessment Model
+Quality Score
+Database
+Online Ranking System
+new feature ranking
+list of each webpage
+......
+Ranking
+System
+new ranking results
+Input Data Construction
+Online System Workflow
+Figure 3: The overview of deployment in online ranking system.
+which is shown in the lower left area of the online component. Note
+that, the quality assessment scores of all webpages are calculated
+offline and are independent of the online search query, thus are
+inefficient for the online search query.
+6
+OFFLINE EVALUATION
+In this section, we conduct an offline evaluation of the proposed
+layout-aware webpage quality assessment model on the manually-
+labeled dataset from the search engine serves through the offline
+experiments.
+6.1
+Dataset
+To evaluate the proposed method, we first collect a set of webpages
+from our database, which stores the real webpages that our search
+engine serves. Next, we manually label all the collected webpages on
+our crowdsourcing platform, where a group of experts are required
+to assign low-quality (0) or high-quality (1) to each of the given
+webpage. In our experiments, we use 600,000 webpages for training
+and 20,000 webpages for testing.
+6.2
+Evaluation Metrics
+Positive-Negative Ratio (PNR). We use PNR to measure the con-
+sistency between manual quality labels and the scores estimated by
+the model. In particular, by enumerating all the pairs of webpages
+in the dataset (i.e., 𝐷), PNR can be formally defined as
+𝑃𝑁𝑅 =
+�
+𝑑𝑖,𝑑𝑗 ∈𝐷 I �𝑦𝑖 > 𝑦𝑗
+� · I �𝑓 (𝑑𝑖) > 𝑓 �𝑑𝑗
+��
+�
+𝑑𝑖′,𝑑𝑗′ ∈𝐷 I �𝑦𝑖′ > 𝑦𝑗′� · I �𝑓 (𝑑𝑖′) < 𝑓 �𝑑𝑗′�� ,
+(9)
+where I is an indicator function, i.e., I (𝑎 > 𝑏) = 1, if 𝑎 > 𝑏, and 0
+otherwise. Here, 𝑓 (𝑑𝑖) represents the quality score of a webpage 𝑑𝑖
+estimated by the model. Higher PNR value indicates better perfor-
+mance of the model.
+Area Under Curve, Precision, Recall, F1-Score. We also report
+Area Under Curve (AUC), Precision (P), Recall (R) and F1-Score (F1)
+to evaluate our proposed model. Precision and recall are often in
+tension, that is, improving precision typically reduces recall and
+vice versa. F1-Score combines them to one performance metric. Area
+under curve summarizes the trade-off between the true positive
+rate and false positive rate for a predictive model using different
+probability thresholds.
+6.3
+Compared Baselines and Our Approach
+To validate the effectiveness of our layout-aware webpage quality
+model, we conduct experiments on several related baseline mod-
+els: TreeLSTM [29], a standard LSTM architecture designed for
+tree-structured network topologies. GIN [33] introduces a learnable
+parameter to adjust the weight of the central node. GAT [30] lever-
+ages the attention mechanism to improve neighbor aggregation
+scheme. Our proposed models: Virt-GIN has a more expressive
+readout mechanism by adding the virtual node �ℎ𝑣 to GIN model.
+Virt-GAT is our approach similar to virt-GIN model, i.e., a GAT
+model with virtual node. Models-NC: Note that all the above-
+mentioned models use category information as proposed in Section
+4.3. To further clarify the influence of category in the model, we also
+include four variants without using category information, which is
+denoted with a suffix Non-Category (-NC).
+In addition, we also compare our proposed method with Online
+Baseline, which is the quality assessment model that was previ-
+ously served online in our search engine. This can clearly illustrate
+the improvement brought by the proposed solution for our search
+engine.
+6.4
+Experimental Settings
+In our experiments, Adam is selected as the optimizer. We use the
+following hyper-parameters: embedding size (64), number layers
+(5), dropout probability (0.2), batch size (32), learning rate (0.0001)
+for GNN models, train epochs (25). As for the TreeLSTM model,
+we set the embedding size (64), dropout probability (0.5), batch size
+(128), learning rate (0.0001), epochs (25) for it. We run 5 experiments
+with different random seeds for all models mentioned above. The
+
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+Cheng and Liu, et al.
+Table 4: Offline experimental results of different models.
+Model
+PNR
+AUC (%)
+label 0
+label 1
+P (%)
+R (%)
+F1 (%)
+P (%)
+R (%)
+F1 (%)
+Online Baseline
+1.51
+60.10
+73.44
+63.99
+68.39
+40.09
+58.57
+47.60
+TreeLSTM
+2.91 ± 0.01
+74.93 ± 0.07
+79.69 ± 0.01
+81.86 ± 0.07
+80.76 ± 0.03
+57.64 ± 0.06
+54.19 ± 0.06
+55.86 ± 0.05
+GIN
+4.27 ± 0.05
+81.26 ± 0.20
+83.82 ± 0.39
+81.04 ± 1.17
+82.40 ± 0.44
+61.22 ± 0.96
+65.64 ± 1.45
+63.34 ± 0.26
+GAT
+4.43 ± 0.06
+81.94 ± 0.23
+84.87 ± 1.03
+79.93 ± 2.21
+82.30 ± 0.69
+61.00 ± 1.52
+68.65 ± 3.40
+64.53 ± 0.75
+Our Approach
+Virt-GIN
+4.62 ± 0.03
+82.47 ± 0.10
+85.23 ± 0.41
+78.95 ± 1.49
+81.96 ± 0.61
+60.26 ± 1.20
+69.95 ± 1.56
+64.72 ± 0.14
+Virt-GAT
+5.22 ± 0.10
+84.18 ± 0.24
+86.81 ± 0.57
+80.17 ± 1.13
+83.35 ± 0.35
+62.75 ± 0.79
+73.24 ± 1.71
+67.57 ± 0.29
+Non-Category (-NC)
+GIN-NC
+4,15 ± 0.07
+80.80 ± 0.30
+83.36 ± 1.42
+81.29 ± 3.57
+82.25 ± 1.19
+61.24 ± 2.62
+64.20 ± 5.15
+62.50 ± 1.17
+GAT-NC
+4.26 ± 0.04
+81.27 ± 0.15
+83.85 ± 0.23
+81.11 ± 0.75
+82.46 ± 0.31
+61.32 ± 0.67
+65.70 ± 0.87
+63.43 ± 0.25
+Virt-GIN-NC
+4.48 ± 0.04
+82.05 ± 0.14
+84.70 ± 0.53
+79.49 ± 1.27
+82.01 ± 0.44
+60.35 ± 0.86
+68.45 ± 1.77
+64.13 ± 0.33
+Virt-GAT-NC
+5.03 ± 0.03
+83.66 ± 0.08
+85.99 ± 0.48
+81.40 ± 1.23
+83.62 ± 0.43
+63.47 ± 1.04
+70.86 ± 1.60
+66.94 ± 0.25
+final result we reported is the mean test AUC, Precision, Recall,
+F1-Score and their corresponding standard deviation. All the above
+mentioned GNN models are implemented by Paddle Graph Learning
+(PGL)1, an efficient and flexible graph learning framework.
+6.5
+Offline Experimental Results
+We report the offline experimental results of the proposed model
+and all baseline models. Besides, we also include a baseline method,
+i.e., the model that is used in the system before deploying the layout-
+aware webpage quality assessment model.
+All results are shown in Table 4, from where we have the follow-
+ing key findings:
+• We can clearly see that our layout-aware webpage qual-
+ity model can beat the online baseline by large margins on
+all metrics e.g., Δ𝐴𝑈𝐶 = 24.08, Δ𝐹1 = 14.96 (label0) and
+Δ𝐹1 = 19.97 (label1). Especially for PNR, where the value is
+improved from 1.51 to 5.22. These tell us that the proposed
+model prefers high-quality results.
+• By applying the proposed readout function, the model can
+have a significant improvement on all metrics. Especially,
+the new readout mechanism is able to improve PNR by a
+margin of 0.38 and 0.96 based on GIN and GAT, respectively.
+Moreover, we also observe that the relative improvement
+of both virt-GIN and virt-GAT over GIN and GAT is consid-
+erable for high-quality webpage (label1), in terms of recall
+(Δ(𝑉𝑖𝑟𝑡_𝐺𝐴𝑇,𝐺𝐴𝑇) = 4.59%, Δ(𝑉𝑖𝑟𝑡_𝐺𝐼𝑁,𝐺𝐼𝑁 ) = 2.22%). All
+these phenomena show that our readout mechanism is capa-
+ble of improving the model’s performance.
+• Comparing the results of the two models whether apply
+the category-aware optimization strategy (w,r,t., GIN-NC
+vs. GIN, Virt-GIN-NC vs. Virt-GIN, GAT-NC vs. GAT, Virt-
+GAT-NC vs. Virt-GAT), we can come to the conclusion that
+all methods with the proposed category-aware optimization
+have better performance than their backbone models, in
+terms of PNR and AUC. Although a few models obtain lower
+1https://github.com/PaddlePaddle/PGL
+values on a few metrics (e.g., the F1-score of Virt-GAT-NC on
+label0 is 83.62 while Virt-GAT is 83.35, the precision of Virt-
+GAT-NC is 63.74% but Virt-GAT is 62.75%), the models with
+category-aware optimization show more robust performance
+considering all metrics.
+• The performance on different GNN models is better than
+TreeLSTM, model Virt_GAT is the most significant, Com-
+pare with Virt_GAT and TreeLSTM, Δ𝑃𝑁𝑅 = 2.31, Δ𝐴𝑈𝐶 =
+9.25%. For high-quality webpage (label1) Δ𝑅 = 14.67%. These
+large margins suggest that our model is more expressive than
+TreeLSTM, although TreeLSTM is specifically designed for
+tree-structured network topologies.
+Overall, our proposed model is able to gain superior performance
+on webpage assessment task through the improved readout mech-
+anism and category-aware optimization and can beat the online
+baseline by a significant margin.
+6.6
+Varying the number of GNN layer
+In general, a webpage is represented as a DOM tree. Its depth deter-
+mines how many layers of GNN are needed to obtain information
+from the root node to the leaf nodes. However, as the number of
+GNN layers increases, the computational efficiency will be lower.
+Therefore, we provide an experiment to verify the influence of the
+number of layers on the experimental results, as shown in Table
+5. As seen from the table, the more layers, the higher the AUC
+score can be reached. However, compared with the 5-layer virt-
+GAT model, the improvement of 7-layer virt-GAT model is not
+significant. As it is important to trade off the efficiency and effec-
+tiveness for large search system, we use 5-layer GNN models on
+online evaluation which can maintain the experimental effect while
+reducing the amount of calculation.
+7
+ONLINE EVALUATION
+To investigate the impact of our proposed quality assessment model
+to the search engine, we deploy the new model and conduct online
+experiments to compare it with the old retrieval system. Specifically,
+
+Layout-aware Webpage Quality Assessment
+SIGKDD ’23, August 06–10, 2023, Long Beach, CA
+Table 5: The influence of layer number on virt-GAT.
+#Layers
+AUC (%)
+label 0
+label 1
+P (%)
+R (%)
+F1 (%)
+P (%)
+R (%)
+F1 (%)
+1
+80.77 ± 0.23
+83.38 ± 0.92
+80.46 ± 2.49
+81.87 ± 0.85
+60.26 ± 1.82
+64.72 ± 3.42
+62.33 ± 0.66
+3
+83.80 ± 0.27
+86.25 ± 0.64
+79.80 ± 2.06
+82.89 ± 0.86
+61.98 ± 1.79
+72.05 ± 2.16
+66.59 ± 0.45
+5
+84.18 ± 0.24
+86.81 ± 0.57
+80.17 ± 1.13
+83.35 ± 0.35
+62.75 ± 0.79
+73.24 ± 1.71
+67.57 ± 0.29
+7
+84.25 ± 0.22
+86.91 ± 0.80
+80.53 ± 1.77
+83.58 ± 0.61
+63.23 ± 1.41
+73.32 ± 2.43
+67.86 ± 0.42
+we conduct a manual evaluation on the final ranking results with
+some real user-generated queries. This directly reflects the quality
+of the results exposed to the end users.
+We log a set of (million-scale) online queries and the correspond-
+ing final impressions, i.e., the top-ranked web documents in the
+final ranking stage, by individually using the layout-aware web-
+page quality assessment model and the old retrieval systems. Note
+that the data logging is conducted by multiple rounds to eliminate
+randomness. We filter out examples in which queries have identical
+impressions between the two systems, and then utilize the rest for
+the manual evaluation. Note that, considering the extremely high
+cost of the manual evaluation, we randomly generate thousands of
+data and eventually send it to experts for evaluation, so as to control
+costs while validating the effectiveness of the proposed model.
+7.1
+Online Experimental Metrics
+As mentioned in Section 5, our proposed quality assessment model
+works in Baidu retrieval system. The online experiments major
+focus on the end-to-end evaluation, the metrics are often used to
+measure the effectiveness of information retrieval system. Details
+are as follows:
+Discounted Cumulative Gain (DCG). We first log a dataset
+and manually label the data with 0 to 4 grades, and then report
+the relative improvement w.r.t. the average DCG over the top-4
+final results of all queries. The formula of DCG accumulated at a
+particular rank position p is defined as
+DCGp =
+𝑝
+∑︁
+𝑖=1
+2𝑟𝑒𝑙𝑖 − 1
+log2(𝑖 + 1) ,
+(10)
+where 𝑟𝑒𝑙𝑖 indicates the manually label of 𝑖-th webpage.
+Additionally, we also report the relative improvement of DCG
+for the low quality ranking result w.r.t., manually label is 0/1.
+Side-by-side Comparison. Besides, we also conduct a side-by-
+side comparison between the two systems. We log another dataset
+and require the human experts to judge whether the new system or
+the base system gives better results that satisfy intentions of users.
+Here, the relative gain is measured Good vs. Same vs. Bad (GSB) as
+Δ𝐺𝑆𝐵 =
+#Good − #Bad
+#Good + #Same + #Bad,
+(11)
+where #Good (or #Bad) indicates the number of queries that the
+new system provides better (or worse) final results.
+Table 6: Discounted cumulative gain on manual evaluation.
+Rand-Query
+Tail-Query
+Same-Quality
+Δ𝐷𝐶𝐺
++0.19%
++0.42%
+-
+DCG_0/1 ratio
+-0.63%
+-0.56%
+-
+Table 7: Side-by-side comparison on manual evaluation.
+Rand-Query
+Tail-Query
+Same-Quality
+Δ𝐺𝑆𝐵
++4.10%
++0.52%
++5.13%
+Node that we not only measure the final results but also measure
+the webpage quality when the relative result of two webpage is
+Same.
+7.2
+Online Experimental Results
+The relative improvement validated by manual evaluation is given
+in Table 6 and 7, where we can summarize observations as below:
+• By applying our quality assessment model, the system can
+significantly outperform the base system. Especially for DCG_0/1
+ratio, the relative improvement values are respectively −0.63%,
+−0.56% for rand query and tail query. This shows that our
+proposed method can better filtrate retrieval results with
+low DCG scores, which is very helpful in improving the user
+experience for real-world search engine.
+• The conventional case-by-case comparison also has signifi-
+cant improvement over the base system, especially for the
+rand query (Δ𝐺𝑆𝐵 = +4.1%). This tells us that user experi-
+ence can be improved by taking into account the web page
+quality in search system.
+• In addition, we can observe that with comparable relevance,
+the GSB value of the quality improvement is Δ𝐺𝑆𝐵 = +5.13%.
+This intuitively shows that our new system can provide
+higher quality search results based on the guaranteed rele-
+vance of search results.
+Moreover, we perform the statistical test to estimate whether
+the experimental results is statistically significant. The p-value of
+DCG rand and tail query are 0.0613 and 0.1276, respectively. The p-
+value approximates the significance level that is set in our retrieval
+
+SIGKDD ’23, August 06–10, 2023, Long Beach, CA
+Cheng and Liu, et al.
+(a) Offline quality assessment
+(b) Online position changes
+Figure 4: The overview of case study.
+system, which can demonstrate that our experimental results are
+statistically significant.
+Overall, the online experimental results show that our proposed
+layout-aware quality assessment model can effectively improve the
+performance of real-world ranking system.
+8
+CASE STUDY
+In this section, we present an illustration that includes the offline
+quality assessment score of webpage and online position changes
+of web pages. These typically cases are shown in Figure 4.
+8.1
+Offline Quality Assessment
+In Figure 4(a), we present three webpages with different layout
+styles and their quality assessment scores.
+The first webpage has a chaotic layout, elements in this web-
+page are unreasonable. It affects the user’s normal browsing and
+is very difficult for user to obtain information from this webpage.
+Our quality assessment model marks this webpage as low quality
+(𝑠𝑐𝑜𝑟𝑒 = 0.0068). This extremely low score will be considered by
+the ranking system to lower its ranking position.
+The second webpage also has low quality, different with the
+chaotic layout of the first webpage, it has a normal layout. How-
+ever, considering that it contains very small amount of information
+(almost no valuable information), it should be presented to the user
+with a very small probability. The ranking system can judge this
+by our quality assessment model score 0.1653.
+Unlike the previous two webpages, the third one is high-quality.
+It is carefully laid out and informative, and quality score is 0.9788,
+which will help the ranking system raise its ranking position.
+8.2
+Online Position Changes
+The case shown in Figure 4(b) comes from Section 7. Under the same
+query, these two webpages swapped positions in the new and old
+systems, The position of the left webpage in new system is 3-th but 4-
+th in the old system. Comparing the two webpages, we can observe
+that the left webpage (quality score is 0.5623) contains a rich amount
+of information but the right one (quality score is 0.2415) does not.
+This phenomenon demonstrates that online ranking system has
+adopted our model’s recommendations to provide users with higher
+quality webpage, which can greatly improve the user experience.
+9
+CONCLUSION AND FUTURE WORK
+In this paper, we propose a layout-aware webpage assessment model
+to suggest ranking system providing webpages with higher quality.
+We not only enhance GAT with the read mechanism but also care-
+fully design the features for improving the quality assessment on
+the webpages. In addition, taking into account the particularity of
+real-world data, we utilize the category of webpage for optimiza-
+tion. Both input data construction and model calculation are offline,
+which guarantees the efficiency of the ranking system. We devel-
+oped and deployed the layout-aware webpage assessment model in
+Baidu Search, which is highly effective in conducting high-quality
+ranking for web search. Extensive offline and online experiments
+have shown that the ranking system can significantly improve the
+effectiveness and general usability of the search engine.
+In future work, we will explore the heterogeneous GNN architec-
+ture to model the multiple graph-based information of webpages.
+It is interesting to improve the construction method of layout and
+enhance the representation of nodes/edges with self-supervised
+contrastive pre-training techniques.
+REFERENCES
+[1] Armen Avetisyan, Tatiana Khanova, Christopher Bongsoo Choy, Denver Dash,
+Angela Dai, and Matthias Nießner. 2020. SceneCAD: Predicting Object Align-
+ments and Layouts in RGB-D Scans. ArXiv abs/2003.12622 (2020).
+[2] Federico Baldassarre, David Ménendez Hurtado, Arne Elofsson, and Hossein
+Azizpour. 2021. GraphQA: protein model quality assessment using graph convo-
+lutional networks. Bioinformatics 37 (2021), 360 – 366.
+
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+haoyunlai2188的文章
+ApP内打开Layout-aware Webpage Quality Assessment
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+

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+Call for Papers - The BabyLM Challenge: Sample-efficient pretraining
+on a developmentally plausible corpus
+https://babylm.github.io/
+Alex Warstadt
+ETH Zürich
+Leshem Choshen
+IBM Research
+Aaron Mueller
+Johns Hopkins University
+Ethan Wilcox
+ETH Zürich
+Adina Williams
+Meta AI
+Chengxu Zhuang
+MIT
+Abstract
+We present the call for papers for the BabyLM
+Challenge: Sample-efficient pretraining on a
+developmentally plausible corpus. This shared
+task is intended for participants with an in-
+terest in small scale language modeling, hu-
+man language acquisition, low-resource NLP,
+and cognitive modeling. In partnership with
+CoNLL and CMCL, we provide a platform for
+approaches to pretraining with a limited-size
+corpus sourced from data inspired by the input
+to children. The task has three tracks, two of
+which restrict the training data to pre-released
+datasets of 10M and 100M words and are ded-
+icated to explorations of approaches such as
+architectural variations, self-supervised objec-
+tives, or curriculum learning. The final track
+only restricts the amount of text used, allowing
+innovation in the choice of the data, its domain,
+and even its modality (i.e., data from sources
+other than text is welcome). We will release a
+shared evaluation pipeline which scores mod-
+els on a variety of benchmarks and tasks, in-
+cluding targeted syntactic evaluations and nat-
+ural language understanding.
+1
+Motivation
+Huge efforts have been put into optimizing LM pre-
+training at massive scales in the last several years
+(Raffel et al., 2020; Brown et al., 2020; Chowdhery
+et al., 2022; Hoffmann et al., 2022). While grow-
+ing parameter counts often get the most attention,
+datasets have also grown by orders of magnitude.
+These increasingly larger pretraining datasets are
+visualized, to scale, in Figure 1. At the same time,
+there has been almost no progress in pretraining at
+smaller human-like data scales.
+Focusing on scaled-down pretraining has several
+potential benefits: First, small-scale pretraining can
+be a sandbox for developing novel techniques that
+improve data efficiency. These techniques have the
+potential to then scale up to larger datasets com-
+monly seen in applied NLP, and could be used
+Figure 1: Data Scale: Modern Language Models are
+trained on data multiple orders of magnitude larger than
+the amount available to a typical human child. Image
+based off Fig. 1 from Warstadt and Bowman (2022)
+to enhance current approaches to modeling low-
+resource languages. Second, improving our ability
+to train LMs on the same types and quantities of
+data that humans learn from will give us greater
+access to more plausible cognitive models of hu-
+mans and help us understand what allows humans
+to acquire language so efficiently (Keller, 2010;
+Dupoux, 2018). That is, even model failure can
+help in developing hypotheses about the differences
+between human and LM language learning.
+The goal of this shared task will be to incentivize
+researchers with an interest in pretraining and/or
+cognitive modeling to focus their efforts on opti-
+mizing pretraining given data limitations inspired
+by human development. Additionally, we hope to
+democratize research on pretraining—which is typ-
+ically thought to be practical only for large industry
+groups—by drawing attention to open problems
+that can be addressed on a university budget.
+2
+Key Dates
+• January 2023: Training data released
+• March 2023: Evaluation pipeline released
+• July 15, 2023: Results due
+• August 1, 2023: Paper submissions due
+• Date TBA: Presentation at CoNLL
+arXiv:2301.11796v1  [cs.CL]  27 Jan 2023
+
+200
+1.4
+Billion
+Trillion
+30
+3
+<100
+Billion
+Billion
+Million
+13 y.0.
+BERT
+RoBERTa
+GPT-3
+Chinchilla
+Human
+(2018)
+(2019)
+(2020)
+(2022)# Words
+Dataset
+Domain
+STRICT-SMALL
+STRICT
+Proportion
+CHILDES (MacWhinney, 2000)
+Child-directed speech
+0.44M
+4.21M
+5%
+British National Corpus (BNC),1 dialogue portion
+Dialogue
+0.86M
+8.16M
+8%
+Children’s Book Test (Hill et al., 2016)
+Children’s books
+0.57M
+5.55M
+6%
+Children’s Stories Text Corpus2
+Children’s books
+0.34M
+3.22M
+3%
+Standardized Project Gutenberg Corpus (Gerlach and Font-Clos, 2018)
+Written English
+0.99M
+9.46M
+10%
+OpenSubtitles (Lison and Tiedemann, 2016)
+Movie subtitles
+3.09M
+31.28M
+31%
+QCRI Educational Domain Corpus (QED; Abdelali et al., 2014)
+Educational video subtitles
+1.04M
+10.24M
+11%
+Wikipedia3
+Wikipedia (English)
+0.99M
+10.08M
+10%
+Simple Wikipedia4
+Wikipedia (Simple English)
+1.52M
+14.66M
+15%
+Switchboard Dialog Act Corpus (Stolcke et al., 2000)
+Dialogue
+0.12M
+1.18M
+1%
+Total
+–
+9.96M
+98.04M
+100%
+Table 1: The datasets we release for the STRICT and STRICT-SMALL tracks of the BabyLM Challenge. We present
+the number of words in the training set of each corpus that we include. 1http://www.natcorp.ox.ac.uk
+2https:
+//www.kaggle.com/datasets/edenbd/children-stories-text-corpus
+3https://dumps.wikimedia.org/
+enwiki/20221220/
+4https://dumps.wikimedia.org/simplewiki/20221201/
+3
+Tracks
+This shared task includes three tracks: STRICT,
+STRICT-SMALL, and LOOSE.
+The STRICT and STRICT-SMALL tracks require
+that submissions are trained exclusively on a fixed
+dataset, which we provide. The main difference be-
+tween these tracks is the size of the dataset (∼10M
+words vs. ∼100M words). Both datasets con-
+tain child-directed speech, transcribed speech from
+multiple sources, children’s books, and Wikipedia,
+among other datasets. The STRICT-SMALL dataset
+is an approximately 10% uniform subsample of the
+STRICT dataset. See §4 for a full description of the
+fixed datasets. Winners will be determined based
+on performance on the shared evaluation set.
+The LOOSE track relaxes these restrictions. Sub-
+missions must still be trained on a maximum of
+100M words, and will be tested on the shared eval-
+uation set. However, they are permitted to use
+unlimited non-linguistic data or text which differs
+from the restricted shared task. Training on addi-
+tional text is allowed without limits if that text is
+generated by a model trained following the above
+restrictions. For this track, winners will be selected
+holistically based on evaluation performance, rel-
+evance to the shared task goals, potential impact,
+and novelty.
+4
+Dataset
+We distribute a developmentally plausible pretrain-
+ing dataset inspired by the input to children.1 Sub-
+missions must use only this training data to be con-
+1Clicking on the following link will download the dataset
+(240MB zipped, 700MB unzipped): https://github.com/
+babylm/babylm.github.io/raw/main/babylm_data.zip
+sidered for the STRICT(-SMALL) tracks, but may
+use different data for the LOOSE track. The dataset
+has two key properties:
+• Under 100M words: Children are exposed
+to 2M-7M words per year (Gilkerson et al.,
+2017). Choosing the beginning of adolescence
+(age 12) as a cutoff, the dataset should be
+between 24M-84M words.
+• Mostly transcribed speech: Most of the in-
+put to children is spoken. Thus, we include a
+higher proportion of transcribed speech in our
+dataset.
+The datasets we release are mixed domain, taken
+from multiple sources. Table 1 summarizes the
+composition of the datasets.
+5
+Evaluation
+We will distribute a shared evaluation pipeline
+based in Google Colab. Colab provides access
+to relatively small GPUs; this will allow users from
+various research settings of varying resources to ef-
+ficiently evaluate their submissions. Our evaluation
+code will also be public, such that those wishing to
+use their own computational resources may do so.
+More details about the evaluation pipeline and the
+set of tasks will be released subsequently.
+The pipeline assumes all models can be loaded
+and queried in HuggingFace’s transformers li-
+brary (Wolf et al., 2020).2 Additionally, all mod-
+els must be able to score a sequence—e.g., assign
+2While discouraged, participants whose models are not
+compatible with the transformers library can still conduct
+the necessary evaluation through their own pipeline.
+
+a log-likelihood or pseudo log-likelihood (Wang
+and Cho, 2019; Salazar et al., 2020)—and must
+be able to be fine-tuned to perform classification
+tasks. Models do not need to be able to generate se-
+quences. Submissions must include model outputs
+for each of the core evaluations in a format that we
+specify in our evaluation pipeline.
+We choose evaluations that represent the core
+interests of this shared task, focusing on efficiency
+and applied NLP, as well as cognitive science, lin-
+guistics and language acquisition. Especially good
+performance in one but not both of these areas may
+be acknowledged with a special award.
+5.1
+Baselines
+We will also release a series of baseline models
+with the evaluation pipeline. To train these, we sim-
+ply take the hyperparameters from a series of estab-
+lished large language models and train them from
+scratch on our fixed datasets. We use hyperparame-
+ters from OPT (decoder-only; Zhang et al., 2022),
+RoBERTa (encoder-only; Liu et al., 2019), and T5
+(encoder-decoder; Raffel et al., 2020). These are
+not meant to be strong baselines, but rather to pro-
+vide a naïve starting point for improving language
+models for this domain.
+6
+Submissions
+What you Need to Submit
+• A link where we can download the model
+• A .zip of predictions (from our eval pipeline)
+• A short description of the approaches taken
+• If LOOSE track: a link where we can down-
+load any additional data
+Although scaled-down pretraining is more ac-
+cessible to research groups with limited resources,
+pretraining is still expensive from a computational,
+energy, and financial perspective. To help groups
+plan for total costs, we will release an estimate of
+the resources required to pretrain on 10M words
+and 100M words. For the LOOSE track, evaluation
+of submissions may take into consideration compu-
+tational efficiency as part of the holistic evaluation.
+7
+FAQs
+Can papers be submitted to multiple tracks?
+Yes. For example, a single paper can describe mod-
+els which are submitted separately to the STRICT
+and STRICT-SMALL tracks.
+Can I submit a paper about my work?
+Yes,
+we encourage all participants to submit their re-
+ports, which will be published in the proceedings
+of CoNLL. You may also describe any additional
+experiments beyond those required for the shared
+task evaluation.
+Can I submit additional evaluation metrics?
+Yes, if you wish to submit your own evaluation
+metrics, along with model performance, alongside
+our standardized evaluation results these can be
+considered as part of the holistic evaluation in the
+LOOSE track.
+What training regimes are permitted?
+For the
+STRICT/STRICT-SMALL tracks, any kind of train-
+ing objective/regime is permitted, as long as the
+data restrictions are followed. Pretrained models
+may not be used for any purpose such as reranking
+or data augmentation.
+We do however require for evaluation purposes
+that the model provides a function to score a
+sequence—e.g., log-likelihood for autoregressive
+models or pseudo-log-likelihood for masked lan-
+guage models—without the need for additional
+fine-tuning.
+Are there any limits on hyperparameters?
+No.
+In the LOOSE track, parameter efficiency and train-
+ing efficiency may be considered along with other
+factors in ranking submissions.
+Are there any limits on the number of epochs?
+No. We put no restrictions on the number of epochs,
+for several reasons: First, from an engineering per-
+spective, training LMs with SGD tends to require
+multiple epochs at these scales to achieve peak per-
+formance. Second, from a cognitive perspective,
+humans have a memory of linguistic experience,
+and can continue to access and learn from these
+memories. Third, we try not to make a stand on
+implementations to allow the most freedom for in-
+novation.
+8
+Organizing Committee
+Leshem Choshen
+Aaron Mueller
+Ryan Cotterell
+Alex Warstadt
+Kundan Krishna
+Ethan Wilcox
+Tal Linzen
+Adina Williams
+Haokun Liu
+Chengxu Zhuang
+
+Questions? Feel free to contact us at the following
+email addresses:
+leshem.choshen@mail.huji.ac.il
+haokunl@cs.unc.edu
+amueller@jhu.edu
+alexwarstadt@gmail.com
+ewilcox@ethz.ch
+chengxuz@mit.edu
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+

CNFKT4oBgHgl3EQfXy51/content/tmp_files/load_file.txt

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+page_content='io/  Alex Warstadt  ETH Zürich  Leshem Choshen  IBM Research  Aaron Mueller  Johns Hopkins University  Ethan Wilcox  ETH Zürich  Adina Williams  Meta AI  Chengxu Zhuang  MIT  Abstract  We present the call for papers for the BabyLM  Challenge: Sample-efficient pretraining on a  developmentally plausible corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' This shared  task is intended for participants with an in-  terest in small scale language modeling, hu-  man language acquisition, low-resource NLP,  and cognitive modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' In partnership with  CoNLL and CMCL, we provide a platform for  approaches to pretraining with a limited-size  corpus sourced from data inspired by the input  to children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' The task has three tracks, two of  which restrict the training data to pre-released  datasets of 10M and 100M words and are ded-  icated to explorations of approaches such as  architectural variations, self-supervised objec-  tives, or curriculum learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
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+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=', data from sources  other than text is welcome).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' We will release a  shared evaluation pipeline which scores mod-  els on a variety of benchmarks and tasks, in-  cluding targeted syntactic evaluations and nat-  ural language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  1  Motivation  Huge efforts have been put into optimizing LM pre-  training at massive scales in the last several years  (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Chowdhery  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Hoffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' While grow-  ing parameter counts often get the most attention,  datasets have also grown by orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  These increasingly larger pretraining datasets are  visualized, to scale, in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' At the same time,  there has been almost no progress in pretraining at  smaller human-like data scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  Focusing on scaled-down pretraining has several  potential benefits: First, small-scale pretraining can  be a sandbox for developing novel techniques that  improve data efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' These techniques have the  potential to then scale up to larger datasets com-  monly seen in applied NLP, and could be used  Figure 1: Data Scale: Modern Language Models are  trained on data multiple orders of magnitude larger than  the amount available to a typical human child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Image  based off Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' 1 from Warstadt and Bowman (2022)  to enhance current approaches to modeling low-  resource languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Second, improving our ability  to train LMs on the same types and quantities of  data that humans learn from will give us greater  access to more plausible cognitive models of hu-  mans and help us understand what allows humans  to acquire language so efficiently (Keller, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  Dupoux, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' That is, even model failure can  help in developing hypotheses about the differences  between human and LM language learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  The goal of this shared task will be to incentivize  researchers with an interest in pretraining and/or  cognitive modeling to focus their efforts on opti-  mizing pretraining given data limitations inspired  by human development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Additionally, we hope to  democratize research on pretraining—which is typ-  ically thought to be practical only for large industry  groups—by drawing attention to open problems  that can be addressed on a university budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  2  Key Dates  January 2023: Training data released  March 2023: Evaluation pipeline released  July 15, 2023: Results due  August 1, 2023: Paper submissions due  Date TBA: Presentation at CoNLL  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='11796v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='CL]  27 Jan 2023  200  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='4  Billion  Trillion  30  3  <100  Billion  Billion  Million  13 y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  BERT  RoBERTa  GPT-3  Chinchilla  Human  (2018)  (2019)  (2020)  (2022)# Words  Dataset  Domain  STRICT-SMALL  STRICT  Proportion  CHILDES (MacWhinney, 2000)  Child-directed speech  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='44M  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='21M  5%  British National Corpus (BNC),1 dialogue portion  Dialogue  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='86M  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='16M  8%  Children’s Book Test (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=', 2016)  Children’s books  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='57M  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='55M  6%  Children’s Stories Text Corpus2  Children’s books  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='34M  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='22M  3%  Standardized Project Gutenberg Corpus (Gerlach and Font-Clos, 2018)  Written English  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='99M  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='46M  10%  OpenSubtitles (Lison and Tiedemann, 2016)  Movie subtitles  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='09M  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='28M  31%  QCRI Educational Domain Corpus (QED;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Abdelali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=', 2014)  Educational video subtitles  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='04M  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='24M  11%  Wikipedia3  Wikipedia (English)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='99M  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='08M  10%  Simple Wikipedia4  Wikipedia (Simple English)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='52M  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='66M  15%  Switchboard Dialog Act Corpus (Stolcke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=', 2000)  Dialogue  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='12M  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='18M  1%  Total  –  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='96M  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='04M  100%  Table 1: The datasets we release for the STRICT and STRICT-SMALL tracks of the BabyLM Challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' We present  the number of words in the training set of each corpus that we include.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' 1http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='natcorp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
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+page_content='uk  2https:  //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='com/datasets/edenbd/children-stories-text-corpus  3https://dumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='wikimedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='org/  enwiki/20221220/  4https://dumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='wikimedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='org/simplewiki/20221201/  3  Tracks  This shared task includes three tracks: STRICT,  STRICT-SMALL, and LOOSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  The STRICT and STRICT-SMALL tracks require  that submissions are trained exclusively on a fixed  dataset, which we provide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' The main difference be-  tween these tracks is the size of the dataset (∼10M  words vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' ∼100M words).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Both datasets con-  tain child-directed speech, transcribed speech from  multiple sources, children’s books, and Wikipedia,  among other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' The STRICT-SMALL dataset  is an approximately 10% uniform subsample of the  STRICT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' See §4 for a full description of the  fixed datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Winners will be determined based  on performance on the shared evaluation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  The LOOSE track relaxes these restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Sub-  missions must still be trained on a maximum of  100M words, and will be tested on the shared eval-  uation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' However, they are permitted to use  unlimited non-linguistic data or text which differs  from the restricted shared task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Training on addi-  tional text is allowed without limits if that text is  generated by a model trained following the above  restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' For this track, winners will be selected  holistically based on evaluation performance, rel-  evance to the shared task goals, potential impact,  and novelty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  4  Dataset  We distribute a developmentally plausible pretrain-  ing dataset inspired by the input to children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='1 Sub-  missions must use only this training data to be con-  1Clicking on the following link will download the dataset  (240MB zipped, 700MB unzipped): https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='com/  babylm/babylm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='io/raw/main/babylm_data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='zip  sidered for the STRICT(-SMALL) tracks, but may  use different data for the LOOSE track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' The dataset  has two key properties:  Under 100M words: Children are exposed  to 2M-7M words per year (Gilkerson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=',  2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Choosing the beginning of adolescence  (age 12) as a cutoff, the dataset should be  between 24M-84M words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  Mostly transcribed speech: Most of the in-  put to children is spoken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Thus, we include a  higher proportion of transcribed speech in our  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  The datasets we release are mixed domain, taken  from multiple sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Table 1 summarizes the  composition of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  5  Evaluation  We will distribute a shared evaluation pipeline  based in Google Colab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Colab provides access  to relatively small GPUs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' this will allow users from  various research settings of varying resources to ef-  ficiently evaluate their submissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Our evaluation  code will also be public, such that those wishing to  use their own computational resources may do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  More details about the evaluation pipeline and the  set of tasks will be released subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  The pipeline assumes all models can be loaded  and queried in HuggingFace’s transformers li-  brary (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='2 Additionally, all mod-  els must be able to score a sequence—e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=', assign  2While discouraged, participants whose models are not  compatible with the transformers library can still conduct  the necessary evaluation through their own pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  a log-likelihood or pseudo log-likelihood (Wang  and Cho, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Salazar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=', 2020)—and must  be able to be fine-tuned to perform classification  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Models do not need to be able to generate se-  quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Submissions must include model outputs  for each of the core evaluations in a format that we  specify in our evaluation pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  We choose evaluations that represent the core  interests of this shared task, focusing on efficiency  and applied NLP, as well as cognitive science, lin-  guistics and language acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Especially good  performance in one but not both of these areas may  be acknowledged with a special award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='1  Baselines  We will also release a series of baseline models  with the evaluation pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' To train these, we sim-  ply take the hyperparameters from a series of estab-  lished large language models and train them from  scratch on our fixed datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' We use hyperparame-  ters from OPT (decoder-only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=', 2022),  RoBERTa (encoder-only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=', 2019), and T5  (encoder-decoder;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' These are  not meant to be strong baselines, but rather to pro-  vide a naïve starting point for improving language  models for this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  6  Submissions  What you Need to Submit  A link where we can download the model  A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='zip of predictions (from our eval pipeline)  A short description of the approaches taken  If LOOSE track: a link where we can down-  load any additional data  Although scaled-down pretraining is more ac-  cessible to research groups with limited resources,  pretraining is still expensive from a computational,  energy, and financial perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' To help groups  plan for total costs, we will release an estimate of  the resources required to pretrain on 10M words  and 100M words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' For the LOOSE track, evaluation  of submissions may take into consideration compu-  tational efficiency as part of the holistic evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  7  FAQs  Can papers be submitted to multiple tracks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  Yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' For example, a single paper can describe mod-  els which are submitted separately to the STRICT  and STRICT-SMALL tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  Can I submit a paper about my work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  Yes,  we encourage all participants to submit their re-  ports, which will be published in the proceedings  of CoNLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' You may also describe any additional  experiments beyond those required for the shared  task evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  Can I submit additional evaluation metrics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  Yes, if you wish to submit your own evaluation  metrics, along with model performance, alongside  our standardized evaluation results these can be  considered as part of the holistic evaluation in the  LOOSE track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  What training regimes are permitted?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  For the  STRICT/STRICT-SMALL tracks, any kind of train-  ing objective/regime is permitted, as long as the  data restrictions are followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Pretrained models  may not be used for any purpose such as reranking  or data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  We do however require for evaluation purposes  that the model provides a function to score a  sequence—e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=', log-likelihood for autoregressive  models or pseudo-log-likelihood for masked lan-  guage models—without the need for additional  fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  Are there any limits on hyperparameters?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  In the LOOSE track, parameter efficiency and train-  ing efficiency may be considered along with other  factors in ranking submissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  Are there any limits on the number of epochs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' We put no restrictions on the number of epochs,  for several reasons: First, from an engineering per-  spective, training LMs with SGD tends to require  multiple epochs at these scales to achieve peak per-  formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Second, from a cognitive perspective,  humans have a memory of linguistic experience,  and can continue to access and learn from these  memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Third, we try not to make a stand on  implementations to allow the most freedom for in-  novation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  8  Organizing Committee  Leshem Choshen  Aaron Mueller  Ryan Cotterell  Alex Warstadt  Kundan Krishna  Ethan Wilcox  Tal Linzen  Adina Williams  Haokun Liu  Chengxu Zhuang  Questions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Feel free to contact us at the following  email addresses:  leshem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='choshen@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='huji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='il  haokunl@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='unc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='edu  amueller@jhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='edu  alexwarstadt@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='com  ewilcox@ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='ch  chengxuz@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='edu  References  Ahmed Abdelali, Francisco Guzman, Hassan Sajjad,  and Stephan Vogel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  The AMARA corpus:  Building parallel language resources for the educa-  tional domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  In Proceedings of the Ninth Inter-  national Conference on Language Resources and  Evaluation (LREC’14), Reykjavik, Iceland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Euro-  pean Language Resources Association (ELRA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  Tom Brown, Benjamin Mann, Nick Ryder, Melanie  Subbiah,  Jared  D  Kaplan,  Prafulla  Dhariwal,  Arvind Neelakantan, Pranav Shyam, Girish Sastry,  Amanda Askell, Sandhini Agarwal, Ariel Herbert-  Voss, Gretchen Krueger, Tom Henighan, Rewon  Child, Aditya Ramesh, Daniel Ziegler, Jeffrey Wu,  Clemens Winter, Chris Hesse, Mark Chen, Eric  Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess,  Jack Clark, Christopher Berner, Sam McCandlish,  Alec Radford, Ilya Sutskever, and Dario Amodei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Language models are few-shot learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' In  Advances in Neural Information Processing Systems,  volume 33, pages 1877–1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Curran Associates,  Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  Aakanksha Chowdhery,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Sharan Narang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Jacob Devlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  Maarten Bosma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Gaurav Mishra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Adam Roberts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  Paul Barham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Hyung Won Chung,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Charles Sutton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  Sebastian Gehrmann,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Parker Schuh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Kensen Shi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  Sasha Tsvyashchenko,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Joshua Maynez,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Abhishek  Rao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Parker Barnes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Yi Tay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Noam Shazeer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Vin-  odkumar Prabhakaran,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Emily Reif,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Nan Du,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Ben  Hutchinson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Reiner Pope,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' James Bradbury,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Jacob  Austin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Michael Isard,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Guy Gur-Ari,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Pengcheng  Yin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Toju Duke,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Anselm Levskaya,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Sanjay Ghe-  mawat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Sunipa Dev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Henryk Michalewski,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Xavier  Garcia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Vedant Misra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Kevin Robinson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Liam Fe-  dus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Denny Zhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Daphne Ippolito,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' David Luan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  Hyeontaek Lim,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Barret Zoph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Alexander Spiridonov,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  Ryan Sepassi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' David Dohan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Shivani Agrawal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Mark  Omernick,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Andrew M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' Dai, Thanumalayan Sankara-  narayana Pillai, Marie Pellat, Aitor Lewkowycz,  Erica Moreira, Rewon Child, Oleksandr Polozov,  Katherine Lee, Zongwei Zhou, Xuezhi Wang, Bren-  nan Saeta, Mark Diaz, Orhan Firat, Michele Catasta,  Jason Wei, Kathy Meier-Hellstern, Douglas Eck,  Jeff Dean, Slav Petrov, and Noah Fiedel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  PaLM: Scaling language modeling with pathways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  Computing Research Repository, arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='02311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content='  Emmanuel Dupoux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNFKT4oBgHgl3EQfXy51/content/2301.11796v1.pdf'}
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+Myths and Legends in High-Performance
+Computing
+arXiv preprints
+©The Author(s) 2023
+Reprints and permission:
+sagepub.co.uk/journalsPermissions.nav
+DOI: 10.1177/ToBeAssigned
+www.sagepub.com/
+SAGE
+Satoshi Matsuoka1, Jens Domke1, Mohamed Wahib1, Aleksandr Drozd1, Torsten Hoefler2
+Abstract
+In this humorous and thought provoking article, we discuss certain myths and legends that are folklore among members
+of the high-performance computing community. We collected those myths from conversations at conferences and
+meetings, product advertisements, papers, and other communications such as tweets, blogs, and news articles within
+(and beyond) our community. We believe they represent the zeitgeist of the current era of massive change, driven by the
+end of many scaling laws such as Dennard scaling and Moore’s law. While some laws end, new directions open up, such
+as algorithmic scaling or novel architecture research. However, these myths are rarely based on scientific facts but often
+on some evidence or argumentation. In fact, we believe that this is the very reason for the existence of many myths
+and why they cannot be answered clearly. While it feels like there should be clear answers for each, some may remain
+endless philosophical debates such as the question whether Beethoven was better than Mozart. We would like to see
+our collection of myths as a discussion of possible new directions for research and industry investment.
+Keywords
+Quantum; zettascale; deep learning; clouds; HPC myths
+This manuscript is intended for the “CCDSC Special Issue”.
+Introduction
+Any human society has their myths and legends—this also
+applies to the high-performance computing (HPC) community.
+HPC drives the largest and most powerful computers and
+latest computing and acceleration technologies forward. One
+may think that it’s scientific reasoning all the way down in
+such an advanced field. Yet, we find many persistent myths
+revolving around trends of the moment.
+Since it’s late 2022, we started our analysis by asking the
+all-knowing intelligence ChatGPT “Create myths or legends
+in high performance computing”. In a HAL 9000 manner, it
+refused to make up something for us: “I’m sorry [Dave], but
+as an AI language model, I am not programmed to generate or
+share myths or legends. My primary function is to assist users
+with information and general knowledge, and I do not have
+the ability to create or share fictional content.”. So, even the
+smartest of internet parrots (Bender et al. 2021) that was itself
+created with massive high-performace computation running
+on a large accelerator system still has a long way to go. Thus,
+we fall back to reasoning among the authors of this work.
+We discuss 12 of today’s HPC myths, a number customary
+in our community, similar to a panel statement where we
+debate supporting and contradicting facts with a healthy
+exaggeration in one of those directions. We attempt to neither
+judge nor prove folklore right or wrong but instead try to
+stipulate an intensive discussion in the community that drives
+our future thinking.
+Myth 1: Quantum Computing Will Take Over
+HPC!
+Numerous articles are hyping the quantum computing
+revolution affecting nearly all aspects of life ranging from
+quantum artificial intelligence to even quantum gaming.
+The whole IT industry is following the quantum trend
+and conceives quickly growing expectations. The actual
+development of quantum technologies, algorithms, and use-
+cases is on a very different time-scale. Most practitioners
+would not expect quantum computers to outperform classical
+computers within the next decade. Yet, we have constantly
+been surprised by advances in device scaling as well as, more
+recently, artificial intelligence. Thus, the fear of missing out
+on getting rich is driving the industry to heavily invest into
+quantum technologies pushing the technology forward.
+With all this investment, it seems reasonable to expect that
+quantum computation, which promises to deliver exponential
+speedups, will replace high-performance computation as
+we know it today with its meager linear speedup through
+parallelism. Yet, the nature of quantum computation poses
+some severe limitations: First, reading unstructured data into
+a quantum state seems very challenging. Reasonable future
+quantum computer designs can read in the order of Gigabit/s
+while modern single-chip processors are already achieving
+1RIKEN Center for Computational Science, Japan
+2Eidgen¨ossische Technische Hochschule Z¨urich, Switzerland
+Corresponding author:
+Torsten
+Hoefler,
+ETH
+Z¨urich,
+Inst.
+f.
+Hochleistungsrechnersyst.,
+Universit¨atstrasse 6, 8092 Z¨urich, Switzerland
+Prepared using sagej.cls [Version: 2017/01/17 v1.20]
+arXiv:2301.02432v1  [cs.DC]  6 Jan 2023
+
+2
+arXiv preprints
+Terabit/s—many orders of magnitude more (Hoefler et al.
+2023).
+Furthermore, once a quantum state is constructed, it can
+often be “used” only once because measurements destroy
+superposition. A second limitation stems from the lack of
+algorithms with high speedups. Most algorithms achieve
+quadratic speedups for a wide range of use-cases using
+amplitude amplification at their core. While this technique is
+extremely versatile and can search any unstructured quantum
+state (cf. Grover’s algorithm), its limited speedup is unlikely
+to make it practical for quantum computers that may be
+constructed in the next decades (Hoefler et al. 2023).
+Thus, it seems unlikely that quantum computation is going
+to replace a significant fraction of traditional HPC. It is more
+likely that it will start as quantum acceleration with a small set
+of use-cases that may grow in the future. To determine which
+use-cases can realistically benefit from quantum acceleration,
+resource estimation techniques (Beverland et al. 2022)
+become crucial. But unlikely does not mean impossible—
+we believe that now is the right time to begin a discussion
+about the role of quantum computation in HPC. Furthermore,
+it is crucial to guide the resources we invest into the right
+directions.
+We close with these questions. . .
+x When will quantum computing be commercially
+profitable? y What will be the first useful algorithm?
+z What will be the next break-through area enabled by a
+new quantum algorithm?
+Myth 2: Everything Will Be Deep Learning!
+Simultaneously with the quantum hype, we are in the midst
+of the deep learning revolution. Indeed, in recent years
+there has been a plethora of papers replacing traditional
+simulation methods, or whole computational kernels with
+data-driven models. Most of those employ deep neural
+network architectures. Impressive results fire up expectations
+equally high to the quantum world. Data-driven weather and
+climate predictions apparently beat the best models (Pathak
+et al. 2022; Bi et al. 2022) and output data can be compressed
+by three orders of magnitude (Huang and Hoefler 2022).
+Similar successes are touted in literally any application area.
+There is no doubt that deep learning models can learn to
+approximate complex functions used in scientific simulations
+in a specific input domain. The issue is, as always, the trade-
+offs: between speed on one hand, and accuracy on the other—
+and we have to be very careful with these comparisons. In fact,
+any result can be skewed into any of the extremes (Hoefler
+2022).
+Sometimes even very simple models (and they have to be
+simple to be compute-performance competitive) such as multi-
+layer perceptrons (MLPs) can work well enough in place
+of exact mathematical expression, e.g., Rasp et al. (2018);
+Brenowitz and Bretherton (2018). One wonders sometimes
+whether the latter could have been simplified in the first
+place. A possible explanation is that neural nets, rather than
+learning to approximate a given function in some abstract
+sense, learn to decompose the input space into polyhedra
+with corresponding simple mappings (Aytekin 2022). In other
+words, neural nets can exploit the fact that typical input values
+in many tasks are concentrated in particular ranges, which,
+in turn, raises concerns about accuracy guarantees for out-of-
+distribution inputs, and a possibility of some sort of hybrid /
+fall-back mechanism.
+An independent question is whether the architectures used
+for machine learning tasks, like classification, are a good
+match to serve as surrogate models in the first place? A new
+line of research is addressing this by using neural architecture
+search for such models (Kasim et al. 2021). In an extreme case,
+the objective is to find a purely symbolic (and thus hopefully
+more robust to out-of-distribution inputs) formulation for
+cases where an exact mathematical expression for the problem
+is not a-priori known (Liu and Tegmark 2021). Uncertainty
+quantification and explainability are also two main aspects of
+high importance in the scientific domain where DL is lacking
+(due to its black-box optimization nature).
+Overall the jury is still out as to which extent surrogate
+models can replace first-principles simulations. However,
+one thing is clear: there is a whole spectrum of simulation
+tasks (Lavin et al. 2021)—ranging from ones where exact
+mathematical expressions are not available in the first place
+(e.g., contribution of specific vegetation to weather dynamics)
+and learning it from data could not only be more efficient
+but also more accurate; to those where utmost accuracy and
+precision guarantees are required and can only be provided
+by specialized error-controlling numerical methods.
+We close with these questions. . .
+x Will ML models replace or just augment traditional
+simulations? y Where will ML models fail to deliver?
+z How can we classify (pieces of) an application as ML-
+acceleratable or not?
+Myth 3: Extreme Specialization as Seen in
+Smartphones Will Push Supercomputers
+Beyond Moore’s Law!
+AI, like Stable Diffusion, is now in the palm of everyone’s
+hand. These modern smartphones typically are driven by a
+System on Chip (SoC) that consists of a plethora of special
+function units (SFUs) and/or special purpose processors that
+accelerate various aspects of smartphone workloads. The main
+purpose of such a composition is to achieve low power for
+longer battery life while maintaining acceptable performance.
+The success of GPUs, growing demands for lower power and
+highest performance, and the end of Moore’s law created
+a myth that future supercomputer architectures will be just
+like smartphones in that there will be multitudes of hardware
+customization per each facet of the entire workload.
+However, such a claim misses the point in the analogy, and
+entirely ignores multiple drawbacks of such an approach as
+described below. In fact, the only successful “accelerator”
+in the recent history of HPC is a GPU. The primary
+reason for its success is high memory bandwidth, a feature
+known since the vector supercomputer days, which is now
+adopted by mainstream CPUs such as Fujitsu A64FX and
+Intel Sapphire Rapids. The reason for the acceleration
+is primarily that the majority of the HPC workloads are
+memory bandwidth bound (Domke et al. 2021). Thus, modern
+Prepared using sagej.cls
+
+Matsuoka, Domke, Wahib, Drozd, Hoefler
+3
+reincarnations of vector processors, such as vector units and
+fast memory with HBM/GDDR variants, have been sufficient
+to accelerate such workloads beyond CPUs with slower
+DDR memory (Matsuoka 2008). So, to claim that multitudes
+of special accelerators will constitute a supercomputers is
+stretching the success of GPUs somewhat unfoundedly.
+In fact, there are mainly three reasons why the plethora
+of customized accelerated hardware approach would fail.
+The first is the most important, in that acceleration via SoC
+integration of various SFU is largely to enable strong scaling
+at a compute node level, and will be subject to the limitations
+of the Amdahl’s law, i.e., reducing the time to solution, the
+potential speedup is bound by the ratio of accelerated and
+non-accelerable fractions of the algorithm, which quickly
+limits the speedup. Modern supercomputing is rather driven
+by weak scaling as explained by Gustafson (1988), where the
+speedup is based on how well the parallelizable or accelerable
+fraction can be scaled on many nodes. This is often achieved
+by linearly increasing the overall workload and maintaining
+a constant amount of work per node, so the time to solution
+remains constant but performance gain is proportional to
+the number of nodes in an ideal case. This was exactly how
+massive performance gain was obtained, despite skepticisms
+from the then experts, towards massively parallel computing,
+culminating in the first awarding of the Gordon Bell prize in
+1987 (Bell et al. 2017).
+Combination of strong and weak scaling has been
+instrumental in utilizing massive parallelism and performance
+speedup in modern supercomputers such as Frontier and
+Fugaku, but the contribution of the latter has been greater
+in absolute speedup terms*. Now, weak scaling to large
+number of nodes require that the workload can be subdivided
+to achieve extremely good load balancing, i.e., (amount of
+work) / (processing capability) is uniform among all nodes.
+For homogeneous systems, if the workload domain is easily
+compostable, then simple uniform partitioning will suffice,
+and multitudes of studies have been conducted to achieve
+proper domain decomposition for more complex algorithms.
+Such load balancing work can be readily be applied even
+for nodes that are composed of heterogeneous elements,
+provided that (a) the architecture of the nodes are largely
+uniform (homogeneous) across the entire machine, and (b)
+during execution, the codes will be running simultaneous on
+one of the processors within the node, all at the same time
+within the machine. Practically all successful ‘accelerated’
+supercomputers and their applications, e.g., GPU machines
+such as Frontier, follow this pattern.
+However, once the nodes would be composed of plethora
+of customized hardware, and expected to be utilized in a
+more random, heterogeneous fashion as in a smartphone, load
+balancing becomes extremely difficult, and thus weak scaling
+speedup will flatten quickly, especially in a large parallel
+system. There have been efforts to alleviate this by creating
+a task graph of the workload and conduct dynamic load
+balancing, but have not really achieved success for very large
+systems, let alone for numerous heterogeneous accelerators.
+This is why, even for GPU-based machines, not only
+the node architectures are homogeneous, but also, in any
+given workload only GPUs or CPUs are used dominantly,
+but not typically both. Contrastingly, that large parallel
+program decomposed into a smaller task/dataflow graph and
+executed on-demand basis heterogeneously on a plethora of
+accelerators is only largely beneficial for small programs on
+a small machine, but not for HPC where parallelism will
+continue to increase to exploit weak scaling
+The second reason is the increasing difficulty of dark
+silicon being available in the system to be utilized for
+heterogeneously specialized hardware, for cost reasons. In the
+past, dark silicon was projected to be abundant with reduced
+lithography, thus justifying the “plethora of accelerators” view,
+as they were available for very low cost. However, with the
+slowing down of Moore’s law, coupled with high cost of
+manufacturing due to more advanced fab technologies such
+as EUV, transistor cost over time is flattening, or may even
+increase. Thus, the chip cost will become largely proportional
+to the number of transistors irrespective of the lithography, so
+every transistor has to contribute to the overall performance
+improvements in a major fashion, turning dark silicon into
+expensive unused silicon.
+For smartphones, the major cost of the phone is not the
+SoC but rather in the peripherals such as screen, camera, flash
+memory, etc., and the battery life is premium in the cost metric
+so extra cost incurred by dark silicon may be tolerable. For
+supercomputers, however, the major cost of the machine is the
+processors themselves, dominating over 50% of the overall
+CapEx. So unless the acceleration could benefit some major
+proportion of the workload, dark silicon would become an
+intolerable waste. That is why, over generations, accelerators
+such as GPUs tend to become more general purpose to
+cover an increasing proportion of the workload, ultimately
+becoming general purpose as the CPUs (or, GPGPUs).
+The third reason is software and productivity. Unless the
+accelerator usage is extremely easy, e.g., hidden under a set
+of very simple APIs, expecting the programmers to adopt
+an arcane programming model is not viable. In fact, this is
+more serious for HPCs where the market for applications
+is much smaller than major commodity ecosystems such
+as smartphones, with a less performance-conscious but
+extremely large market. Thus, for example, a large consumer-
+oriented IT company such as Apple can afford to replace a
+part of its API for a phone with hardware because it will sell
+more than 100 million iPhones, but not for supercomputers
+that have a much narrower market and thus do not warrant
+such investment.
+We close with these questions. . .
+x Will extreme heterogeneity happen? y Are supercom-
+puter workloads worth extreme specialization? z When
+will we have production supercomputers with more than
+one accelerator type?
+∗If one considers power efficiency for system scaling, massive weak
+scaling would not have been possible without dramatic increase in
+power/performance of compute nodes. However, such improvements usually
+allow increase in the number of nodes and/or processor units, thus helping to
+push weak scaling; as such, in terms of algorithmic scalability, weak scaling
+is still the dominating factor.
+Prepared using sagej.cls
+
+4
+arXiv preprints
+Figure 1. Classification of Compute Kernels and Supercomputing Architecture
+Myth 4: Everything Will Run on Some
+Accelerator!
+Related to our previous myth, even if one accepts that there
+will not be a plethora of accelerators, there could be a few
+such as GPUs or FPGAs, where the dominant portion of
+the workload will run. Indeed, for GPU-based machines
+that would be an assumption, lest the extra investment will
+not make sense. However, one could question, would some
+superchip such as GPUs largely replace the CPUs, the latter
+be degraded to second class citizens? It is not trivial as it may
+seem, as such statements are rather dogmatic and not based
+on candid analysis of the workloads. By proper analysis of
+the workloads, we may find that CPUs may continue to play
+a dominant role, with accelerator being an important but less
+dominant sidekick.
+From the hardware perspective, workloads can be largely
+divided into three classes, (C) compute bound, (B) memory
+bandwidth bound, and (L) memory latency bound. Any
+application will be composed of multiple compute kernels,
+each one being able to be largely classified into one of the
+three in Figure 1. Over time, supercomputer architectures
+have evolved in an attempt to cover all three in effective ways.
+Up until the 90s, special-purpose vector machines such
+as Cray and NEC SX accelerated largely (B), and (C) to
+some extent. This was largely due to the dominant workload
+that was CFD which was largely (B). Then in the 90s
+the microprocessor evolution for HPC happened, utilizing
+the commodity one-chip CPUs which had become very
+powerful due to high end applications such as engineering
+and multimedia needs, starting with workstation/server RISC
+then later x86 processors in massively parallel fashion,
+e.g., DoE ASCI Red. Individual processors were mediocre
+in performance but attained performance via massive
+parallelism, exercising weak-scaling, cf. Myth 3.
+Then in the late 2000s, although achieving Petascale
+performance was pioneered with the DoE Roadrunner and
+Jaguar machines, there was an ambition to achieve exascale
+by the late 2010s, achieving 1000x scaling in performance
+in 10 years. The roadblock was power/performance
+using conventional CPUs. However by the late 2000s
+the GPUs were evolving from their graphics-specific
+purpose to become general purpose compute processors,
+as they were architectural descendents of classical vector
+processors Matsuoka (2008). Different from classical vectors
+were that the floating point performance had been significantly
+enhanced, motivated by graphical workloads, and when
+generalized, the GPUs were now covering (C) and (B), while
+(L) was left for CPUs as GPU vector pipeline had very long
+latency. CPUs that facilitated SIMD vector units with high
+bandwidth memory such as the Intel Xeon Phi and Fujitsu
+A64FX brought in classical vector properties back into the
+CPUs, so in a sense homogeneous system composed of such
+chips were not direct reincarnations of simple commodity
+CPU based massively parallel machines, but rather, can be
+more regarded as converging the GPU and CPU properties.
+Circa 2022, the top machines are either homogeneously
+configured heterogeneous CPU-GPU nodes, or ‘converged’
+nodes such as RIKEN Fugaku or forthcoming machines with
+Intel Sapphire Rapids CPUs with HBM. However, this is not
+the only possible combination, and other configurations have
+not been properly explored.. For example, one could conceive
+of a machine with the latter configuration, with purpose built
+matrix-based accelerators for compute intensive kernels as a
+separate chip (or chiplet). In such a machine, the CPU would
+cover workloads (B) and (L), while the matrix accelerator will
+cover (C), . The benefit of such a machine would be ease of
+programming of (B) workloads which often involve complex
+memory access patterns, and thus porting to GPU codes has
+proven to be challenging.
+For further acceleration of (L) workloads, there is a limit to
+acceleration, such as molecular dynamics that require strong
+scaling. The best strategy seen for such workloads is fully
+customized data pipelines such as Anton (Shaw et al. 2008)
+with hardware design time synthesis. One could almost mimic
+such customization with cost but make it programmable
+by FPGAs or CGRAs. Such dataflow customization could
+also be useful for compute bound workloads such as DL
+Transformers, if small matrix engines as special function units
+can be conjoined in a larger macro dataflow as seen in modern
+FPGAs and CGRA chips. As such, in such a machine, (B)
+will be covered by CPUs, while (C) and (L) will be covered
+by a ‘strong scaling accelerator’.
+As we observe here, we find that we have not even covered
+the possible configurations of divergence/convergence of
+Prepared using sagej.cls
+
+Matsuoka, Domke, Wahib, Drozd, Hoefler
+5
+processing units, as the only mainstream ‘accelerated’
+machines are GPUs with the second property, while other
+design spaces have not been properly explored.
+We close with these questions. . .
+x Will CPUs become pure “servants” to the accelerators?
+y Are accelerators actually more than just better balanced
+processors? z Will reconfigurable accelerators see a
+renaissance?
+Myth 5: Reconfigurable Hardware Will Give
+You 100X Speedup!
+In a “fool me once...” fashion, one accelerator in particular
+has taken the HPC community by storm with lofty promises
+of 100x speedup (Lee et al. 2010) ever since the first
+ported matrix-multiplication by Larsen and McAllister (2001).
+Fueled by NVIDIA’s gross margin of over 50% (Macrotrends
+LLC 2022), and supported by billions of dollars from US
+DOE for ECP and similar programs in other parts of the world,
+the HPC community eventually migrated to a well designed
+and broadly adopted GPU/CUDA ecosystem. Consequently,
+164 systems of the TOP500 list utilize accelerators from
+NVIDIA. Nearly two decades later, Fugaku has shown that
+it only took long vectors and high-bandwidth memory to
+match GPU performance and energy-efficiency for many
+workloads. One positive aspect is that that much code has
+been “modernized”, i.e., rewritten in CUDA or languages and
+frameworks promising portability to utilize new devices. But
+the open question is how portable are these modernized codes
+really? Can they run seamlessly on all new devices?
+The global FPGA market was recently valued at about
+one-third of the global GPU market (Allied Market Research
+2020, 2022). Major chip vendors buying the leading FPGA
+hardware vendors, AMD acquired Xilinx and Intel bought
+Altera, respectively, indicate an interest for FPGA integration
+into future mainstream products. However, so far this has not
+materialized. Whether FPGA can replace or complement the
+mainstream GPUs in the HPC and data center market hinges
+on the questions regarding the cost-to-performance ratio,
+an existing software ecosystem, and most importantly the
+productivity of programmers. Unfortunately, we see hurdles
+in all these areas, which the community and industry might
+be able to solve with enough time and money. Without
+offering at least a factor of 10x performance gain at moderate
+porting costs, “FPGAs are not a factor in our current planning,
+because of their unprogrammability” (Sorensen et al. 2019).
+The question whether reconfigurable logic can replace
+or ament GPUs as accelerators is interesting. FPGAs will
+certainly have a harder time due to their high flexibility that
+comes at a cost. Units built from reconfigurable logic are
+10–20x less energy and performance efficient in silicon area.
+This issue can be addressed by hardening certain blocks, e.g.,
+floating point units, as some FPGA companies do. However,
+even then, the whole control path would be much less efficient
+and it is unclear whether program-driven execution is that
+much less efficient compared to reconfigurable dataflow. A
+new line of reconfigurable accelerators as materialized in
+Xilinx’ adaptive compute acceleration platform are similar
+to coarse-grained reconfigurable arrays (CGRAs) and offer
+more programmable blocks with a configurable dataflow
+interconnect. But if now 90% of the chip are hardened units,
+are those devices just GPUs with a less mature ecosystem?
+We close with these questions. . .
+x Will the HPC community embrace FPGAs as
+alternatives to GPUs in large-scale production systems?
+y Can the community afford a “Fool me twice...”
+moment? z Will CGRA-style reconfigurable dataflow
+accelerators take the place of FPGAs to compete?
+Myth 6: We Will Soon Run at Zettascale!
+Maybe FPGAs are the way to zettascale. With Aurora still
+under construction, Intel ignited the debate about zettascale
+in late 2021. While the HPC community initially smirked
+at their plans, Intel continued pushing the zettascale agenda,
+culminating in the latest claims to achieve 1 zettaflop/s by the
+end of the decade (Cutress 2022a). This proposition needs to
+be addressed, and we try to put their claims into perspective
+and predict a realistic timeline. Obviously, we cannot rule
+out that Intel has a secret, revolutionary technology which
+they plan to commercialize in due time, however let us not
+speculate now and instead build on publicly available data.
+But first we have to distinguish the terms. We assume
+in the following, that (1) “zettaflop system” refers to
+any computer capable of achieving over 1021 double-
+precision floating-point operations (“FP64”) per second
+on the Linpack benchmark; (2) “zettaop system” refers
+to any computer theoretically capable of performing 1021
+operations† per second, and (3) “zettascale system” denotes
+any computer executing a scientific application with a
+sustained performance of over 1 zettaflop/s in fp64.
+Before we extrapolate, we look at historical trends
+by Strohmaier et al. (2022). The HPC community achieved
+1.068 teraflop/s with Sandia/IBM’s ASCI Red in summer
+1997, 1.026 petaflop/s with Los Alamos/IBM’s Roadrunner
+in summer 2008, and achieved (unofficially) 1.05 exaflop/s
+in spring of 2021 with China’s OceanLight system and
+1.1 exaflop/s with OakRidge/HPE’s Frontier in summer
+2022. Not only do 11 and 13 years lie in between these
+achievements, respectively, but also multiple megawatt. ASCI
+Red consumed “only” 0.850 MW, Roadrunner increased that
+to 2.35 MW, and OceanLight and Frontier now consume
+35 MW and 21.1 MW, respectively. This and Figure 2 show
+that the energy efficiency of modern chips cannot keep up
+with the demand for increasing compute.
+Back to Intel claiming to manage 2x performance
+improvements year-over-year which would yield zettaflop/s
+by 2032—but at a power requirement of the entire
+system of 50–100 MW (Cutress 2022b). Hence, this 1,000x
+in performance comes at the cost of 3–5x in power;
+and reformulated: the energy efficiency to perform fp64
+operations needs to increase by 200–350x, from ≈50 to
+over 10.000 Gflop/s
+Watt . Even under idealized conditions and
+using Frontier’s Rpeak as baseline, this goal requires a
+†An exact and consistent definition of “operation” in this context is still
+debated in the HPC community.
+Prepared using sagej.cls
+
+6
+arXiv preprints
+Figure 2. Historical fp64 power efficiency [in Gflop/s
+Watt ] extrapolated until 2038 to put Intel’s zettaflop/s claims into perspective.
+125x improvement in 10 years, and all of that while
+other big players slowly acknowledge the end of practical
+silicon scaling laws (White 2022). If we believe the IEEE
+IRDS™ (2021) roadmap, we might gain 5x in power
+density (optimistically rounded from 4.27x) by 2034 at 7 ˚A
+compared to 5 nm. This leaves 25x, which we could split
+into 5x from increased transistor count per chip and 5x from
+increased node count per system. Can we cool the former,
+yes (Wu et al. 2021), and can we interconnect the latter?
+Sure, but doing so, at 2.5 GW, comes down to the will to
+invest more than anything else, and without revolutions in
+memory and interconnect technologies, we might see Linpack
+transition into memory- or I/O-bound territory, nullifying any
+computational advances.
+On the other hand, a zettaop/s system at 100 MW in 2032
+is far more likely, since low-precision units (such as tensor
+cores) can boost the op/s
+Watt metric, e.g., currently fp16 tensor
+cores demonstrate an 8x advantage over fp64 vector units.
+Lowering the precision further from fp16 to 3-bit operands
+could allow for another 5x improvement (Frantar et al. 2022),
+but only if the industry (and HPC community) sees the need
+for adding these low-precision units, as we discuss in Myth 11.
+Considering the above, our more realistic, yet optimistic,
+timeline for zetta is zettaop/s in 2032 at 50 MW, zettaflop/s
+in 2037 at 200 MW, and zettascale by 2038. Can Intel or
+anybody else pull it off before then? Only time will tell.
+We close with these questions. . .
+x Will we reach zettaflop/s performance or will fp64
+lose relevance before? y Will we continue to build
+more power-hungry supercomputers as we did in the
+past? z Which one will happen first: zettascale, practical
+quantum advantage, or all internal combustion-based
+engines cease to be produced?
+Myth 7: Next-Generation Systems Need
+More Memory per Core!
+Before, on the road to peta- and exascale, application
+scientists continuously raised alarms that the memory per
+core is decreasing with each new computer generation.
+This was mainly due to the quick growth in the number
+of cores while the performance per core was stagnating.
+Yet, many workloads can keep those cores utilized with a
+relatively small working set while staging larger amounts of
+data remotely and/or recomputing parts. Much of this large
+memory requirement seemingly turns out to be legacy and
+somewhat wasteful design from times where memory space
+was abundant compared to other resources.
+Simplistic arguments along the lines of “we need more
+of X” seem to have a solid tradition in our community. For
+example, the HPC community spent the first decades to hunt
+more floating point computations per second. Recently, a
+demand for larger and faster memory replaced this main goal.
+The community nearly made a complete 360-degree turn,
+with Haus (2021) saying “computation is free” and Ivanov
+et al. (2021) showing “data movement is all you need”.
+Some even argue that this turn was taken too late due
+to the fixation on flop/s. While this was all true at the
+time, the general discussion should really be about the
+intricate relation between the application requirements and
+the system capabilities in terms of balance, i.e., ratio between
+the different resources such as memory size/bandwidth and
+compute (Czechowski et al. 2011).
+These ratios usually shift with chip technology and
+architectural choices. For example, Moore’s law drove the
+costs for compute on chip down over decades but off-chip
+communication was limited by Rent’s rule. This led to the
+recent data movement crisis. Newly emerging optical off-
+chip connectivity, see Myth 8, as well as 3D integrated
+memory (Domke et al. 2022) shifts the balance again and
+may alleviate many of these aspects, at least at the scale of
+a single chip. It seems key to understand the malleability of
+application, i.e., which resources can be traded for which
+other resources (e.g., memory capacity for computation
+bandwidth using recomputation or caching as techniques).
+Prepared using sagej.cls
+
+Matsuoka, Domke, Wahib, Drozd, Hoefler
+7
+Here, specifically I/O complexity analysis is a tool to deeply
+understand this trade-off. Once all trade-offs are understood,
+requirements models (Calotoiu et al. 2018) could be used to
+fix trade-offs into designs. These models could then inform
+architectural choices as well as hardware developments.
+One area to highlight in this context is embedded design
+where such trade-offs have long been used to build real
+systems due to resource scarcity (e.g., battery). While those
+designs were initially limited to very narrow application
+domains (e.g., radio signal, audio, or video processing),
+embedded devices have recently been expanded towards more
+diverse workloads (“apps”). We believe that HPC can learn
+from this field by defining clear system design methodologies
+based on a solid combination of empirical and analytical
+modeling. More particularly, systems design in HPC can
+benefit from the embedded systems doctrine of accounting for
+over-engineering just as one accounts for under-engineering.
+We close with these questions. . .
+x When will the current “data movement” focus end?
+y What will be the next bottleneck resource? z Will
+our community be able to adopt a performance modeling
+discipline to discuss bottlenecks scientifically?
+Myth 8: Everything Will Be Disaggregated!
+To stop the waste of memory resources, the academic com-
+munity is advancing on the Silicon Photonics front (Gonzalez
+et al. 2022) and industry is pursuing scale-out technologies (Li
+et al. 2022), such as Compute Express LinkTM (CXL), a
+cache-coherent interconnect for data centers. But a few
+players seem to push the idea over the edge with their
+plans to disaggregate everything (NTT R&D 2020; Shan
+et al. 2022). As Gonzalez et al. (2022) stated: “An optical
+interconnect is more appealing than an electrical interconnect
+for memory disaggregation due to three properties: its (1)
+high bandwidth density significantly reduces the number of
+IO lanes, (2) power consumption and crosstalk do not increase
+with distance, and (3) propagation loss is low.” However,
+several barriers remain before we can fully replace copper-
+based interconnects in our supercomputers.
+Generally, we see two remaining challenges for a broad
+adoption of Silicon Photonics and all-optical interconnects:
+low-cost manufacturing and optical switching. The former
+is obvious, because after all, the data center and HPC
+community relies on inexpensive components to optimize the
+overall system performance-to-cost ratio. The latter challenge
+is less obvious for the uninitiated. Current electrically
+switched networks can operate in “packet switching” mode
+to effectively lower the observable latency and utilize the
+available link bandwidth. The alternative to this mode
+is “circuit-switching” and it was abandoned by the HPC
+community long ago in favor of packet-switching. However,
+without (cost-)effective means to buffer light, process photon
+headers in-flight, or reverting to electric switches with
+expensive optical-electrical-optical conversions, we would
+have to resort to circuit-switching (Bergman et al. 2022)
+with all the inherent deficiencies: complex traffic steering
+calculations, switching delays, latency increase due to lack of
+available paths, under-utilization of links, just to name some.
+For HPC, an extensive or extreme disaggregation yields
+another challenge, specifically the speed of light. Photons
+travel at a maximum speed of 3.3 ns/m in hollow fibers
+(or slower in other transport media). This is equivalent to
+a level-2 cache access of a modern CPU, but does not yet
+include the disaggregation overhead, such as from the CXL
+protocol itself, switching, or optical-electrical conversions at
+the endpoints. At 3–4 m distance, the photon travel time alone
+exceeds the first-word access latency of modern DDR memory.
+Therefore, if main memory would be disaggregated beyond
+rack boundaries, it will become noticeable for memory-
+latency sensitive applications, cf. Myth 4. The more sensible
+solution, in line with Myth 7, for future HPC systems are
+smaller node-local memory configurations (e.g., HBM3)
+paired with rack-local, CXL-based memory pools if the
+capacity- and performance-to-cost ratios of the memory pool
+plus required interconnect can outperform node-local SSD
+solutions.
+We close with these questions. . .
+x Will CXL be deployed widely in HPC? y Will large-
+scale supercomputers be disaggregated beyond rack-
+scale? z Should we disaggregate main memory?
+Myth 9: Applications Continue to Improve,
+Even on Stagnating Hardware!
+Modernizing hardware, with Silicon Photonics, Tensor Cores,
+or simply shrinking transistors, has too long been the primary
+method of accelerating legacy software. More than half
+of this improvement was based on Moore’s law and its
+observation that transistors will continue to become smaller
+every few years (originally 18 months). The remaining
+hardware improvements came from architectural innovations,
+such as deeper cache hierarchies, the migration to more
+specialized architectures (e.g., GPUs), or the utilization of
+larger and wider vector-units (SIMD), as well as scaling the
+HPC systems up by giving them more processors and cores.
+Unfortunately, we are no longer seeing the consistent
+technology scaling that Moore observed. Consequently, in
+the so-called Post-Moore era, the “performance road” forks
+three-ways, yielding the following options: (1) architectural
+innovations will attempt to close the performance gap, and
+an explosion of diverging architectures tailored for specific
+science domains will emerge, or (2) alternative materials and
+technologies (e.g., non-CMOS technologies) that allow the
+spirit of Moore’s law to continue for a foreseeable future,
+or (3) we abandon the von-Neumann paradigm together and
+move to a neuromorphic or quantum-like computer (which,
+in time, might or might not become practical as discussed in
+Myth 1). One major aspect that reflects the uncertainty about
+the future is the initiatives of unprecedented scale: CHIPS act
+in the US and similar initiatives in other countries in the order
+of 100s Billion USD, quantum computing initiatives in the
+order of 10s Billion USD, etc.
+But one point that is often overlooked is that algorithmic
+improvements in HPC (dubbed as “Algorithmic Moore’s
+Law” by Keyes (2022)) have over time provided exponential
+improvement in key areas of HPC, see Figure 3. Similar
+reports attribute a significant portion of the performance
+Prepared using sagej.cls
+
+8
+arXiv preprints
+higher 
+order AMR
+1
+10
+100
+1000
+10000
+100000
+1000000
+10000000
+100000000
+1980
+1990
+2000
+2010
+2020
+Effective Sustained Speedup
+Algorithmic Moore's Law Examples
+100
+101
+102
+103
+104
+105
+106
+107
+108
+Sustained Speed in Gflop/s 
+Combustion Simulation
+(Complex Kinetics) 
+Combustion Simulation
+(CFD) 
+COSMO Climate Model
+Fusion Energy Simulation 
+(Global MHD) 
+Moore’s Law
+Fusion Energy Simulation 
+(Micro-turbulence) 
+improved
+linear solver
+ARK integrator
+complex chem
+AMR
+semi-implicit
+high-order
+elements
+gyro-
+kinetics
+delta-f,
+magnetic
+coordinates
+improved
+electron
+models
+low Mach
+auto-code
+high order
+improved 
+explicit/implicit 
+solvers
+Figure 3. Examples of “Algorithmic Moore’s Law” for different areas in HPC; Fusion energy and combustion simulations data
+by Keyes (2022) and climate simulation data by Schulthess (2016)
+improvement in many legacy codes to be from numerical
+solvers, algorithms, low-precision numerics, system software,
+etc Schulthess (2016). However, we have to be cautious that—
+just as hardware improvements have physics and engineering
+limits—the “Algorithmic Moore’s Law” also has its own
+limits: numerical stability, hitting asymptotic limits, etc. That
+being said, those limits might not be as clear and quantifiable
+as the limits on hardware. That is since even if one numerical
+method hits its limit, domain experts can often reduce/pre-
+condition their problem to another numerical method that is
+more efficient.
+We close with these questions. . .
+x As the performance improvements from hardware
+technologies drop, should the HPC community dramat-
+ically increase the investment in software? y Will the
+“Algorithmic Moore’s Law” end soon as well? z To what
+extent is the HPC community willing to refactor/rewrite
+legacy codebases when/if hardware stagnates?
+Myth 10: Fortran Is Dead, Long Live the DSL!
+Applications might have limits, but what about languages.
+How often have we heard “Fortran is dead, long live X”?
+Slogans like this have been resonating in the community for
+nearly 40 years (Post 1982). X has been everything from
+C to C++, and more recently Python or Domain-Specific
+Languages (DSLs). Yet, Fortran remains in wide use in
+important communities such as weather and climate even
+for newly written codes. Other languages, such as COBOL
+were indeed replaced with more modern alternatives such
+as Java. Why is this? Are some parts of our community just
+stubborn to follow the youngsters? Or are old languages not
+necessarily bad for the task? Indeed, Fortran is a very well
+designed language for its purpose of expressing mathematical
+programs at highest performance. It seems hard to replace it
+with C or other languages and outperform it or even achieve
+the same baseline. This may be due to the highly optimized
+Fortran compilers or the limited language features (e.g., no
+pointer aliasing) that enable more powerful optimizations.
+Fortran and other general-purpose languages remain
+competitive with many DSLs on CPUs (Ben-Nun et al.
+2022) and are recently also adopted to GPUs, albeit often
+less elegant. General-purpose portability approaches such as
+SYCL (Keryell et al. 2015), also powering Intel’s oneAPI,
+or OpenMP provide flexibility as well as some portability
+across devices. High-productivity general-purpose languages
+are hard to accelerate in practice. For example, Python’s
+flexibility (e.g., monkey patching and flexible typing) disables
+many static optimizations. However, when restricting the
+syntax to high-performance Python (much of NumPy), then
+optimizations become simpler (Ziogas et al. 2021). Any
+language becomes more complex over time—Fortran 66
+evolved into the complex Fortran 2018 language standard.
+Similar trends affect DSLs that are widening their scope over
+time. Do we require this generality? If yes, then DSLs are
+doomed to fail or they morph into general-purpose languages.
+Another argument is that the lower levels usually remain
+C/C++ and programmers interested in highest performance
+are often happy to dig into the lower levels. Then the question
+remains—where should the portability layer be located? At a
+(virtualized) Instruction Set Architecture (ISA) as in LLVM’s
+IR (Lattner and Adve 2004), some lower-level language
+such as C/C++ as in SYCL/oneAPI, or even dataflow graph
+representations as in DaCe (Ben-Nun et al. 2019)?
+We close with these questions. . .
+x When will programmers stop using Fortran for new
+applications? y Will we ever have more application codes
+written in DSLs than general-purpose languages? z What
+will be the next big DSL?
+Myth 11: HPC Will Pivot to Low or Mixed
+Precision!
+A high-performance language is nothing without proper data
+types, but high-precision operations such as fp64 come at a
+significant cost in terms of silicon area, energy and speed,
+according to Myth 6. Lowering this precision can save costs
+Prepared using sagej.cls
+
+Matsuoka, Domke, Wahib, Drozd, Hoefler
+9
+but may reduce accuracy of the results and, in the worst case,
+break the application (e.g., convergence). But there is more
+to this trade-off: what if a more clever implementation could
+maintain convergence properties of high precision numerics,
+while enjoying computational efficiency of low precision?
+One common trick is using mixed precision on the algorithmic
+level, for example, using low precision for individual particles
+and only using high precision for aggregated values (Kutzner
+et al. 2019). Some processors offer mixed precision tricks
+at the hardware level in the form of instructions with low
+precision inputs but higher precision accumulations.
+There is however more to reduced precision than using
+fewer bits—the question is how to optimally distribute bits
+between mantissa and exponent (Tesla, Inc. 2021), or even if
+to use an entirely different (not IEEE-754) way to represent
+numbers (Gustafson and Yonemoto 2017). The story of
+reduced precision in AI hardware is quite telling: In early
+days of the field, predominantly the IEEE fp32 format was
+used, but knowing that in deep neural nets the weights and
+activations are typically distributed on a small range of values,
+researchers began to explore the fp16 format. Soon the Pascal
+generation of GPUs with fp16 performance—at a factor of
+two compared to fp32 was released—and the magic did not
+happen by itself. Exploding and vanishing gradients, outlier
+weights, etc., made training large deep neural nets require
+extra effort to stabilize (incurring corresponding overhead) or
+just did not converge at all. The next generation of devices
+came with bfloat16 format—same 16 bits, but more bits
+allocated to range, less for precision. It worked better, but
+still once in a while a model would collapse. Finally, the
+recent generation of GPUs came with a 19-bit numeric format,
+misleadingly called TensorFloat-32. So far it seems to be at
+the sweet spot for artificial intelligence workloads—allowing
+for noticeably faster arithmetics than fp32, while maintaining
+enough numeric stability for the models to reliably converge
+without extra programming effort.
+Now that mixed precision is a de-facto standard in the AI
+domain, more hardware support is being implemented. So
+far there is no general clarity on the limits—how few bits
+can we get away with in different HPC areas. The following
+factors in particular are important to consider as we move
+forward. A fully transparent solution for the problem is to
+simulate higher precision using low precision operations,
+e.g., as shown by Ootomo and Yokota (2022). Our Myth 4’s
+memory-bound problems in particular are good candidates
+for exploiting “simulated” high precision, since the overhead
+can be masked by data transfers. It is not clear however
+if this incurred overhead is an acceptable price that HPC
+agrees to pay for remaining in higher precision. A less
+transparent method is to approach the problem as precision
+auto-tuning task by adapting the precision to a minimum
+while bounding the error, e.g., as demonstrated by Menon et al.
+(2018). One main limitation of that method is the reliance
+on automatic differentiation (AD) to track error propagation,
+which is not practical for large codebases. Finally, the least
+transparent approach requires domain experts in HPC to study
+the numerical stability of solvers to identify, on a case-by-case
+basis, the susceptibility of solvers to lower/mixed precision.
+While this approach is viable for solvers that are wrapped in
+libraries to be consumed by HPC domain experts, it is unclear
+whether domain experts writing their own solvers (common
+in HPC) would be willing to take on this burden.
+We close with these questions. . .
+x Is the HPC community ready (or already late?) to react
+to the new low precision formats driven by deep learning?
+y Will HPC navigate itself into a high-precision niche?
+z When, if ever, will the industry drop fp64 support?
+Myth 12: All HPC Will Be Subsumed by the
+Clouds!
+The rapidly advancing AI and new precision options has
+reignited the cloud discussion. The question whether clouds
+will subsume supercomputing has been ongoing for more
+than a decade, since the late 2000s Deelman et al. (2008), but
+remains inconclusive. Today’s cloud offerings offer a wide
+spectrum for HPC customers, ranging from low-cost standard
+virtual machines to specialized top-gear HPC equipment in
+the cloud. It is not surprising that cloud providers offer exactly
+the same performance as on-prem supercomputing centers
+in practice De Sensi et al. (2022). After all, they simply buy
+the same hardware! Thus, this discussion is more of a fiscal
+argument with an interesting economy-of-scale twist.
+There are actually bi-directional aspects to the cloud-vs-
+supercomputer argument. One is the so-called “cloudification
+of supercomputers”, and the latter being “supercomputifica-
+tion of clouds”, but they often get mixed-up leading to the
+confusions in the discussions. We must look at both aspects,
+and it is in fact the latter where such subsumption may happen
+or not.
+The former, “cloudification of supercomputers”, is an
+unmistakable trend, in that various software stack features
+and APIs are added so that supercomputers effectively
+become high end compute resources in the same manner as
+commercial clouds. Indeed, many major supercomputers are
+already facilitating cloud features, so that they are effectively
+clouds themselves, and interoperable with commercial clouds.
+However, this assumes that there is already a supercomputing
+resource facilitated for themselves, and does not directly affect
+the subsumption argument.
+The latter, or “supercomputification of clouds”, is where
+subsumption may happen, in that clouds nowadays can
+support features as well as performances of dedicated
+supercomputers directly, such that they are directly amenable
+as their replacement. Certainly, there are now multiple
+cloud services that facilitate virtual compute clusters in
+the cloud. However, although Intersect 360 reports that
+HPC-in-the-cloud CAGR has been dramatic, over 80% in
+2021 Intersect360 Research (2022), it also reports the overall
+high growth in the HPC market, especially in the high end,
+and also projects that, the growth in the cloud HPC market
+will flatten over time to be consistent with the overall HPC
+industry growth. Continued investments by all major global
+regions in exascale machines and beyond, coupled with
+companies facilitating their own top-ranked machines, will
+likely continue to fuel the on-prem infrastructure growth.
+In fact, for enterprise IT infrastructures, there has always
+been a swing between on-prem and public clouds, largely
+Prepared using sagej.cls
+
+10
+arXiv preprints
+driven by economics. While standing up comprehensive
+internal IT has become less attractive with multitudes of
+cloud services readily available in the cloud, so the CAPX for
+clouds would be cheaper, especially for small enterprises and
+startups, for large enterprises there is a tendency to move back
+to on-prem infrastructures, as the OPEX of clouds could be
+expensive. The same could be the case of HPC increasingly as
+the whole field would pose continuous uprisings in economic
+viability for industry and societal benefits, thus being driven
+by economic metrics.
+However, the variant of the subsumption scenario is
+that, although on-prem supercomputers continue to exist,
+processors and other hardware developments will be largely
+driven by enterprise HPC needs, currently dominated by
+AI / deep learning workloads. The R&D expenditures of
+hyperscalers in IT now outclass the government investments,
+and increasingly the hyperscalers are investing in high end
+computing. If the commercial cloud hyperscalers can work
+out the scale of economy in their own hardware manufacturing
+to the extent that, it could build and operate large scale
+HPC infrastructures cheaper than on-prem supercomputers
+of any size, then the swing could totally happen towards
+full subsumption— although somewhat unlikely, this could
+compromise the ability to cover some of the traditional HPC
+workloads that do not meet main industrial needs, such as the
+requirement for dense 64 bit linear algebra capabilities.
+We close with these questions. . .
+x What could be a defining development to decide
+between cloud and on-prem HPC? y When will more
+than half of the HPC cycles be spent in the cloud? z Will
+on-prem systems be a niche or remain with a significant
+fraction of HPC cycles spent?
+Conclusions
+Many myths shape the discussions in the HPC community
+today—in this work, we debate some of those and hope to
+stir up arguments. While we present them in an exaggerated
+and humorous way, many of those myths form the core of
+thinking in our community. Some may be more divisive than
+others but it seems that many are hard to answer definitively.
+Maybe some points will settle in the future while others will
+not. Yet, their sheer importance mandates a serious treatment
+in order to help guide future directions for academic research
+but also industry and government investment.
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+arXiv:2301.03123v1  [math.AG]  9 Jan 2023
+Lax colimits of posets with structure sheaves:
+applications to descent
+Javier Sánchez González
+Universidad de Salamanca, Department of Mathematics
+javier14sg@usal.es
+January 10, 2023
+Abstract
+We consider categories of posets with C-valued structure sheaves for
+any category C and see how they possess poset-indexed lax colimits that
+are both easy to describe and "weakly equivalent" to their ordinary
+colimits in a certain sense.
+We employ this construction to study
+descent problems on schematic spaces—a particular scheme-like kind
+of ringed poset—, proving a general Seifert-Van Kampen Theorem
+for their étale fundamental group that recovers and generalizes the
+homonym result for schemes to the topology of flat monomorphisms.
+The techniques are general enough to consider their applications in
+many other frameworks.
+1
+Introduction
+In the geometric world, categorical colimits can be thought of as a general
+way of expressing gluing of spaces. For example, any scheme is the colimit,
+in the category of locally ringed spaces, of the components of any of its affine
+coverings. Furthermore, for any reasonable scheme, one might assume that
+these colimits are indexed by posets: the nerve of the corresponding affine
+covering with its redundancies removed. This idea is simply a generalization
+to locally ringed spaces of the construction of finite models of topological
+spaces, an old technique of McCord to study homotopy types [2].
+To better understand the situation, one may consider these recollement
+data of schemes as functors from some poset X, understood as a category,
+which we may assume finite under mild compactness hypothesis on the
+original scheme S; to the category of commutative rings with unit, i.e.
+1
+
+X → CRing. It is also very classical that the category of such functors
+coincides with the category of sheaves of rings on the poset X, understood
+as a topological space, so we may study these collections of data as (non-
+locally) ringed spaces themselves. A major advantage of this point of view
+is that many sheaf-theoretic notions admit a simple description, like quasi-
+coherent modules or the Čech resolution of an abelian sheaf; and in many
+cases they coincide with their scheme-theoretic analogues on S. From these
+observations, Sancho first axiomatized in [6] a—non-full—subcategory of
+finite ringed posets that behaves suitably well with respect to quasi-coherent
+modules and that, not only contains all finite models of (quasi-compact and
+quasi-separated) schemes, but also that non-trivially generalizes them: the
+category of (finite) schematic spaces, SchFin. After spending some time
+with these objects, it is easy to convince oneself of their geometric interest
+compared to similarly-purposed constructions such as those of simplicial
+schemes. A brief summary of this non-standard approach will be provided
+in Section 2.
+Taking a step back from the previous discussion, we also note that
+computing colimits in the category of schemes—or simply determining if
+they exist—is a very general and difficult problem: even when existence is
+guaranteed—for example, colimits via open immersions—, obtaining explicit
+expressions for them is no easy task. The situation is not much better if
+we attempt to compute colimits of schematic spaces as described in the
+previous paragraph; however, due to their combinatorial nature, there is
+an alternative: given a poset-indexed "datum" U : P → SchFinop, we may
+construct a space, which we call the cylinder of U, by simply turning the set
+�
+p∈P |U(p)| into a poset with the structure inherited from the underlying
+posets of each |U(p)| and the transition morphisms |U(p)| → |U(q)| for q ≤ p.
+If we endow the resulting poset with the structure sheaf—which is just a
+functor—induced by the structure sheaves of each U(p), we obtain a ringed
+poset—which will be another schematic space under certain conditions—that
+represents—will be weakly equivalent to—the desired colimit in a precise
+sense and which has been computed without performing any complicated
+categorical operations on either commutative rings or posets.
+In this paper, we will study this cylinder space in its most general
+formulation, replacing CRing for any category C. The category of finite
+posets with a C-values structure sheaf will be constructed and called the
+category of C-data, denoted C -data. If C = pos is the category of posets,
+it turns out that the cylinder space is just an incarnation of a poset-indexed
+lax colimit, understanding pos as a strict 2-category in the natural way. We
+prove this and see how it generalizes to posets admiting structure sheaves
+2
+
+with values in C, detailing the different 2-categorical structures that one may
+consider and how they interact with each other. For this purpose, we will
+need to consider D -data with D being a strict 2-category, which also proves
+useful in other applications. Our language of choice will keep things analogue
+to the category of ringed posets that we use as a reference.
+These constructions, of C -data and lax colimits on it, are general enough
+to model many descent problems, which appear in a natural and functorial
+way.
+For the sake of keeping the discussion focused, we will specialize
+everything to the previously-discussed schematic case: we will characterize
+when a cylinder of schematic spaces remains schematic and will apply such
+characterization to give a very general descent Theorem for data on schematic
+spaces, which even admits a topos-theoretic interpretation—only sketched
+here due to space limitations—.
+We will see some examples, with the
+main one being the Seifert-Van Kampen Theorem for the étale fundamental
+group of schematic spaces, as constructed in [4].
+It is worth noting that
+a variation of the homonym result for schemes and their topology of flat
+monomorphisms—rather than Zariski or étale—follows purely from the formal
+descent result developed in this paper and the classical case, which exemplifies
+how these techniques automatically extend any "reasonable" Zariski-local
+statement to the aforementioned topology.
+2
+Motivation: schematic spaces
+Let us give a rather informal introduction to the objects of study for the
+applications, which were the original motivation to develop the theory of
+C -data that we will introduce in the following sections. In prose: schematic
+spaces arise as the largest subcategory of ringed finite posets that behaves
+"like quasi-compact and quasi-separated (qc-qs)" schemes with respect to
+categories of quasi-coherent sheaves.
+The basic example comes from the
+construction of finite models of schemes, see [7], which is a generalization
+of an earlier topological technique, see [2]: given a qc-qs scheme S and a
+finite covering {Ui} of S, one may define a poset X as the T0-fication of the
+topology generated by the covering. Explicitly, if for any s ∈ S we denote
+U s = ∩s∈UiUi, one sets X = S/ ∼ with s ∼ s′ whenever U s = U s′ and
+[s] ≤ [s′] if and only if U s′ ⊆ U s. The result is a morphism of ringed spaces
+π: S → (X, π∗OS).
+If the covering is chosen so that the U s are affine, π induces an adjoint
+equivalence (π∗ ⊣ π∗): Qcoh(S) ∼
+→ Qcoh(X), so S can be studied from X.
+3
+
+Schematic spaces were first introduced in [6] and studied in [5], [8] or [4].
+In [9] one can find an extensive compilation of characterizations, and the
+author of this paper has expanded upon this and exhaustively explored their
+role in algebraic geometry in his PhD thesis, which includes the contents of
+this paper and more to come. Morally speaking, we recommend to think
+of a schematic space X as a model or "structured descent data" of the
+locally affine locally ringed space Spec(X) = colimx∈X Spec(OX,x), but the
+schematic condition forces such discrete incarnation X to be "nice enough"
+to reflect the geometry of Spec(X) to some extent. One shows that these
+Spec(X) have to be locally affine in the topology of flat monomorphisms of
+affine schemes and, thus, contain all qc-qs schemes; but schematic spaces do
+not model, for example, algebraic spaces, for which the associated locally
+ringed space does not preserve enough useful algebraic information. There
+are also "geometric" arguments for considering ringed posets as the basis
+for our combinatorial models over other, more classical, alternatives like
+simplicial schemes.
+While perhaps not the most enlightening approach, it will be convenient
+for our purposes to consider the following definitions. Assume for simplicity
+that all stalk rings of our ringed posets are Noetherian.
+Definition 2.1. A finite ringed poset X is a (finite) schematic space if
+• For any x ≤ y, the morphism rxy : OX,x → OX,y is flat.
+• For any t ≤ x, y, the morphism
+OX,x ⊗OX,t OX,y →
+�
+z≥x,y
+OX,z
+is faithfully flat.
+A morphism f : X → Y between schematic spaces is schematic if
+• For any x ∈ X and y ≥ f(x), the morphism
+OX,x ⊗OY,f(x) OY,y →
+�
+z∈Ux∩f−1(Uy)
+OX,z
+induces a surjection between the prime spectra.
+Remark 2.2. A simple descent argument for faithfully flat morphisms shows
+that OX,x ⊗OX,y OX,y ≃ OX(Ux ∩Uy) for all t ≤ x, y. If x = y, this condition
+implies that the restriction morphisms rxy of any schematic space are flat
+epimorphisms of rings, hence local isomorphisms.
+4
+
+Let SchFin denote the category of schematic spaces and morphisms.
+All ringed posets and morphisms will be considered schematic unless stated
+otherwise. The Spec construction outlined in the introduction of this section
+defines a functor to the category of locally ringed spaces which is neither full
+or faithful:
+Spec: SchFin → LRS
+X �→ Spec(X) := colimx∈X Spec(OX,x).
+It can be shown that the schematic category has finite fibered products and
+that are preserved by both the forgetful to CRing -data and by Spec.
+Remark 2.3. Heuristically, the restriction maps of X being flat epimorphisms
+implies that the information in X can be recovered from Spec(X). The other
+schematicity conditions can be shown to be equivalent to the existence of a
+certain map πX : Spec(X) → X, i.e. to X being essentially a "finite model"
+of Spec(X) in a topological sense.
+Definition 2.4. A morphism f : X → Y is said to be a qc-isomorphism if
+Spec(f) is an isomorphism.
+The class of qc-isomorphisms is a multiplicative system of arrows in
+SchFin that is maximal by definition, so the corresponding localization—
+Verdier quotient—defines a faithful—but not full—functor
+Spec: SchFinqc → LRS.
+To study properties P of schematic spaces we will ask for two requisites:
+• A "rigorous" requisite: that P factors through the localization, i.e.
+that any representative of the qc-isomorphism class of a space or arrow
+determines is the whole class verifies the property or not.
+In other
+words, P is geometric.
+• A "moral" requisite:
+P can be studied in terms of finite models,
+without applying the Spec functor. In other words, P is discretizable.
+In certain cases, one can "rigidify" poorly-behaved properties by studying
+them on certain reflective subcategories of SchFin that induce equivalences
+after localizing by qc-isomorphisms. As an example of this, see the discussion
+about connectedness in [4]. In this paper we will tacitly assume that all our
+definitions work in this nice way, but let it be known that more technical
+considerations are needed for a full exposition—and that is the reason why we
+employ quotation marks so often to highlight seemingly ordinary notions—.
+5
+
+Finally, the main result in [4]—which, with enough work, can be written
+in much more geometric and elegant terms that the ones presented there—is
+concerned with the existence of a Galois category of "finite étale covers" for
+any "connected" schematic space X that, when X models a qc-qs scheme S,
+is naturally equivalent to the homonym Galois category of S.
+Theorem 2.5. [4] For a schematic space X and a schematic morphism
+x: Spec(Ω) → X with Ω an algebraically closed field—a geometric point—,
+there exists a category RÉt(X) and a functor Fibx : RÉt(X) → FinSet
+such that, when X is "connected", the pair (RÉt(X), Fibx) is a Galois
+category. We denote its fundamental group by πet
+1 (X, x). If S = Spec(X)
+is a scheme, the Spec functor induces a equivalence of Galois categories
+(RÉt(X), Fibx) ≃ (FEt(S), FibSpec(x)), where FEt(S) is the category of
+finite étale covers of S, and thus πet
+1 (X, x) ≃ πet
+1 (S, Spec(x)).
+Remark 2.6. In [4] we also showed that qc-isomorphic spaces have equivalent
+Galois categories of finite étale covers: the construction is geometric.
+Of course, for a general schematic space X, one can consider the set of
+all its geometric points and define the étale fundamental (Stone) groupoid
+Πet
+1 (X).
+In its general version, the Galois Theorem states that the fiber
+functors induce an equivalence of categories
+RÉt(X) ≃ [Πet
+1 (X), FinSet] ≡ Πet
+1 (X)-FinSet
+where the action of this groupoid is continuous. As always, this is just a
+particular case of more general topos-theoretic results.
+2.1
+The topology of flat immersions
+We begin by introducing the natural (pre)topology on SchFin.
+Definition 2.7. Let f : X → Y be a schematic morphism. We say that f
+is flat if f ♯
+x: OY,f(x) → OX,x is flat for all x ∈ X. Such flat morphism is a
+flat immersion if its diagonal ∆f : X → X ×Y X is a qc-isomorphism. A flat
+morphism f is faithfully flat if Spec(f) is surjective.
+These three types of maps can be characterized in terms of the adjoint
+pair (f ∗, f∗) for quasi-coherent sheaves. We remark that a flat immersion
+is, by definition, a flat monomorphism in SchFinqc; and Spec(f) for such
+f is a flat monomorphism of locally ringed spaces. One can show that qc-
+isomorphisms are exactly faithfully flat immersions.
+6
+
+Remark 2.8. It can be shown that a morphism Ux → Uy is a flat immersion
+if and only if OY,y → OX,x is a flat epimorphism of rings. Since schematic
+spaces have flat epimorphisms of rings as restriction maps, the restriction
+morphisms between their basic open subsets are flat immersions—actually,
+between all their open subsets—.
+In other words, schematic spaces are
+colimits of (certain) affine schematic spaces via flat immersions. This class
+of morphisms was first shown to be important in the context of descent
+problems in [3].
+Lemma 2.9. If f : X → Y is a flat immersions, f ♯
+x : OY,f(x) → OX,x are flat
+epimorphisms of rings for all x ∈ X.
+Proof. They are flat by definition and the condition on the diagonal trivially
+translates to OY,f(x) ⊗OX,x OY,f(x) → OY,f(x) being an isomorphism.
+Recall that an open immersion of schemes is a flat monomorphism (locally)
+of finite presentation. As such, flat immersions are like "open immersions",
+but without the finite presentation condition. The reader might notice the
+analogy with the étale and pro-étale topologies for schemes. This justifies
+the following notation:
+Definition 2.10. Let X be a schematic space. We define XwZar to be the site
+of flat immersions with target X, whose covers are given by finite and jointly
+faithfully flat families of flat immersions. Similarly, we define SchFinwZar
+to be the "big" site of flat immersions.
+These sites present a number of interesting pathologies that we will
+describe in more detail in future papers. We shall remark a few of them:
+• The category XwZar is not small, only its localization (XwZar)qc. Each
+qc-isomorphism class of open immersions is identified with a subset of
+Spec(X), but before localization, the collection of representatives is as
+large as the entire class of finite posets.
+• Since qc-isomorphisms are both flat immersions and covers, yet they
+are not isomorphisms, a standard descent argument shows that sheaves
+map qc-isomorphisms to isomorphisms—Category theorists sometimes
+call morphisms with such property local isomorphisms—. In particular,
+functors of points are not sheaves, because they determine spaces up
+to isomorphism, so the site XwZar and its bigger analogue are not
+7
+
+subcanonical. However, one can show that, for any Y ∈ XwZar and
+sheaf F ∈ XwZar, there are natural bijections
+HomPSh(XwZar)(F, HomXwZar(−, Y )) ≃
+≃ HomSh(XwZar)(F, Hom(XwZar)qc(−, Y ))
+In other words, the functor of points in the localization satisfies the
+universal property of sheafification.
+• We have avoided talking about sheafification in the previous points
+because, due to potential size issues, we cannot guarantee that such
+functor exists in XwZar—this is related to the inability to find bounds
+for refinements of covers, which may lead to pathologies, as it happens
+with the fpqc topology of schemes, see [12, Theorem 5.5]—; but the
+good news is that it does exist (XwZar)qc. I.e. we can sheafify presheaves
+that factor through qc-isomorphism, which will be enough in all natural
+situations.
+• Endowing XwZar with the natural sheaf of rings, it is possible to show
+that Qcoh(XwZar) ≃ Qcoh(X).
+As it happens with schemes and open immersions, it is obvious that if
+X = {Xi}i∈I is a diagram of schematic spaces and the transition morphisms
+Xi → Xj (for any i → j) are flat immersions, taking colimi Xi in the
+category of ringed posets yields Spec(colimi Xi) = colimi Spec(Xi) and the
+resulting space is a gluing of affine schemes via flat monomorphisms of affine
+schemes. The problem is that, in general, it is very difficult to determine
+if colimi Xi is schematic or not, due to the combinatorial nature of the
+definition of schematicity and the surprisingly subtle description of colimits
+of finite posets (see [1, Proposition 2.4]).
+Our solution will be defining an object "equivalent" to colimi Xi in the
+sense of representing the same locally ringed space, but whose combinatorial
+nature is elementary. This will be done in Sections 5 and 6. The result will
+be called cylinder space, denoted Cyl(X).
+This construction is central in the theory of schematic spaces will have
+applications that are beyond our purposes here, but the goal for this paper is
+to study descent properties with respect to the topology of flat immersions.
+For instance, let us consider the case of the étale fundamental groupoid. It
+clearly defines a functor
+Πét
+1 : SchFin → GpdStone
+8
+
+valued in the strict 2-category of Stone groupoids. Proving the Seifert-Van
+Kampen Theorem in its general form—for the topology of flat immersions—
+essentially amounts to saying that Πét
+1 maps colimits to 2-colimits.
+This
+will be the same as saying that (Πét
+1 )op is a 2-sheaf —thus it maps qc-
+isomorphisms to equivalences—.
+By the properties of these sites, this is
+equivalent to proving that it maps objects "qc-equivalent to colimits"—our
+cylinder spaces—to 2-colimits. However, our abstract descent result for the
+topology of flat immersions and cylinders will show that it is enough to prove
+that it is a 2-sheaf in the combinatorial topology. Such statement amounts
+to showing that Πét
+1 maps a very specific kind of cylinders to 2-colimits; and
+in some particular cases, this will even be formal.
+3
+Categories of C -data
+Without further ado, let C be a 1-category and pos be the category of finite
+posets—or arbitrary posets, being careful in that case with set-theoretic size
+considerations—. For a given poset X and x ∈ X, let Ux = {x′ ≥ x} denote
+the minimal open neighborhood of the point X. The following is well-known:
+Lemma 3.1. If C has finite limits and X ∈ pos, there is an equivalence
+Sh(X, C) ≃ [X, C]
+between the categories of C-valued sheaves on X and functors X → C.
+Proof. Each sheaf gives a functor defined by its stalks—sections at the
+minimal open neighborhoods—and restrictions morphisms.
+The converse
+follows from the sheaf condition and the fact that the {Ux}x∈X are a basis
+for the topology, so for any open U ⊆ X and functor F : X → C, one defines
+its "sections" on U as F(U) = limx∈U F(x).
+Now let us consider the functor to the 1-category of categories—big
+enough so that C ∈ Cat—
+C -data: pos → Cat
+X �→ [X, C]
+f �→ f −1.
+Definition 3.2. For any C, the cateory of C -data is the fibered category
+over C defined by the Grothendieck construction applied to the previous
+functor. Explicitly:
+9
+
+• Ob(C -data) = {F
+not
+≡ (X, F) : X ∈ pos and F ∈ [X, C]};
+• HomC -data((X, F), (Y, G)) = {f : X → Y and f ♯: f −1G → F};
+• | − |: C -data → pos is the "underlying poset" structure functor.
+Notation 3.3. We will usually denote F
+not
+≡ (X, F) and X = |F|, unless
+C = CRing is the category of commutative rings, in which case C -data
+is the category of ringed posets and we will keep the traditional notation
+(X, OX).
+Furthermore, for any F and x ≤ y ∈ |F|, we will denote its
+"restriction morphisms" by Fxy : F(x) → F(y).
+Remark 3.4. Note that the construction of C -data is functorial on the
+category: if Φ: C → D is a functor, we have Φ∗ : C -data → D -data induced
+by post-composition.
+This category comes with a natural inclusion functor
+iC : Cop → C -data
+c �→ (⋆, c)
+analogue to the "diagonal inclusion" in categories of diagrams of a fixed
+shape. Due to the choice of ⋆ as the final object in pos, we have the following:
+Lemma 3.5. If C has finite limits (resp. colimits), the functor iC has a left
+(resp. right) adjoint Γ ≡ ΓC : C -data → Cop (resp. L) called the sections
+(resp. cosections) functor. Explicitly, Γ(F) = lim F (resp. Γ(F) = colim F).
+Remark 3.6. The terminology of Lemma 3.5 comes from the equivalence
+of Lemma 3.1. Of course, one may assume no hypothesis on C and define
+sections via Yoneda at the level of [Cop, Set] -data, only to ask if these
+"sheaves of sections" are representable on a case-by-case basis. One may
+also interpret sections via projections to the terminal poset π: X → ⋆ by
+constructing π∗ right adjoint to π−1.
+Example 3.7 (Locally representable functors). As a simple application of
+this terminology, we will give a "structured" interpretation of the concept of
+locally representable functor. Indeed, let Y : C → [Cop, Set] be the Yoneda
+embedding for C and Y∗ : C -data → [Cop, Set] -data the—fully faithful—
+induced functor. One may think of an object in the image of Y∗ as a "locally
+representable functor".
+Note that, if C has finite limits, the sections of
+such an object are representable by the sections of the original C-datum.
+Additionally, we shall consider the Yoneda embedding for C -data, that is
+Y ′ : C -data → [C -dataop, Set]. At this stage, we define a third functor
+D: [Cop, Set] -data → [C -dataop, Set]
+X �→ Hom[Cop,Set] -data(Y∗(−), X)
+10
+
+such that D ◦ Y∗ = Y ′—since Y∗ is fully faithful—. We leave as an exercise
+to the reader checking that D is fully faithful itself—recall that categories
+of presheaves are compactly generated by their representable functors—. In
+particular, if X is such that D(X) is representable by some F ∈ C -data,
+one has that Y∗(F) ≃ X, in other words, "representing each X(p) by some
+Fp ∈ C for each p ∈ |X| in a compatible way is equivalent to representing X
+by a C-datum F with F(p) = Fp".
+One of the main advantages of considering C -data over categories of
+diagrams of fixed shape is that it inherits the natural 2-categorical structure
+of pos.
+More precisely, recall that pos is a strict 2-category with its 2-
+morphisms being, for each X, Y ∈ pos,
+HomHompos(X,Y )(f, g) =
+�
+⋆ if f ≤ g
+∅ otherwise.
+If f, g: F → G are morphisms in C -data and |f| ≤ |g| in pos, we
+have a natural transformation rfg : f −1G → g−1G given, at each x ∈ |F|,
+by the restriction morphisms of G. We simply ask this arrow to induce a
+commutative triangle, i.e. we define our 2-morphisms to be:
+HomHomC -data(F,G)(f, g) =
+�
+⋆ if f ≤ g and g♯ = rfg ◦ f ♯
+∅ otherwise.
+We note that this structure generalizes the partial order defined in [7] to
+study naif homotopy types of ringed posets. We also remark that C -data is
+actually a pos-enriched category.
+It is easy to check that, if C has finite limits (resp. colimits), then C -data
+has finite colimits (resp. limits), described in an analogous way as in the
+category of ringed posets (or spaces). To approach descent problems, we are
+interested in computing colimits of C -data, or in other words, describing
+the sections functor of the inclusion
+iC -dataop : C -data → (C -data)op -data;
+but it turns out that we can obtain, up to a certain to-be-introduced notion
+of weak equivalence, a more explicit description of these colimits that does
+not require us to perform any 1-categorical operations on either pos or C.
+We will call this construction the "cylinder functor". The context in which
+it arises naturally employs the 2-categorical structure of C -data, hence, for
+this and other reasons, we shall devote the next section to briefly describe
+D -data for D a strict 2-category.
+11
+
+4
+The 2-categorical case
+Let D be a strict 2-category and endow posets with the trivial 2-categorical
+structure. Among other possibilities, we shall consider the categories of
+• pseudofunctors X → D and pseudonatural transformations, [X, D];
+• pseudofunctors X → D and lax natural transformations, [X, D]Lax.
+The Grothendieck construction for each of these possibilities now yields,
+as in Definition 3.2, two different 1-categories, denoted for emphasis as
+D -data and D -dataLax respectively. In both cases, their objects are pairs
+(X, F) of a finite poset and a pseudofunctor, with the only difference being
+that a morphism (f, f ♯): F → G is defined by a pseudonatural transformation
+f ♯ when considering it in D -data and by a Lax natural transformation
+when considering it in D -dataLax.
+Note that, if D is pos-enriched—as
+is the case when D = C -data for a 1-category C—, defining such a lax
+natural transformation amounts to giving, for each p ≤ q ∈ |F|, 1-morphisms
+αp : F(p) → G(f(p)) such that
+Gf(p)f(q) ◦ αp ≤ αq ◦ Fpq,
+rather than asking for strict equality.
+Furthermore, in order to turn the inclusion functors
+iD : Dop → D -data,
+iLax
+D
+: Dop → D -dataLax,
+D -data → D -dataLax
+into pseudofunctors, we need to endow both categories of data with the same
+lax 2-categorical structure, whose 2-morphisms are:
+HomHomD -data(F,G)(f, g) =
+�
+η: rfg ◦ g♯ → f ♯ when |f| ≤ |g|
+∅ otherwise.
+Again, if D is pos-enriched, giving this lax natural transformation amounts
+to asking that, for |f| ≤ |g|, we only have
+rfg ◦ f ♯ ≤ g♯.
+With this structure, iD and iLax
+D
+are pseudofunctors that map any 2-morphism
+η: s → t in D to the 2-morphism defined by the natural transformation η,
+since riD(s)iD(t) is the identity and the underlying posets are singletons.
+Finally, as in the 1-categorical case, and almost by definition, we have:
+12
+
+Proposition 4.1. The left 2-adjoint of iD (resp. iLax
+D ) is, if it exists, the
+pseudolimit (resp.
+lax limit) of the structure pseudofunctor.
+We call it
+sections (resp. lax sections) functor and denote it by Γ ≡ ΓD (resp. LaxΓ).
+5
+The Cylinder Functor
+Now we construct the lax sections functor for the 2-category D = C -dataop
+with C a 1-category, that is, the lax colimit functor in C -data. We begin
+with the explicit description:
+Definition 5.1. For any X ∈ (C -data)op -data, we define the cylinder of
+X as the C-datum Cyl(X) such that:
+• As a set, |Cyl(X)| = �
+p∈|X |X(p)|. We endow it with the partial order
+induced by those of |X(p)| and setting that xp ≤ yq—with xp ∈ |X(p)|
+and yq ∈ |X(q)|—whenever xp ≤ Xpq(yq).
+• The structure functor is Cyl(X)(xp) = X(p)(xp) on objects, and its
+restriction morphisms are given by X(p)xpx′p in each X(p) and by
+(Xpq)♯
+yq : Cyl(X)(yq) → Cyl(X)(Xpq(yq))
+when p ≤ q.
+It is easy to check that this construction is functorial, thus we have
+Cyl: (C -data)op -data → C -data .
+Lemma 5.2. If C = ⋆, hence C -data = pos, the functor Cyl coincides up to
+natural isomorphism with the lax sections functor of the inclusion i⋆ -dataop.
+In other words, pos has pos-indexed lax colimits, described by Cyl.
+Proof. We will check that, for any Y ∈ C -data and X ∈ (C -data)op -data,
+there are functorial isomorphisms of categories
+Hompos(Cyl(X), Y ) ∼
+→ Homposop -dataLax(X, Y ).
+Since Y ≡ iC -dataop(Y ) has the terminal category ⋆ as underlying poset, there
+is an isomorphism Homposop -dataLax(X, Y ) ≃ Hom[X,pos]Lax(X, Y ), where
+X ≡ |X|. Now, given a morphism f : Cyl(X) → Y , we have, by construction,
+a family of morphisms {fp : X(p) → Y }p∈X that verify fp◦Xpq ≤ fq for p ≥ q.
+This is exactly the information that defines a lax natural transformation
+X → Y : giving, for each p ∈ X, an arrow X(p) → Y (p) = Y in pos and,
+13
+
+for each p ≤ q, a 2-morphism on the corresponding diagram, which amounts
+to asking that the previous inequalities hold. The converse follows from the
+same argument: given g: X → Y , the g♯
+p are exactly the morphisms fp.
+Finally, saying that two morphisms f, g: Cyl(X) → Y verify f ≤ g is
+just saying that fp ≤ gq for all p ∈ X—with the previous notations—. This
+is precisely the notion of 2-morphism in posop -dataLax.
+Proposition 5.3. For any category C, the functor Cyl coincides up to
+natural isomorphism with the lax sections functor of the inclusion iC -dataop.
+I.e. C -data has pos-indexed lax colimits and they are described by Cyl.
+Proof. Again, we check that for Y ∈ C -data and X ∈ (C -data)op -data,
+there are functorial isomorphisms of categories
+HomC -data(Cyl(X), Y ) ∼
+→ Hom(C -data)op -dataLax(X, Y ).
+The topological part of the proof has been taken care of in Lemma 5.2, so we
+only need to check that such isomorphism extends to the level of C-valued
+functors.
+Given f : Cyl(X) → Y , using the same notations as in the aforementioned
+Lemma, we have morphisms fp such that fp◦Xpq ≤ fq : X(q) → Y topologically.
+This is a 2-morphism of C -data because, for each yq ∈ |X(q)|,
+Y(fp◦Xpq)(yq) ◦ (fp ◦ Xpq)♯
+yq = Y(fp◦Xpq)(yq) ◦ (Xpq)♯
+yq ◦ (fp)♯
+Xpq(yq) = (fq)♯
+yq;
+but by the definition of Cyl(X) and f, for all p ≤ q and xp = Xpq(yq),
+Cyl(X)xpyq ◦ (fp)♯
+xp = (fq)♯
+yq,
+(5.1)
+where Cyl(X)xpyq = (Xpq)♯
+yq, as desired. The converse follows from the same
+relations.
+At the level of morphisms, if we have arrows f, g: Cyl(X) → Y with
+f ≤ g in C -data, they verify |f| ≤ |g| in pos and, for all xp ∈ Cyl(X),
+g♯
+xp ◦ Yf(xp)g(xp) = f ♯
+xp.
+(5.2)
+If {fq : X(p) → Y } and {gp : X(p) → Y } are their corresponding families of
+morphisms in [X, (C -data)op]Lax, there only remains to check that fp ≤ gp
+for all p ∈ X. Once again, |fp| ≤ |gp| by Lemma 5.2, so we complete the
+proof by remarking that, for each xp ∈ |X(p)|, the fact that the equation 5.2
+holds is equivalent to fp ≤ gp in C -data.
+14
+
+Note that C -data is actually a pos-enriched category, hence the universal
+property of Cyl is necessarily given by an isomorphism of categories, rather
+than an equivalence. This means that, provided that colimits of C -data also
+exist, there is a natural transformation to the 1-categorical sections:
+Cyl → ΓC -dataop.
+One can make a case for this natural transformation being a "weak
+equivalence" relative to certain descent problems for information codified
+in a given collection of C-datum. We will not introduce the full terminology
+here, since that would be a technical exercise far past our aim, but Sections
+2 and 6 will put us in a particular case that hints towards this direction.
+Example 5.4. A very important remark is that, not only Cyl ◦ iC -dataop is
+trivially the identity, but that every C-datum is the "cilinder of its points".
+More precisely, for any C, there is a second "obvious" inclusion functor given
+by post-composition with iop
+C :
+(iop
+C )∗ : C -data → (C -data)op -data;
+such that (iop
+C )∗(F) has the same underlying poset as F, but we "replace"
+each F(p) by the constant datum (⋆, F(p)). It is obvious that Cyl ◦ (iop
+C )∗ is
+also the identity. Furthermore, there is a natural transformation
+ηC : (iop
+C )∗ → iC -dataop
+given by the natural projections to the terminal poset and identities in C,
+which will be relevant when dealing with descent problems.
+Proposition 5.5. The functor Cyl commutes with finite fibered products.
+Proof. Exercise to the reader: it follows from the explicit construction.
+6
+The schematic cylinder
+The schematic category introduced in Section 2 is a non-full subcategory
+of CRing -data, where CRing denotes the category of commutative rings
+with unit. In particular, the cylinder functor restricts to
+Cyl: SchFinop -data → CRing -data .
+The next few pages are devoted to characterizing SchFinop-data whose
+cylinder spaces are schematic. The first justification is that such lax colimit
+represents up to "qc-isomorphism"—see discussion after the next Lemma—
+the same locally ringed space:
+15
+
+Lemma 6.1. Given X ∈ SchFinop -data, the natural morphism of ringed
+spaces Cyl(X) → Γ(X) induces an isomorphism Spec(Cyl(X)) ∼
+→ Spec(Γ(X)).
+Proof. This follows from the fact that colimits commute with colimits.
+We would like to say that Cyl(X) → Γ(X) is a qc-isomorphism, but
+note that we have not checked—and will not check—whether or not Γ(X) is
+schematic. However, it will be sufficient to check schematicity of Cyl(X) for
+our applications—and crucial, since we would not be able to guarantee the
+stability under qc-isomorphisms of the properties and constructions we are
+interested in dealing with otherwise—.
+Definition 6.2. A ringed poset X is said to be pseudo-schematic if it has
+flat epimorphisms of rings as restriction maps.
+Definition 6.3. A ringed poset X is Mod-affine if π: X → (⋆, OX(X))
+induces an adjoint equivalence (π∗ ⊣ π∗): Qcoh(X) → Mod(OX(X)). We
+say that X is affine if it is schematic and Mod-affine.
+Example 6.4. Any ringed poset with a minimum X = Ux is Mod-affine.
+Remark 6.5. If X is pseudo-schematic, Qcoh(X) is a Grothendieck abelian
+category. In particular, if X is also Mod-affine, π∗ is exact.
+Lemma 6.6. If X is pseudo-schematic and Mod-affine, the natural morphism
+OX(X) → �
+x∈X OX,x is faithfully flat.
+Proof. It suffices to see that �
+x∈X Spec(OX,x) → Spec(OX(X)) is surjective.
+Given a prime p ⊆ OX(X) with non-zero residue field κ(p), the equivalence
+gives a non-zero module π∗κ(p) ̸= 0, thus there is some x ∈ X such that
+(π∗κ(p))x ≃ κ(p)⊗OX(X) OX,x ̸= 0. Geometrically, this means that the fiber
+of p via Spec(OX,x) → Spec(OX(X)) is non-empty, so we win.
+Definition 6.7. A morphism of ringed spaces f : X → Y between pseudo-
+schematic spaces will be called a qc-isomorphism if f −1(Uy) is Mod-affine
+for all y ∈ Y and f♯: OY → f∗OX is an isomorphism.
+Example 6.8. Any ringed poset with a minimum X = Ux is qc-isomorphic
+to (⋆, OX,x) via the natural projection.
+In the schematic category, Definition 6.7 restricts to the usual one. In
+this generality, we cannot even guarantee that the notion is stable under
+composition and base change, so the reader must think of it as an abbreviated
+way of storing information whose purpose will soon become clear.
+We
+16
+
+would like to remark, however, that the notion of Mod-affinity and the
+concept of qc-isomorphism it produces are particular cases of more abstract
+constructions for C -data.
+Lemma 6.9. Given X ∈ SchFinop -data whose restriction morphisms are
+flat immersions, Cyl(X) is pseudo-schematic.
+Proof. It follows from the construction, Lemma 2.9 and Remark 2.2.
+Now, given X ∈ SchFinop -data and p ∈ |X|, denote by Up the datum
+induced on the open subset Up ⊆ |X|. We have qc-isomorphisms of ringed
+spaces
+πp: Cyl(Up) → X(p).
+In general, given an open subset U ⊂ |X| and endowing it with the induced
+structure functor, we have open subsets
+iU : Cyl(U) ֒→ Cyl(X);
+so, for every p, q ∈ |X| and fixed t ≤ p, q, we have natural morphisms
+ip
+pq : Cyl(Up ∩ Uq) → Cyl(Up),
+iq
+pq : Cyl(Up ∩ Uq) → Cyl(Uq);
+which, composing with the previous projections, induce
+πt
+pq : Cyl(Up ∩ Uq) → X(p) ×X(t) X(q).
+Note that the space on the right hand side is always schematic and that, for
+every (xp, yq) ∈ |X(p) ×X(t) X(q)|, we have
+(πt
+pq)−1(U(xp,yq)) = Uxp ∩ Uyq ⊆ |Cyl(Up ∩ Uq)| ⊆ |Cyl(X)|.
+Theorem 6.10. Given X ∈ SchFinop -data whose restriction morphisms
+are flat immersions, Cyl(X) is schematic if and only if for every t ≤ p, q in
+|X|, the natural morphism πt
+pq is a qc-isomorphism (a priori of ringed posets,
+a posteriori of schematic spaces).
+Proof. With the technology introduced in this paper, we can only prove the
+"if" part, which will be the one used in our applications. Indeed, if πt
+pq is
+a qc-isomorphism, Uxp ∩ Uyq is Mod-affine for every (xp, yq) as before and
+its global sections are isomorphic to OX(p),xp ⊗OX(t),zt OX(q),yq, with zt the
+common image of xp and yq. Now, Lemma 6.6 translates exactly into the
+conditions of Definition 2.1.
+17
+
+For morphisms f : X → Y in SchFinop -data, we can modify the previous
+construction to obtain, for each p ∈ |X| and q ≥ f(p),
+ρf
+pq : Cyl(Up ∩ f −1(Uq)) → X(p) ×Y(f(p)) Y(q).
+Theorem 6.11. Given a morphism f : X → Y in SchFinop -data and such
+that Cyl(X) and Cyl(Y) are schematic, Cyl(f) is schematic if and only if for
+every p, q ≥ f(p), the map ρf
+pq is a qc-isomorphism.
+Proof. We only prove the "if" part, which follows from the same results as
+Theorem 6.10 and the fact that, for (xp, yq) ∈ X(p) ×Y(f(p)) Y(q), one has
+ρ−1
+pq (U(xp,yq)) = Uxp ∩ Cyl(f)−1(Uyq).
+Remark 6.12. Note that, applied to a datum X with X(p) = (⋆, Ap) for all
+p, Theorems 6.10 and 6.11 restrict to the usual Definition of schematicity.
+See this in view of Example 5.4.
+Definition 6.13. Given a finite family of flat immersions {Ui → X}i∈I,
+we define the Nerve datum associated to it as U ∈ SchFinop -data with
+underlying poset |U| = P∗(I)—non-empty parts of I—and U(∆) = �
+i∈∆ Ui
+—fibered product over X—.
+Note that U comes equipped with a morphism U → X ≡ iC -data(X).
+Corollary 6.14. If {Ui → X} a finite family of flat immersions, Cyl(U)
+is schematic and the morphism Cyl(U) → X is a schematic flat immersion,
+which is a qc-isomorphism if and only if the family is a covering.
+Proof. First, we check the condition of Theorem 6.10: for ∆1, ∆2 ∈ |U|,
+U∆1 ∩ U∆2 = U∆1∪∆2; but Cyl(U∆1∪∆2) → U(∆1 ∪ ∆2) is a qc-isomorphism,
+with U(∆1 ∪ ∆2) ≃ U(∆1) ×U(∆1∩∆2) U(∆2) by definition. Schematicity of
+Cyl(f) follows from Theorem 6.11 and a similar argument.
+The morphism Cyl(f) is flat by the local construction and its diagonal
+is a qc-isomorphism because, by Proposition 5.5,
+Cyl(U) → Cyl(U) ×X Cyl(U) ≃ Cyl(U ×X U),
+and a morphism of SchFinop -data that is topologically the identity and a
+qc-isomorphism at each point, induces a qc-isomorphism between cylinder
+spaces (as shown by an easy computation).
+Finally, Cyl(f) being faithfully flat (hence a qc-isomorphism) is clearly
+equivalent to {f∆ : U(∆) → X}∆ being a covering family, which happens if
+and only if the original family was a covering.
+18
+
+7
+Descent and the topos of flat immersions
+Now we use the technology of the previous section to describe colimits in
+a sheaf-theoretic manner. In the following definition—if appropriate—, one
+shall consider SchFin as a 1-category with the trivial 2-categorical structure.
+Definition 7.1. Let C be a 1-category (resp. strict 2-category). A geometric
+datum is a functor (resp. pseudofunctor) Dat: SchFin → C that maps qc-
+isomorphisms to isomorphisms (resp. equivalences); in other words, one that
+factors through SchFinqc.
+Example 7.2. The functors Spec: SchFin → LRS, Qcoh: SchFin → Catop
+—with values in the 2-category of categories—and Πét
+1 : SchFin → GpdStone
+—with values in the 2-category of Stone groupoids—are all geometric data.
+In the discussion that follows, let us assume that C is a 1-category; the
+argument also works for 2-categories, replacing isomorphisms by equivalences.
+In Example 5.4 we saw that there are two natural immersions of any category
+of C-data into its category (C -data)op -data. In this case, there is a natural
+transformation between functors in [SchFin, SchFinop
+qc -data]:
+ηC : (iop
+C )∗ → iC -dataop.
+Remark 7.3. For general ringed posets, this natural transformation is induced
+by the morphisms (⋆, OX,x) → X, which are not schematic. That is one of
+the reasons to consider the localized category, where it is induced by the
+triangles (⋆, OX,x) ← Ux → X.
+Now, given a geometric datum Dat: SchFinqc → C, we define
+Dat := Dat∗ ◦ (iop
+C )∗
+Dat ≡ Dat ◦ iC -dataop;
+where Dat(X) is the Cop-datum with |Dat(X)| = |X| and structure functor
+Dat(X)(x) = Dat(⋆, OX,x). These induce a natural transformation
+ηDat : Dat → Dat
+between functors in [SchFin, Cop -data], given by the projection to the point
+at the topological level and by the morphisms in C
+Dat(⋆, OX,x) ∼
+← Dat(Ux) → Dat(X).
+Composing with the sections functor Γ: Cop -data → C—always assuming
+that C has enough limits—, one arrives to the following definition:
+19
+
+Definition 7.4. We say that a geometric datum Dat satisfies internal descent
+if Γ(ηDat): Γ∗ ◦ Dat → Γ∗ ◦ Dat ≡ Dat is an isomorphism in [SchFin, C].
+Example 7.5. The datum Qcoh: SchFin → Catop satisfies internal descent.
+Indeed, since Qcoh(⋆, OX,x) = Mod(OX,x), this amounts to proving that
+the natural functor
+Qcoh(X) → 2-limx∈X Mod(OX,x)
+is an equivalence of categories. This holds because quasi-coherent modules
+on ringed posets are collections of {Mx}x∈X with Mx an OX,x-module such
+that, for all x ≤ y, the natural morphisms Mx ⊗OX,x OX,y → My are
+isomorphisms; which coincides with the description of this pseudolimit in
+Cat. The reader may notice that this result holds for arbitrary ringed posets,
+but that it tacitly requires the tensor-Hom adjunction for modules to hold.
+If one wants to extend the result to quasi-coherent sheaves of algebras, it is
+necessary to assume that X is, at least, pseudo-schematic. This is because
+base changes by flat epimorphisms of rings satisfy said adjunction (left as an
+algebra exercise to the reader).
+Proposition 7.6 (External descent for nerves). If X is a schematic space,
+{fi : Ui → X} is a covering by flat immersions with associated nerve datum
+U and Dat is a geometric datum satisfying internal descent, then there is a
+natural isomorphism
+colim∆∈|U| Dat(U(∆)) ∼
+→ Dat(X).
+Proof. By Corollary 6.14, Cyl(U) → X is a qc-isomorphism, and since Dat is
+geometric, one has that Dat(Cyl(U)) ≃ Dat(X). Since Dat satisfies internal
+descent—applied at each U(∆)—and colimits commute with colimits,
+Dat(Cyl(U)) ≃ colimx∆∈Cyl(U) Dat(⋆, OU(∆),x∆) ≃
+≃ colim∆∈|U| colimx∆∈|U(∆)| OU(∆),x∆) ≃ colim∆∈|U| Dat(U(∆)),
+which completes the proof.
+Example 7.7. In the situation of Proposition 7.6 and thanks to Example 7.5,
+we obtain that Qcoh(X) ≃ 2-lim∆∈|U| Qcoh(U(∆)). In particular, being
+quasi-coherent is local in the topology of flat immersions.
+Theorem 7.8 (External descent for topoi). If SchFinτ denotes the (big)
+site of schematic spaces with the combinatorial topology and SchFinwZar
+20
+
+denotes the (big) site of flat immersions, the natural inclusion defines an
+equivalence of topoi
+Sh((SchFinqc)τ) ≃ Sh(SchFinwZar).
+Similarly, it induces equivalences between respective categories of C-valued
+sheaves (resp. stacks) for any 1-category (resp. 2-category) that has finite
+poset-indexed colimits.
+Proof. This is simply a reinterpretation of Proposition 7.6 in terms of the
+language of Section 2.1: sheaves in Sh(SchFinwZar) map qc-isomorphisms
+to isomorphisms, so they are geometric data in the sense fo this section. We
+shall remark that the analogous equivalence between small topoi does not
+hold—a priori—because cylinders change the base space.
+Remark 7.9. Thanks to the sheaf condition, it can be shown that a sheaf F in
+Sh(SchFinτ) maps qc-isomorphisms to isomorphisms if and only if, for every
+affine schematic space X, the natural morphism F(⋆, OX(X)) → F(X) is
+an isomorphism.
+In other words, to prove that a presheaf in the schematic category is a
+sheaf in the topology of flat immersions, it is enough to see that it maps
+qc-isomorphisms to isomorphisms and that it is a sheaf in the combinatorial
+topology for every poset. This is similar to what happens in the category of
+schemes for the set-theoretic topology and the Zariski site. A consequence
+for qc-qs schemes is the following slogan:
+In the category of qc-qs schemes, any Zariski sheaf that can be
+studied through finite models is a sheaf in the topology of flat
+monomorphisms of schemes and finite coverings.
+Theorem 7.8 makes the meaning of can be studied through precise: such a
+sheaf F must induce a geometric datum on the schematic category that is a
+sheaf in the combinatorial topology.
+8
+Example: Seifert-Van Kampen Theorem
+A less trivial application comes from the étale fundamental groupoid—and
+group—, as promised. Let us consider the pseudofunctor
+Πét
+1 : SchFin → GpdStone
+21
+
+to the 2-category of Stone groupoids. By Remark 2.6 it is a geometric datum,
+so to apply the results of the previous section it is enough to see that it is
+a sheaf in the combinatorial topology. This follows quite easily in two steps.
+Before that, we highlight that the category of finite étale covers defined
+without any detail in Theorem 2.5 can be described in terms of quasi-coherent
+sheaves of algebras, as done in [4]; more precisely:
+RÉt(X) =
+�
+"opposite category of quasi-coherent algebras A
+such that OX,x → Ax is a finite étale ring map".
+(8.1)
+Lemma 8.1. The pseudofunctor RÉt: SchFinqc → Cat defined on objects
+by1 X �→ RÉt(X) satisfies internal descent.
+Proof. We have to show that RÉt(X) ≃ 2-limx∈X RÉt(⋆, OX,x). Since it is
+a subcategory of the category of quasi-coherent algebras, this follows from
+Example 7.5—bearing in mind the remark at the end—and the fact that the
+property of being finite étale at stalks is obviously local in this sense.
+Proposition 8.2. The pseudofunctor Πét
+1 : SchFin → GpdStone satisfies
+internal descent.
+Proof. Following the notations of Section 7. By Lemma 8.1 we know that
+the natural transformation ηRÉt : RÉt → RÉt induces an equivalence after
+taking sections. Composing Πét
+1 with the 2-functor Φ: GpdStone → Catop
+such that G �→ G-FinSet—with continuous action—, we obtain a commutative
+square of functors [SchFin, Catop]
+RÉt
+�
+�
+RÉt
+�
+Πét
+1 -FinSet
+ηΦ◦Πét
+1 � Πét
+1 -FinSet;
+where the vertical arrows are isomorphisms after taking sections by the
+Galois Theorem for fundamental groupoids, hence Γ(ηΦ◦Πét
+1 ) an isomorphism.
+Finally, since Φ well known to commute with pseudocolimits, one has
+that Γ(ηΦ◦Πét
+1 ) ≃ Φ ◦ Γ(ηΠét
+1 ); and since this map is an equivalence and Φ
+is (2-)conservative by [10, 3.11], Γ(ηΠét
+1 ) is an equivalence, which proves the
+statement.
+1On 1-morphisms, we send each f : X → Y to the inverse image functor; since SchFin
+is considered as a 1-category, it only remains to specify invertible equivalences in Cat that
+make all suitable diagrams commute, but we can and do choose those to be the ones given
+by the universal property of tensor products.
+22
+
+Example 8.3. If a schematic space X satisfies that Πét
+1 ((⋆, OX,x)) = {⋆}—the
+trivial 2-category—for all x ∈ X, Proposition 8.2 yields
+Πét
+1 (X) ≃ 2-colimx∈|X|{⋆} ≃
+�
+Π1(|X|),
+where the hat denotes the profinite completion.
+Theorem 8.4. Let X be a schematic space and {Ui → X} be a covering by
+flat immersions with associated nerve datum U. Then, the natural morphism
+2-colim∆∈|U| Πét
+1 (U(∆)) → Πét
+1 (X)
+is an equivalence of topological groupoids. In other words, the functor Πét
+1
+is a (co)stack in the topology of flat immersions.
+Proof. It follows from 8.2 and Theorem 7.8.
+Remark 8.5. Note that the topological fundamental groupoid of |U| is always
+trivial, since any space of parts has generic point and thus is contractible to
+a point. One can give the statement of the Theorem in greater generality,
+for any X ∈ SchFinop-datum such that Cyl(X) is schematic; and in that
+case the topological fundamental groupoid of |X| plays a role.
+Corollary 8.6. If S is a qc-qs scheme and {Vj → S}j∈J is a finite cover by
+flat monomorphisms with associated nerve codatum V : P∗(J) → Schop—
+with Sch the category of schemes—, the natural morphism
+2-colim∆∈|V| Πét
+1 (V(∆)) → Πét
+1 (S)
+is an equivalence of Stone groupoids, i.e. the étale fundamental groupoid of
+schemes is a costack in the topology of flat monomorphisms and finite covers.
+Finally, we can very easily specialize this result to fundamental groups,
+which a formulation that we deem more natural than that of [11].
+Definition 8.7. Given a schematic space X and a cover by flat immersions
+with associated nerve datum U extended to P(I) by U(∅) = X, a system of
+base points x⋆ is an object
+x⋆ ∈ Ob(2-lim∆∈|U|(Πét
+1 (U(∆)))).
+In other words: x⋆ is given by a collection geometric points x∆ of U(∆)
+for each ∆ and a collection of Tannaka paths
+ϕ∆∆′ : Fibx∆ ◦ RÉt(X)(∆ → ∆′) ∼
+→ Fibx∆′
+for each ∆ ≤ ∆′. Let us denote by x = x∅ the geometric point of X given
+by this collection.
+23
+
+Theorem 8.8. Let X be schematic and connected, U the nerve codatum
+associated to some covering by flat immersions such that U(∆) is connected,
+and x⋆ a system of base points. Then there is an isomorphism of topological
+groups
+colim∆∈|U| πét
+1 (U(∆), x∆) ∼
+→ πét
+1 (X, x)
+induced by conjugation the ϕ∆∆′.
+Proof. Since X is connected, the natural inclusion πét
+1 (X, x) → Πét
+1 (X) is an
+equivalence. Let GrStone ⊆ GpdStone be the category of profinite groups,
+which one may think set-theoretically or as Top-enriched categories. Define
+the datum
+πét
+1 (−, x⋆): |U| → Grop
+Stone
+∆ → πét
+1 (U(∆), x∆)
+whose restriction morphisms given by conjugation with the ϕ∆∆′.
+Since
+U(∆) is connected for every ∆, the natural transformation
+πét
+1 (−, x⋆) → Πét
+1 ◦ U
+is an isomorphism of GpdStone-valued pseudofunctors, hence it induces an
+isomorphism after taking sections. From this fact and Theorem 8.4, there
+are equivalences
+2-colim∆∈|U| πét
+1 (U(∆), x∆) ∼
+→ 2-colim∆∈|U| Πét
+1 (U(∆)) ≃ Πét
+1 (X);
+where the first groupoid is identified with the 1-colimit of abstract profinite
+groups colim∆∈|U| πét
+1 (U(∆), x∆) and the last one is equivalent to πét
+1 (X, x)
+as remarked before. Since any equivalence between one-object categories is
+an isomorphism, the proof ends.
+Remark 8.9. Note that the topological Seifert-Van Kampen Theorem can be
+written in terms of C -data: if S is a quasi-compact topological space and
+π: S → X
+is a finite model, we can turn X into a Topop-datum—with Top being the
+category of topological spaces—by setting that X(x) = π−1(Ux). If each one
+of these fibers is simply connected and we assume connectedness, the result
+recovers the classical one of McCord for π1. For the higher homotopy groups,
+we are positive that should be a consequence of a Seifert-Van Kampen
+Theorem for fundamental homotopy groupoids thought as strict n-categories.
+24
+
+References
+[1] Codara, P. PhD thesis:
+A theory of partitions of partially ordered
+sets; O.M. D’Antona, V. Marra. Milano:
+Università degli studi di
+Milano. Dipartimento di Matematica, Dipartimento di Informatica e
+Comunicazione, 2008 Nov 21. 20. ciclo, Anno Accademico 2006/2007.
+[2] McCord, M. C. Singular homology groups and homotopy groups of finite
+topological spaces, Duke Math. J. 33 (1966), 465-474.
+[3] Raynaud, M. Un critère de effectivité de descente. In: Séminaire Samuel,
+Algèbre Conmutative, vol. 2, pp. 1-22 (1967-1967).
+[4] Sánchez González, J.; Tejero Prieto, C. Étale Covers and Fundamental
+Groups of Schematic Finite Spaces. Mediterr. J. Math. 19 (2022), no. 5,
+229.
+[5] Sancho de Salas, F.; Sancho de Salas, P. Affine ringed spaces and Serre’s
+criterion. Rocky Mountain J. Math. 47 (2017), no. 6, 2051–2081.
+[6] Sancho de Salas, F. Finite spaces and schemes. J. Geom. Phys. 122
+(2017), 3–27.
+[7] Sancho de Salas, F. Homotopy of finite ringed spaces. J. Homotopy Relat.
+Struct. 13 (2018), no. 3, 481–501.
+[8] Sancho de Salas, F.; Torres Sancho, J.F. Derived categories of finite
+spaces and Grothendieck duality. Mediterr. J. Math. 17 (2020), no. 3,
+Paper No. 80, 22 pp.
+[9] Sancho de Salas, F.; Sancho de Salas, P. Notes on schematic finite spaces.
+arXiv:2102.09263v1 [math.AG].
+[10] Pirashvili, I. The étale fundamental groupoid as a 2-terminal costack.
+Kyoto J. Math. 60 (2020), no. 1, 379–403.
+[11] Stix,
+J.
+A
+general
+Seifert-Van
+Kampen
+theorem
+for
+algebraic
+fundamental groups. Publ. Res. Inst. Math. Sci. 42 (2006), no. 3,
+763–786.
+[12] Waterhourse, W.C. Basically bounded functors and flat sheaves. Pacific
+J. Math. 57 (1975), no. 2, 597-610.
+25
+

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+Data-driven discovery and extrapolation of parameterized pattern-forming dynamics
+Zachary G. Nicolaou,1 Guanyu Huo,1 Yihui Chen,1 Steven L. Brunton,2 and J. Nathan Kutz1
+1Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
+2Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, USA
+Pattern-forming systems can exhibit a diverse array of complex behaviors as external parameters
+are varied, enabling a variety of useful functions in biological and engineered systems. First-principle
+derivations of the underlying transitions can be characterized using bifurcation theory on model sys-
+tems whose governing equations are known. In contrast, data-driven methods for more complicated
+and realistic systems whose governing evolution dynamics are unknown have only recently been de-
+veloped. Here, we develop a data-driven approach sparse identification for nonlinear dynamics with
+control parameters (SINDyCP) to discover dynamics for systems with adjustable control parameters,
+such as an external driving strength. We demonstrate the method on systems of varying complexity,
+ranging from discrete maps to systems of partial differential equations. To mitigate the impact of
+measurement noise, we also develop a weak formulation of SINDyCP and assess its performance
+on noisy data. We demonstrate applications including the discovery of universal pattern-formation
+equations, and their bifurcation dependencies, directly from data accessible from experiments and
+the extrapolation of predictions beyond the weakly nonlinear regime near the onset of an instability.
+Data-driven approaches to system identification are
+undergoing a revolution, spurred by the increasing avail-
+ability of computational resources, data, and the develop-
+ment of novel and reliable machine learning algorithms
+[1–3].
+The sparse identification of nonlinear dynamics
+(SINDy) is a particularly simple and flexible mathemat-
+ical approach that leverages efficient sparse optimization
+algorithms in the automated discovery of complex sys-
+tem dynamics and governing equations [4]. In this work,
+we leverage the SINDy model discovery framework to
+understand parametric dependencies and underlying bi-
+furcations in pattern forming systems. Specifically, we
+develop the SINDY with control parameters (SINDyCP)
+to discover such parameterized dynamics.
+It has been thirty years since Cross and Hohenberg’s
+seminal and authoritative review consolidating an excep-
+tionally large body of work on pattern formation across a
+broad range of physical systems [5]. Universal equations
+determined by normal forms of canonical bifurcations [6],
+such as the complex Ginzburg-Landau equation [7], gov-
+ern the formation of patterns near the onset of instabili-
+ties across scientific disciplines. Such equations continue
+to reveal insights into complex systems, including in the
+study of, for example, synchronization, biophysics, active
+matter, and quantum dynamics [8, 9].
+Despite the success of pattern-formation theory in
+modeling complex dynamics, ongoing challenges remain
+in applying such model equations more broadly. First-
+principle derivations and the computation of normal-
+form parameters in terms of physical driving parameters
+are tedious, costly, and error-prone.
+Furthermore, the
+normal-form approach is only theoretically justified in
+the weakly-nonlinear regime near the onset of an insta-
+bility, while interesting and important pattern-forming
+processes often occur far from the instability threshold.
+Recent advances in data-driven system identification are
+opening new avenues of research to address these chal-
+lenges, including a paradigm for modeling strongly non-
+linear regimes beyond the asymptotic approximations re-
+viewed by Cross and Hohenberg [5].
+The SINDy model discovery framework is particularly
+well-suited to the modern analysis of bifurcations and
+normal forms, as it generates interpretable models that
+have as few terms as possible, balancing model complex-
+ity and descriptive capability. A variety of extensions of
+the SINDy approach have been developed since its in-
+troduction. For example, SINDYc enables discovery of
+systems subject to external control signals [11, 12], while
+PDEFind [13, 14] enables discovery of spatio-temporal
+dynamics characterized by partial differential equations
+(PDEs). SINDy can also learn to disambiguate between
+parametric dependency and governing equations [15].
+Model pattern formation equations typically encode the
+effects of external drive through a number of driving pa-
+rameters, which characterize the bifurcation leading to
+the onset of instability. Several recent works establish
+system identification on pattern-forming systems rang-
+ing from closure models for fluid turbulence [16–18] to
+biochemical reactions and active active matter systems
+[19–21]. These approaches show promise, but crucially,
+they have not demonstrated the ability to extrapolate
+by detecting pattern-forming instabilities that may de-
+velop when driving parameter differ the training data.
+While there has been success for discrete maps and or-
+dinary differential equations (ODEs) [4, 22], combining
+the PDEFind and SINDYc approaches to discover pa-
+rameterized spatio-temporal dynamics poses a significant
+challenge, as we detail below.
+The key insight underlying SINDyCP is recognizing
+the need to introduce distinct libraries of possible de-
+pendencies for the dependent variables and the control
+parameters. Our approach is implemented in the open-
+source PySINDy repository [23, 24], enabling other pow-
+erful methods to be used in conjunction.
+In particu-
+arXiv:2301.02673v1  [nlin.PS]  6 Jan 2023
+
+2
+Construct library 
+and derivatives
+ from samples
+Sparse 
+regression
+Parameterized 
+equation
+Trajectories with 
+varying parameters
+Feature 
+Library,
+Parameter
+Library,
+Time
+Derivatives,
+FIG. 1. Schematic of the SINDyCP approach. Data collected from sample trajectories collected under various driving parame-
+ters are processed to construct a matrix of time derivatives, a feature library Θfeat of possible governing terms, and a parameter
+library Θpar of parametric dependencies. Sparse regression is applied on the library coefficients ξ to identify a parameterized
+governing equation.
+lar, we develop and assess a weak formulation [25–28] of
+SINDyCP, which shows excellent performance on noisy
+data. We demonstrate that the method can be easily and
+effectively employed to discover accurate parameterized
+models from the kind of data available in typical pat-
+tern formation experiments and that these parameterized
+models enable extrapolation beyond the conditions under
+which they were developed.
+Building
+the
+library.—Figure
+1
+illustrates
+the
+SINDyCP approach applied to the spatio-temporal
+evolution of four trajectories of the complex Ginzburg-
+Landau equation
+˙A = A + (1 + ib)∇2A − (1 − ic)|A|2A,
+(1)
+which
+is
+described
+by
+a
+complex
+dependent
+vari-
+able A(x, t) in two spatial dimensions x
+=
+(x, y).
+Ginzburg-Landau exhibits a stunning variety of pat-
+terns, depending on the bifurcation parameters b and
+c.
+We generate four trajectories with parameters val-
+ues (b, c) = (2.0, 1.0), (2.0, 0.75), (0.5, 0.5) and (1.0, 0.75),
+which exhibit differing dynamical phases, corresponding
+to amplitude turbulence, phase turbulence, stable waves,
+and frozen spiral glasses, respectively [7]. Our goal is to
+discover the partial differential equation for the real and
+imaginary components A = X + iY parameterized by b
+and c from time series data.
+As with most SINDy algorithms, we first form a ma-
+trix of the input data X, whose columns correspond to
+the dependent variables and whose rows correspond to
+the sample measurements of the dependent variables. In
+the case of Fig. 1, for example, X consists of two columns
+corresponding to the real and imaginary parts of A and
+4NxNyNt rows, where Nx, Ny, and Nt are the number
+of sample points in the corresponding spatio-temporal di-
+mensions; again, there are four trajectories. We then de-
+termine the temporal derivative ˙X for each sample point,
+either through numerical differentiation or through direct
+measurements.
+In basic SINDy, we define a matrix of library terms
+Θ = Θ(X) depending on the input data, which includes
+all possible terms that may be present in the differen-
+tial equation that describes the temporal derivatives.
+These terms may be built from polynomial combina-
+tions of the dependent variables and their spatial deriva-
+tives, for example, although more general libraries are
+possible. In the SINDYc approach, we augment the li-
+brary dependence with an external control signal U, i.e.,
+Θ = Θ(X, U). The library terms are typically determined
+by appending the control variables to the dependent vari-
+ables and again forming polynomials and derivatives. In
+the case in Fig. 1, we can treat the parameters as exter-
+nal control signals, U = (b, c) and apply SINDYc, but
+the traditional implementation of this approach will fail
+for PDEs, as we show.
+SINDYc aims to find a sparse linear combination of
+the library terms determined by the vector of coefficients
+ξ which minimizes the fit error
+ξ∗ = argminξ
+��� ˙X − Θ(X, U)ξ
+��� + λ |ξ|0 .
+(2)
+Crucially, all SINDy methods employ sparse regression
+(with appropriate regularization) to determine a sparse
+set of nonzero coefficients ξ∗. Such sparsity is expected in
+physically-relevant dynamics and produces parsimonious
+and interpretable models.
+A significant challenge arises when applying the tradi-
+tional SINDYc to control parameters in PDEs with ex-
+isting implementations such as PySINDy.
+The matrix
+of library terms Θ is traditionally formed by computing
+all polynomial combinations of spatial derivatives of the
+dependent and control variables. However, since the con-
+trol parameters are spatially constant, the spatial deriva-
+tives will vanish identically, leading to a singular matrix
+Θ.
+To overcome this challenge, we propose construct-
+ing a more general library through products of a feature
+
+3
+library Θfeat(X) and a parameter library Θpar(U), as
+Θ(X, U) = Θfeat(X) ⊗ Θpar(U),
+(3)
+where the product ⊗ here is defined to give the ma-
+trix consisting of all combinations of products of columns
+(computed component-wise across the row elements) be-
+tween the libraries.
+By distinguishing the feature and
+parameter library dependencies with this SINDyCP ap-
+proach, we can construct much more targeted and well-
+conditioned libraries.
+Using a feature library consisting of spatial derivatives
+up to third order and polynomials up to third order along
+with a linear parameter library, the SINDyCP approach
+easily discovers Eq. (1) in Cartesian coordinates. Details
+of the numerical integration, an animation illustrating
+the temporal evolution of the sample trajectories, and
+additional demonstrations for maps and ODEs are avail-
+able in the Supplemental Materials [29].
+Beyond weakly nonlinear theory.—SINDyCP enables
+discovery of nonlinear corrections to weakly nonlin-
+ear theory directly from data that can be gathered in
+pattern-formation experiments. To illustrate this result,
+we implement an in silico experiment of the Belousov-
+Zhabotinksy chemical reaction system. We numerically
+integrate the Oregonator model [30],
+˙CX = k1CAC2
+HCY − k2CHCXCY + k3CACHCX
+− 2k4C2
+X + DX∇2CX,
+(4a)
+˙CY = −k1CAC2
+HCY − k2CHCXCY + νk5CBCZ
++ DY ∇2CY
+(4b)
+˙CZ = 2k3CACHCX − k5CBCZ + DZ∇2CZ,
+(4c)
+which describes the evolution of oscillating chemical con-
+centrations CX, CY , and CZ for given supplied concen-
+trations CA, CB, and CH and stoichiometric coefficient
+ν, which depends on the experimental setup. We vary
+the concentration of CB and define a control parame-
+ter µ ≡ CB − Cc
+B, where Cc
+B is the critical value where
+the Hopf bifurcation occurs (see Supplementary Mate-
+rials [29] for parameter values and other details in the
+Oregonator model) to generate six trajectories with µ
+ranging from 0.02 to 0.12.
+We use a SINDyCP feature library with polynomial
+terms up to fifth order and second order spatial deriva-
+tives and a parameter library with polynomial terms up
+to second order for the control parameter µ1/2 in con-
+junction with implicit SINDy [31] to discovers a highly
+nonlinear parameterized model from time-series measure-
+ments of CX and CZ. Figure 2(a) shows the R2 score of
+the model on test trajectories corresponding to the pa-
+rameter values that the model was trained on (a value
+of R2 = 1 means that the fit perfectly predicts the tem-
+poral derivatives of the data). While the score decreases
+modestly as µ increases, the model remains very accurate
+(a)
+(b)
+(c)
+FIG. 2.
+Corrections to the weakly nonlinear theory of
+the Oregonator model.
+(a) R2 score for the parameterized
+SINDyCP model on test trajectories collected at the param-
+eter values used to train the model. (b) Corrected normal-
+form parameter values relative to the weakly nonlinear values
+b0 and c0 as a function of the bifurcation parameter µ1/2. (c)
+Limit cycles in the homogeneous system exhibiting the highly-
+nonlinear canard explosion with increasing µ. The pattern
+formation above the canard explosion (upper inset) is quali-
+tatively different than for smaller driving (lower inset), with
+more extreme spatio-temporal variation that does not emerge
+in the weakly nonlinear theory.
+on all the testing trajectories, accounting for 99% of the
+variation in the data in each case.
+A nonlinear change of coordinates transforms the dis-
+covered model into the normal-form in Eq. (1) with
+parameter-dependent values of b(µ) and c(µ) and small
+quintic corrections. These normal-form parameters agree
+with the analytic values derived [32] from the original
+model as µ → 0, but here we are able to discover them di-
+rectly from data without any knowledge of the governing
+equations. Furthermore, as shown in Fig. 2(b), the pa-
+rameters vary with µ, representing additional corrections
+to the weakly nonlinear theory. This variation becomes
+extreme for µ1/2 > 0.35, which we were able to discover
+via the implicit version of SINDy. In fact, as shown in
+Fig. 2(c), the Oregonator model exhibits a canard explo-
+sion (in which the limit cycle amplitude expands abruptly
+due to highly nonlinear effects) [30] around µ1/2 ≈ 0.39,
+where the weakly nonlinear theory breaks down.
+The
+SINDyCP model reflects this breakdown and enables the
+development of higher-order corrections to account for it.
+Weak formulation.—The weak formulation utilizes in-
+tegration against compactly supported “test functions”
+to defined the SINDy problem. The weak method shows
+excellent performance for noisy data, owing to its ability
+to minimize the need for computing numerical deriva-
+tives.
+Rather than forming samples (rows in Fig. 1)
+from spatio-temporal points for each trajectory, the weak
+method constructs the system rows by projecting the
+data onto weak samples such as
+wν
+ik ≡
+�
+Ωk
+φ(x; t)X(ν)
+i
+(x; t) dDxdt,
+(5)
+
+4
+where Ωk is a compactly-supported spatio-temporal do-
+main, φ is the test function, and X(ν)
+i
+denotes the νth
+partial derivative the ith dependent variable. By moving
+derivatives off of the data and onto the test functions via
+integration by parts,
+wν
+ik = (−1)|ν|
+�
+Ωk
+φ(ν)(x; t)Xi(x; t) dDxdt,
+(6)
+the weak method significantly reduces the impact of mea-
+surement noise on the SINDy library and improves the
+fit results [33].
+To maximize the performance for the weak method,
+we have optimized and fully vectorized numerical inte-
+gration for the weak formulation in PySINDy, which can
+be easily combined with the SINDyCP library class. De-
+tails about our efficient numerical implementation are
+available in the Supplemental Material [29]. Products of
+weak features do not generally form reasonable samples
+for a SINDy model, since multiplication and integration
+do not commute, so on first sight, it is not clear how to
+combine weak form feature and parameter libraries with
+SINDyCP. However, when computing the weak samples
+corresponding to constant functions, such as those that
+form the parameter library, the integrals simply repre-
+sent the spatio-temporal volume of the domain Ωk. Our
+implementation thus rescales the weak features for the
+temporal derivatives by the same volumetric factors.
+Performance.—Using 500 randomly distributed sam-
+ple domains (measuring 1/10th the spatio-temporal do-
+main size in each direction), the weak SINDyCP easily
+identifies the complex Ginzburg-Landau equation using
+the same data used for the traditional differential form
+shown in Fig. 1. Furthermore, it can do so in just a few
+seconds of run-time on a modern processor in this case
+(over five times faster than the differential form).
+To assess the impact of noise, we inject random Gaus-
+sian noise of varying intensity [34] into the four trajecto-
+ries used as the training data for the complex Ginzburg-
+Landau equation. We then generate two new sample tra-
+jectories to use as testing data, with b = 2.0, 1.5 and
+c = 1.5, 1.0, respectively.
+Using the training data, we
+perform the SINDyCP fits using both the differential for-
+mulation and the weak formulation and evaluate the R2
+score on our test trajectories. Figure 3(a) shows the re-
+sults for the R2 score on the test trajectories. While both
+methods provide good fits for low noise intensity, only
+the weak method exhibits a robust fit for parameterized
+equations for large noise intensities.
+The SINDyCP fit also requires a sufficient amount
+of data to identify governing equations.
+Figure 3(b)
+shows the performance of SINDyCP on the testing data
+for fits performed with a varying number of trajectories
+nt = 2, 3, 4, 5 and of varying length corresponding to a
+number of time samples Nt = 25, 50, 75, 100, with an in-
+jected noise intensity of 10−3.
+Unlike the trajectories
+in Fig. 1, the parameters for trajectories were randomly
+(a)
+(b)
+FIG. 3. Performance of SINDyCP for the fit of the complex
+Ginzburg-Landau equation with noisy data. (a) Model score
+vs noise intensity using the differential and weak forms of
+SINDyCP with nt = 4 trajectories. (b) Model score vs num-
+ber of samples for varying number of randomly generated tra-
+jectories, varying trajectory length, and noise intensity 10−3.
+generated, with (b, c) distributed as Gaussian random
+variables with means (1.5, 1.0) and standard deviations
+(0.5, 0.25).
+For too little data, the fit fails to identify
+the correct model, and the value of 1 − R2 is O(1). The
+models improve moderately with an increasing number
+of samples per trajectory (the product of Nt with the
+number of spatial grid points). More importantly, a suf-
+ficiently large number of trajectories nt is required to
+achieve a good fit (at least 3 in this case). The amount
+of data required will further increase when including a
+larger number of possible library terms and when identi-
+fying a larger number of parameters. These requirements
+should be carefully assessed in order to achieve successful
+SINDyCP fits for more general pattern forming systems.
+Parameter extrapolation.—As a final demonstration
+(Fig. 4), we consider the one-dimensional cubic-quintic
+Swift-Hohenberg equation
+˙u = du − uxxxx − 2uxx − u + eu3 − fu5,
+(7)
+with parameters d, e, and f describing the linear, cu-
+bic, and regularizing quintic terms, respectively.
+This
+model pattern formation equation has been used to study
+defect dynamics incorporating corrections beyond the
+weakly nonlinear approximation and has revealed uni-
+versal snaking bifurcations leading to the formation of
+localized states for e > 0 and d < 0 [35].
+The parameters d, e and f are the minimal and natu-
+ral set to describe the possible dynamics in the Swift-
+Hohenberg equation derived from normal-form theory.
+However, in typical pattern formation applications, one
+does not have direct control over such parameters. In-
+stead, experimentally accessible parameters will have a
+complicated and nonlinear relationship with the normal-
+form parameters, which requires detailed knowledge and
+
+5
+(b)
+(a)
+FIG. 4. Extrapolation of localized states in the cubic-quintic
+Swift-Hohenberg equation. (a) The randomly generated re-
+lationships between the normal-form parameters (d, e, f) and
+the experimental parameter ε (bottom panel) gives rise to
+snaking bifurcations (top panel) near ε = 0. Red dotted lines
+show the values used to train the SINDyCP fit, and dashed
+colored lines show the coefficients derived from the fit. (b)
+Localized states extrapolated from numerical simulations of
+the SINDyCP fit with ε = 0.1, corresponding to the black
+dotted line in (a).
+tedious calculations to derive, e.g., an expansion and cen-
+ter manifold transformation around a bifurcation point.
+The SINDyCP approach enables an automated discovery
+of such relationships, which can be used to extrapolate
+system behavior beyond a set of measurements.
+To illustrate this idea, we generate random quadratic
+relationships between an experimental parameter ε and
+the normal-form parameters (d, e, f), and we create three
+training trajectories using random values of the param-
+eter 1 < ε < 3 [Fig. 4(a)]. To determine the possible
+dynamics, we numerically continue the solution branches
+corresponding to the trivial state and localized and pe-
+riodic states using the AUTO package [36].
+For all of
+the training trajectories, ε is sufficiently large that no
+localized or periodic states are exist, and all trajecto-
+ries decay to the trivial u = 0 solution.
+We perform
+the weak SINDyCP fit using these trajectories subject
+to 1% injected noise with a quadradic parameter library.
+To test the ability of SINDyCP to extrapolate beyond
+the parameter regime given in the input data, we sim-
+ulate the identified model for the experimental param-
+eter value ε = 0.1. Remarkably, even with limited and
+noisy training data, the method identifies an accurate
+relationship between ε and the normal-form parameters.
+Thus, simulations of the identified model with random
+initial conditions converge to localized states [Fig. 4(b)]
+for ε = 0.1 despite the significant extrapolation of the
+parameter value beyond the input data.
+Discussion—The SINDyCP approach represents a
+simple but powerful generalization of SINDy with con-
+trol. By disambiguating the feature and parameter com-
+ponents of the SINDy libraries, the method enables dis-
+covery of systems of partial differential equations param-
+eterized by driving parameters. Such equations arise nat-
+urally in the context of pattern formation, where the
+normal forms of bifurcations lead to parameterized equa-
+tions near the onset of instabilities. The approach can be
+easily applied with the data available in typical pattern
+formation experiments and promises to enable true ex-
+trapolation beyond the regime that can be theoretically
+described with weakly nonlinear theory. Combining the
+SINDyCP approach with autoencoder-assisted discovery
+of physical coordinates [37–39] will further enable re-
+searchers to discover nonlinear equations governing com-
+plex systems directly from data gathered through ex-
+periments conducted under various driving parameters.
+This approach may also help inform universal mecha-
+nisms leading to the formation of localized states beyond
+the snaking bifurcations of the Swift-Hohenberg equation
+[40, 41].
+This work benefited from insightful discussions with
+Alan Kaptanoglu. Zachary G. Nicolaou is a WRF post-
+doctoral fellow. We acknowledge support from the Na-
+tional Science Foundation AI Institute in Dynamic Sys-
+tems (grant number 2112085).
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+integration, additional demonstrations, the oregonator
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+the weak formulation, but the maximum order of deriva-
+tives can generally be reduced to at most half the original
+order for the library.
+[34] Noise intensity here refers to the pointwise standard devi-
+ation on the spatio-temporal grid employed in the simu-
+lations. True white noise has a Dirac delta variance, and
+intensity should thus scale with grid spacing and time
+step to 1/2 power.
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+07P: Continuation and bifurcation software for ordinary
+differential equations. (2007).
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+tions. Proc. Natl. Acad. Sci. U.S.A. 116, 22445-22451
+(2019).
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+Du, and H. Lipson, Automated discovery of fundamental
+variables hidden in experimental data, Nat. Comput. Sci.
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+states in driven media. Nat. Comm. 12, 4486 (2021).
+
+1
+Supplementary Material for “Data-driven discovery and extrapolation
+of parameterized pattern-forming dynamics”
+Zachary G. Nicolaou, Guanyu Huo,Yihui Chen, Steven L. Brunton, and J. Nathan Kutz
+S1.
+NUMERICAL INTEGRATION
+For the complex Ginzburg-Landau equation, we use a pseudospectral integration method. We take a
+periodic domain of size of size L = 32π in each direction and discretize using Nx = Ny = 128 grid points in
+each spatial direction. Derivatives are calculated using fast Fourier transforms, and the discretized system
+is integrated with a 4(5) Runge-Kutta-Fehlberg method (which is also used for the other equations, with
+relative and absolute error tolerances of 10−6). To produce states in the dynamical phases of interest, we
+take random initial conditions A0 = �
+nm αnmeiknm·x + ϵeik2 2·x, where αnm are complex random Gaussian
+amplitudes with mean zero and variance σ2/(1 + n2 + m2), knm = 2π(nˆx + mˆy)/L, the sum ranges over
+−2 ≤ n, m ≤ 2, and ϵ is the scale of an initial plane wave perturbation with wavevector k2 2. The mode
+amplitudes are determined by σ = 0.1, 0.1, 0.1, 1.0 and ϵ = 0.01, 0.01, 1.0, 0.01 for the four trajectories used
+in the main text. The system is allowed to approach an attractor for the first 90 time units, then the
+trajectory is formed by the next 10 time units, in steps of 0.01. We also provide an animation showing the
+phase and amplitude for longer runs of 100 time units (Fig. S1). A similar pseudospectral approach was
+used for the Oregonator and Swift-Hohenberg examples, but, in the Swift-Hohenberg case, with Nx = 256
+discretization points, a domain of size L = 64π, an integration time of 5 time units, and random initial
+condition given by the real part of u0 = �20
+n=−20 αneiknx with kn = 2πn/L and αn complex random Gaussian
+amplitudes with mean zero and variance 1.0/(1 +
+�
+|n|)2.
+FIG. S1. Snapshot of the animation showing the phase φ and amplitude r of the trajectories, where A = reiφ.
+
+0
+r/2
+3π/2
+2rl
+L/2
+1.2
+0
+-L/2
+0.9
+-L
+0.6 r
+L
+L/2 
+0.3
+0
+0.0
+-L/2 
+-L
+-L/2
+0
+L/2
+7-7
+-L/2
+0
+L/2
+7
+x
+x2
+S2.
+DEMONSTRATIONS
+Demonstrations of SINDyCP in discrete maps, ODEs and PDEs are shown in Fig. S2. The left panels
+illustrate the logistic map,
+xn+1 = rxn(1 − xn),
+(S1)
+which is a discrete-time system with a single dependent variable xn and a single parameter r. This equation
+is the model for a universal period-doubling route to chaos as the parameter r increases past 3.56995. We
+perform the SINDyCP fit using four sample trajectories of 1000 iterations, corresponding to parameter
+values r = 3.6, 3.7, 3.8, 3.9 (red dotted lines in Fig. S2). We employ a library consisting of polynomials up to
+third order in the dependent variable xn and linear functions of the control parameter r, and the SINDyCP
+approach correctly identifies the parameterized equation. The middle panels illustrate the Lorenz system,
+˙x = σ(y − x), ˙y = x(ρ − z) − y, ˙z = xy − βz,
+(S2)
+which consists of three ordinary differential equations in three dependent variables x, y, and z and three
+parameters σ, ρ and β. This equation exhibits the iconic butterfly-shaped Lorenz attractor for certain
+parameter values.
+We perform the SINDyCP fit using five sample trajectories that have converged to
+their attractors, corresponding to the randomly selected parameter values σ = 10.0, 9.8, 9.9, 10.3, 9.5, ρ =
+27.6, 28.2, 28.3, 27.6, 28.1, and β = 3.1, 2.4, 2.4, 2.3, 2.4, respectively. We use feature and parameter libraries
+consisting of polynomials up to fourth order in the dependent variables (x, y, z) and linear functions in
+the parameters (σ, ρ, β), and the SINDyCP approach again correctly identifies the parameterized equation.
+Finally, the right panels illustrate the CGLE described in the main text.
+SINDyCP ft
+Input data
+Model
+Logistic map
+Lorenz system
+Complex Ginzburg-Landau equation
+FIG. S2.
+Demonstrations of the SINDyCP approach for three models (top row) of nonlinear dynamics.
+Several
+trajectories produced from different parameter values (middle row) are supplied as input, and the SINDyCP fit
+(bottom row) correctly identifies the governing equations in each case.
+
+3
+S3.
+OREGONATOR MODEL AND NORMAL FORM TRANSFORMATION
+We mainly follow the analyses of the Oregonator model in Refs. [30,32], with realistic parameter values
+shown in Table I. The fixed point (CX, CY , CZ) = (C0
+X, C0
+Y , C0
+Z) undergoes a Hopf bifurcation as µ increases
+from zero, leading to oscillatory chemical dynamics. For small µ, the weakly nonlinear theory follows from
+a perturbative expansion of the model. Take x ≡ (CX, CY , CZ) − (C0
+X, C0
+Y , C0
+Z) and express Eqs. (4)-(6)
+as ˙x = F(x). Define the multilinear operators of partial derivatives Fxn(ei1, · · · , ein) = ∂nF/∂xi1 · · · ∂xin
+with ei the ith component unit vector. Then the Taylor expansion for the system is
+˙x = (∂F/∂µ) µ + Fx1(x) + (∂F/∂µ)x1 (x)µ + 1
+2Fx2(x, x) + 1
+6Fx3(x, x, x) + D · ∇2x + · · · ,
+(S3)
+where D is a diagonal matrix with elements DX, DY and DZ. We develop a transformation x = y+h(y, µ)
+perturbatively, where y ≡ Aeiω0tu + ¯Ae−iω0t¯u. Here u is one of the critical eigenvectors of the Jacobian
+matrix Fx1 with eigenvalue λ = iω0 (with zero real part for µ = 0) and overbars represent complex
+conjugates, and we also define the corresponding left eigenvector at u⊥. The near-identity transformation
+function h(y, µ) is selected so as to eliminate the non-resonant terms in the evolution equation of A, which
+can be accomplished under general conditions. This results in an amplitude equation ˙A = µσA + g|A|2A +
+d∇2A, where
+σ = u⊥ · (∂F/∂µ)x1 (u) − u⊥ · Fx2
+�
+u, (Fx1)−1 (∂F/∂µ)
+�
+,
+(S4)
+g = 1
+2u⊥ · Fx3 (u, u, ¯u) − u⊥ · Fx2
+�
+u, [Fx1]−1 [Fx2 (u, ¯u)]
+�
+− 1
+2u⊥ · Fx2
+�
+¯u,
+�
+Fx1 −
+�
+λ − ¯λ
+�
+I
+�−1 [Fx2 (u, u)]
+�
+,
+(S5)
+d = u⊥ · D · u.
+(S6)
+By rescaling the amplitude by a factor of µ1/2, time by a factor of 1/µ, and space by a factor of 1/µ1/2 and
+employing additional rescalings to unitize the real components and eliminate the mean rotation, we can
+arrive at the CGLE in Eq.(2), where b ≡ Im(d)/Re(d) = b0 = 0.173 and c ≡ −Im(g)/Re(g) = c0 = 2.379.
+As expected, these parameter values correspond to the amplitude turbulence regime of the CGLE.
+For our numerical simulations, we use a spatial domain of length L = 0.4/µ1/2 cm and an integration
+time of T = 200/µ s, where we scaled by µ to ensure the trajectories have corresponding scales.
+We
+strobe the time in steps of 5.94804 s, which corresponds to the critical frequency of the instability. We
+then interpolate the time series in steps of T/1000 to generate the trajectories. The first 200 time steps
+are discarded as the trajectories relax to their attractors. The next 400 time steps are used to train the
+SINDyCP model, while the remaining 400 steps are used as test trajectories to evaluate the R2 scores. We
+finally employ the normal form transformation described above for the SINDyCP model to evaluate the
+parameterized b(µ) and c(µ) shown in Fig. 2(b) of the main text. Consistently, the normal form parameters
+very closely approximate the analytic results b(0) ≈ b0 and c(0) ≈ c0, but significant variations emerge for
+larger µ.
+k1 k2 k3
+k4
+k5 DX
+DY
+DZ
+CH CA CB/(1 − µ) ν
+2 106 10 2 × 103 1 10−5 1.6 × 10−5 0.6 × 10−5 0.5
+1
+0.787
+1
+TABLE I. Parameter values for the Oregonator model, in cgs units (suppressed for brevity).
+
+4
+S4.
+WEAK FORMULATION IMPLEMENTATION
+We refer the reader to Refs. [25-28] for the theory of the weak formulation of SINDy. Here, we only
+briefly describe our efficient numerical integration method for the weak formulation used in pysindy. We
+suppose that the spatial grid is one-dimensional, for the moment, and the values of the coordinates on the
+grid points are xi. The weak form requires us to calculate the integral of interpolated data f(x) weighted
+by the dth derivatives of test function φ(x),
+I(d) ≡
+� xN
+x0
+f(x)φ(d)(x)dx.
+(S7)
+We choose to use test functions φ(x) = (x2 − 1)p in our implementation, and thus their dth derivatives are
+φ(d)(x) =
+∂
+∂xd (x2 − 1)p =
+p
+�
+k=0
+�
+p
+k
+�
+(−1)k
+(2(p − k))!
+(2(p − k) − d)!x2(p−k)−d.
+(S8)
+We are provided with some feature values ui at the grid points, and we consider the value of a library
+function f applied to that feature, fi ≡ f(ui).
+We linearly interpolate the function as f(x) = fi +
+x−xi
+xi+1−xi (fi+1 − fi) where xi ≤ x ≤ xi+1. Expanding the interpolation, and integrating the xφ(d)(x) terms
+by parts,
+I(d) =
+N−1
+�
+i=0
+� xi+1
+xi
+�
+fi +
+x − xi
+xi+1 − xi
+(fi+1 − fi)
+�
+φ(d)(x)dx
+=
+N−1
+�
+i=0
+fixi+1 − fi+1xi
+xi+1 − xi
+�
+Φ(d)(xi+1) − Φ(d)(xi)
+�
++ fi+1 − fi
+xi+1 − xi
+�
+Φ(d−1)(xi+1) − Φ(d−1)(xi)
+�
+,
+(S9)
+where Φ(d)(x) are the antiderivatives of φ(d) [i.e. Φ(d)(x) = φ(d−1)(x) for d > 0].
+By relabelling the dummy summation variables, we can recast Eq. (S9) as a dot product between the
+input data fj and a weight wj
+I(d) =
+N−1
+�
+j=0
+wj · fj,
+(S10)
+with
+wj ≡ xj+1
+�
+Φ(d)(xj+1) − Φ(d)(xj)
+�
+xj+1 − xj
+− xj−1
+�
+Φ(d)(xj) − Φ(d)(xj−1)
+�
+xj − xj−1
++ Φ(d−1)(xj) − Φ(d−1)(xj−1)
+xj − xj−1
+− Φ(d−1)(xj+1) − Φ(d−1)(xj)
+xj+1 − xj
+,
+(S11)
+where 0 < j < N − 1. At the left and right sides of the domain (for j = 0 and j = N − 1), we must adjust
+the weights to correct for boundary effects,
+w0 ≡ x1
+�
+Φ(d)(x1) − Φ(d)(x0)
+�
+x1 − x0
+− Φ(d−1)(x1) − Φ(d−1)(x0)
+x1 − x0
+,
+(S12)
+wN−1 ≡ −xN−2
+�
+Φ(d)(xN−1) − Φ(d)(xN−2)
+�
+xN−1 − xN−2
++ Φ(d−1)(xN−1) − Φ(d−1)(xN−2)
+xN−1 − xN−2
+.
+(S13)
+
+5
+Expressing the integrals along each dimension as dot products [Eq. (S10)] enables efficient vectorization
+with BLAS operations, and the integration weights [Eq. (S11)-(S13)] only need to be evaluated a single
+time when the library is first initialized (in a vectorized fashion). We further vectorize the code by forming
+tensor products over all integration dimensions to calculate multidimensional integrals using a single tensor
+dot product.
+

I9AzT4oBgHgl3EQfVPxY/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf,len=155
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='01280v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='NA]  3 Jan 2023  An asymptotic formula for Aldaz-Kounchev-Render  operators on the hypercube  Ana-Maria Acua, Ioan Ra¸sab  aLucian Blaga University of Sibiu, Department of Mathematics and Informatics, Romania,  e-mail: anamaria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='acu@ulbsibiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='ro  bTechnical University of Cluj-Napoca, Faculty of Automation and Computer Science,  Department of Mathematics, Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Memorandumului nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' 28, 400114 Cluj-Napoca, Romania  e-mail: ioan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='rasa@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='utcluj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='ro  Abstract  We prove a version of a conjecture concerning the asymptotic behavior of the  Aldaz-Kounchev-Render operators on the hypercube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  Keywords:  Aldaz-Kounchev-Render operators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Bernstein operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  Voronovskaja-type formula;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  2010 MSC: 41A36  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Introduction  Let B[1]  n : C[0, 1] → C[0, 1] be the classical Bernstein operator defined as  B[1]  n f(x) =  n  �  i=0  f  � i  n  �  pn,i(x),  where pn,i(x) =  �n  i  �  xi(1 − x)n−i, x ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  For a fixed j ∈ N, j ≥ 2 and for n ≥ j, Aldaz, Kounchev and Render [2]  introduced a polynomial operator B[1]  n,j : C[0, 1] → C[0, 1] that fixes e0 and ej,  investigated its approximation properties and gave applications to CAGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' The  operator is explicitly given by  B[1]  n,jf(x) =  n  �  k=0  f  �  tj  n,k  �  pn,k(x),  where  tj  n,k =  � k(k − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' (k − j + 1)  n(n − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' (n − j + 1)  �1/j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  The Voronovskaja-type formula for the sequence (B[1]  n,j)n≥1 was conjectured in  [4] and proved in [3], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  Preprint submitted to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  January 4, 2023  For f ∈ C([0, 1]2), the tensor product B[1]  n ⊗ B[1]  n is given by  B[2]  n f(x, y) := (B[1]  n ⊗ B[1]  n )f(x, y) =  n  �  k=0  n  �  l=0  f  �k  n, l  n  �  pn,k(x)pn,l(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='1)  Let B[1]  n,j : C[0, 1] → C[0, 1] be the AKR operator and (x, y) ∈ [0, 1]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Then, for  f ∈ C([0, 1]2), the tensor product B[1]  n,j ⊗ B[1]  n,j is given by  B[2]  n,jf(x, y) := (B[1]  n,j ⊗ B[1]  n,j)f(x, y)  =  n  �  k=0  n  �  l=0  f  �  tj  n,k, tj  n,l  �  pn,k(x)pn,l(y), (x, y) ∈ [0, 1]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='2)  A conjecture concerning the Voronovskaja-type formula for the sequence  (B[2]  n,j) was formulated in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' The aim of this paper is to prove a version of this  conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Proof of Conjecture  For the sake of conciseness we consider only the case j = 2, but obviously  the proof can be extended to arbitrary j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  Let k and n be integers, n ≥ 2, 0 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Define  R(n, k) := k  n −  �  k(k − 1)  n(n − 1) − 1  2n +  k  2n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  It is elementary to prove that  R(n, 0) = − 1  2n,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='1)  R(n, k) ≥ 0, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' , n,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='2)  0 ≤ k  n −  �  k(k − 1)  n(n − 1) ≤ 1  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='3)  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' If 0 < x ≤ 1, then  lim  n→∞ n  n  �  k=1  pn,k(x)R(n, k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='4)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Let x ∈ (0, 1] and f ∈ C2[0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' It is known (see [3], [5]) that  lim  n→∞ n(B[1]  n,2f(x) − f(x)) = x(1 − x)  2  f ′′(x) − 1 − x  2  f ′(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  It is also well known that  lim  n→∞ n(B[1]  n f(x) − f(x)) = x(1 − x)  2  f ′′(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  2  It follows that  lim  n→∞ n  �  B[1]  n,2f(x) − B[1]  n f(x)  �  = −1 − x  2  f ′(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  In particular, for the function f(t) = t, we get  lim  n→∞ n  n  �  k=1  pn,k(x)  ��  k(k − 1)  n(n − 1) − k  n  �  = −1 − x  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  This can be written as  lim  n→∞ n  n  �  k=1  pn,k(x)  � 1  2n  �  1 − k  n  �  + R(n, k)  �  = 1 − x  2  ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=',  1  2 lim  n→∞  n  �  k=1  pn,k(x)  �  1 − k  n  �  + lim  n→∞ n  n  �  k=1  pn,k(x)R(n, k) = 1 − x  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='5)  Let us remark that  1  2 lim  n→∞  n  �  k=1  pn,k(x)  �  1 − k  n  �  = 1  2 lim  n→∞  �  B[1]  n (1 − t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' x) − (1 − x)n�  = 1  2(1 − x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  Combined with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='5) this leads to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='4), and the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Let 0 < x ≤ 1, 0 < y ≤ 1, f ∈ C2([0, 1]2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Then  lim  n→∞ n  �  B[2]  n,2f(x, y) − f(x, y)  �  = x(1 − x)  2  f ′′  x2(x, y) + y(1 − y)  2  f ′′  y2(x, y) − 1 − x  2  f ′  x(x, y) − 1 − y  2  f ′  y(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='6)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' First we have  n  �  B[2]  n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='2f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' y) − B[2]  n f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' y)  �  = n  n  �  k=0  n  �  l=0  pn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='k(x)pn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='l(y)  �  f  ��  k(k − 1)  n(n − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  �  l(l − 1)  n(n − 1)  �  − f  �k  n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' l  n  ��  = Enf(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' y) + Fnf(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' y) + Gnf(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  3  where  Enf(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' y) := n  n  �  k=0  n  �  l=0  pn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='k(x)pn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='l(y)  ��  k(k − 1)  n(n − 1) − k  n  �  f ′  x  �k  n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' l  n  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  Fnf(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' y) := n  n  �  k=0  n  �  l=0  pn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='k(x)pn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='l(y)  ��  l(l − 1)  n(n − 1) − l  n  �  f ′  y  �k  n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' l  n  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  Gnf(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' y) := n  2  n  �  k=0  n  �  l=0  pn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='k(x)pn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='l(y)  \uf8f1  \uf8f2  \uf8f3  ��  k(k − 1)  n(n − 1) − k  n  �2  f ′′  x2(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' η)  + 2  ��  k(k − 1)  n(n − 1) − k  n  � ��  l(l − 1)  n(n − 1) − l  n  �  f ′′  xy(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' η)  +  ��  l(l − 1)  n(n − 1) − l  n  �2  f ′′  y2(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' η)  \uf8fc  \uf8fd  \uf8fe ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  for suitable (ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' η) furnished by Taylor’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='3) we see that  lim  n→∞ Gnf(x, y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='7)  Moreover,  lim  n→∞ Enf(x, y)  = − lim  n→∞ n  n  �  k=0  n  �  l=0  pn,k(x)pn,l(y)  � 1  2n  �  1 − k  n  �  + R(n, k)  �  f ′  x  �k  n, l  n  �  = −1  2 lim  n→∞  n  �  k=0  n  �  l=0  pn,k(x)pn,l(y)  �  1 − k  n  �  f ′  x  �k  n, l  n  �  − lim  n→∞ n  n  �  k=0  n  �  l=0  pn,k(x)pn,l(y)R(n, k)f ′  x  �k  n, l  n  �  = −1  2 lim  n→∞ B[2]  n ((1 − s)f ′  x(s, t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' (x, y))  − lim  n→∞ n  n  �  k=1  n  �  l=0  pn,k(x)pn,l(y)R(n, k)f ′  x  �k  n, l  n  �  + lim  n→∞ n  n  �  l=0  (1 − x)npn,l(y) 1  2nf ′  x  �  0, l  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  The first term equals −1  2(1 − x)f ′  x(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  4  Moreover, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='2) we have  �����n  n  �  k=1  n  �  l=0  pn,k(x)pn,l(y)R(n, k)f ′  x  �k  n, l  n  ������  ≤  n  �  l=0  �  n  n  �  k=1  pn,k(x)R(n, k)∥f ′  x∥∞  �  pn,l(y)  = n  n  �  k=1  pn,k(x)R(n, k)∥f ′  x∥∞,  and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='4) shows that the second term is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' The third one is also zero, and so  lim  n→∞ Enf(x, y) = −1 − x  2  f ′  x(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='8)  Similarly,  lim  n→∞ Fnf(x, y) = −1 − y  2  f ′  y(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='9)  Now (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='7), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='8), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='9) yield  lim  n→∞ n  �  B[2]  n,2f(x, y) − B[2]  n f(x, y)  �  = −1 − x  2  f ′  x(x, y) − 1 − y  2  f ′  y(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='10)  On the other hand, it is well known that  lim  n→∞ n(B[2]  n f(x, y) − f(x, y)) = x(1 − x)  2  f ′′  x2(x, y) + y(1 − y)  2  f ′′  y2(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='11)  From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='11) we get (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='6) and the theorem is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  References  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Acu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' De Marchi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Ra¸sa, Aldaz–Kounchev–Render Operators and  Their Approximation Properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Results Math 78, 21 (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Aldaz, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Kounchev, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Render, Shape preserving properties of gener-  alized Bernstein operators on extended Chebyshev spaces, Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=',  2009, 114(1), 1–25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Birou, A proof of a conjecture about the asymptotic formula of a Bern-  stein type operator, Results Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' 72 (2017), 1129–1138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' C´ardenas-Morales, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Garrancho, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Rasa, Asymptotic Formulae via a  Korovkin-Type Result, Abstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Volume 2012, Article ID 217464,  12 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  [5] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Gavrea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Ivan, Complete asymptotic expansions related to conjecture  on a Voronovskaja-type theorem, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content=' 458 (1) (2018),  452-463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}
+page_content='  5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AzT4oBgHgl3EQfVPxY/content/2301.01280v1.pdf'}

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+arXiv:2301.04481v1  [physics.optics]  11 Jan 2023
+“Analytical Continuation” of Flattened Gaussian Beams
+Riccardo Borghi
+Dipartimento di Ingegneria Civile, Informatica e delle Tecnologie Aeronautiche,
+Universit`a “Roma Tre”, Via Vito Volterra 62, I-00146 Rome, Italy
+A purely analytical extension of the flattened Gaussian beams [Opt. Commun. 107, 335 (1994)]
+to any values of the beam order, is here proposed. Thanks to it, the paraxial propagation problem of
+axially symmetric, coherent flat-top beams through arbitrary ABCD optical systems can definitely
+be closed in terms of a particular bivariate confluent hypergeometric function.
+I.
+INTRODUCTION
+Flat-top beams continue to attract a considerable at-
+tention in optics: during the last five years more than
+sixty papers have been published on the subject. In or-
+der to model flat-top axially symmetric distributions, two
+classes of different scenarios appeared: in the first one,
+simple analytical profiles were employed, the most known
+of them being the superGaussian (SG) [1, 2], which is for-
+mally defined by
+SGν(ξ) = exp(−ξ2ν) ,
+(1)
+where ν denotes a real parameter which controls the“flat-
+ness” of the profile, with the particular case ν = 1 giving
+the Gaussian profile. The symbol ξ denotes a normal-
+ized radial transverse position.
+Despite its mathemat-
+ical simplicity, it is well known that Eq. (1) does not
+allow the wavefield of paraxially propagated superGaus-
+sian (i.e., for ν ̸= 1) beams to be analytically evaluated,
+even within the simplest scenario, namely free space.
+To overcome such a difficulty, which two or three
+decades ago could represent a considerable computational
+bottleneck in several practical situations, alternative ap-
+proaches were proposed in 1994 and in 2002 by Gori and
+Li, respectively, to conceive analytical models able to
+solve the free space propagation problem. The former
+was called flattened Gaussian (FG henceforth) [3], and,
+differently from SG, is expressed through an explicit fi-
+nite sum of terms, namely
+FGN(ξ) = exp(−Nξ2)
+N−1
+�
+m=0
+(Nξ2)m
+m!
+,
+(2)
+where the integer parameter N will be referred to as the
+FG order. Scaling the ξ variable by the factor
+√
+N gives
+the FG transverse profile a flat-topped shape which, for
+N = 1, reduces to a Gaussian distribution, whereas for
+N → ∞ tends to the characteristic function of the uni-
+tary disk [4]. The model is computationally exact, since
+the initial distribution (2) can be recast in terms of a su-
+perposition of N standard Laguerre-Gauss (sLG hence-
+forth) beams. Accordingly, in order to evaluate the field
+propagated in free space, it was enough to sum up the
+N propagated sLG, a job which can exactly be done, al-
+ways [5]. In [6], a different superposition scheme of the
+profile (2) was proposed, in which the sLG family was
+replaced by the so- called elegant Laguerre-Gauss (eLG
+henceforth) set. In this way, not only free-space propa-
+gation, but also the interaction of FG beams with any
+axially symmetric paraxial optical system can be dealt
+with in exact terms, always through finite sums.
+In 2002, Yaijun Li proposed an analytical model al-
+ternative to the FG one. The idea was to impose a lo-
+cal “flatness” condition, which required the first 2N ξ-
+derivatives of the profile to be null at the origin ξ = 0 [7].
+On using such condition, Li conceived the following ana-
+lytical model:
+LiGN(ξ) =
+N
+�
+m=1
+(−1)m−1
+�N
+m
+�
+exp(−mξ2) =
+=
+�
+1 −
+�
+1 − exp
+�
+−ξ2���N
+N
+,
+(3)
+which, differently from FG, is based on the superposi-
+tion of N fundamental Gaussian beams having variable
+widths.
+Both Gori’s and Li’s models provide exact solutions
+to the paraxial propagation problem of coherent, axially
+symmetric flat-topped beams. From a merely mathemat-
+ical perspective, their only own limit is represented by
+the fact that, differently from SG, only positive integer
+orders N can be dealt with to describe the initial flat-top
+distribution. It is important to mention that, for 1D ge-
+ometry (or rectangular 2D geometries), general analyt-
+ical solutions were already provided, at least upon free
+propagation, by modeling the flat-top profile via an error
+function [8]. An attempt to extend the 2D circular FG
+model to noninteger orders was also proposed in [9], but
+only approximate estimates of the free space propagated
+field were found within the asymptotic limit N ≫ 1.
+The aim of the present paper is to solve exactly the
+propagation problem of FG beams of any order (real or
+even complex) through typical axially symmetric parax-
+ial optical systems. To this end, the right side of Eq. (2)
+will first be identified as an incomplete Gamma func-
+tions, which is known to be defined onto the whole com-
+plex plane, as far as both arguments are concerned. An
+immediate byproduct of such identification will be the
+closed form expression of the M 2 factor of FG beams
+of any order, an interesting generalization of the result
+found in [5]. This is shown in Sec. II of the present pa-
+per. The most important results are indeed presented
+
+2
+in Secs. III and IV. In the former, the free- space prop-
+agation problem will be solved thanks to an important
+class of integrals recently closed by Yuri Brychkov. Al-
+though the more general propagation problem will be
+solved in Sec. IV, the analysis presented in Sec. III should
+be viewed as an important propaedeutical step. There, it
+will be shown that a very important, but nevertheless not
+so much known, class of special functions, called bivariate
+hypergeometric functions, together with the correspond-
+ing confluent versions, form the mathematical skeleton of
+the paraxially diffracted wavefield. Bivariate hypergeo-
+metric were first introduced in 1880 by Paul Appell [10],
+their confluent version forty years later by Paul Hum-
+bert [11]. The results we are going to present would also
+give readers a partial answer about the lack, for more
+than thirty years, of purely analytical solutions to the
+problem of the paraxial propagation of coherent 2D flat-
+topped beams.
+The present work has a clear mathematical character:
+for instance, dimensionless quantities will be used wher-
+ever possible.
+Moreover, the number of mathematical
+appendices have been considerably limited, because we
+strongly believe that following all most important math-
+ematical steps could greatly help readers to fully grasp
+the essence of our analysis, as well as the importance of
+such still mysterious special functions, which will lead to
+analytical, elegant, and exact solutions.
+II.
+PRELIMINARIES
+A.
+“Analytical continuation” of the FG model
+Already in 1996, Sheppard & Saghafi [12] pointed out
+that Eq. (2) can be given the closed form
+FGN(ξ) = Γ(N, Nξ2)
+Γ(N)
+,
+(4)
+where Γ(·) and Γ(·, ·) denote Gamma and incomplete
+Gamma functions, respectively [13].
+Differently from
+Eq. (2), Eq. (4) is not limited to integer FG orders, but
+rather it can be analytically continued to real and also
+complex values of N.
+As a preliminary result of the extended definition into
+Eq. (4), an analytical check of Li’s“flatness condition”[7]
+will now be carried out. To this end, it is sufficient to use
+formulas 1.1.1.1 and 1.8.1.17 of [14] to prove, with long
+but simple algebra, that
+dn
+dξn Γ(N, Nξ2) = −2nn!N N exp(−Nξ2) ξ2N−n
+×
+[n/2]
+�
+k=0
+(n − k − 1)!
+4kk!(n − 2k)! L(N−n+k)
+n−k−1
+(Nξ2) ,
+(5)
+which gives at once
+� dn
+dξn Γ(N, Nξ2)
+�
+ξ=0
+= 0 ,
+0 ≤ n < 2 Re{N} ,
+(6)
+thus implying the real part of N to be chosen greater
+than one.
+B.
+Spreading properties: closed form expression of the M 2
+factor
+An interesting byproduct of the extended Γ-based def-
+inition into Eq. (4) is the evaluation of the M 2 factor of
+FG beams, first established in [5] for N ∈ N, for nonin-
+teger orders. To this end, consider an initial field distri-
+bution across the plane z = 0 of a cylindrical reference
+frame (r, z), say ψ0(r), given by
+ψ0(r) = FGN
+�r
+a
+�
+=
+Γ
+�
+N, N r2
+a2
+�
+Γ(N)
+,
+(7)
+where an overall amplitude constant has been set to one
+and the symbol a denotes the “width” of lat-top distri-
+bution field distribution.
+For simplicity, it will be set
+a = 1.
+The evaluation of the M 2 factor, which is defined as the
+product of the normalized second order moments across
+the z = 0 and the spatial frequency planes is detailed
+in Appendix A, where it is proved the following closed-
+form expression:
+M 2 =
+�
+(N + 1) Γ(N + 1/2)
+√π Γ(N + 1)
+�
+1 −
+Γ(N + 3/2)
+√π Γ(N + 2)
+�
+1 −
+Γ(N + 1/2)
+√π Γ(N + 1)
+,
+(8)
+which extends the 1996 analysis of [5] to N /∈ N. It is
+worth comparing Eq. (8) with the corresponding expres-
+sion of SG beam M 2 factor, namely [2]
+M 2 =
+�
+Γ(2/ν)
+Γ(1/ν)/ν ,
+(9)
+deceptively simpler. In the next two sections, our exten-
+sion of the FG model will further reveal its powerfulness
+and mathematical elegance.
+III.
+FREE-SPACE PARAXIAL PROPAGATION OF FG
+BEAMS
+A.
+Preliminaries
+Suppose the initial field distribution given by Eq. (7)
+is allowed to propagate in free space. The corresponding
+
+3
+field, say ψ(r; z), can be expressed, apart from an overall
+phase factor exp(ikz), as follows:
+ψ(r; z) = −i U
+2π
+�
+R2 d2ρ ψ0(ρ) exp
+�iU
+2 |r − ρ|2
+�
+,
+(10)
+where the Fresnel number U = ka2/z has been intro-
+duced and the beam width a has been used as unit for
+measuring all transverse sizes. This means that the quan-
+tity r should be meant as the ratio between the trans-
+verse position vector of the observation point and a. For
+integer FG orders, the free space propagation problem
+has already been solved in [3] by expanding the initial
+field distribution ψ0 as the linear combination of a finite
+number of sLG beams.
+It is then sufficient to propa-
+gate each sLG beam up to the observation plane and to
+recombine all of them with the initial expanding coeffi-
+cients for the correct value of ψ(r; z) to be retrieved. As
+we are going to show in a moment, the Γ-based model
+into Eq. (4) allows an exact evaluation of the propagated
+wavefield (10) also for N /∈ N. It is worth recalling that,
+from a mere practical perspective, the present section
+could seem somewhat redundant, as in Sec. IV the more
+general propagation problem within ABCD systems will
+be solved.
+Nevertheless, we believe what is contained
+in the present section could help nonspecialist readers
+to familiarize with the main notations and mathemati-
+cal tools which will constitute the basis of the general
+results presented into Sec. IV. In other words, it should
+be considered as a useful, propaedeutical material.
+We start on substituting from Eqs. (7) into Eq. (10),
+which after simple algebra gives
+ψ(r; z) = − i U
+Γ(N) exp
+�iU r2
+2
+�
+×
+� ∞
+0
+dρ ρ exp
+�iU
+2 ρ2
+�
+Γ
+�
+N, N ρ2�
+J0(Ur ρ) ,
+(11)
+where J0 denotes the 0th-order Bessel function of the first
+kind. It is worth recasting the incomplete Γ function as
+Γ(N, Nξ)
+Γ(N)
+= 1 − γ(N, Nξ)
+Γ(N)
+,
+(12)
+where γ(·, ·) denotes the “lower”incomplete gamma func-
+tion. Then Eq. (11) takes on the form
+ψ(r; z) =
+= −i U exp
+�iU r2
+2
+� � ∞
+0
+dρ ρ exp
+�
+−U
+2i ρ2
+�
+J0(Ur ρ)
++
+i U exp
+�iU r2
+2
+�
+Γ(N)
+×
+� ∞
+0
+dρ ρ exp
+�
+−U
+2i ρ2
+�
+γ
+�
+N, Nρ2�
+J0(Ur ρ) .
+(13)
+The first term is identically equal to one (it is nothing
+but a unitary plane wave propagating along the z-axis).
+As far as the second is concerned, the following notable
+formula has recently been published by Brychkov [15,
+formula 9.2.20]:
+� ∞
+0
+dx xα−1 exp(−a x2) γ(µ, bx2) Jν(c x) =
+=
+2−ν−1bµcνΓ
+�
+µ + α + ν
+2
+�
+µaµ+(α+ν)/2Γ(ν + 1)
+Ψ1
+�
+µ + α + ν
+2
+, µ
+µ + 1, ν + 1
+����� − b
+a, − c2
+4a
+�
+.
+(14)
+Then, on using Eqs. (13) and (14), long but straightfor-
+ward algebra gives
+ψ(r; z) = 1 − exp
+�iU r2
+2
+� �2iN
+U
+�N
+× Ψ1
+�
+N + 1, N
+N + 1, 1
+���� − 2iN
+U , −iU r2
+2
+�
+.
+(15)
+B.
+A short Tour on Bivariate Hypergeometric Functions
+The symbol Ψ1 into Eq. (15) denotes a special func-
+tion called bivariate confluent hypergeometric. It is worth
+briefly describing the principal definitions and properties
+which are important for our scopes. Function Ψ1 is for-
+mally defined through the following double series power
+expansion:
+Ψ1
+�
+a, b
+c, c′
+���� z, w
+�
+=
+∞
+�
+k=0
+∞
+�
+ℓ=0
+(a)k+ℓ (b)k
+(c)k(c′)ℓ
+zk
+k!
+wℓ
+ℓ! ,
+(16)
+valid for |z| ≤ 1.
+The symbol (·)n denotes Pochham-
+mer’s symbol. Another bivariate confluent hypergeomet-
+ric function which will be meet in the present paper is
+the function Φ1, defined by
+Φ1
+�
+a, b
+c
+���� z, w
+�
+=
+∞
+�
+k=0
+∞
+�
+ℓ=0
+(a)k+ℓ (b)k
+(c)k+ℓ
+zk
+k!
+wℓ
+ℓ! ,
+(17)
+valid for |z| ≤ 1.
+Functions Ψ1 and Φ1 are members
+of a family of functions that generalize Kummer’s con-
+fluent hypergeometric function 1F1. In particular, Φ1 is
+obtained from the so-called Appell function F1, defined
+by
+F1
+�
+a, b1, b2
+c
+���� z, w
+�
+=
+∞
+�
+k=0
+∞
+�
+ℓ=0
+(a)k+ℓ (b1)k (b2)ℓ
+(c)k+ℓ
+zk
+k!
+wℓ
+ℓ! ,
+(18)
+(again valid for |z| ≤ 1), through the following limiting
+definition:
+Φ1
+�
+a, b
+c
+���� z, w
+�
+= lim
+ǫ→0 F1
+�
+a, b, 1
+ǫ
+c
+����� z, ǫw
+�
+,
+(19)
+
+4
+which can be proved on first substituting the identity
+lim
+ǫ→0
+�1
+ǫ
+�
+ℓ
+ǫℓ = 1 ,
+(20)
+directly into Eq. (19), then on interchanging the limit
+with the double series.
+Multivariate hypergeometric and confluent hypergeo-
+metric functions play a role of considerable importance
+in theoretical physics and applied math. In optics, the
+role of bivariate confluent hypergeometric functions in de-
+scribing a large class of paraxial optical disturbances has
+recently been pointed out [16, 17]. Moreover, it is worth
+stressing that, from a practical viewpoint, Appell’s func-
+tion F1 is nowadays part of the symbolic platform Math-
+ematica, where it is computable with arbitrarily high ac-
+curacies. Also the whole family of Appell functions, in-
+cluding F1 as well as its three sisters F2, F3, and F4,
+are currently implemented in the latest release of Maple.
+It is then highly desirable that in a near future also the
+set of bivariate confluent hypergeometric functions, in-
+cluding Ψ1 and Φ1, could become part of such family of
+“evaluable”special functions. In the meanwhile, someone
+might rightly object to the practical usefulness of func-
+tions that are defined through double infinite series like
+those into Eqs. (16) - (18). To overcome such difficulties,
+some tricks will be implemented in the rest of the paper,
+tricks which are aimed at extending the validity domain
+of Ψ1 and Φ1 beyond the series definitions, and then to
+improve the practical usefulness of our analytical results.
+C.
+Free-space propagation formula
+Function Ψ1 can be continued by using the following
+transformation [18, formula 2.54]:
+Ψ1
+�
+α, β
+γ1, γ2
+���� z, w
+�
+=
+=
+1
+(1 − z)α Ψ1
+�
+α, γ1 − β
+γ1, γ2
+����
+z
+z − 1,
+w
+1 − z
+�
+,
+(21)
+which, once substituted into Eq. (15), gives a new, closed-
+form, expression of the paraxial propagated field
+ψ(r; z) = 1 −
+exp
+�iU r2
+2
+�
+1 + 2iN
+U
+
+
+
+1
+1 +
+U
+2iN
+
+
+
+N
+× Ψ1
+
+
+ N + 1, 1
+N + 1, 1
+����
+1
+1 +
+U
+2iN
+, −
+iU r2
+2
+1 + 2iN
+U
+
+
+ ,
+(22)
+indubitably one of the main results of the present paper.
+Waiting for Mathematica or Maple to develop their own
+built-in version of Ψ1, it is worth working on the expres-
+sion into Eq. (22) by using a notable integral represen-
+tation found again in [18].
+For the sake of clarity, all
+mathematical steps are confined into Appendix B, where
+it is proved that
+Ψ1
+�
+N + 1, 1
+N + 1, 1
+���� x, y
+�
+=
+= N
+� 1
+0
+dξ (1 − ξ)N−1
+(1 − xξ)N+1 1F1
+�
+N + 1; 1;
+y
+1 − xξ
+�
+.
+(23)
+Equation (23) appears to be somewhat intriguing: the
+wavefield of a free-space paraxially propagated FG beam
+of any order can be represented via a 1D integral defined
+over a finite integral. This could seem a somewhat pe-
+culiar situation, due to the fact that the initial field dis-
+tribution (7) has an infinite support, namely the whole
+plane z = 0.
+But what is, in our opinion, even more
+important is that the integral representation (23) would
+hardly be reachable starting from Fresnel’s integral (10),
+without passing through the Ψ1 function and its trans-
+formation rules. In the next section, a similar scenario
+will also be found as far as the more general problem is
+concerned.
+IV.
+PARAXIAL PROPAGATION THROUGH ABCD
+SYSTEMS
+A.
+Preliminaries
+The free-space paraxial propagation formula derived in
+the previous section will now be extended to the general
+case of the paraxial propagation of FG beams of any or-
+der through typical paraxial optical systems with axial
+symmetry, characterized by the so-called ABCD optical
+matrices. For FG beams of integer order, it was found
+in [6] that the propagation problem can be dealt with
+in exact terms by expanding the initial field distribution
+given into Eqs. (7) and (2) as a finite superposition of
+so-called elegant Laguerre (eLG henceforth) beams as fol-
+lows:
+ψ0(r) =
+N−1
+�
+n=0
+(−)n
+�
+N
+n + 1
+�
+eLGn
+�ikr2
+2qN
+�
+,
+(24)
+where the symbol eLGn(x) = exp(x)Ln(−x) will be re-
+ferred to as the elegant Laguerre function of order n and
+the complex radius of curvature qN = ka2
+2iN has also been
+introduced. The initial distribution ψ0 is then recast as
+follows:
+ψ0(r) = exp
+�ikr2
+2qN
+�
+GN
+�
+1, −ikr2
+2qN
+�
+,
+(25)
+where the function GN (·, ·) is defined, for integer N, as
+GN (t, s) =
+N−1
+�
+n=0
+(−t)n
+�
+N
+n + 1
+�
+Ln(s) ,
+(26)
+
+5
+In [6] it was proved that, if the initial field distribution
+given by Eq. (25) feeds an axially symmetric paraxial
+optical system described by the optical matrix M
+M =
+
+
+A B
+C D
+
+ ,
+(27)
+then the wavefield at the output plane of the system, say
+ψ1(r), takes on the following form:
+ψ1(r) =
+=
+exp
+� ikr2
+2QN
+�
+A
+1
+1 +
+B
+A qN
+GN
+
+
+
+
+1
+1 +
+B
+A qN
+,
+kr2
+2iA2 qN
+1 +
+B
+A qN
+
+
+
+ ,
+(28)
+where an overall phase factor exp(ikℓ) (with ℓ being the
+optical lenght) will be tacitly assumed and QN denotes
+the complex quantity
+QN = A qN + B
+C qN + D .
+(29)
+The problem of extending the function GN (t, s) to N /∈ N
+will now be addressed.
+B.
+Extension of the function GN (t, s) to N /∈ N
+The starting point is the following Laplace transform
+representation of GN(t, s) established in [9]:
+GN(t, s) = exp(s)
+� ∞
+0
+dξ exp(−ξ) J0
+�
+2
+�
+s ξ
+�
+L(1)
+N−1(ξ t) .
+(30)
+For N ∈ N, the Laguerre polynomials L(1)
+N−1 can be writ-
+ten as
+L(1)
+N−1(ξ t) =
+N−1
+�
+n=0
+Ln(ξ t) ,
+(31)
+so that, on substituting from Eq. (31) into Eq. (30), it is
+found
+GN(t, s) =
+= exp(s)
+N−1
+�
+n=0
+� ∞
+0
+dξ exp(−ξ) J0
+�
+2
+�
+s ξ
+�
+Ln(ξ t) =
+=
+N−1
+�
+n=0
+(1 − t)n Ln
+� st
+t − 1
+�
+,
+(32)
+where in the last passage, [22, formula 3.24.6.2] has been
+used. Equation (32) is a valid alternative, for N ∈ N, to
+the definition given into Eq. (26). For the scopes of the
+present paper, its importance stems from the fact that
+the quantity GN can also be thought of as function of
+two new variables, namely
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+1 − t =
+1
+1 + AqN
+B
+,
+st
+t − 1 = ikr2
+2AB
+1
+1 +
+B
+AqN
+,
+(33)
+and this will reveal of a certain importance in the rest of
+our analysis.
+In order to extend the integral into Eq. (30) to N /∈
+N, the following notable formula, again established by
+Brychkov [15], will be employed:
+� ∞
+0
+xα−1 exp(−ax) Jν(b√x) L(λ)
+n (cx) dx =
+=
+� b
+2
+�ν Γ
+�
+α + ν
+2
+�
+(λ + 1)n
+n! aα+ν/2Γ(ν + 1) Ψ1
+�
+α + ν
+2 , −n
+λ + 1, ν + 1
+�����
+c
+a, − b2
+4a
+�
+.
+(34)
+In particular, on letting α = 1, a = 1, ν = 0, b = 2√s,
+t = c, n = N − 1, and λ = 1, Laplace’s transform into
+Eq. (30) takes on the form
+GN(t, s) = N exp(s) Ψ1
+�
+1, 1 − N
+2, 1
+���� t, −s
+�
+.
+(35)
+Again, it can be appreciated how the confluent hyperge-
+ometric function Ψ1 constitutes the mathematical skele-
+ton of the propagated field. But there is more. In Ap-
+pendix C, the following relationship has been established:
+Ψ1
+�
+1, 1 − N
+2, 1
+���� t, −s
+�
+=
+=
+exp(−s)
+(1 − t)1−N Φ1
+�
+1 − N, 1
+2
+����
+t
+t − 1,
+st
+t − 1
+�
+,
+(36)
+where Φ1 is the confluent hypergeometric function de-
+fined by Eq. (17). On substituting from Eq. (36) into
+Eq. (35), we have
+GN(t, s) = N (1 − t)N−1 Φ1
+�
+1 − N, 1
+2
+����
+t
+t − 1,
+st
+t − 1
+�
+(37)
+so that Eq. (28) eventually becomes
+ψ1(r) = exp
+� ikr2
+2QN
+� qNN
+B
+
+
+
+1
+1 + A qN
+B
+
+
+
+N
+× Φ1
+
+
+
+
+1 − N, 1
+2
+���� − A qN
+B
+, ikr2
+2AB
+1
+1 +
+B
+AqN
+
+
+
+ .
+(38)
+
+6
+Equation (38) summarizes the main result of the present
+paper: the general FG beam paraxial propagation prob-
+lem is reduced to the evaluation of the bivariate confluent
+hypergeometric Φ1.
+Again, it is possible to give Eq. (38) a different dress on
+using the following integral representation of Φ1, estab-
+lished in 2012 by Brychkov and Saad [19, formula 3.4]:
+Φ1
+�
+a, 1
+2
+���� w, z
+�
+=
+= (1 − w)1−a
+� 1
+0
+dξ (1 − w ξ)a−2
+1F1(a; 1; zξ) ,
+(39)
+which eventually leads to
+ψ1(r) = exp
+� ikr2
+2QN
+� qNN
+B
+
+
+
+1
+1 + A qN
+B
+
+
+
+N
+×
+� 1
+0
+dξ
+�
+1 + A qN
+B
+ξ
+�N+1 1F1
+
+
+
+1 − N; 1; ikr2
+2AB
+ξ
+1 +
+B
+AqN
+
+
+
+ .
+(40)
+Similarly as it was found for the free-space propagation
+into Eq. (23), also the integral representation of ψ1 given
+by Eq. (40) turns out to be defined onto a finite interval
+[0, 1], despite the infinite support of both the initial field
+distribution ψ0, as well as its Fourier transform. In the
+present case, however, at least a qualitative explanation
+of such a mathematical counterintuitive behavior can be
+grasped by estimating the right side of Eq. (40) within the
+asymptotic limit N → ∞, which corresponds to replace
+the initial FG beam distribution ψ0 by that emerging
+from a circular hole of radius a.
+In particular, the asymptotics can be carried out in an
+elementary way, by first noting that QN → B/D and
+that
+lim
+N→∞
+1
+�
+1 + A qN
+B
+ξ
+�N+1 = exp
+�
+iA ka2
+2B
+ξ
+�
+.
+(41)
+As far as Kummer’s function inside the integral is
+concerned, the following asymptotics holds [13, for-
+mula 13.8.13]:
+1F1(1 − N; 1; z) ∼ exp(z/2) J0
+�
+2
+√
+N z
+�
+,
+N ≫ 1 ,
+(42)
+which, once substituted into Eq. (40) together with
+Eq. (41), leads to
+ψ1(r) ∼ U
+2i exp
+�
+iUD
+2
+�r
+a
+�2�
+×
+� 1
+0
+dξ exp
+�
+iA U
+2
+ξ
+�
+J0
+�
+U r
+a
+�
+ξ
+�
+,
+N ≫ 1 ,
+(43)
+where now U = ka2/B.
+Finally, it is not difficult to convince that Eq. (43)
+is nothing but von Lommel’s integral [23], namely, the
+result of Collins’ integral for an incident wavefield ψ0 =
+circ(r/a), as it should be expected.
+V.
+CONCLUSIONS
+Even today, the term“superGaussian beam”is synony-
+mous of flat-topped beam, despite the indisputable lim-
+its, both practical and theoretical, of the SG model and
+the availability of more efficient analytical approaches.
+For rectangular geometries, Sedukhin’s work should have
+contributed to identify flat-topped profiles with an error
+function. For two-dimensional, axially symmetric geome-
+tries, Gori’s and Li’s models, despite allowing to solve ex-
+actly the paraxial propagation problem, to date continue
+struggling to supplant the obsolete SG model.
+In the present paper, the FG model has been general-
+ized to any values, no longer necessarily integer, of the or-
+der N. In doing this, use has been made of the suggestion,
+dating back more than twenty-five years ago, by Shep-
+pard & Saghafi to mathematically identify the model FG
+through an incomplete Gamma function. From a merely
+technical viewpoint, our work rests on some beautiful re-
+sults recently established by Brychkov and co-workers. In
+this way, it has been possibile to analytically express the
+optical wavefield generated by the propagation of such
+flat-topped “Γ-beams”of any order through arbitrary ax-
+ially symmetric paraxial optical system (free space in-
+cluded) in terms of a single bivariate confluent hyperge-
+ometric function.
+Our model is purely analytical and provided purely an-
+alytical closed expressions of the paraxially propagated
+wavefield.
+It is a rare situation in physics in general
+and in optics in particular.
+The ubiquitous presence
+of less and less known special functions, such as bivari-
+ate hypergeometric ones certainly are, also constitutes
+in our opinion an added value of the present work. We
+strongly encourage our readers to go through an interest-
+ing paper written more than twenty years ago by Michael
+Berry [24], whose content seems nowadays more than ever
+more relevant. In particular, the current availability of
+powerful computational platforms, such as Mathematica
+and Maple, will allow in the future to increase the set of
+special functions whose evaluation could be implemented
+at arbitrarily high accuracies. We hope bivariate con-
+fluent hypergeometric functions, including of course Ψ1
+and Φ1, could soon become part of such a mathematical
+weaponry.
+Acknowledgements
+I wish to thank Turi Maria Spinozzi for his help during
+the preparation of the manuscript.
+
+7
+Appendix A: Proof of Eq. (8)
+The M 2 factor is defined by
+M 2 = 2π σr σp ,
+(A1)
+where σr and σp denote the widths across the plane z = 0
+and the plane of spatial frequencies, respectively, both of
+them normalized to the beam energy. Due to the axial
+symmetry, σr can then be expressed (in units of a) as
+follows:
+σ2
+r =
+� ∞
+0
+dr r3 ψ2
+0(r)
+� ∞
+0
+dr r ψ2
+0(r)
+.
+(A2)
+The denominator turns out to be
+� ∞
+0
+dr r ψ2
+0(r) = π
+
+1 −
+Γ
+�
+N + 1
+2
+�
+√π Γ(N + 1)
+
+ ,
+(A3)
+while the numerator is
+� ∞
+0
+dr r3 ψ2
+0(r)π
+2
+
+1 + 1
+N − (2N + 1)
+N
+Γ
+�
+N + 1
+2
+�
+√π Γ(N + 1)
+
+ .
+(A4)
+The spectral width σp can also be expressed in terms
+of quantities defined across the plane z = 0, being (in
+units of 1/a)
+σ2
+p =
+1
+2π
+� ∞
+0
+dr r
+�∂ψ0
+∂r
+�2
+� ∞
+0
+dr r ψ2
+0(r)
+,
+(A5)
+where the numerator turns out to be
+� ∞
+0
+dr r
+�∂ψ0
+∂r
+�2
+= 21−2N Γ(2N) ,
+(A6)
+so that, on using again Eq. (5),
+σ2
+p =
+1
+π2 22N Γ(N)2
+√π Γ(N + 2) Γ(2N)
+√π Γ(N + 1) − Γ
+�
+N + 1
+2
+� .
+(A7)
+Finally, on substituting from Eqs. (A2) and (A7) into
+Eq. (A1), Eq. (8) follows.
+Appendix B: Proof of Eq. (23)
+Thanks to the 2011 paper by Choi and Hasanov [18],
+the following integral representation of Ψ1 can be estab-
+lished:
+Ψ1
+�
+N + 1, 1
+N + 1, 1
+���� x, y
+�
+=
+Γ(ǫ)
+Γ(N)Γ(ǫ − N − 1) ×
+� 1
+0
+� 1
+0
+dξ dη ηN(1 − ξ)N−1(1 − η)ǫ−N−2
+(1 − xξ)N+1
+× exp
+�
+−
+yη
+xξ − 1
+�
+1F1
+�
+1 − ǫ; 1;
+yη
+xξ − 1
+�
+(B1)
+where ǫ denotes an arbitrary complex parameters which
+must only satisfy the condition Re{ǫ} > Re{N} + 1. In
+particular, on letting ǫ = N + 2, Eq. (B1) yields
+Ψ1
+�
+N + 1, 1
+N + 1, 1
+���� x, y
+�
+= Γ(N + 2)
+Γ(N)Γ(1) ×
+� 1
+0
+dξ (1 − ξ)N−1
+(1 − xξ)N+1
+×
+� 1
+0
+dη ηN exp
+�
+−
+yη
+xξ − 1
+�
+1F1
+�
+−N − 1; 1;
+yη
+xξ − 1
+�
+=
+= Γ(N + 2)
+Γ(N)
+×
+� 1
+0
+dξ (1 − ξ)N−1
+(1 − xξ)N+1
+� 1
+0
+dη ηN 1F1
+�
+N + 2; 1;
+yη
+1 − xξ
+�
+,
+(B2)
+where, in the last step, Kummer’s transformation has
+been employed. The inner η integral can be evaluated by
+using [21, formula 2.21.1.4], which yields
+� 1
+0
+dη ηN
+1F1
+�
+N + 2; 1;
+yη
+1 − xξ
+�
+=
+=
+1
+N + 1 1F1
+�
+N + 1; 1;
+y
+1 − xξ
+�
+.
+(B3)
+Finally, on substituting from Eq. (B3) into Eq. (B2), after
+simple algebra Eq. (23) follows.
+
+8
+Appendix C: Proof of Eq. (36)
+From the very definition into Eq. (16) we have
+Ψ1
+�
+1, β
+2, 1
+���� t, −s
+�
+=
+∞
+�
+k=0
+∞
+�
+ℓ=0
+(1)k+ℓ (β)k
+(2)k(1)ℓ
+tk
+k!
+(−s)l
+ℓ!
+=
+=
+∞
+�
+k=0
+(1)k (β)k
+(2)k
+tk
+k!
+∞
+�
+ℓ=0
+(1 + k)ℓ
+(1)ℓ
+(−s)l
+ℓ!
+=
+=
+∞
+�
+k=0
+(1)k (β)k
+(2)k
+tk
+k! 1F1(1 + k; 1; −s) =
+= exp(−s)
+∞
+�
+k=0
+(β)k
+(2)k
+tkLk(s) .
+(C1)
+Last series can be expressed in closed form via [20,
+5.11.2.7], i.e.,
+∞
+�
+k=0
+(a)k tk
+(α + β)k
+Lα
+k(x) = (1 − t)−aΦ1
+�
+a, β − 1
+α + β
+����
+t
+t − 1,
+tx
+t − 1
+�
+,
+(C2)
+from which, on letting a = β, α = 0, β = 2, and x = s,
+after straightforward algebra Eq. (36) follows.
+[1] S. De Silvestri, P. Laporta, V. Magni, and 0. Svelto,
+“Solid-state laser unstable resonators with tapered reflec-
+tivity mirrors: the super-Gaussian approach,” IEEE J.
+Quant. El. 24, 1172 - 1177 (1988).
+[2] A. Parent, M. Morin, and P. Lavigne, “Propagation of
+super-Gaussian field distributions,” Opt. Quant. El. 24,
+S1071 - S1079 (1992).
+[3] F. Gori, “Flattened Gaussian beams,” Opt. Commun.
+107, 335-341 (1994).
+[4] Equation (2) was originally derived starting from the
+identity 1 = exp(−ξ2) exp(ξ2) and on truncating the
+Taylor expansion of the second exponential up to N. In
+the present paper, however, we restrict the expansion to
+the first N terms. With such choice the case N = 1 cor-
+respond to the Gaussian beam. But in this
+[5] V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santar-
+siero, D. Ambrosini, and G. Schirripa Spagnolo, “Prop-
+agation of axially symmetric flattened Gaussian beams,”
+J. Opt. Soc. Am. A 13, 1385-1394 (1996).
+[6] R. Borghi, “Elegant Laguerre-Gauss beams as a new tool
+for describing axisymmetric flattened Gaussian beams,”
+J. Opt. Soc. Am. A 18, 1627-1633 (2001).
+[7] Y. Li, “Light beams with flat-topped profiles,” Opt. Lett.
+27, 1007-1009 (2002).
+[8] A.G. Sedukhin, “Rectangular symmetrical mesa beams
+and their comparison with flattened Gaussian and multi-
+Gaussian beams,” Optics Communications, 335, 284 - 292
+(2015).
+[9] R.
+Borghi,
+“Uniform
+approximation
+of
+flat-topped
+beams,” J. Opt. Soc. Am. A (2013)
+[10] P. Appell,“Sur les s´eries hyperg´eom´etriques de deux vari-
+ables et sur des ´equations diff´erentielles lin´eaires aux
+d´eriv´ees partielles,” Comptes rendus hebdomadaires des
+s´ances de l’Acad´emie des sciences 90, 296 - 298 (1880).
+[11] P. Humbert, “The Confluent Hypergeometric Functions
+of Two Variables,” Proceedings of the Royal Society of
+Edinburgh, IX, 73 - 96 1922.
+[12] C. J. R. Sheppard and S. Saghafi, “Flattened light
+beams,” Opt. Commun. 132, 144 -152 (1996).
+[13] Digital
+Library
+of
+Mathematical
+Functions,
+Na-
+tional
+Institute
+of
+Standards
+and
+Technology
+http://dlmf.nist.gov/.
+[14] Y. A. Brychkov, Handbook of Special Functions (CRC
+Press, London, 2008).
+[15] Y. A. Brychkov, New Indefinite and Definite Integrals
+of Elementary and Special Functions (A. A. Dorodnicyn
+Computing Center of the Russian Academy of Sciences,
+Moscow, 2014).
+[16] E.M. El Halba, H. Nebdi, M. Boustimi, and A. Belafhal,
+“On the Humbert confluent hypergeometric function used
+in laser field,” Phys. Chem. News 73, 90 - 93 (2014).
+[17] A. Belafhal and F. Saad,“Conversion of circular beams by
+a spiral phase plate: Generation of Generalized Humbert
+beams,” Optik 138, 516 - 528 (2017).
+[18] J. Choi and A. Hasanov, “Applications of the operator
+H(α, β) to the Humbert double hypergeometric func-
+tions,” Computers and Mathematics with Applications
+61, 663 - 671 (2011).
+[19] Y. A. Brychkov and N. Saad, “Some formulas for the
+Appell function F1(a, b, b′; c; w, z).” Integral Transforms
+and Special Functions 23, 793 - 802 (2012).
+[20] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev,
+Integrals and Series (Gordon Breach, 1986), Vol. II.
+[21] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev,
+Integrals and Series (Gordon Breach, 1986), Vol. III.
+[22] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev,
+Integrals and Series (Gordon Breach, 1986), Vol. IV.
+[23] M. Born and E. Wolf, Principles of Optics (Cambridge
+University Press, Cambridge, 1999).
+[24] M. V. Berry, “Why are special functions special?,” Phys.
+Today, 11-12 (2001)
+

KNE3T4oBgHgl3EQfXwo4/content/tmp_files/load_file.txt

@@ -0,0 +1,398 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf,len=397
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='04481v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='optics]  11 Jan 2023  “Analytical Continuation” of Flattened Gaussian Beams  Riccardo Borghi  Dipartimento di Ingegneria Civile, Informatica e delle Tecnologie Aeronautiche,  Universit`a “Roma Tre”, Via Vito Volterra 62, I-00146 Rome, Italy  A purely analytical extension of the flattened Gaussian beams [Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' 107, 335 (1994)]  to any values of the beam order, is here proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Thanks to it, the paraxial propagation problem of  axially symmetric, coherent flat-top beams through arbitrary ABCD optical systems can definitely  be closed in terms of a particular bivariate confluent hypergeometric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  INTRODUCTION  Flat-top beams continue to attract a considerable at-  tention in optics: during the last five years more than  sixty papers have been published on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' In or-  der to model flat-top axially symmetric distributions, two  classes of different scenarios appeared: in the first one,  simple analytical profiles were employed, the most known  of them being the superGaussian (SG) [1, 2], which is for-  mally defined by  SGν(ξ) = exp(−ξ2ν) ,  (1)  where ν denotes a real parameter which controls the“flat-  ness” of the profile, with the particular case ν = 1 giving  the Gaussian profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' The symbol ξ denotes a normal-  ized radial transverse position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  Despite its mathemat-  ical simplicity, it is well known that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (1) does not  allow the wavefield of paraxially propagated superGaus-  sian (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=', for ν ̸= 1) beams to be analytically evaluated,  even within the simplest scenario, namely free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  To overcome such a difficulty, which two or three  decades ago could represent a considerable computational  bottleneck in several practical situations, alternative ap-  proaches were proposed in 1994 and in 2002 by Gori and  Li, respectively, to conceive analytical models able to  solve the free space propagation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' The former  was called flattened Gaussian (FG henceforth) [3], and,  differently from SG, is expressed through an explicit fi-  nite sum of terms, namely  FGN(ξ) = exp(−Nξ2)  N−1  �  m=0  (Nξ2)m  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  ,  (2)  where the integer parameter N will be referred to as the  FG order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Scaling the ξ variable by the factor  √  N gives  the FG transverse profile a flat-topped shape which, for  N = 1, reduces to a Gaussian distribution, whereas for  N → ∞ tends to the characteristic function of the uni-  tary disk [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' The model is computationally exact, since  the initial distribution (2) can be recast in terms of a su-  perposition of N standard Laguerre-Gauss (sLG hence-  forth) beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Accordingly, in order to evaluate the field  propagated in free space, it was enough to sum up the  N propagated sLG, a job which can exactly be done, al-  ways [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' In [6], a different superposition scheme of the  profile (2) was proposed, in which the sLG family was  replaced by the so- called elegant Laguerre-Gauss (eLG  henceforth) set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' In this way, not only free-space propa-  gation, but also the interaction of FG beams with any  axially symmetric paraxial optical system can be dealt  with in exact terms, always through finite sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  In 2002, Yaijun Li proposed an analytical model al-  ternative to the FG one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' The idea was to impose a lo-  cal “flatness” condition, which required the first 2N ξ-  derivatives of the profile to be null at the origin ξ = 0 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  On using such condition, Li conceived the following ana-  lytical model:  LiGN(ξ) =  N  �  m=1  (−1)m−1  �N  m  �  exp(−mξ2) =  =  �  1 −  �  1 − exp  �  −ξ2���N  N  ,  (3)  which, differently from FG, is based on the superposi-  tion of N fundamental Gaussian beams having variable  widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  Both Gori’s and Li’s models provide exact solutions  to the paraxial propagation problem of coherent, axially  symmetric flat-topped beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' From a merely mathemat-  ical perspective, their only own limit is represented by  the fact that, differently from SG, only positive integer  orders N can be dealt with to describe the initial flat-top  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' It is important to mention that, for 1D ge-  ometry (or rectangular 2D geometries), general analyt-  ical solutions were already provided, at least upon free  propagation, by modeling the flat-top profile via an error  function [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' An attempt to extend the 2D circular FG  model to noninteger orders was also proposed in [9], but  only approximate estimates of the free space propagated  field were found within the asymptotic limit N ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  The aim of the present paper is to solve exactly the  propagation problem of FG beams of any order (real or  even complex) through typical axially symmetric parax-  ial optical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' To this end, the right side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (2)  will first be identified as an incomplete Gamma func-  tions, which is known to be defined onto the whole com-  plex plane, as far as both arguments are concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' An  immediate byproduct of such identification will be the  closed form expression of the M 2 factor of FG beams  of any order, an interesting generalization of the result  found in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' This is shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' II of the present pa-  per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' The most important results are indeed presented  2  in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' III and IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' In the former, the free- space prop-  agation problem will be solved thanks to an important  class of integrals recently closed by Yuri Brychkov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Al-  though the more general propagation problem will be  solved in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' IV, the analysis presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' III should  be viewed as an important propaedeutical step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' There, it  will be shown that a very important, but nevertheless not  so much known, class of special functions, called bivariate  hypergeometric functions, together with the correspond-  ing confluent versions, form the mathematical skeleton of  the paraxially diffracted wavefield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Bivariate hypergeo-  metric were first introduced in 1880 by Paul Appell [10],  their confluent version forty years later by Paul Hum-  bert [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' The results we are going to present would also  give readers a partial answer about the lack, for more  than thirty years, of purely analytical solutions to the  problem of the paraxial propagation of coherent 2D flat-  topped beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  The present work has a clear mathematical character:  for instance, dimensionless quantities will be used wher-  ever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  Moreover, the number of mathematical  appendices have been considerably limited, because we  strongly believe that following all most important math-  ematical steps could greatly help readers to fully grasp  the essence of our analysis, as well as the importance of  such still mysterious special functions, which will lead to  analytical, elegant, and exact solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  PRELIMINARIES  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  “Analytical continuation” of the FG model  Already in 1996, Sheppard & Saghafi [12] pointed out  that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (2) can be given the closed form  FGN(ξ) = Γ(N, Nξ2)  Γ(N)  ,  (4)  where Γ(·) and Γ(·, ·) denote Gamma and incomplete  Gamma functions, respectively [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  Differently from  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (2), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (4) is not limited to integer FG orders, but  rather it can be analytically continued to real and also  complex values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  As a preliminary result of the extended definition into  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (4), an analytical check of Li’s“flatness condition”[7]  will now be carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' To this end, it is sufficient to use  formulas 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='17 of [14] to prove, with long  but simple algebra, that  dn  dξn Γ(N, Nξ2) = −2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='N N exp(−Nξ2) ξ2N−n  ×  [n/2]  �  k=0  (n − k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  4kk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (n − 2k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' L(N−n+k)  n−k−1  (Nξ2) ,  (5)  which gives at once  � dn  dξn Γ(N, Nξ2)  �  ξ=0  = 0 ,  0 ≤ n < 2 Re{N} ,  (6)  thus implying the real part of N to be chosen greater  than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  Spreading properties: closed form expression of the M 2  factor  An interesting byproduct of the extended Γ-based def-  inition into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (4) is the evaluation of the M 2 factor of  FG beams, first established in [5] for N ∈ N, for nonin-  teger orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' To this end, consider an initial field distri-  bution across the plane z = 0 of a cylindrical reference  frame (r, z), say ψ0(r), given by  ψ0(r) = FGN  �r  a  �  =  Γ  �  N, N r2  a2  �  Γ(N)  ,  (7)  where an overall amplitude constant has been set to one  and the symbol a denotes the “width” of lat-top distri-  bution field distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  For simplicity, it will be set  a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  The evaluation of the M 2 factor, which is defined as the  product of the normalized second order moments across  the z = 0 and the spatial frequency planes is detailed  in Appendix A, where it is proved the following closed-  form expression:  M 2 =  �  (N + 1) Γ(N + 1/2)  √π Γ(N + 1)  �  1 −  Γ(N + 3/2)  √π Γ(N + 2)  �  1 −  Γ(N + 1/2)  √π Γ(N + 1)  ,  (8)  which extends the 1996 analysis of [5] to N /∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' It is  worth comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (8) with the corresponding expres-  sion of SG beam M 2 factor, namely [2]  M 2 =  �  Γ(2/ν)  Γ(1/ν)/ν ,  (9)  deceptively simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' In the next two sections, our exten-  sion of the FG model will further reveal its powerfulness  and mathematical elegance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  FREE-SPACE PARAXIAL PROPAGATION OF FG  BEAMS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  Preliminaries  Suppose the initial field distribution given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (7)  is allowed to propagate in free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' The corresponding  3  field, say ψ(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' z), can be expressed, apart from an overall  phase factor exp(ikz), as follows:  ψ(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' z) = −i U  2π  �  R2 d2ρ ψ0(ρ) exp  �iU  2 |r − ρ|2  �  ,  (10)  where the Fresnel number U = ka2/z has been intro-  duced and the beam width a has been used as unit for  measuring all transverse sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' This means that the quan-  tity r should be meant as the ratio between the trans-  verse position vector of the observation point and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' For  integer FG orders, the free space propagation problem  has already been solved in [3] by expanding the initial  field distribution ψ0 as the linear combination of a finite  number of sLG beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  It is then sufficient to propa-  gate each sLG beam up to the observation plane and to  recombine all of them with the initial expanding coeffi-  cients for the correct value of ψ(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' z) to be retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' As  we are going to show in a moment, the Γ-based model  into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (4) allows an exact evaluation of the propagated  wavefield (10) also for N /∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' It is worth recalling that,  from a mere practical perspective, the present section  could seem somewhat redundant, as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' IV the more  general propagation problem within ABCD systems will  be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  Nevertheless, we believe what is contained  in the present section could help nonspecialist readers  to familiarize with the main notations and mathemati-  cal tools which will constitute the basis of the general  results presented into Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' In other words, it should  be considered as a useful, propaedeutical material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  We start on substituting from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (7) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (10),  which after simple algebra gives  ψ(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' z) = − i U  Γ(N) exp  �iU r2  2  �  ×  � ∞  0  dρ ρ exp  �iU  2 ρ2  �  Γ  �  N, N ρ2�  J0(Ur ρ) ,  (11)  where J0 denotes the 0th-order Bessel function of the first  kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' It is worth recasting the incomplete Γ function as  Γ(N, Nξ)  Γ(N)  = 1 − γ(N, Nξ)  Γ(N)  ,  (12)  where γ(·, ·) denotes the “lower”incomplete gamma func-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (11) takes on the form  ψ(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' z) =  = −i U exp  �iU r2  2  � � ∞  0  dρ ρ exp  �  −U  2i ρ2  �  J0(Ur ρ)  +  i U exp  �iU r2  2  �  Γ(N)  ×  � ∞  0  dρ ρ exp  �  −U  2i ρ2  �  γ  �  N, Nρ2�  J0(Ur ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  (13)  The first term is identically equal to one (it is nothing  but a unitary plane wave propagating along the z-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  As far as the second is concerned, the following notable  formula has recently been published by Brychkov [15,  formula 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='20]:  � ∞  0  dx xα−1 exp(−a x2) γ(µ, bx2) Jν(c x) =  =  2−ν−1bµcνΓ  �  µ + α + ν  2  �  µaµ+(α+ν)/2Γ(ν + 1)  Ψ1  �  µ + α + ν  2  , µ  µ + 1, ν + 1  ����� − b  a, − c2  4a  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  (14)  Then, on using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (13) and (14), long but straightfor-  ward algebra gives  ψ(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' z) = 1 − exp  �iU r2  2  � �2iN  U  �N  × Ψ1  �  N + 1, N  N + 1, 1  ���� − 2iN  U , −iU r2  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  (15)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  A short Tour on Bivariate Hypergeometric Functions  The symbol Ψ1 into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (15) denotes a special func-  tion called bivariate confluent hypergeometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' It is worth  briefly describing the principal definitions and properties  which are important for our scopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Function Ψ1 is for-  mally defined through the following double series power  expansion:  Ψ1  �  a, b  c, c′  ���� z, w  �  =  ∞  �  k=0  ∞  �  ℓ=0  (a)k+ℓ (b)k  (c)k(c′)ℓ  zk  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  wℓ  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' ,  (16)  valid for |z| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  The symbol (·)n denotes Pochham-  mer’s symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Another bivariate confluent hypergeomet-  ric function which will be meet in the present paper is  the function Φ1, defined by  Φ1  �  a, b  c  ���� z, w  �  =  ∞  �  k=0  ∞  �  ℓ=0  (a)k+ℓ (b)k  (c)k+ℓ  zk  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  wℓ  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' ,  (17)  valid for |z| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  Functions Ψ1 and Φ1 are members  of a family of functions that generalize Kummer’s con-  fluent hypergeometric function 1F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' In particular, Φ1 is  obtained from the so-called Appell function F1, defined  by  F1  �  a, b1, b2  c  ���� z, w  �  =  ∞  �  k=0  ∞  �  ℓ=0  (a)k+ℓ (b1)k (b2)ℓ  (c)k+ℓ  zk  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  wℓ  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' ,  (18)  (again valid for |z| ≤ 1), through the following limiting  definition:  Φ1  �  a, b  c  ���� z, w  �  = lim  ǫ→0 F1  �  a, b, 1  ǫ  c  ����� z, ǫw  �  ,  (19)  4  which can be proved on first substituting the identity  lim  ǫ→0  �1  ǫ  �  ℓ  ǫℓ = 1 ,  (20)  directly into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (19), then on interchanging the limit  with the double series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  Multivariate hypergeometric and confluent hypergeo-  metric functions play a role of considerable importance  in theoretical physics and applied math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' In optics, the  role of bivariate confluent hypergeometric functions in de-  scribing a large class of paraxial optical disturbances has  recently been pointed out [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Moreover, it is worth  stressing that, from a practical viewpoint, Appell’s func-  tion F1 is nowadays part of the symbolic platform Math-  ematica, where it is computable with arbitrarily high ac-  curacies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Also the whole family of Appell functions, in-  cluding F1 as well as its three sisters F2, F3, and F4,  are currently implemented in the latest release of Maple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  It is then highly desirable that in a near future also the  set of bivariate confluent hypergeometric functions, in-  cluding Ψ1 and Φ1, could become part of such family of  “evaluable”special functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' In the meanwhile, someone  might rightly object to the practical usefulness of func-  tions that are defined through double infinite series like  those into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (16) - (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' To overcome such difficulties,  some tricks will be implemented in the rest of the paper,  tricks which are aimed at extending the validity domain  of Ψ1 and Φ1 beyond the series definitions, and then to  improve the practical usefulness of our analytical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  Free-space propagation formula  Function Ψ1 can be continued by using the following  transformation [18, formula 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='54]:  Ψ1  �  α, β  γ1, γ2  ���� z, w  �  =  =  1  (1 − z)α Ψ1  �  α, γ1 − β  γ1, γ2  ����  z  z − 1,  w  1 − z  �  ,  (21)  which, once substituted into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (15), gives a new, closed-  form, expression of the paraxial propagated field  ψ(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' z) = 1 −  exp  �iU r2  2  �  1 + 2iN  U  \uf8eb  \uf8ec  \uf8ed  1  1 +  U  2iN  \uf8f6  \uf8f7  \uf8f8  N  × Ψ1  \uf8eb  \uf8ec  \uf8ed N + 1, 1  N + 1, 1  ����  1  1 +  U  2iN  , −  iU r2  2  1 + 2iN  U  \uf8f6  \uf8f7  \uf8f8 ,  (22)  indubitably one of the main results of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  Waiting for Mathematica or Maple to develop their own  built-in version of Ψ1, it is worth working on the expres-  sion into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (22) by using a notable integral represen-  tation found again in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  For the sake of clarity, all  mathematical steps are confined into Appendix B, where  it is proved that  Ψ1  �  N + 1, 1  N + 1, 1  ���� x, y  �  =  = N  � 1  0  dξ (1 − ξ)N−1  (1 − xξ)N+1 1F1  �  N + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  y  1 − xξ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  (23)  Equation (23) appears to be somewhat intriguing: the  wavefield of a free-space paraxially propagated FG beam  of any order can be represented via a 1D integral defined  over a finite integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' This could seem a somewhat pe-  culiar situation, due to the fact that the initial field dis-  tribution (7) has an infinite support, namely the whole  plane z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  But what is, in our opinion, even more  important is that the integral representation (23) would  hardly be reachable starting from Fresnel’s integral (10),  without passing through the Ψ1 function and its trans-  formation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' In the next section, a similar scenario  will also be found as far as the more general problem is  concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  PARAXIAL PROPAGATION THROUGH ABCD  SYSTEMS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  Preliminaries  The free-space paraxial propagation formula derived in  the previous section will now be extended to the general  case of the paraxial propagation of FG beams of any or-  der through typical paraxial optical systems with axial  symmetry, characterized by the so-called ABCD optical  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' For FG beams of integer order, it was found  in [6] that the propagation problem can be dealt with  in exact terms by expanding the initial field distribution  given into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (7) and (2) as a finite superposition of  so-called elegant Laguerre (eLG henceforth) beams as fol-  lows:  ψ0(r) =  N−1  �  n=0  (−)n  �  N  n + 1  �  eLGn  �ikr2  2qN  �  ,  (24)  where the symbol eLGn(x) = exp(x)Ln(−x) will be re-  ferred to as the elegant Laguerre function of order n and  the complex radius of curvature qN = ka2  2iN has also been  introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' The initial distribution ψ0 is then recast as  follows:  ψ0(r) = exp  �ikr2  2qN  �  GN  �  1, −ikr2  2qN  �  ,  (25)  where the function GN (·, ·) is defined, for integer N, as  GN (t, s) =  N−1  �  n=0  (−t)n  �  N  n + 1  �  Ln(s) ,  (26)  5  In [6] it was proved that, if the initial field distribution  given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (25) feeds an axially symmetric paraxial  optical system described by the optical matrix M  M =  \uf8eb  \uf8ed  A B  C D  \uf8f6  \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  (27)  then the wavefield at the output plane of the system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' say  ψ1(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' takes on the following form:  ψ1(r) =  =  exp  � ikr2  2QN  �  A  1  1 +  B  A qN  GN  \uf8eb  \uf8ec  \uf8ec  \uf8ed  1  1 +  B  A qN  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  kr2  2iA2 qN  1 +  B  A qN  \uf8f6  \uf8f7  \uf8f7  \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  (28)  where an overall phase factor exp(ikℓ) (with ℓ being the  optical lenght) will be tacitly assumed and QN denotes  the complex quantity  QN = A qN + B  C qN + D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  (29)  The problem of extending the function GN (t, s) to N /∈ N  will now be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  Extension of the function GN (t, s) to N /∈ N  The starting point is the following Laplace transform  representation of GN(t, s) established in [9]:  GN(t, s) = exp(s)  � ∞  0  dξ exp(−ξ) J0  �  2  �  s ξ  �  L(1)  N−1(ξ t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  (30)  For N ∈ N, the Laguerre polynomials L(1)  N−1 can be writ-  ten as  L(1)  N−1(ξ t) =  N−1  �  n=0  Ln(ξ t) ,  (31)  so that, on substituting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (31) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (30), it is  found  GN(t, s) =  = exp(s)  N−1  �  n=0  � ∞  0  dξ exp(−ξ) J0  �  2  �  s ξ  �  Ln(ξ t) =  =  N−1  �  n=0  (1 − t)n Ln  � st  t − 1  �  ,  (32)  where in the last passage, [22, formula 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='2] has been  used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Equation (32) is a valid alternative, for N ∈ N, to  the definition given into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' For the scopes of the  present paper, its importance stems from the fact that  the quantity GN can also be thought of as function of  two new variables, namely  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  1 − t =  1  1 + AqN  B  ,  st  t − 1 = ikr2  2AB  1  1 +  B  AqN  ,  (33)  and this will reveal of a certain importance in the rest of  our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  In order to extend the integral into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (30) to N /∈  N, the following notable formula, again established by  Brychkov [15], will be employed:  � ∞  0  xα−1 exp(−ax) Jν(b√x) L(λ)  n (cx) dx =  =  � b  2  �ν Γ  �  α + ν  2  �  (λ + 1)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' aα+ν/2Γ(ν + 1) Ψ1  �  α + ν  2 , −n  λ + 1, ν + 1  �����  c  a, − b2  4a  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  (34)  In particular, on letting α = 1, a = 1, ν = 0, b = 2√s,  t = c, n = N − 1, and λ = 1, Laplace’s transform into  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (30) takes on the form  GN(t, s) = N exp(s) Ψ1  �  1, 1 − N  2, 1  ���� t, −s  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  (35)  Again, it can be appreciated how the confluent hyperge-  ometric function Ψ1 constitutes the mathematical skele-  ton of the propagated field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' But there is more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' In Ap-  pendix C, the following relationship has been established:  Ψ1  �  1, 1 − N  2, 1  ���� t, −s  �  =  =  exp(−s)  (1 − t)1−N Φ1  �  1 − N, 1  2  ����  t  t − 1,  st  t − 1  �  ,  (36)  where Φ1 is the confluent hypergeometric function de-  fined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' On substituting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (36) into  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (35), we have  GN(t, s) = N (1 − t)N−1 Φ1  �  1 − N, 1  2  ����  t  t − 1,  st  t − 1  �  (37)  so that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (28) eventually becomes  ψ1(r) = exp  � ikr2  2QN  � qNN  B  \uf8eb  \uf8ec  \uf8ed  1  1 + A qN  B  \uf8f6  \uf8f7  \uf8f8  N  × Φ1  \uf8eb  \uf8ec  \uf8ec  \uf8ed  1 − N, 1  2  ���� − A qN  B  , ikr2  2AB  1  1 +  B  AqN  \uf8f6  \uf8f7  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  (38)  6  Equation (38) summarizes the main result of the present  paper: the general FG beam paraxial propagation prob-  lem is reduced to the evaluation of the bivariate confluent  hypergeometric Φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  Again, it is possible to give Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (38) a different dress on  using the following integral representation of Φ1, estab-  lished in 2012 by Brychkov and Saad [19, formula 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='4]:  Φ1  �  a, 1  2  ���� w, z  �  =  = (1 − w)1−a  � 1  0  dξ (1 − w ξ)a−2  1F1(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' zξ) ,  (39)  which eventually leads to  ψ1(r) = exp  � ikr2  2QN  � qNN  B  \uf8eb  \uf8ec  \uf8ed  1  1 + A qN  B  \uf8f6  \uf8f7  \uf8f8  N  ×  � 1  0  dξ  �  1 + A qN  B  ξ  �N+1 1F1  \uf8eb  \uf8ec  \uf8ec  \uf8ed1 − N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' ikr2  2AB  ξ  1 +  B  AqN  \uf8f6  \uf8f7  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  (40)  Similarly as it was found for the free-space propagation  into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (23), also the integral representation of ψ1 given  by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (40) turns out to be defined onto a finite interval  [0, 1], despite the infinite support of both the initial field  distribution ψ0, as well as its Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' In the  present case, however, at least a qualitative explanation  of such a mathematical counterintuitive behavior can be  grasped by estimating the right side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (40) within the  asymptotic limit N → ∞, which corresponds to replace  the initial FG beam distribution ψ0 by that emerging  from a circular hole of radius a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  In particular, the asymptotics can be carried out in an  elementary way, by first noting that QN → B/D and  that  lim  N→∞  1  �  1 + A qN  B  ξ  �N+1 = exp  �  iA ka2  2B  ξ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  (41)  As far as Kummer’s function inside the integral is  concerned, the following asymptotics holds [13, for-  mula 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='13]:  1F1(1 − N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' z) ∼ exp(z/2) J0  �  2  √  N z  �  ,  N ≫ 1 ,  (42)  which, once substituted into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (40) together with  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (41), leads to  ψ1(r) ∼ U  2i exp  �  iUD  2  �r  a  �2�  ×  � 1  0  dξ exp  �  iA U  2  ξ  �  J0  �  U r  a  �  ξ  �  ,  N ≫ 1 ,  (43)  where now U = ka2/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  Finally, it is not difficult to convince that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (43)  is nothing but von Lommel’s integral [23], namely, the  result of Collins’ integral for an incident wavefield ψ0 =  circ(r/a), as it should be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  CONCLUSIONS  Even today, the term“superGaussian beam”is synony-  mous of flat-topped beam, despite the indisputable lim-  its, both practical and theoretical, of the SG model and  the availability of more efficient analytical approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  For rectangular geometries, Sedukhin’s work should have  contributed to identify flat-topped profiles with an error  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' For two-dimensional, axially symmetric geome-  tries, Gori’s and Li’s models, despite allowing to solve ex-  actly the paraxial propagation problem, to date continue  struggling to supplant the obsolete SG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  In the present paper, the FG model has been general-  ized to any values, no longer necessarily integer, of the or-  der N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' In doing this, use has been made of the suggestion,  dating back more than twenty-five years ago, by Shep-  pard & Saghafi to mathematically identify the model FG  through an incomplete Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' From a merely  technical viewpoint, our work rests on some beautiful re-  sults recently established by Brychkov and co-workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' In  this way, it has been possibile to analytically express the  optical wavefield generated by the propagation of such  flat-topped “Γ-beams”of any order through arbitrary ax-  ially symmetric paraxial optical system (free space in-  cluded) in terms of a single bivariate confluent hyperge-  ometric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  Our model is purely analytical and provided purely an-  alytical closed expressions of the paraxially propagated  wavefield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  It is a rare situation in physics in general  and in optics in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  The ubiquitous presence  of less and less known special functions, such as bivari-  ate hypergeometric ones certainly are, also constitutes  in our opinion an added value of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' We  strongly encourage our readers to go through an interest-  ing paper written more than twenty years ago by Michael  Berry [24], whose content seems nowadays more than ever  more relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' In particular, the current availability of  powerful computational platforms, such as Mathematica  and Maple, will allow in the future to increase the set of  special functions whose evaluation could be implemented  at arbitrarily high accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' We hope bivariate con-  fluent hypergeometric functions, including of course Ψ1  and Φ1, could soon become part of such a mathematical  weaponry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  Acknowledgements  I wish to thank Turi Maria Spinozzi for his help during  the preparation of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  7  Appendix A: Proof of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (8)  The M 2 factor is defined by  M 2 = 2π σr σp ,  (A1)  where σr and σp denote the widths across the plane z = 0  and the plane of spatial frequencies, respectively, both of  them normalized to the beam energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Due to the axial  symmetry, σr can then be expressed (in units of a) as  follows:  σ2  r =  � ∞  0  dr r3 ψ2  0(r)  � ∞  0  dr r ψ2  0(r)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  (A2)  The denominator turns out to be  � ∞  0  dr r ψ2  0(r) = π  \uf8ee  \uf8ef\uf8ef\uf8f01 −  Γ  �  N + 1  2  �  √π Γ(N + 1)  \uf8f9  \uf8fa\uf8fa\uf8fb ,  (A3)  while the numerator is  � ∞  0  dr r3 ψ2  0(r)π  2  \uf8ee  \uf8ef\uf8ef\uf8f01 + 1  N − (2N + 1)  N  Γ  �  N + 1  2  �  √π Γ(N + 1)  \uf8f9  \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  (A4)  The spectral width σp can also be expressed in terms  of quantities defined across the plane z = 0, being (in  units of 1/a)  σ2  p =  1  2π  � ∞  0  dr r  �∂ψ0  ∂r  �2  � ∞  0  dr r ψ2  0(r)  ,  (A5)  where the numerator turns out to be  � ∞  0  dr r  �∂ψ0  ∂r  �2  = 21−2N Γ(2N) ,  (A6)  so that, on using again Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (5),  σ2  p =  1  π2 22N Γ(N)2  √π Γ(N + 2) Γ(2N)  √π Γ(N + 1) − Γ  �  N + 1  2  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  (A7)  Finally, on substituting from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (A2) and (A7) into  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (A1), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (8) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  Appendix B: Proof of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (23)  Thanks to the 2011 paper by Choi and Hasanov [18],  the following integral representation of Ψ1 can be estab-  lished:  Ψ1  �  N + 1, 1  N + 1, 1  ���� x, y  �  =  Γ(ǫ)  Γ(N)Γ(ǫ − N − 1) ×  � 1  0  � 1  0  dξ dη ηN(1 − ξ)N−1(1 − η)ǫ−N−2  (1 − xξ)N+1  × exp  �  −  yη  xξ − 1  �  1F1  �  1 − ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  yη  xξ − 1  �  (B1)  where ǫ denotes an arbitrary complex parameters which  must only satisfy the condition Re{ǫ} > Re{N} + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' In  particular, on letting ǫ = N + 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (B1) yields  Ψ1  �  N + 1, 1  N + 1, 1  ���� x, y  �  = Γ(N + 2)  Γ(N)Γ(1) ×  � 1  0  dξ (1 − ξ)N−1  (1 − xξ)N+1  ×  � 1  0  dη ηN exp  �  −  yη  xξ − 1  �  1F1  �  −N − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  yη  xξ − 1  �  =  = Γ(N + 2)  Γ(N)  ×  � 1  0  dξ (1 − ξ)N−1  (1 − xξ)N+1  � 1  0  dη ηN 1F1  �  N + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  yη  1 − xξ  �  ,  (B2)  where, in the last step, Kummer’s transformation has  been employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' The inner η integral can be evaluated by  using [21, formula 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='4], which yields  � 1  0  dη ηN  1F1  �  N + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  yη  1 − xξ  �  =  =  1  N + 1 1F1  �  N + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  y  1 − xξ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  (B3)  Finally, on substituting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (B3) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (B2), after  simple algebra Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (23) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  8  Appendix C: Proof of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (36)  From the very definition into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (16) we have  Ψ1  �  1, β  2, 1  ���� t, −s  �  =  ∞  �  k=0  ∞  �  ℓ=0  (1)k+ℓ (β)k  (2)k(1)ℓ  tk  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  (−s)l  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  =  =  ∞  �  k=0  (1)k (β)k  (2)k  tk  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  ∞  �  ℓ=0  (1 + k)ℓ  (1)ℓ  (−s)l  ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  =  =  ∞  �  k=0  (1)k (β)k  (2)k  tk  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' 1F1(1 + k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' −s) =  = exp(−s)  ∞  �  k=0  (β)k  (2)k  tkLk(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  (C1)  Last series can be expressed in closed form via [20,  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='7], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=',  ∞  �  k=0  (a)k tk  (α + β)k  Lα  k(x) = (1 − t)−aΦ1  �  a, β − 1  α + β  ����  t  t − 1,  tx  t − 1  �  ,  (C2)  from which, on letting a = β, α = 0, β = 2, and x = s,  after straightforward algebra Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' (36) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' De Silvestri, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Laporta, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Magni, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Svelto,  “Solid-state laser unstable resonators with tapered reflec-  tivity mirrors: the super-Gaussian approach,” IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' El.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' 24, 1172 - 1177 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Parent, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Morin, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Lavigne, “Propagation of  super-Gaussian field distributions,” Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' El.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' 24,  S1071 - S1079 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  [3] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Gori, “Flattened Gaussian beams,” Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  107, 335-341 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content='  [4] Equation (2) was originally derived starting from the  identity 1 = exp(−ξ2) exp(ξ2) and on truncating the  Taylor expansion of the second exponential up to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' In  the present paper, however, we restrict the expansion to  the first N terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' With such choice the case N = 1 cor-  respond to the Gaussian beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' But in this  [5] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Bagini, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Borghi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Gori, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Pacileo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
+page_content=' Santar-  siero, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE3T4oBgHgl3EQfXwo4/content/2301.04481v1.pdf'}
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+Condensed Matter Physics, 2022, Vol. 25, No. 4, 43708: 1–12
+DOI: 10.5488/CMP.25.43708
+http://www.icmp.lviv.ua/journal
+Path integral Monte Carlo simulations of the
+geometrical effects in KDP crystals
+F. Torresi
+, J. Lasave
+, S. Koval
+∗
+Instituto de Física Rosario, Universidad Nacional de Rosario and CONICET, 27 de Febrero 210 Bis, 2000 Rosario,
+Argentina
+Received July 10, 2022
+Path integral Monte Carlo (PIMC) simulations with very simple models were used in order to unveil the physics
+behind the isotope effects in H-bonded ferroelectrics. First, we studied geometrical effects in the H-bonds caused
+by deuteration with a general three-site model based on a back-to-back double Morse potential plus a Morse
+potential between oxygens, fitted to explain different general features for a wide set of H-bonded compounds.
+Our model results show the Ubbelohde or geometrical effect (GE), i.e., the expansion of the H-bond with deute-
+ration, in agreement to what is observed in H-bonded ferroelectrics with short H-bonds. Moreover, adjusting the
+potential parameters to ab initio results, we have developed a 1D model which considers the bilinear proton-
+proton interaction in mean-field to study nuclear quantum effects that give rise to the GE in KDP crystals. PIMC
+simulations reveal that protons tunnel more efficiently than deuterons along the 1D chain, giving rise to a strong
+attraction center that pulls the oxygens together. This mechanism, which is based on the correlation between
+tunneling and geometrial modifications of the H-bonds, leads to a strong GE in the ordered phase of the chain
+at low temperature which is in good agreement with the experimental data.
+Key words: ferroelectric phase transition, H-bonded ferroelectrics, path integral Monte Carlo
+1. Introduction
+KH2PO4 or KDP is the prototype of a wide family of H-bonded ferroelectric compounds which has
+extensive applications as a key component in optoelectronic devices [1]. Besides the technological interest,
+KDP has also attracted much attention due to its rich, complex and intriguing phenomenology, e.g., the
+huge isotope effect that displays associated to its ferroelectric-paraelectric (FE-PE) phase transition. With
+deuteration, the critical temperature 𝑇𝑐 changes from ≈ 122 K to ≈ 210 K. The saturated polarization 𝑃𝑠
+at low 𝑇 also shows a large isotope effect, increasing from ≈ 5.0 µC/cm2 for KDP to ≈ 6.2 µC/cm2 for a
+sample with 98% of deuteration [2].
+The origin of these strong isotope effects is still controversial. The first explanation of the large
+increase of 𝑇𝑐 upon deuteration was given by the quantum tunneling model [3], which focuses purely
+on mass-dependent effects. However, increasing experimental evidence since the late nineteen eighties
+showed that the large isotope effect is mainly driven by geometrical modifications of the H bonds [4, 5]
+(Ubbelohde effect [6]). The recent observation of tunneling in the PE phase of KDP by neutron Compton
+scattering experiments added even more controversy to the problem [7], although in deuterated KDP
+(DKDP), tunneling could not be detected [8].
+Ab initio calculations have recently shown that tunneling and geometric effects are complementary
+aspects of the same phenomenon[9, 10]. With a simple selfconsistent model based on ab initio results, it
+is demonstrated that the wave function solution of the nonlinear Schrödinger equation for deuteron/proton
+clusters evolves from a double peak to a broad single peak located at the center of the H-bonds as the
+cluster mass diminishes. This is explained by a strong nonlinear feedback between proton delocalization
+(tunneling) and the effective proton potential barrier in the H-bonds, which changes concomitantly with
+∗Corresponding author: koval@ifir-conicet.gov.ar.
+This work is licensed under a Creative Commons Attribution 4.0 International License. Further distribution
+of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
+43708-1
+arXiv:2301.01536v1  [cond-mat.mtrl-sci]  4 Jan 2023
+
+F. Torresi, J. Lasave, S. Koval
+the H-bond geometry. It is concluded that such a large mass dependence can explain the large isotope effect
+found in KDP, via an amplified and selfconsistent geometric modification of the H bond in agreement
+with experiments. On the other hand, these results are in striking contrast with the very weak dependence
+obtained at fixed potential and geometry. Thus, the proton tunneling subunit and the host lattice are
+strongly coupled and the host-and-tunneling system is not separable.
+Many models were successfully developed in the past to shed light into the general phenomenology of
+H-bonded ferroelectric materials [11–18]. In this paper, we address with very simple models the problem
+of geometrical effects in KDP crystals by performing path integral Monte Carlo (PIMC) simulations.
+First, we develop a three-site model for the H-bond to study local quantum geometric effects. This simple
+model already serves us to gain knowledge about the interplay between proton tunneling and H-bond
+geometric modifications such as the O–O distance variation. After this first insight, we develop a 1D
+chain model of concatenated H-bonds to study in the ordered phase the geometrical effects caused by
+deuteration. The model parameters are fitted using recent ab initio results [19]. We demonstrate that this
+simple linear model can account for the geometrical effects observed in real H-bonded ferroelectrics,
+which are at the root of the giant isotope effect in the critical temperature observed in the FE phase
+transitions of these materials. The paper is organized as follows: in the next section we explain the models
+used and describe details of the PIMC calculations. Section 3 describes and discusses the results obtained
+for the three-site model and for the linear chain. Finally, we elaborate a summary and our conclussions
+in section 4.
+2. Models and calculation details
+2.1. Three-site model
+���
+���
+���
+���
+�
+�
+Figure 1. (Colour online) H-bond parameters in the three-site model. 𝑅 ≡ 𝑅OO is the distance between
+oxygen nuclei. 𝑟OH is the proton-oxygen distance. The variable 𝛿 = 𝑅OO −2𝑟OH is defined as the distance
+between the two possible equilibrium positions of the proton. Then, 𝑥 = 𝑅OO/2 − 𝑟OH is the proton
+coordinate relative to the H-bond center. This parameter definition is also used in the linear chain model.
+We developed a three-site (3S) model which represents a single O–H–O cluster embedded in the H-
+bonded ferroelectric as it is sketched in figure 1. With the aim to model linear H-bonds, a Double Morse
+(or back-to-back) potential (see e.g., [20–24]) is usually used, which is essentially the superposition of
+two Morse potentials representing what the proton feels while interacting with both oxygens:
+𝑉OH (𝑥, 𝑅) = 𝑉𝑀
+�
+𝑥 + 𝑅
+2
+�
++ 𝑉𝑀
+� 𝑅
+2 − 𝑥
+�
+= 𝐷
+�
+1 − exp
+�
+−𝑎
+� 𝑅
+2 + 𝑥 − 𝑟0
+���2
++ 𝐷
+�
+1 − exp
+�
+−𝑎
+� 𝑅
+2 − 𝑥 − 𝑟0
+���2
+− 2𝐷,
+(2.1)
+where 𝑅 is the O–O distance, and 𝑥 represents the H position relative to the H-bridge center (see figure 1).
+If we assume that 𝑅 is fixed, there is a critical value 𝑅𝑐 = 2(𝑎−1 ln 2+𝑟0) such that for 𝑅 < 𝑅𝑐 the potential
+profile is a single well with a minimum at 𝑥 = 0. On the contrary, for 𝑅 > 𝑅𝑐 we have a symmetric double-
+well potential, with a local maximum at 𝑥 = 0 and minima at 𝑥 = ±𝑎−1 cosh−1{1/2 exp[𝑎(𝑅/2 − 𝑟0)]}.
+Notice that the energy barrier for the proton jump from one side to the other of the H-bond diminishes
+concomitantly with the O–O distance 𝑅, vanishing for 𝑅 < 𝑅𝑐. Actually, we are interested in the
+43708-2
+
+Quantum geometrical effects in KDP crystals
+proton/deuteron tunneling regime, thus we would need that the equilibrium distance 𝑅 remains in the
+region where the proton barrier exists, that is 𝑅 > 𝑅𝑐. However, simulations at low temperature with the
+potential described in equation 2.1, relaxing both variables 𝑥 and 𝑅, yield to a collapse of the potential
+barrier and the equilibrium energy profile displays one minimum only. Therefore, it is mandatory to
+introduce a new interaction which preserves the system from the O–O distance collapse. This O–O
+potential will represent the interaction between both oxygens and the lattice. The following Morse
+potential between oxygens is chosen [19]:
+𝑉OO (𝑅) = 𝐷OO
+�
+1 − e−𝑎OO(𝑅−𝑅0)�2
+− 𝐷OO.
+(2.2)
+We adopted a Morse potential to describe the O–O interaction with the lattice because this kind of
+anharmonic potential enables the system to explore with sufficient probability O–O distances larger
+than 𝑅0, in such a way that the collapse tendency to a single well is drastically diminished. This is in
+contrast to the case of a harmonic potential for the O–O interaction, where in this case the O–O collapse
+is inevitable. The complete potential for the 3S model is as follows:
+𝑉3𝑆 (𝑥, 𝑅) = 𝑉OH (𝑥, 𝑅) + 𝑉OO (𝑅) = 𝐷
+�
+1 − e−𝑎[(𝑅/2)+𝑥−𝑟0]�2
++ 𝐷
+�
+1 − e−𝑎[(𝑅/2)−𝑥−𝑟0]�2
+− 2𝐷 + 𝐷OO
+�
+1 − e−𝑎OO(𝑅−𝑅0)�2
+− 𝐷OO.
+(2.3)
+The correlation between the H displacement 𝑥 and the O–O distance 𝑅 observed in experiments and ab
+initio calculations is reflected by the anharmonic potential of equation (2.3): when the H approaches one
+of the O’s in the covalent bond O–H (increasing 𝑥), the hydrogen-bond with the other O weakens and
+the O–O distance (𝑅) increases. Moreover, 𝑅 diminishes with decreasing 𝑥, which is the inverse situation.
+This correlation is precisely the important ingredient necessary for the existence of the Ubbelohde or the
+geometrical effect observed in compounds with strong H-bonds.
+2.2. 1D model of concatenated H-bonds
+Going a step beyond the simple three-site model, we have developed a one dimensional chain model
+of concatenated H-bonds to study the GE in a more realistic way in the ordered phase. This 1D linear
+model consists of a chain ...O–H...O–H...O–H...O–H..., which is built as a supercell containing 𝑁 = 200
+unit cells of linear dimension 𝑅, the O–O distance, as shown schematically in figure 2. There are two
+atoms, one oxygen and one hydrogen in each unit cell (O–H...). The supercell of dimension 𝐿 = 200𝑅 is
+subjected to periodic boundary conditions. In the simulation, 𝐿 is allowed to relax at zero stress, as well
+as each coordinate 𝑥𝑖 and 𝑅𝑖 of each unit cell 𝑖. For instance, this chain represents a model approximation
+to the 1D H-bonded ferroelectric CsH2PO4 (CDP) if the model chain oxygen is interpreted as a PO4 unit
+plus an ordered hydrogen covalently bonded to the phosphate at any temperature, and the model hydrogen
+is the one that is disordered at high temperature in CDP [25]. Then, the global motion of hydrogens in our
+linear model in the ordered phase, from one minimum to the other along the H-bonds of the chain, could
+be related to the FE mode that accounts for the spontaneous polarization arising along the 𝑏 direction at
+low 𝑇 in CDP [25]. Alternatively, the chain model may also represent an approximation to the study of
+the GE in KH2PO4 (KDP) if the model effective oxygen now represents a KDP cluster of two phosphate
+units including seven protons moving coordinately as a local FE mode [9, 10]. In all these cases, we must
+adopt a convenient effective mass for the effective model hydrogen/deuteron considering that the real
+displacements of H(D) are accompanied with the heavier atom motions [9, 10, 19].
+The total potential energy for the linear chain (1D) model is defined as:
+𝑉1𝐷 (𝑅) =
+∑︁
+𝑖
+𝑉3𝑠 (𝑥𝑖, 𝑅𝑖) − 1
+2
+∑︁
+⟨𝑖 𝑗⟩
+𝐽𝑥𝑖𝑥 𝑗,
+(2.4)
+where 𝑉3𝑠 is the unit cell local potential defined exactly in the same way for the 3S model, as is shown
+in equation (2.3), and the last term is the short-range interaction energy between protons/deuterons
+43708-3
+
+F. Torresi, J. Lasave, S. Koval
+Figure 2. (Colour online) Schematic representation of the 1D chain model in the ordered phase. Each
+unit cell is formed with one oxygen (red sphere) and one hydrogen (white sphere). Our model consists of
+a supercell subjected to periodic boundary conditions containing 200 unit cells (for better visualization
+only 8 unit cells are shown).
+stemming from the ice rules restrictions, i.e., in this 1D model, only one proton is attached to each
+oxygen. The last sum in equation 2.4 is restricted to nearest neighbours for each index ⟨𝑖𝑗⟩. There is no
+long-range part in this model, which precludes a phase transition in one dimension. However, the last
+bilinear term is treated in mean-field, which enables the system to have a second order phase transition
+at finite temperature [26]. Therefore, the 1D model total potential, is written in the following way [27]:
+𝑉1𝐷 (𝑅) =
+∑︁
+𝑖
+𝑉3𝑠 (𝑥𝑖, 𝑅𝑖) − 𝐽⟨𝑥⟩
+∑︁
+𝑖
+𝑥𝑖 + 1
+2 𝑁𝐽⟨𝑥⟩2,
+(2.5)
+where ⟨𝑥⟩ ≡ 1/𝑁 �
+𝑖 𝑥𝑖 is the time and lattice average of the 𝑥𝑖 positions for each unit cell 𝑖 taken at each
+MC step in the simulation.
+2.3. Path integral Monte Carlo simulations
+In the PIMC simulations [28], the effective short-time propagator for two adjacent points in the dis-
+cretized imaginary-time path describing each quantum particle was evaluated to fourth-order accuracy
+with the Takahashi-Imada approximation [28–30]. The effective action in this case allows us to signifi-
+cantly reduce the Trotter number 𝑀 required for convergence. In all the simulations performed we have
+used 𝑀 = 128 beads for the quantum polymer associated with each atom in the O–H...O bonds, which
+yielded well-converged results [19, 25, 28]. Additionally, a normal-mode representation of the quantum
+polymers was used in order to ensure ergodicity in the MC sampling [28, 30]. The PIMC simulations were
+performed at low 𝑇 = 50 K such that the quantum nuclear effects were predominant compared to entropic
+contributions in the 3S model and also with the aim to obtain GE in the ordered phase for the 1D model
+(the classical version of this model has a transition to a disordered paraelectric phase at ≈ 350 K). The
+simulations for the 3S model consisted of 1 × 106 MC steps preceded by 5 × 105 steps of thermalization.
+In the 1D chain model simulations, we took 3 × 104 steps of thermalization plus 1 × 105 MC steps for
+computing averages. In this case, each calculation performed was an average of 20 runs with different
+random number generator seeds.
+To characterize the degree of particle delocalization in the PIMC simulations, we studied the centroid
+and radius of gyration (RG) distributions for the quantum polymers [31]. The centroid is defined as the
+center of mass (CM) of the polymer and represents the average position of the quantum particle. The
+radius of gyration represents the variance of the quantum path and is a quantitative measure of how
+far away are the beads or monomers from the polymer center, and therefore, provides a measure of the
+quantum delocalization of the particle [31].
+3. Results and discussion
+3.1. Geometrical effect study using the three-site model
+The six potential parameters of equation (2.3) have been fitted in order to perform the GE study
+with the 3S model. First, we fixed the values of 𝑎 = 2.89 Å
+−1 [20, 21] and 𝐷 = 3.12 eV of the model
+parameters for the proton potential defined in equation 2.1, such that the stretching frequency for the O–H
+bond in the limit 𝑅 → ∞ coincides with the experimental average value 𝜔∞ ≈ 3750 cm−1 [20, 21, 32] for
+43708-4
+
+Quantum geometrical effects in KDP crystals
+(a) Proton
+(b) Deuteron
+Figure 3. (Colour online) Proton/Deuteron probability distribution contours for the three-site PIMC
+simulations at 𝑇 = 50 K.
+different H-bonded compounds. There is a strong correlation between the OH and OO distances for the
+family of H-bonded compounds. The equilibrium distance 𝑟OH diminishes systematically with increasing
+𝑅 for 𝑅 > 𝑅𝑐 [33, 34], reaching a saturated value around 𝑟∞
+OH ≈ 0.95 Å for very large 𝑅. Therefore, we
+took the parameter value 𝑟0 = 0.93 Å so that the values 𝑥 that minimize 𝑉OH (𝑥, 𝑅) in equation (2.1) for
+different values of 𝑅 give a curve 𝑟min
+OH = 𝑅OO/2 − 𝑥min as a function of 𝑅 that is a lower bound for the
+set of experimental points spread in the OH–OO correlation [20, 21, 33, 34]. With this choice, when the
+nuclear quantum effects are included in the PIMC calculations, we observe a very good agreement with
+the experimental correlation curve using the model of equation (2.1) with the OO distance 𝑅 fixed [35].
+On the other hand, the parameter values for the OO interaction 𝑉OO (𝑅) [see equation (2.2)], were
+initially taken from reference [23]. They were further adjusted, especially the value of 𝐷OO, due to the
+important correlation between 𝑟OH and 𝑅OO, such that the classic potential profile has the minimum at
+𝑅cl
+OO ≈ 2.55 Å. We considered this condition because the most important geometrical effects are observed
+in H-bonded crystals with strong H-bonds which have distances 𝑅 in a range between 2.5 and 2.6 Å [36],
+with 𝑅cl
+OO lying precisely in the middle of that window. The final parameter values for the 3S model are
+shown in table 1.
+Table 1. Potential parameters used in the 3S model.
+𝐷 [eV]
+𝑎 [ Å−1]
+𝑟0 [ Å]
+𝐷OO [eV]
+𝑎OO [ Å−1]
+𝑅0 [ Å]
+3.12
+2.89
+0.93
+0.55
+2.28
+2.76
+We have verified that the 3S-model PIMC simulations performed at 𝑇 = 50 K with 𝑀 = 128
+beads for the quantum polymer representing each atom yielded probability distributions for the H-bond
+parameters (𝑥 and 𝑅) and energies well converged. The low temperature of 50 K for the simulation was
+chosen because we are interested in the nuclear quantum effects for the H-bonds and the geometrical
+changes with deuteration without most of the influence of entropic contributions in the particle dynamics.
+The 3S model results for the probability density contours to find the system in a given (𝑥, 𝑅) configuration
+are shown in figure 3 for the proton and deuteron cases. The curves are qualitatively different but both
+cases are found to have symmetric distributions around 𝑥 = 0 in the 𝑥 coordinate with two prominent
+peaks with maximum probability, which are clearly shifted in the deuterated case. The OO distance for the
+peak positions are in each case: 𝑅peak
+OO (𝐻) = 2.527 Å and 𝑅peak
+OO (𝐷) = 2.543 Å, which represents a distance
+enlargement for the OO bond of Δ𝑅OO = 0.016 Å, evidencing the geometrical or Ubbelohde effect of
+the H-bond expansion with deuteration. Moreover, the corresponding average values also increase with
+43708-5
+
+F. Torresi, J. Lasave, S. Koval
+-0,3
+-0,2
+-0,1
+0
+0,1
+0,2
+0,3
+xCM [Å] 
+0,05
+0,1
+0,15
+0,2
+0,25
+0,3
+rG [Å]
+(a) Proton
+-0,3
+-0,2
+-0,1
+0
+0,1
+0,2
+0,3
+xCM [Å]
+0,05
+0,1
+0,15
+0,2
+0,25
+0,3
+rG [Å]
+(b) Deuteron
+Figure 4. (Colour online) Distribution of the radius of gyration 𝑟𝐺 vs. centroid coordinate 𝑥𝐶𝑀 for the
+three-site simulations at 𝑇 = 50 K.
+deuteration: ⟨𝑅OO(𝐻)⟩ = 2.525 Å and ⟨𝑅OO(𝐷)⟩ = 2.540 Å.
+The PIMC simulations also show a change in the variable 𝛿 with deuteration for the peaks observed
+in figure 3. The variation is: Δ𝛿 = 𝛿𝐷 − 𝛿𝐻 = 0.079 Å, where 𝛿𝐻 = 0.417 Å and 𝛿𝐷 = 0.496 Å. This
+is also reflected in a shrinking of the O–H bonds: Δ𝑟 = 𝑟OH − 𝑟OD = 0.032 Å. The overall changes in
+the variables 𝛿 and 𝑅 with deuteration in the simulations are in agreement with what is observed in the
+experimental data for different H-bonded compounds with strong H-bonds [36, 37]. Thus, our simple 3S
+model satisfactorily reproduces the isotopic geometrical effects for these systems.
+It is worth to notice that if the OO distance is not allowed to relax, then the GE is smaller. For
+instance, we have fixed the value 𝑅OO = 2.527 Å, which corresponds to the peak in the probability
+distribution for the protonic system (see figure 3), and the simulations gave a change with deuteration in
+the OH bond of only Δ𝑟 = 0.021 Å. Comparing this result with that considering the oxygen dynamics
+(Δ𝑟 = 𝑟OH − 𝑟OD = 0.032 Å), we observe an increment of ≈ 50% in the isotopic geometrical effect in the
+case where the oxygens are allowed to relax. This can be understood in the following way: first, when
+the oxygens are fixed, protons, being more delocalized than deuterons, have more probability to stay
+closer to the middle of the O–O bond. Second, when the oxygen dynamics is included, the protons act
+as a strong attraction center that pulls the two bridge oxygens together, more effectively than deuterons
+which are more localized near the oxygen. This proton-mediated O–O contraction lowers the potential
+barrier, which delocalizes even more the proton, and so on, giving rise to a nonlinear selfconsistent
+mechanism [9, 10]. For the deuteron, being less delocalized than the proton, the selfconsistent effect is
+weaker. This mechanism leads to an isotopic geometrical effect which is stronger than that generated by
+the proton/deuteron quantum delocalization at fixed potential (fixed oxygens) [9, 10].
+To further illustrate the microscopic mechanism that rules the GE, we have analyzed the behavior
+of the quantum polymers for the proton/deuteron in the simulation via an analysis of the center of mass
+of the quantum polymer or centroid position 𝑥𝐶𝑀 and the radius of gyration 𝑟𝐺 representing a measure
+of the quantum delocalization of the particle (i.e., the extension of the quantum polymer) [31]. We plot
+in figure 4 the instantaneous values of 𝑟𝐺 as a function of the proton/deuteron centroids 𝑥𝐶𝑀, taken
+every 100 MC steps in the PIMC simulation. As can be seen in the figure, the density of points reveals
+that the deuteron prefers to be localized at both sides and far from the bond middle with small values
+of 𝑟𝐺, indicating a more classical behavior in these cases. When the deuteron centroid takes the values
+of 𝑥𝐶𝑀 closer to 0 (the bond middle), it is observed an increase of 𝑟𝐺 indicating that the quantum polymer
+is delocalized and is spread through both sides of the potential barrier, signaling the presence of tunneling
+in this case. Notice that the largest values of 𝑟𝐺 are found at 𝑥𝐶𝑀 ≈ 0 where delocalization is maximum.
+On the other hand, in the proton case, tunneling is much more frequent because the region with larger
+density of points appears near 𝑥𝐶𝑀 ≈ 0 with large values of 𝑟𝐺, as shown in figure 4. This is precisely
+43708-6
+
+Quantum geometrical effects in KDP crystals
+an important ingredient for the GE: the proton spends much more time delocalized with the quantum
+polymer center of mass near the middle of the O–O bond, which finally produces a strong contraction
+of the O–O distance. On the contrary, the deuteron is much more localized at both sides and far from
+the bond middle which leads to a weakening of the O–O bond and to an increase of the O–O distance.
+This yields the isotopic geometrical effect, which is observed in the calculated probability distribution of
+figure 3.
+3.2. Isotope effects obtained with the 1D model simulations
+The previous analysis of the 3S model results, which has clearly shown the isotopic GE, was carried
+out based on the parametrization of the model which reproduces the universal OH–OO correlation
+observed for a family of diverse H-bonded compounds. In this sense, this model is quite simple and
+general, accounting for the geometrical effects with deuteration of a set of H-bonded ferroelectrics with
+strong H-bonds. Now, we focus on the development of a 1D chain model, described in section 2.2 [see
+equation (2.5)], which was specifically designed to explain the isotope effects in the phase transition of
+KDP and was fitted to ab initio results [19]. This more realistic 1D model has, in the classical nuclei
+version, a ferroelectric-paraelectric transition at 𝑇 ≈ 350 K [35]. In this paper, we have used it in the
+ordered phase of KDP at 𝑇 = 50 K to analyze the isotopic GE which is at the root of the microscopic
+mechanism that leads to the giant isotope effect in the critical temperature.
+We start from equation 2.5 for the 1D model, which has seven parameters to be adjusted for the
+KDP case. The six model parameters of the local proton potential 𝑉3𝑆 for each unit cell in the chain,
+which is just the same that was used in the 3S model (see equation 2.3), have been adjusted to reproduce
+six magnitudes obtained from ab initio calculations for KDP. These magnitudes were the global energy
+barrier between the PE and FE states, the O–O and 𝛿 distances in the FE phase, the O–O distance in
+the PE phase, the ab initio vibrational frequency of the PO4 rotation mode, which is equivalent to the
+stretching mode in the 3S model, and the energy barrier between the energy minimum and the transition
+state in the FE phase keeping the O–O distance fixed (see reference [19]). We adopted the model fit to the
+ab initio calculations that includes dispersion corrections at the vdW-DF level, which exhibit, compared
+to other methods, the best agreement with the experimental geometry for both KDP and deuterated KDP
+(DKDP) [19].
+Finally, we have fitted the remaining parameter 𝐽 that corresponds to the proton-proton interaction
+term in equations (2.4) and (2.5). To this end, 𝐽 was adjusted to 0.55 eV/Å
+2 so that the critical temperature
+𝑇𝑐 for the FE-PE transition obtained by the 1D model simulation with classical nuclei reaches the value
+of ≈ 350 K, similar to the value obtained by ab initio molecular dynamics calculations with dispersion
+corrections at the vdW-DF level for DKDP [38].
+The final values for the parameters used in the 1D model are listed in table 2.
+Table 2. Potential parameters used in the 1D model.
+𝐷 [eV]
+𝑎 [Å−1]
+𝑟0 [Å]
+𝐷OO [eV]
+𝑎OO [Å−1]
+𝑅0 [Å]
+𝐽 [ eV/Å
+2]
+8.838
+3.027
+0.966
+10.542
+0.831
+2.917
+0.55
+The motion of the proton/deuteron is strongly correlated with that of the heavy ions, and its mass is
+dressed accordingly as discussed in reference [10]. Therefore, instead of using the bare proton (deuteron)
+masses 𝑚 𝑝 (2𝑚 𝑝), we have used in the PIMC simulations the effective masses for H and D: 𝜇𝐻 = 2.3𝑚 𝑝
+and 𝜇𝐷 = 3𝑚 𝑝, respectively, with 𝑚 𝑝 the proton mass [9, 10, 19].
+We plot in figure 5 the probability distribution contours for the PIMC simulation with the 1D model,
+obtained in the ordered phase at 𝑇 = 50 K. Due to the ordered phase, only one peak is observed in the
+proton and deuteron distributions, which is in contrast to the symmetrical double peaks around 𝑥 = 0 found
+in the 3S model distribution results (see figure 3). The calculated distribution for the chain of protons
+in figure 5 is asymmetric around the peak position due to the potential anharmonicity and quantum
+delocalization, which is in qualitative agreement with the experimental diffraction pattern measured near
+43708-7
+
+F. Torresi, J. Lasave, S. Koval
+(a) Proton
+(b) Deuteron
+Figure 5. (Colour online) Proton/Deuteron probability distribution contours in the H-bonds for the linear-
+chain PIMC simulation at 𝑇 = 50 K.
+𝑇𝑐 in the FE phase of KDP [39]. The asymmetry around the peak is less pronounced in the deuterated
+case as shown in figure 5, because the deuteron is less delocalized than the proton.
+The prominent single peak found in the distribution results for the 1D simulation is clearly shifted
+in the deuterated case towards larger 𝑥 and 𝑅, revealing the existence of the isotopic geometrical effect,
+i.e., the expansion of the H-bonds in the chain with deuteration. The O–O distance for the peak positions
+are in each case: 𝑅peak
+OO (𝐻) = 2.515 Å and 𝑅peak
+OO (𝐷) = 2.542 Å, which represents a distance enlargement
+for the O–O bond of Δ𝑅OO ≡ 𝑅OO(𝐷) − 𝑅OO(𝐻) = 0.027 Å. The 𝑥 coordinate of the peak position also
+expands with deuteration, from 𝑥peak
+𝐻
+= 0.188 Å to 𝑥peak
+𝐷
+= 0.218 Å, with a net increase of Δ𝑥 = 0.030 Å
+or similarly Δ𝛿 ≡ 𝛿𝐷 − 𝛿𝐻 = 0.060 Å. These results are summarized in table 3 and compared with the
+available experimental data for KDP and DKDP [40]. We observe a good agreement between theory
+and experiment, although the GE is a little bit underestimated, with difference values under deuteration
+≈ 25% lower than the experimental data.
+Table 3. Nuclear quantum calculations of the H-bond geometries for KDP and DKDP using the 1D
+linear model. The results, which correspond to the peak positions of figure 5, are contrasted with the
+experimental data of reference [40]. Distances are in Å.
+PIMC
+KDP (𝜇𝐻 = 2.3 𝑚 𝑝)
+DKDP (𝜇𝐷 = 3.0 𝑚 𝑝)
+Δ𝑅OO
+Δ𝛿
+results
+𝑅OO
+𝛿
+𝑅OO
+𝛿
+1D model
+2.515
+0.376
+2.542
+0.436
+0.027
+0.060
+Expt. [40]
+2.497
+0.385
+2.533
+0.472
+0.036
+0.087
+To get a deeper insight into the microscopic mechanism of the geometrical effect in the linear chain
+model, we plot in figure 6 the distribution of the instantaneous radius of gyration 𝑟𝐺 as a function of the
+centroid positions 𝑥𝐶𝑀 for all H-bonds in the chain, where the points are taken every 100 MC steps along
+the PIMC simulation. The region with largest density of points in figure 6 coincides with the position of
+the peaks in both proton and deuteron cases (see figure 5). We again observe an asymmetric distribution
+centered in one of the sides of the H-bond consistent with the (𝑥, 𝑅) distribution pattern of figure 5. The
+asymmetry observed in figure 6 is more pronounced in the proton case, indicating that protons jump more
+often than deuterons to the other side of the O–H–O bond. The mechanism to pass through the potential
+barrier is to increase the radius of gyration near 𝑥𝐶𝑀 ≈ 0 which means that the particle tunnels through
+the barrier. This is helped by a strong contraction of the 𝑅 distance, which diminishes concomitantly with
+43708-8
+
+Quantum geometrical effects in KDP crystals
+(a) Proton
+(b) Deuteron
+Figure 6. Distribution of the radius of gyration 𝑟𝐺 vs. centroid coordinate 𝑥𝐶𝑀 of the quantum polymer
+representing the protons (a) and deuterons (b) relative to the center of the H-bonds, for the linear-chain
+PIMC simulation at 𝑇 = 50 K.
+the potential barrier, to a lower bound of 𝑅min ≈ 2.3 Å near 𝑥 = 0 as shown in figure 5. Thus, we conclude
+that tunneling is assisted by the 𝑅 distance modulation. However, in this ordered phase at 𝑇 = 50 K,
+the proton spends more time in one of the sides of the O–H–O bond where the behavior is more classic
+(low value of 𝑟𝐺). On the other hand, in the deuteron case, the particle remains localized practically all
+the time, with a general classical behavior with low values of 𝑟𝐺. In other words, the tunneling for the
+deuteron is very scarce. These results are consistent with the general assumption in the tunneling model:
+protons are capable of tunelling while deuterons are not [3]. However, there is an essential difference:
+protons tunnel being assisted by the strong correlation with the O–O distance, which is the behavior that
+originates the geometrical effect [9, 10]. Therefore, the proton has a larger probability than the deuteron
+to spend more time tunneling through the barrier near the middle of the O–H–O bond, and this generates
+a strong attraction center that pulls the two oxygens together, much more efficiently than deuterons. This
+“tunneling – geometrical effect” interrelation gives rise to the final geometrical effect observed in KDP
+crystals, that is, the H-bond expansion with deuteration, which is crucial for the isotope effects in the
+FE-PE phase transitions [9, 35].
+4. Summary and conclusions
+We have carried out PIMC simulations with simple models to account for the geometrical effects (GE)
+with deuteration in H-bonded ferroelectrics such as KDP crystals. Firstly, we have developed a general
+three-site (3S) model consisting in a back-to-back double Morse potential for the O–H interaction and
+a Morse potential which represents the interaction between the oxygens and the lattice. The model was
+fitted to reproduce general features for a large set of different H-bonded compounds. The computed
+probability distribution contours in the (𝑅, 𝑥) configuration space, with 𝑅 the O–O distance and 𝑥 the
+proton/deuteron distance to the middle of the O–O bond, reveal a symmetric distribution around 𝑥 = 0
+with two peaks on either side, for both proton and deuteron cases. The results show an increase with
+deuteration of 𝑅 and 𝑥 for the observed peaks, i.e., a GE, which is in agreement with that observed in
+H-bonded compounds with strong H-bonds. Moreover, if the oxygens are not allowed to relax during the
+simulation, the GE in the 𝑥 coordinate is much smaller, which means that there is a strong correlation
+between 𝑅 and 𝑥 that is important for the GE. During the PIMC simulations we have also plotted the
+instantaneous radius of gyration 𝑟𝐺 vs. the centroid position 𝑥𝐶𝑀 of the quantum polymer representing
+the proton/deuteron. The results show that the proton tunnels more frequently than the deuteron (that is,
+it spends more time with the center of mass near 𝑥𝐶𝑀 = 0 with large values of 𝑟𝐺), while the deuteron is
+43708-9
+
+0,25
+0,2
+0,15
+rG
+0,1
+0,05
+-0,4
+-0,2
+0
+0,2
+0,4
+XcM [A]0,25
+0,2
+0,15
+rG
+0,1
+0,05
+-0,4
+-0,2
+0
+0,2
+0,4
+XcM [A]F. Torresi, J. Lasave, S. Koval
+more localized in both sides and far from the O–H–O bond center, with small values of 𝑟𝐺 (i.e., a more
+classsical behavior). These features yield a more effective contraction of the O–O bond in the proton
+case, explaining the GE observed.
+Secondly, we have developed a more realistic 1D model, with the same local potential for the H-bonds
+as that used in the 3S model, but adding also a bilinear proton-proton interaction treated in mean-field.
+The parameters of the 1D model were fitted to ab initio results for KDP. The bilinear interaction parameter
+of the model was adjusted such that the classical nuclei version of the model has a second order FE-PE
+phase transition at 𝑇 = 350 K in agreement with ab initio molecular dynamics simulations for DKDP.
+In this paper, by means of PIMC simulations of the 1D model, we have studied the GE caused by
+deuteration in the ordered phase at 𝑇 = 50 K. The calculated probability distribution contours show
+only one peak in the (𝑅, 𝑥) configuration space for both proton/deuteron cases. The distribution is more
+asymmetric in the proton case due to the anharmonicity of the potential and the quantum delocalization.
+The distribution pattern is in qualitative agreement with the experimental distribution determined by high-
+resolution neutron diffraction studies [39]. The probability distribution contours show a peak which shifts
+substantially with deuteration. The changes in H-bond geomentry caused by the GE observed in the 1D
+model simulations are in good agreement with the corresponding experimental data. The distribution of
+the radius of gyration vs. the quantum path centroids shows that the protons tunnel through the potential
+barrier frequently while the deuterons are much more localized in one of the sides of the O–H–O bond
+and practically do not tunnel, in agreement with the well-known tunneling model [3], and also with
+recent neutron Compton scattering experiments [7, 8]. We have shown that proton tunneling is assisted
+by a strong contraction of the O–O distance in the 1D model. Thus, there is a strong correlation between
+instantaneous tunneling and geometrical effects of the H-bond that is much more efficient in the proton
+case than in the deuterated system, which gives in average a strong GE for the whole simulation. This
+mechanism is expected to be at the root of the huge isotope effect observed in H-bonded ferroelectrics of
+the KDP type [9, 10].
+Acknowledgements
+We acknowledge support from Consejo Nacional de Investigaciones Científicas y Técnicas (CON-
+ICET), Argentina.
+References
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+Quantum geometrical effects in KDP crystals
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+Phys., 1999, 2, 93, doi:10.5488/CMP.2.1.93.
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+18. Lasave J., Kohanoff J., Migoni R. L., Koval S., Physica B, 2009, 404, 2736, doi:10.1016/j.physb.2009.06.143.
+19. Menchón R., Colizzi G., Johnston C., Torresi F., Lasave J., Koval S., Kohanoff J., Migoni R., Phys. Rev. B, 2018
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+doi:10.1103/PhysRevB.48.12645.
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+32. McKenzie R. H., Bekker C., Athokpam B., Ramesh S. G., J. Chem. Phys., 2014, 140, 174508,
+doi:10.1063/1.4873352.
+33. Ichikawa M., Acta Crystallogr., Sect. B: Struct. Crystallogr. Cryst. Chem., 1978, 34, 2074–2080,
+doi:10.1107/S0567740878007475.
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+2798–2801, doi:10.1107/S0567740882009984.
+35. Torresi F., Lasave J., Koval S., (unpublished).
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+37. Sokolov N. D., Verner M. V., Savel’ev V. A., J. Mol. Struct., 1988, 177, 93–110, doi:10.1016/0022-
+2860(88)80081-4.
+38. Menchón R. E., Ph.D. Thesis, Universidad Nacional de Rosario (UNR), Argentina, 2019.
+39. Nelmes R. J., Kuhs W. F., Howard C. J., Tibballs J. E., Ryan T. W., J. Phys. C: Solid State Phys., 1985, 18, L711,
+doi:10.1088/0022-3719/18/24/001.
+40. Nelmes R. J., Tun Z., Kuhs W. F., Ferroelectrics, 1987, 71, 125, doi:10.1080/00150198708224833.
+43708-11
+
+F. Torresi, J. Lasave, S. Koval
+Метод iнтегралiв за траєкторiями у моделюваннi
+Монте-Карло геометричних ефектiв у кристалах KDP
+Ф. Торрезi, Х. Ласаве, С. Коваль
+Iнститут фiзики Росарiо, Нацiональний унiверситет Росарiо та Нацiональна рада з науково-технiчних
+дослiджень, вул. 27 лютого, 210 Bis, 2000 Росарiо, Аргентина
+Метод iнтегралiв за траєкторiями у моделюваннi Монте-Карло (IТМК) для дуже простих моделей застосо-
+вано для з’ясування фiзичних механiзмiв, що лежать в основi iзотопiчного ефекту в сегнетоелектриках з
+водневими зв’язками. Зумовленi дейтеруванням геометричнi ефекти у водневих зв’язках було дослiдже-
+но за допомогою загальної тривузлової моделi, в якiй використовуються подвiйний потенцiал Морзе та
+потенцiал Морзе мiж киснями; параметри моделi вибрано так, щоб пояснити рiзноманiтнi загальнi влас-
+тивостi низки сполук з водневими зв’язками. З розрахункiв у рамках цiєї моделi випливає виникнення
+геометричного ефекту (ефекту Уббелоде): видовження водневого зв’язка при дейтеруваннi, i це узгоджу-
+ється з тим, що спостерiгається в сегнетоелектриках з короткими водневими зв’язками. Використовуючи
+для параметрiв потенцiалiв результати першопринципних розрахункiв, розвинено одновимiрну модель,
+в якiй бiлiнiйнi протон-протоннi взаємодiї розглядаються в наближеннi середнього поля. Ця модель вико-
+ристовується для дослiдження квантових ефектiв у ядрах, якi призводять до виникнення геометричного
+ефекту в кристалах KDP. Пiдхiд IТМК дає змогу виявити, що протони тунелюють бiльш ефективно вздовж
+одновимiрного ланцюжка, нiж дейтрони; це спричиняє появу сильного притягувального центра, який
+зменшує вiдстань мiж атомами киснiв. Цей механiзм, який ґрунтується на кореляцiї мiж тунелюванням i
+геометричними змiнами водневих зв’язкiв, призводить до виникнення сильного геометричного ефекту
+в ланцюжку у впорядкованiй фазi при низьких температурах, що добре узгоджується з експерименталь-
+ними даними.
+Ключовi слова: сегнетоелектричний фазовий перехiд, сегнетоелектрики з водневими зв’язками, метод
+iнтегралiв за траєкторiями у моделюваннi Монте-Карло
+43708-12
+

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+arXiv:2301.04039v1  [physics.hist-ph]  10 Jan 2023
+A Theory of Theories
+Mich`ele Levi
+Mathematical Institute, University of Oxford,
+Oxford OX2 6GG, United Kingdom
+levi@maths.ox.ac.uk
+Abstract
+We take a tour through the past, present and future of Effective
+Field Theory, with applications ranging from LHC physics to cosmol-
+ogy.
+1
+
+High-energy physics spans a wide range of energies, from a few MeV to
+TeV, that are all relevant. It is therefore often difficult to take all phenomena
+into account at the same time. Effective field theories (EFTs) are designed
+to break down this range of scales into smaller segments so that physicists
+can work in the relevant range. Theorists “cut” their theory’s energy scale
+at the order of the mass of the lightest particle omitted from the theory,
+such as the proton mass.
+Thus, multi-scale problems reduce to separate
+and single-scale problems.EFTs are today also understood to be “bottom-
+up” theories. Built only out of the general field content and symmetries at
+the relevant scales, they allow us to test hypotheses efficiently and to select
+the most promising ones without needing to know the underlying theories in
+full detail. Thanks to their applicability to all generic classical and quantum
+field theories, the sheer variety of EFT applications is striking.
+In hindsight, particle physicists were working with EFTs from as early
+as Fermi’s phenomenological picture of beta decay in which a four-fermion
+vertex replaces the W-boson propagator because the momentum is much
+smaller compared to the mass of the W boson.
+Like so many profound
+concepts in theoretical physics, EFT was first considered in a narrow phe-
+nomenological context. One of the earliest instances was in the 1960s, when
+ad-hoc methods of current algebras were utilised to study weak interactions
+of hadrons.
+This required detailed calculations, and a simpler approach
+was needed to derive useful results. The heuristic idea of describing hadron
+dynamics with the most general Lagrangian density based on symmetries,
+the relevant energy scale and the relevant particles, which can be written in
+terms of operators multiplied by Wilson coefficients, was yet to be known.
+With this approach, it was possible to encode local symmetries in terms of
+the current algebra due to their association with conserved currents.
+For strong interactions, physicists described the interaction between pi-
+ons with chiral perturbation theory, an effective Lagrangian, which sim-
+plified current algebra calculations and enabled the low-energy theory to
+be investigated systematically. This “mother” of modern EFTs describes
+the physics of hadrons and remains valid to an energy scale of the proton
+mass. Heavy-quark effective theory (HQET), introduced by Howard Georgi
+in 1990, complements chiral perturbation theory by describing the interac-
+tions of charm and bottom quarks. HQET allowed us to make predictions on
+B-meson decay rates, since the corrections could now be classified. The more
+powers of energy are allowed, the more infinities appear. These infinities are
+cancelled by available counter-terms.
+Similarly, it is possible to regard the Standard Model as the truncation
+of a much more general theory including non-renormalizable interactions,
+which yield corrections of higher order in energy. This perception of the
+whole Standard Model as an effective field theory started to be formed in
+the late 1970s by Weinberg and others. Among the known corrections to
+the Standard Model that do not satisfy its approximate symmetries are
+2
+
+neutrino masses, postulated in the 1960s and discovered via the observation
+of neutrino oscillations in the late 1990s.
+While the scope of EFTs was
+unclear initially, today we understand that all successful field theories, with
+which we have been working in many areas of theoretical physics, are nothing
+but effective field theories. EFTs provide the theoretical framework to probe
+new physics and to establish precision programmes at experiments.
+The
+former is crucial for making accurate theoretical predictions, while the latter
+is central to the physics programme of CERN in general.
+1
+EFTs in Particle Physics
+More than a decade has passed since the first run of the LHC, in which the
+Higgs boson and the mechanism for electroweak symmetry breaking were
+discovered. So far, there are no signals of new physics beyond the SM. EFTs
+are well suited to explore LHC physics in depth. A typical example for an
+event involving two scales is Higgs-boson production because there is a factor
+10−100 between its mass and transverse momentum. The calculation of each
+Higgs-boson production process leads to large logarithms that can invalidate
+perturbation theory due to the large-scale separation. This is just one of
+many examples of the two-scale problem that arises when the full quantum
+field theory approach for high-energy colliders is applied. Traditionally, such
+two-scale problems have been treated in the framework of QCD factorisation
+and resummation.
+Over the past two decades, it has been possible to recast two-scale prob-
+lems at high-energy colliders with the advent of soft-collinear effective theory
+(SCET). SCET is nowadays a popular framework that is used to describe
+Higgs physics, jets and their substructure, as well as more formal problems,
+such as power corrections to reconstruct full amplitudes eventually.
+The
+difference between HQET and SCET is that SCET considers long-distance
+interactions between quarks and both soft and collinear particles, whereas
+HQET takes into account only soft interactions between a heavy quark and
+a parton. SCET is just one example where the EFT methodology has been
+indispensable, even though the underlying theory at much higher energies
+is known. Other examples of EFT applications include precision measure-
+ments of rare decays that can be described by QCD with its approximate
+chiral symmetry, or heavy quarks at finite temperature and density. EFT
+is also central to a deeper understanding of the so-called flavour anomalies,
+enabling comparisons between theory and experiment in terms of particular
+Wilson coefficients.
+Moreover, precision measurements of Higgs and electroweak observables
+at the LHC and future colliders will provide opportunities to detect new
+physics signals, such as resonances in invariant mass plots, or small devia-
+tions from the SM, seen in tails of distributions for instance at the HL-LHC
+3
+
+– testing the perception of the SM as a low-energy incarnation of a more fun-
+damental theory being probed at the electroweak scale. This is dubbed the
+SMEFT (SM EFT) or HEFT (Higgs EFT), depending on whether the Higgs
+fields are expressed in terms of the Higgs doublet or the physical Higgs bo-
+son. This particular EFT framework has recently been implemented in the
+data-analysis tools at the LHC, enabling the analyses across different chan-
+nels and even different experiments.At the same time, the study of SMEFT
+and HEFT has sparked a plethora of theoretical investigations that have
+uncovered its remarkable underlying features, for example allowing EFT to
+be extended or placing constraints on the EFT coefficients due to Lorentz
+invariance, causality and analyticity.
+2
+EFTs in Gravity
+Since the inception of EFT, it was believed that the framework is applicable
+only to the description of quantum field theories for capturing the physics
+of elementary particles at high-energy scales, or alternatively at very small
+length scales. Thus, EFT seemed mostly irrelevant regarding gravitation,
+for which we are still lacking a full theory valid at quantum scales. The only
+way in which EFT seemed to be pertinent for gravitation was to think of
+general relativity as a first approximation to an EFT description of quantum
+gravity, which indeed provided a new EFT perspective at the time. However,
+in the past decade it has become widely acknowledged that EFT provides a
+powerful framework to capture gravitation occurring completely across large
+length scales, as long as these scales display a clear hierarchy.
+The most notable application to such classical gravitational systems
+came when it was realised that the EFT framework would be ideal to handle
+gravitational radiation emitted at the inspiral phase of a binary of compact
+objects, such as black holes. At this phase in the evolution of the binary, the
+compact objects are moving at non-relativistic velocities. Using the small
+velocity as the expansion parameter exhibits the separation between the
+various characteristic length scales of the system. Thus, the physics can be
+treated perturbatively. For example, it was found that even couplings man-
+ifestly change in classical systems across their characteristic scales, which
+was previously believed to be unique to quantum field theories. The appli-
+cation of EFT to the binary inspiral problem has been so successful that
+the precision frontier has been pushed beyond the state of the art, quickly
+surpassing the reach of work that has been focused on the two-body problem
+for decades via traditional methods in general relativity.
+This theoretical progress has made an even broader impact since the
+breakthrough direct discovery of gravitational waves (GWs) was announced
+in 2016. An inspiraling binary of black holes merged into a single black hole
+in less than a split second, releasing an enormous amount of energy in the
+4
+
+form of GWs, which instigated even greater, more intense use of EFTs for
+the generation of theoretical GW data. In the coming years and decades,
+a continuous increase in the quantity and quality of real-world GW data is
+expected from the rapidly growing worldwide network of ground-based GW
+detectors, and future space-based interferometers, covering a wide range of
+target frequencies.
+3
+EFTs in Cosmology
+Cosmology is inherently a cross-cutting domain, spanning scales over about
+1060 orders of magnitude, from the Planck scale to the size of the observable
+universe. As such, cosmology generally cannot be expected to be tackled
+directly by each of the fundamental theories that capture particle physics
+or gravity. The correct description of cosmology relies heavily on the work
+in many disparate areas of research in theoretical and experimental physics,
+including particle physics and general relativity among many more.
+The development of EFT applications in cosmology – including EFTs of
+inflation, dark matter, dark energy and even EFTs of large-scale structure
+– has become essential to make observable predictions in cosmology. The
+discovery of the accelerated expansion of the universe in 1998 shows our diffi-
+culty in understanding gravity both at the quantum regime and the classical
+one. The cosmological constant problem and dark-matter paradigm might
+be a hint for alternative theories of gravity at very large scales. Indeed, the
+problems with gravity in the very-high and very-low energy range may well
+be tied together. The science programme of next-generation large surveys,
+such as ESA’s Euclid satellite, rely heavily on all these EFT applications
+for the exploitation of the enormous data that is going to be collected to
+constrain unknown cosmological parameters, thus helping to pinpoint viable
+theories.
+4
+The Future of EFTs in Physics
+The EFT framework plays a key role at the exciting and rich interface be-
+tween theory and experiment in particle physics, gravity and cosmology as
+well as in other domains, such as condensed-matter physics, which were
+not covered here. The technology for precision measurements in these do-
+mains is constantly being upgraded, and in the coming years and decades
+we are heading towards a growing influx of real-world data of higher qual-
+ity. Future particle-collider projects, such as the Future Circular Collider
+at CERN, or China’s Circular Electron Positron Collider, are being planned
+and developed.
+Precision cosmology is also thriving, with an upcoming
+next-generation of very large surveys, such as the ground-based LSST, or
+space-based Euclid.
+GW detectors keep improving and multiplying, and
+5
+
+besides those that are currently operating many more are planned, aimed
+at measuring various frequency ranges, which will enable a richer array of
+sources and events to be found.
+Half a century after the concept has formally emerged, effective field
+theory is still full of surprises. Recently, the physical space of EFTs has been
+studied as a fundamental entity in its own right. These studies, by numerous
+groups worldwide, have exposed a new hidden “totally positive” geometric
+structure dubbed the EFT-hedron that constrains the EFT expansion in any
+quantum field theory, and even string theory, from first principles, including
+causality, unitarity and analyticity, to be satisfied by any amplitudes of
+these theories. This recent formal progress reflects the ultimate leap in the
+perception of EFT nowadays as the most fundamental and most generic
+theory concept to capture the physics of nature at all scales. Clearly, in
+the vast array of formidable open questions in physics that still lie ahead,
+effective field theory is here to stay – for good.
+Acknowledgements
+We dedicate this article to the memory of Steven Weinberg, who so gen-
+erously graced us with a spectacular inaugural lecture to the international
+online series hosted at CERN “All Things EFT”, which turned out to be
+his final published lecture.
+We thank Cliff Burgess and HuaXing Zhu for comments and input on a
+preliminary draft. ML has been supported by the Science and Technology
+Facilities Council (STFC) Rutherford Grant ST/V003895 “Harnessing QFT
+for Gravity”, and by the Mathematical Institute University of Oxford.
+References
+[1] S. Weinberg, Eur. Phys. J. H 46 (2021), 6 [arXiv:2101.04241 [hep-th]].
+[2] S. Weinberg, Physica A 96 (1979), 327-340.
+[3] C. P. Burgess, Les Houches100 (2015),148-197 [arXiv:1309.4133 [hep-th]].
+[4] M. Levi, Rept. Prog. Phys. 83 (2020),075901 [arXiv:1807.01699 [hep-th]].
+[5] I. Brivio and M. Trott, Phys. Rept. 793 (2019), 1-98 [arXiv:1706.08945
+[hep-ph]].
+[6] M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98 (2018),
+030001.
+6
+

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf,len=113
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='04039v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='hist-ph]  10 Jan 2023  A Theory of Theories  Mich`ele Levi  Mathematical Institute, University of Oxford,  Oxford OX2 6GG, United Kingdom  levi@maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='uk  Abstract  We take a tour through the past, present and future of Effective  Field Theory, with applications ranging from LHC physics to cosmol-  ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  1  High-energy physics spans a wide range of energies, from a few MeV to  TeV, that are all relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' It is therefore often difficult to take all phenomena  into account at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' Effective field theories (EFTs) are designed  to break down this range of scales into smaller segments so that physicists  can work in the relevant range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' Theorists “cut” their theory’s energy scale  at the order of the mass of the lightest particle omitted from the theory,  such as the proton mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  Thus, multi-scale problems reduce to separate  and single-scale problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='EFTs are today also understood to be “bottom-  up” theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' Built only out of the general field content and symmetries at  the relevant scales, they allow us to test hypotheses efficiently and to select  the most promising ones without needing to know the underlying theories in  full detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' Thanks to their applicability to all generic classical and quantum  field theories, the sheer variety of EFT applications is striking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  In hindsight, particle physicists were working with EFTs from as early  as Fermi’s phenomenological picture of beta decay in which a four-fermion  vertex replaces the W-boson propagator because the momentum is much  smaller compared to the mass of the W boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  Like so many profound  concepts in theoretical physics, EFT was first considered in a narrow phe-  nomenological context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' One of the earliest instances was in the 1960s, when  ad-hoc methods of current algebras were utilised to study weak interactions  of hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  This required detailed calculations, and a simpler approach  was needed to derive useful results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' The heuristic idea of describing hadron  dynamics with the most general Lagrangian density based on symmetries,  the relevant energy scale and the relevant particles, which can be written in  terms of operators multiplied by Wilson coefficients, was yet to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  With this approach, it was possible to encode local symmetries in terms of  the current algebra due to their association with conserved currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  For strong interactions, physicists described the interaction between pi-  ons with chiral perturbation theory, an effective Lagrangian, which sim-  plified current algebra calculations and enabled the low-energy theory to  be investigated systematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' This “mother” of modern EFTs describes  the physics of hadrons and remains valid to an energy scale of the proton  mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' Heavy-quark effective theory (HQET), introduced by Howard Georgi  in 1990, complements chiral perturbation theory by describing the interac-  tions of charm and bottom quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' HQET allowed us to make predictions on  B-meson decay rates, since the corrections could now be classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' The more  powers of energy are allowed, the more infinities appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' These infinities are  cancelled by available counter-terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  Similarly, it is possible to regard the Standard Model as the truncation  of a much more general theory including non-renormalizable interactions,  which yield corrections of higher order in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' This perception of the  whole Standard Model as an effective field theory started to be formed in  the late 1970s by Weinberg and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' Among the known corrections to  the Standard Model that do not satisfy its approximate symmetries are  2  neutrino masses, postulated in the 1960s and discovered via the observation  of neutrino oscillations in the late 1990s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  While the scope of EFTs was  unclear initially, today we understand that all successful field theories, with  which we have been working in many areas of theoretical physics, are nothing  but effective field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' EFTs provide the theoretical framework to probe  new physics and to establish precision programmes at experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  The  former is crucial for making accurate theoretical predictions, while the latter  is central to the physics programme of CERN in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  1  EFTs in Particle Physics  More than a decade has passed since the first run of the LHC, in which the  Higgs boson and the mechanism for electroweak symmetry breaking were  discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' So far, there are no signals of new physics beyond the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' EFTs  are well suited to explore LHC physics in depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' A typical example for an  event involving two scales is Higgs-boson production because there is a factor  10−100 between its mass and transverse momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' The calculation of each  Higgs-boson production process leads to large logarithms that can invalidate  perturbation theory due to the large-scale separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' This is just one of  many examples of the two-scale problem that arises when the full quantum  field theory approach for high-energy colliders is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' Traditionally, such  two-scale problems have been treated in the framework of QCD factorisation  and resummation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  Over the past two decades, it has been possible to recast two-scale prob-  lems at high-energy colliders with the advent of soft-collinear effective theory  (SCET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' SCET is nowadays a popular framework that is used to describe  Higgs physics, jets and their substructure, as well as more formal problems,  such as power corrections to reconstruct full amplitudes eventually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  The  difference between HQET and SCET is that SCET considers long-distance  interactions between quarks and both soft and collinear particles, whereas  HQET takes into account only soft interactions between a heavy quark and  a parton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' SCET is just one example where the EFT methodology has been  indispensable, even though the underlying theory at much higher energies  is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' Other examples of EFT applications include precision measure-  ments of rare decays that can be described by QCD with its approximate  chiral symmetry, or heavy quarks at finite temperature and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' EFT  is also central to a deeper understanding of the so-called flavour anomalies,  enabling comparisons between theory and experiment in terms of particular  Wilson coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  Moreover, precision measurements of Higgs and electroweak observables  at the LHC and future colliders will provide opportunities to detect new  physics signals, such as resonances in invariant mass plots, or small devia-  tions from the SM, seen in tails of distributions for instance at the HL-LHC  3  – testing the perception of the SM as a low-energy incarnation of a more fun-  damental theory being probed at the electroweak scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' This is dubbed the  SMEFT (SM EFT) or HEFT (Higgs EFT), depending on whether the Higgs  fields are expressed in terms of the Higgs doublet or the physical Higgs bo-  son.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' This particular EFT framework has recently been implemented in the  data-analysis tools at the LHC, enabling the analyses across different chan-  nels and even different experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='At the same time, the study of SMEFT  and HEFT has sparked a plethora of theoretical investigations that have  uncovered its remarkable underlying features, for example allowing EFT to  be extended or placing constraints on the EFT coefficients due to Lorentz  invariance, causality and analyticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  2  EFTs in Gravity  Since the inception of EFT, it was believed that the framework is applicable  only to the description of quantum field theories for capturing the physics  of elementary particles at high-energy scales, or alternatively at very small  length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' Thus, EFT seemed mostly irrelevant regarding gravitation,  for which we are still lacking a full theory valid at quantum scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' The only  way in which EFT seemed to be pertinent for gravitation was to think of  general relativity as a first approximation to an EFT description of quantum  gravity, which indeed provided a new EFT perspective at the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' However,  in the past decade it has become widely acknowledged that EFT provides a  powerful framework to capture gravitation occurring completely across large  length scales, as long as these scales display a clear hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  The most notable application to such classical gravitational systems  came when it was realised that the EFT framework would be ideal to handle  gravitational radiation emitted at the inspiral phase of a binary of compact  objects, such as black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' At this phase in the evolution of the binary, the  compact objects are moving at non-relativistic velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' Using the small  velocity as the expansion parameter exhibits the separation between the  various characteristic length scales of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' Thus, the physics can be  treated perturbatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' For example, it was found that even couplings man-  ifestly change in classical systems across their characteristic scales, which  was previously believed to be unique to quantum field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' The appli-  cation of EFT to the binary inspiral problem has been so successful that  the precision frontier has been pushed beyond the state of the art, quickly  surpassing the reach of work that has been focused on the two-body problem  for decades via traditional methods in general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  This theoretical progress has made an even broader impact since the  breakthrough direct discovery of gravitational waves (GWs) was announced  in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' An inspiraling binary of black holes merged into a single black hole  in less than a split second, releasing an enormous amount of energy in the  4  form of GWs, which instigated even greater, more intense use of EFTs for  the generation of theoretical GW data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' In the coming years and decades,  a continuous increase in the quantity and quality of real-world GW data is  expected from the rapidly growing worldwide network of ground-based GW  detectors, and future space-based interferometers, covering a wide range of  target frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  3  EFTs in Cosmology  Cosmology is inherently a cross-cutting domain, spanning scales over about  1060 orders of magnitude, from the Planck scale to the size of the observable  universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' As such, cosmology generally cannot be expected to be tackled  directly by each of the fundamental theories that capture particle physics  or gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' The correct description of cosmology relies heavily on the work  in many disparate areas of research in theoretical and experimental physics,  including particle physics and general relativity among many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  The development of EFT applications in cosmology – including EFTs of  inflation, dark matter, dark energy and even EFTs of large-scale structure  – has become essential to make observable predictions in cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' The  discovery of the accelerated expansion of the universe in 1998 shows our diffi-  culty in understanding gravity both at the quantum regime and the classical  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' The cosmological constant problem and dark-matter paradigm might  be a hint for alternative theories of gravity at very large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' Indeed, the  problems with gravity in the very-high and very-low energy range may well  be tied together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' The science programme of next-generation large surveys,  such as ESA’s Euclid satellite, rely heavily on all these EFT applications  for the exploitation of the enormous data that is going to be collected to  constrain unknown cosmological parameters, thus helping to pinpoint viable  theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  4  The Future of EFTs in Physics  The EFT framework plays a key role at the exciting and rich interface be-  tween theory and experiment in particle physics, gravity and cosmology as  well as in other domains, such as condensed-matter physics, which were  not covered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' The technology for precision measurements in these do-  mains is constantly being upgraded, and in the coming years and decades  we are heading towards a growing influx of real-world data of higher qual-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' Future particle-collider projects, such as the Future Circular Collider  at CERN, or China’s Circular Electron Positron Collider, are being planned  and developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  Precision cosmology is also thriving, with an upcoming  next-generation of very large surveys, such as the ground-based LSST, or  space-based Euclid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  GW detectors keep improving and multiplying, and  5  besides those that are currently operating many more are planned, aimed  at measuring various frequency ranges, which will enable a richer array of  sources and events to be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  Half a century after the concept has formally emerged, effective field  theory is still full of surprises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' Recently, the physical space of EFTs has been  studied as a fundamental entity in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' These studies, by numerous  groups worldwide, have exposed a new hidden “totally positive” geometric  structure dubbed the EFT-hedron that constrains the EFT expansion in any  quantum field theory, and even string theory, from first principles, including  causality, unitarity and analyticity, to be satisfied by any amplitudes of  these theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' This recent formal progress reflects the ultimate leap in the  perception of EFT nowadays as the most fundamental and most generic  theory concept to capture the physics of nature at all scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' Clearly, in  the vast array of formidable open questions in physics that still lie ahead,  effective field theory is here to stay – for good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  Acknowledgements  We dedicate this article to the memory of Steven Weinberg, who so gen-  erously graced us with a spectacular inaugural lecture to the international  online series hosted at CERN “All Things EFT”, which turned out to be  his final published lecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content='  We thank Cliff Burgess and HuaXing Zhu for comments and input on a  preliminary draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
+page_content=' ML has been supported by the Science and Technology  Facilities Council (STFC) Rutherford Grant ST/V003895 “Harnessing QFT  for Gravity”, and by the Mathematical Institute University of Oxford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}
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+page_content='  6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE2T4oBgHgl3EQfqgg6/content/2301.04039v1.pdf'}

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